Nonlinear and Adaptive Control Design (Adaptive and Learning Systems for Signal Processing, Communications and Control Series)

Adaptive and Learning Systems for Signal Processing, Communications. and Control Editor Simon Haykln

Werbos I THE ROOTS OF BACKPROPAGATION: FROM ORDERED DERIVATIVES TO NEURAL NETWORKS AND POLITICAL FORECASTING Krstic. Kanellakopoulos. and Kokotovlc ADAPTIVE CONTROL DESIGN

I

NONLINEAR AND

Nlklas and Shoo / SIGNAL PROCESSING WITH ALPHA-STABLE DISTRIBUTIONS AND APPLICATIONS

NONLINEAR AND ADAPTIVE CONTROL DESIGN Mlroslav Krstlc loannis Kanellakopoulos Petar Kokotovic

A WILEV-JNTERSCIENCE PUBLICATION

JOHN WILEY & SONS, INC. New York I Chichester I

Brisbane

I

Toronto

I

Singapore

This text js printed on acid-free pllpcr. Copyright © 1995 by John Wiley & Sons. Inc. All rights reserved. Published simultaneously in Canada. Rcproduction or translation of Dny part of this work beyond thot permitted in Section 107 or IOH of the 1970 United Stllies Copyright Act without the permissjon of the copyright owner i!'l unluwrul. Requests (or permission or further information should he addressed to the Permissions Department. John Wiley & Sons. Inc•• 605 Third Avenue. New York. NY 10158-0012.

Library oJ Cong,eKs ell/aloging ;" P"blicalioll Data Krstic. Miroslnv. Nonlineor and adaptive control design / MirosJav Krstic. loannis K;anell'lkopoulos. Petar Kokolovic. p. cm. - (Adaptive lind learning systems for signlll processing. communications, and control) "A Wiley-Intcrscience publication." Includes bibliographical references and index,

ISBN 0-471-12732-9 1. Automatic control. 2. Nonlinear conlrol tlteory. 3. Adaptive I. KancllakopouJos. Ionnnis. II. Koklltovic. Petnr III. Title. IV. Series. TJ213.K748 1995 629.R-dc.2U 95-10082 ell' conlrol systems.

Printed in the Unilcd States of America

10 9 R 7 6 S 4 3 .2 I

Preface This book opens a view to the largely une~"'Plored hmdscape of nonlinear systems with un('ertainties, Its main subject is feedback design for sllch syst.pm!;. New design t.oo]s and systcmatic desigll procedures arc developed which guarantee t.hat. t.he designed feedback systems will possess desired properties not. ouly locally, hut. also globally 01' in a Rpecjfjed region of the state spacc.

Backstepping. Compared with other books on nonlim-'ar control, the:' mAjor novelty of this book is a recursive design methodology: backstepP·il1D. \Vith t.his mcthodology the constl'uctioll of bot,h feedback ('out,rol laws aud assodat-ed Lyapullov fUllctions is systenutti('. Strong propf'rt.ies of global 01' regional sta.bility and tracking arc built into the llol1line~tr system in u. number of steps, which is never higher than the system order. While fecdback linellIization methods require precise models and onen cauccl s011le useful llonliuel:tritips. hackstepping designs offpr a choice of design tools for accommodation of tln('el'tain nonlinearit.ies and can a.void wa."teflll cancellations. Classes of systelns. By the very nature of the modeling pr[)('f'SS, llolllineal' models are morc structured dmn their linear offsprillg. IVhlllY llonlinetlrities introduced by physical laws (e.g., centrifugal [orces amI chemical kinetics) CUll not. be confined to a lineal' sector, because, with increasing magnitudes, their growth is po1ynomial 01' even exponentia1. Systems with sitch nonlillenritics are prOllP to explosive iustabilities which 111ust be Rnti['ipaterl and prevented by fl'edbuck control. Non1ille~u-itics usually appear multiplied with physi('al COllstlll1ts, often poorly known or dependent ou the slmvly cbanging environlllent. This commOll form of uneertainty is cttptured by llonlillettr models with lluknown COllsttlnt para,nu.lters. In ~'parametric pure-feedbttck." systems studied in this book t the growth of llonlillearities is ullrestri('tE'd~ whilE' the lIokoO\vn pltrtUlleters tlppear linearly. \\Then only 8U output is measllred the llonlillcaritics are assumed t.o be fuud-iolls of the output. The cousidercd dass of nonliucar systems is broad ttlld Pllcompasses liuetu' systems as a spf'cia] c~tse. t

Adaptive control. The control of nonlinear s.),stP.IlU; with unknown ptU-alllf'ters is traditionally app)'oached as an adaptive control problem. separate from the rest of nonlinear controJ theory. Here we bridge t:his gap and let. adaptive

vi

PREFACE

control be what it is-a paradigm for constructing nonlinear dynamic feedback for both nonlinear and linear systems, We depart from the traditional "certainty equivalence" approach and design stronger nonline81' control laws that achieve the desired objectives either by interacting with parameter update laws or by a.ttenua.ting the effect of parameter estimation errors. In this way we achieve not only stronger stability properties, but also quantifiable improvements of transient performance.

Applications. Strollg regional or global properties achieved by the design methods presented in this book have the potential to expand the operating range of feedback controllers and make tllem applicable to physical plants fol' which linear or other local controllers are inadequate. In this text, application exaJnples of this type include automotive suspensions, jet engine stall and surge control, biochemical processes, aircraft wing rock control. induction motors, robotic manipulators, and magnetic levitation. Organization of the book. Although Illost of the results in this book are new! they are presented at a level accessible to audiences with a standard undergraduate background in control theory and a basic knowledge of stability concepts. Each chapter is written in a pedagogical style, with detailed proofs and illustrative e.xRmples. !vIore inyolved extensions and generalizations are only outlined, Basic tools for nOlladaptive backstepping design with state feedback are given ill Chapter 2, and with output feedback in Chapter 7, while adaptive backstepping is introduced in Chapter 3. CJlapters 1, 2, 3, and 7 are written as a short te..xt for a part of either a course on nonlinear systems or on adaptive control. The remaining chapters, although more advanced, are accessible without Chapters 2, 3, and 7. The core of the controller design is the tuning functions method in Chapters 4 and 8. The modular designs in Chapters 5, 6, and 9 achieve the independence of the controller and the identifier. The new adaptive nonlinear designs are applied to linear systems in Chapter 10 and are shown to outperform the traditional linear designs. Tbis chapter can be read independently from t,Jle rest of the book. Thus, depending on the reader's interest, the book can be read in several ways: • • • • • •

Nonlinear stabilization without adaptation ... , ... . Introduction to adaptive backstepping .. , ... , .... . Adaptive Lyapunov design with tuning fUllctions ., Estimatiou-based design with modularity ... , .... . State feedback adaptive designs ...... , .... ,., .... . Adaptive control of linear systems ............... .

Chapters 2, 7 2,3,7 2,4,8, 10 2, 5, 6, 9, 10 2,4,5,6 10

PREFACE

vii

This book is a joint effort based on recent research of its authors. Chapters 4,5,6,8,9, and 10 and Appendices A-F were written by the first author, and Chapters 2,3, and 7 and Appendix G were written by the second author. Their labor was arduous due to frequent revisions by the third author, their former Ph.D. advisor, who also wrote Chapter 1. Acknowledgments. At the initial stage of this research, Riccardo Ivlarino introduced us to nonlinear geometric methods and collaborated Wit11 us and David Taylor on the early adaptive nonlinear schemes, contributing a key extended matching idea. A separate line of research with Hector Sussmann and Ali Saberi led us to a nonadaptive recursive design. About a year later, ill a joint effort with Steve Morse, we developed the first adaptive recursive design procedure, which we named adaptive backst.epping. We express our deepest gratitude to these colleagues. We are also thankful to Randy Freeman, Petros Ioannou, Laurent Praty, and Andy Teel for continuous exchange and discussion of current results. Among many other colleagues who helped us with frequent debates on broader issues of nonlinear and adaptive control are Brian Anderson, Karl A.strom, Tamer B~ar, Gil Blankenship, Bill Boothby, lvlohammed Dahleh, Reza Ghanadan, Graham Goodwin, Jessy Grizzle, Alberto !sidori, Nlrdjan Jankovic, Hassan Khalil, Art Krener, P. R. Kumar, loan Landau, David Mayne, Rick NIiddleton, Carl Nett, Shankar Sastry, Rodolphe Sepulchre, Eduardo Sontag, !vIar], Spong, Jillg Sun, and Gang Tao. And above all, our warmest thanks go to the most generous cOlltributOl"S t.o this project, our wives, Angela, Georgia, and Anna. The writing of this book was supported in part by the National Science Foundation under Grants ECS-9203401 and RIA ECS-9309402, by the Air Force Office of Scientific Research under Grant F-49620-92-.l-0495, by UCLA through the SEAS Dean's Fund, and also by grants from Ford Motor Company and Rockwell International. lvlIROSLAV KRSTIC IOANNIS KANELLAl(OPOULOS PETAR I(OJnC(-ls

;307 313 315 315 320 32"l

327 32a 329 :332 3~1O

346 :3 0 is a design

=

p~U'a.meter.

±

=

~

=

The resulting feedbark syst.em js of second - (p + ~).l· + fJ.1:

(1.5a) (1.5b)

Its stabilit), pl'operi;ies ean be checked by examining the derivative of the Lyapunov flluctjon l 'J 1( 2 (1.6) lI(:r,~ ) = 2.1:- + 2" ~ - 0) , \Vhit'h t.urns out to be non positive: (1.7) Thus, V·(:I:O) , €(l) evaluated along t.he solut.ions of (1.5) is a nonincreasing fUllrtion of timc. This proves that .r(1) and ~(t) remain bounded for all t ~ O. The proof that limt_oo :t'(t) = 0 is also achieved can be given using Theorem 2.1 in the next chapter. How was the dynamic nonlinear controller (1.4) conceived? Not as a nonlinear controller, but rather as a parameter adaptat.ion scheme! Its dynamic part ~ = !t,2 is, in fact, an update law for ~ as an estimate of O. ConseciuentlYI the estimat.ion error ~ - (J is penalized in the Lyapunov ftllletion (1.6).

1.1.4

Lyapunov-based design

The controller (1.4) is an outcome of a systematic Lyapllflov design procedurc. In I:his pro('edure we seek a. parameter update law for the est.imate 0(1),

8= which, along with a cont.rollaw

11

r(:l', 8) ,

(1.8)

= a(x, Ii), willll1ake the Lyapunov function

~ 11) lA V" (!t',O) = -:r- + -(0 - 0)-

'1

2

a noniucl'easing fuuction

2

(1.9)

or time:

l/(.1:(t),9(t)) ~ V(x(to),O(to)) , 'Vt ~ to! \Ito ~ O.

(1.10)

1.1

5

ADAPTIVE LINEAR CONTROL

1

.s - fJ

I----rx;.;. ,- H :1:

p

Figure 1.1: Lya.punov-baRed adaptive scheme for I.he scnlnr system

x = l' + :cO.

To this end, we express 11 as a funct.ion of u and 8 f:l.nd seek o:(x, 8) and ;(:v, 8) to guarantee that 11" ::; _P:1:2 with p > 0, namely

l~· = :1'(tJ

+ lh) + (0 -

0)0 < _p:r;2.

(1.11)

Rcauallging terms we get :I'll

+ 88 + B (1,2 -

0)

~

-p.r.';!..

Since neitbel' 0:(:1;,0) nor T(:l"1 B) is allowed t.o depend must tl:tke ;(.T t 8) .:t.2 , tlUtt is,

=

iJ

=

(1.12) 011

thC' ulllmowl1 6, we

(1.13)

The remaining condit.ion (1.14)

allows us to select 0:(.1', B) in variolls ways. The choice which results in the dynamic 1l0nJillcm' controller (1.4) is

u == - (p + O):c .

(1.15)

"Ve have thus designed our first Lyapunov-baseu adaptive scheme shown ill Figure 1.1, where s is the complex variable of the Laplace transform. This scheme already exhibits SOlUe features of more general schemes to be designed ill this book.

1.1.5

Estimation-based design

Anothf'r dynamit· nonlinear ('ontroller for the same linoar plant (1.1) wi11 now be desigued starting with tl1e design of 1:1. pnnullcter identifier, Since the signal

6

INTRODUCTION

'II

x

1

s-o IDENTIFIER

I

I I

, I

I I

• t I I I I I

+: I

,-----

--------~------------.

CONTROL LAW

Figure 1.2: Estimation-based adaptive schomo for the scalar system :i; = u + x8.

i is not available for me~1Stlrement, we cannot solve (1.1) for the unknown 8. To overcome this difficulty, we resort to filtering both sides of (1.1) by

.!1:

811 = --u+8--:l:. 8+1 s+1 8+1

--l'

(1.16)

Denoting the filtered versions of the known signals x(t) and u(t} by Xr

=

1

--X, .9+1

'llr

=

1

--1L,

8+1

(1.17)

we can rewrite (1.16) as

.r(t)

=

(0 + l)xr(t)

+ tJr(i).

(1.18)

If instead of the unlmowll 0 we use its estimate fJ I the corresponding predicted value of .1' is (1.19) x(l) = (8(t) + l}xr(t) + ur(t) I and the predictioll error e is related to the estimation error 8 as follo\Vs~

e = x -:i: = (8 - fJ)xr = 8xr.

(1.20)

A parameter update law for 0 call now be derived to aim at a minimum of e2 • To this end, the rate of cluluge of {} is set to be proportional to the negative gradient of e2 with respect to 6: (1.21)

1.2

7

ADAPTIVE NONLINEAR CONTROL

where l' > 0 is the uadapt.ation gain." More commonly used are "normalized" versions of (1.21). One of these is :.. (J

=

"'I

( 1.22)

- 1.)exr· +xf

Once O(t) is available, it call be used in a certainty equivalence control law. For example! the same control law 3..'i in (1.15) would result from the specification t.hat the closed-loop pole be placed at -po This t.ask would indeed be achieved if the estimate 0 were correct, {} == (J. The closed-loop adaptive system with t.he normalized update law (1.21) is shown on Figure 1.2. The dynamic order of the adapt.ive controller is three due to the identifier module which contains two first-order filters and the first-order gradient-t.ype update law. The stability analysis of est.imation-based adapt.ivc schemes is intricate. For gradjent-type schemes, this analysis is conclusive only in the case of normalized update laws. Ouo of its key result.s establishes the bonndedness properties of the identifier module without any restrictions on xr(t) and hence on u(t). This means that. the identifier module can be connected not. only with tho pole-placement cont.roller used in this example, but also with a wide "~U'ioty of other controllers.

1.2 1.2.1

Adaptive Nonlinear Control A nonlinear challenge

To int.roduce the topic of this book, let us apply the estimation-based approach to t.he adaptive control problern for the nonlinear p]a.nt (1.23) We first select, a cert.ainty equivalence control law

'u = - px - Ox2

(1.24)

1

e

which, if iJ were exact, == (J, would result in the closed-loop system x = -p.T, as in the preceding linear example. The construction of an identifier is analogous t.o (1.16)-(1.20). The result.ing estimation error equation is the same, except. for xr(t), which is now the filtered version of x 2 (t):

.1:r

=

1

.,

--x-

s+ l' .

(1.25)

\Ve pr?ceed ~vith the normalized update law (1.22) and, upon t.he substitution of 0 = -8 and e =

8a:r, we obtain

n- =

.,

xf -i---.,(J. 1 + ·'Vf

(1.26)

8

INTRODUC1'ION

This differential equation is linear. It dearly shows that tbe paramet.er Cl'ror O(i) canllot converge to zero faster than cxponent:inlly. Let us consider the most, favorable case where (1.27)

rm·

This rate wonld ha,'e been satisfat'tol'Y the certainty cquivalcncf' l'ont.rol (1.15) of the linear plant (1.1). Is this so witb t,he control (1.24) of the nonlinear plant (1.23)7 The resulting nonlinear closed-loop system is (1.28)

where, for simplicity, p = 1. The substitution of (1.27) wit:h l' yiplds the eqnation

= 1 into (1.28) (1.29)

whose explicit solution is

;J·(I.) =

2x(0)

_

;c(0)8(O)e- t

+ [2 -

_ «c(0)6(O)}c

(1.30)

'

It: is e~lsy t,o see t,hat if :c(0)8(0) < 2, then x(t) will converge to zero as f --+ However. if .1:(0)8(0) > 2, then a.t the time tesc

= ~ In x(~)H(O) 2

;}'(0)8(0) - :2

00.

(1.31)

t.he difference of the t.wo exponential terms ill the denominator becomes zero, t,bat is,

1:1:(1)1

-+ 00 as f -feMe«

(1.32)

The adaptive closed-loop system is not only ullstable, but. worse than that: Us st~ttc J;(t) escapes to infinity ill finite time tf!FlC' The escape time becomes shorter as th£l difference x(O)O(O) - 2 grows, tbat is, a:1.S the initial ('onditions bccome lru«ger. This simple example clearly demonstrates why a tradit.ional estimatiollbased design cannot he applied to nonlinear systems. Normalized ul)date laws arc too slow iu providing t.he estimates 8(t) with which certaint.y equivalence control would be able to prevent: cat.astrophic forms of instability« \Ve nccd either stronger controllers which will be able to achieve stabilization with standard identifiCl's, or fasLer iclellt:ifiers, or a combination arboth. Lyapunov-based designs in Chapters 3 and 4 provide faster ident.ifiers, while ill modular designs ill Chapters 5 and G \ve construct stronger controllers.

1.2

9

ADAPTIVE NONLINEAR CONTROL

1.2.2

A structural obstacle

If t.he estimation-based approach failed because of the slowness of normalized est.imat.ion, wiII a Lyapullov-based design be more succ:essful? Let 11S start. with t.he same Lyapunov function (1.9) as for the linear plant (1.1). Its derivat.ive for the llol1lil1ecu plant (1.28) is

If = :z:{u + 0.1'2) + (0

-

6)9 .

(1.33)

The requirement. t~ ::; _p:v2 imposes the following condit.ion on tl1E~ c:hoice of an updat.e law for {} and a control law for u: (1.34)

To eliminat.e t.he unknown 0, Ule update law must. be

o=

(1.35)

so that (1.34) reduces to (1.36) A control ll:lw which satisfies this condition iF! the certainty equivalence cont.roller 2 'lJ. = -p.l" - fl.-c • (1.37) Since the updat.e law (1.35) and the control law (1.37) yield ,i' = -p:c2 , we achieve the same stability properties and state regulation as in the adaptive control of the linear plant (1.1). Comparing this result with the est.imationbased design, we observe that the control laws (1.24) and (1.37) are the same, whih.:> the Lyapunov update law (1.35) is much faster than the normalized gradient updat.e law (1.26). It would appear that a Lyapunov approach to adaptive nonlinear control is more promising than the estimation-hased approach. However, in the linear case t.he Lyapunov-based design ha..'i heen restricted to plants with transfer functions of relative degree one and two. In the nonlinear state-feedback case, this structural restriction is translated into the "level of uncertainty," that is, the number of integrators between t.he control input and the unknown parameter. In the plant. (1.23) the level of uncm-t.ainty is zero. The uncertainty and tho control are "matched," because they appear in the same equation. In the plant (1.38) the level of uncertaint.y is one. which corresponds to the so-called "ext.cndedmatching" case. As we shaH see, an extension of the Lyapunov design to this

10

INTRODUCTION

case is relatively straightrorward, because 0, which appears in the contro] law, can be substif.uted from the update lmv. For the system with uncertainty level two ~i:l = X2 + (JJ:r j'2 = X3 (1.39)

.1:3 =

U

this is 110 longer SOl because in this case the co 11 t.rol law would have to include 0, which is not available. B~1.Ckstepping designs developed in Chapter 2, 3, and .l.1 remove this structural obFitacle and allow the Lyapunov-based designs to be applied to wide classes of uncertain nonlinear syst.ems.

1.2.8

Early results

[nt.erest in adaptive control of nonlinear systems WI:1S stimulated by major advances in the diffCl'ential-geometric theory of nonlinear feedbtlcl\ ('ontrol in the mid-1980's. A thorough t.reatment of this theory was given by Isidori in Ius seminal book [53} ,vhich unified a decade of results by many researcbers. Porticularly popular were t.he results on '~feedback linearization." that is. the state and feedback tnl.llFiformution of l10nlinear systems into iinem' ones [27, 48. 56, 117, 118]. This methodology helped convert many previously intractable nonlinear problems int.o lUuch simpler pl'oblcms solvable by familiar lineal' methods, It SOOl1 became clear, however, that along with theil' many advantages. the nonlinear geometric methods haye some shortcomings. One of 1.hem is their inability to handle the presence of unknown parameters, This motivated the fil'st series of adaptive nonlinear coutrol schemes. They were ttll restricted to systems satisfying tllO matching condition. Examples of sllch systems are rigid models of rohotic manipulat.ors. While the fhst robot.ic adaptive scheme by Craig [221 required metlSUrement of joint accelerations, t.his impract.ical assumption was soon removed by Slotine and Li [169. 170], I\1iddleton and Goodwin [130], and Orteg~1. and Spong [148], among others. A more general tl'eat;ment of adaptive nonlinear regulation under the Dlat(~hing condition was given by Taylor, !(okotO\Tic, Marino and Kanellakopoulos [186], including U11modeled dynamics which violated the matching condition. The matching rondition was rela.'\:cd to the e:l=f.ended matching condition by Kanellakopoulos, Kokotovic and iVIarillo [65] and Campion and Bastin [8, 15). For a period, the extended matching coudition was t.he frontier which could not b(' crossed by Lyapunov-based designs. This redirected researchers to est.imation-based designs. Nam and Arapostathis [141] and Sastry and Isidori [166] combined feedback linearization with adapbttioll techniques from adaptive linear controL However, to achieve global stability. these schemes reQuired that the non1inearit.ies be restricted by linear growth conditions, Similar

1.3

PREVIEW OF THE IvIAIN TOPICS

11

restrictions on system nonlinearities were imposed by Kancllakopoulos, Kokot.ovie, and NIiddleton [67, 68] and Teel, Kadiya]a, Kokot.ovic, and Sast.ry [190]. The only noulinear estimation-based l'esuIt.s which went. beyond tlle lineal' growt.h constraint.s were obtained by P01l1ct: and Praly [152, 153, 154], who userl Lynpullov funct.ions to ch81'acterize relationships between llolllint.='ar growth const.raints and cont.roller stabilizing prop f"rt ies. In the absence of matching conditions, their schemes st.ill involved SOIllf" growth restrict.ions hut. were able to handle thc bcnchmark third-order example (1.39). The stat.c-oi'-the-art of adaptive cont.rol, including adapt.ive nonlinear control, was reviewed ill t.he 1990 Grainger lecturcs [86]. One of these lectures prcsent.ed the result. of I(anellakopoulos, Kokotovic, and 1v1orsc [69, 87], which finally broke the extended matching barricr. This was achievcd with a new recursive design proccdure ('al1ed (UiC1.pti'tJe bad~steppin9. Adaptive backstepping. developed by Ioannis Kanellakopoulos [63] ill collaboration wit.h Pehtr Kokot.ovic and Steve Iv1orse, cmerged as n conflucnce of the adaptive est.imation idea, on one side, and on the other side, nonlinear ('ontrol idcas cxpres.a;;ed in wor],:s of Tsinias [103], Byrnes and Isidori [12J. Sontag and Sussmann [175], Kokotovie and Susslllann [85), and Sabcri, Kokotovic, and Sussmann flG3]. Adapt.ive backstepping was also strongly influenced by the properties of an early adaptive scheme by Feuer and Ivlorse [36], which, although designed for lincar systems, preserved global stabilit.y under out.put feedback for a dass of out.put. nonlinearitics: as shown by Kanellakopoulos: Kokot;ovic, and I\.forse [70, 71]. Adapt.ivc backstepping influenced further dcvelopment.s in adaptive nonlinear control. ~Ifarino and Tomei [122, 123, 124] combined it with thcir filt.ered transfol'mat.ionM [120, 121] to solve the adaptivc output-feedback problem for a dass of nonlinear systcms that has not since been enlarged, Adaptive hackstepping also st.imulated efforts to l'educc its overparametrization. A partial reduction was achicved by .Jiang and Praly [59]. \Vith the invcnt-ion of tuning junctions, rvIiroslav Krstic [02, 94] int.rodu('cd a new dcsign which completely removed the ovcJ'parametrizat.ion. l

1.3 1.3.1

Preview of the Main Topics Classes of nonlinear systems

The main topic of this book is the design of feedback conlrollers for nonlinear systems wit.h unknown constant. paramet.crs. The most important design speC'ifiC'at.ioll is (:0 achieve asymptotic traC'kil1g of a known reference t.rajectol·Y wit.h the st.rongest possiblc form of stabilit.y. Another key requirelUcnt is that the designed cont.roller should provide effcctive means for Slulpillg the transient. performance and thus allow different per!ormance-robust.ness t.rade-offs. The largest: classes of nonlincar systems for which thc stated design problcm is solvable with eitller stat.e-fecdback or output-feedback controllers are not

L2

INTRODUCTION

mown at. this time. The largest. classes for which solut.ions have bee-ll obtained 11'e those considered in this book. State feedback solutions are given for t.he so-called class of I'pHl'amet.ric )nr~feedback systell1s.~' They are first. present.ed for t.he subclass of '~para­ netrie strict.-feedback systems," for which t.hc achieved st.ability and t.l'tl.cking Jroperties are global. By analogy with linear systems! strict-feedback syst(,lllS ue also caUed "t.riangulru·." Output-fcE"dbtl.ck solutions are rcstrkted to a narrower class of minimulll Jhase systems in which the nonlillearities depend only 011 the output: variable. The class of pure-feedhack systems with unknown parametcrs is wellrcp:~scnt.ed by thv third order syst('1U

.i' l 3:2

l'a

= = =

+ 'Pi ('-Vi' 3~2)8 'It .1:3 + rp2 (,1'1. :r:2, ;l!a)8 u + CPI(J'l, J':!. X:i)(} , :t.:!

(lAO)

\vherc the p x 1 veclor (J is const8Jlt and unknown. (In a more gene-ral cast=>. t.he terms J':!, .1:3, aud '/I ('an be JUultiplied by unknown constant paramct.ers pl'ovided that the signs of these parameters are known.) Apart from t.he re::Juirement that the dependence of the right-hand side of (1.40) on /J be linear, )r, to be precise, affine, pure-feedback systems arc characterized by the struct.ure of t.he known nonlinearitics 'P], cp~, and 'P:}. The function !Pi mnst. not depend on .r:l. and a further implicit. function restriction is imposed on the dependence of 'PIon :l':!l and of 'P'}. on .1.'3- This restriction is automatically satisfied if CPI does not depE"ud on .1:2 and 'P2 does not depend on :1!3, t.hat is, jf we have 'PI (a:d and pillg is that the dynamic order of its overpo.rametrized controller is bigh, On t.he other hand, the order of the tuuiug functions ('outro11er is minimal, but for high-ol'd(,J' systems its nonlinear e}q>l'essions become increasingly complex. The main source of complexity is the built-in int.eraction between t,he identifier and the cont.rol law. Both ofthE'se drawba.cks arc removed by modular' designs in Chapters 5 and 6 which Rre inspired by traditional estimation-based designs, In adaptive line:u control the estimation-based designs achieve a significant levc1 of modularity of the controI1("r-idcntifier pair: Any stabilizing controHer can be combined with an~' identifier. Paramet.er update laws can be of eit.her gradient 0)' loast-sqlUU'es type. The controller module is x(t} E .Al, \:It E lR. A set

(2.9)

n is po.rlitively invariant if this is true for all future time only: (2.10)

Can we guarantee convergence to a desired invariant set? A rewarding answer to this question is provided by LaSalle's Invariance Theorem and its asymptotic stability corollary:

2.1

25

STABILITY

Theorem 2.2 (LaSalle) Let n be a. po.lJiti'lJcly ifl.'lJuriant set of {2.8}. Let V' : n -+ ~ be a continuously difTercntiCl,ble junction \'(:r) .'i·lIch thllt lr(.l·) ~ 0, 'V x En. Let E = {x E n I f! (:J:) = O}, and let il.J be I.he lll7:qesl iml(171aTlt set contained in E. Then, e11e7"Y bounded solution .J:(t) stm'ting in n CD1l.11e1yes to III ll.s t -+ 00. Corollary 2.3 (Asymptotic Stability) Let.1: = 0 be lhe only eq1l.WIJri1l.ln of (2.8). Let l' : R n -+ 1R+ be a continuously diffe1'cntiable, POSiti11C definite, 7'O.dially unbo·unded j1mction ~! (x) such that lr (.J:) ~ 0, 'V J; E IR fl. Let E = {.t: E lRlI Ilr(.t·) = 0 h arJ.d suppose that no solution Of.hc1' t/UI1I 3:(1) ;; 0 can sta,y f07'eYJe1' in E. Then the O1-igin is globtdlll aS1JmptoticllilU stable (GAS). These inVl:lIiance results will motivat.e us t.o closely examine the in"~\l'iallt subsets of E. As we shall see, the convergen(~e properties of the designed syst.em are stronger if the dimension of Al is lower. In the most. favonlble case of tlSyrnptotic stability, the 1argest invariant. subset 1H of E is just the origin x = O. Our aim will thus be to render the dimension of 1'/ as Jow as possible.

Input-to-State Stability. Another st:ability cOllcept which is used t.hroughout the book is Hmt, of illput-to-staf;c stability (1SS) , illj,l'oduc:ed by Sontag [173]. The system :i; = f(x, u) (2.11) is said to be input-to-state stable (ISS) if for any J~(O) and for any input u(·) cont.iU1l011S and bounded on [0,(0) the solution exists for all ., ~ 0 and satisfies

Ix(t)1

~ P(lx(O)I,t) + 1 (sup I'U(T)I) OS'T:$I

1

Vt

~ 0,

(2.12)

=

where {3(,r;, t) and 1'(8) are strictly increasing functiolls of s E ffi+ with 8(0, t) 0, 1(0) 0, while j3 is a decreasing function of t with lilll t _ oo P(s, t) = O. V s E

=

ll4. This definition of iuput-t.o-st.ate stability is appropriate for nonliu(lar systems since it explicitly illcorporates t.he effect of the initial c'onditions :1'(0): (2.12) shows that the norlll of the state x(t) depends not only on the input; U(T), but also includes an asymptotically decaying contribution from :r(0). A more extensive treatment of ISS is given ill Appendix C.

2.1.2

Control Lyapunov functions (clf)

This book is about control design: Our objective is to create closed-loop systems wi1:h desirable stability properties, rather than analyze the properties of a given system. For this reason, we ~lIe interested in an e:x-tensioll of tile Lyapullov function concept, called a control LlIapunov function (elf).

26

DESIGN TOOLS FOR STABJLIZATION

Suppose that our problem for the time-invariant system

.t·

= J(x, u),

.1:

E R",

U

E lR,

J(O,O)

= 0,

{2.13}

is to design a feedback control law 0(.1;) for the control variable u such that. the equilihriullll' = 0 of the dosed-loop syst.em

.1: = /(.1', O-(.T))

(2.14)

,I

is globally aSYlllpt;otically stable. \Ve can pick a funct.ioll (:r:) as a Lyapunov candidllte, and require that its dcrivA,/:ive along the solutiollS of (2.1 1) satisfy \i(:1:) ~ -I-V(x), where H/(x) is a positive definite fUllction. Vle therefore need LO find o(:r) Lo guarantee that for all x E 1R" l

oV

8.1: (x) I(:z:, o-{x))

:s;

(2.15)

-lV(x).

This is a difficult task. A stabilizing control law for (2.13) Illay exist but we may fail t:o satifify (2.15) because of a poor choice of 'I"(.J;) and IF(.1:). A syst,em for which a good cboice of Vex) ~tnd l-l'"(.z:) e.."{ists is said to possess a elf. Let us make this llotion precise.

Definitioll 2.4 A ,'Jmooth positi1Je definite and mdially u.nbounded function. V : JR'I -+ ffi+ is called a cont.fol Lyapunov fUllction (elf) for (2.13) iJ DlI" (x)/{a:,1J) } < 0 1 illf { -8 X

(2.16)

liEU

The elf concept of Artstein [4] and Sontag [172] is a generalization of Lyapunov design results by Jacobson [54] and Judjevic and Quinn [62]. Artstein [4] showed that (2.16) is not only necessary, but also sufficient for the existence of a cont,rollaw satisfying (2.15), tbat is, the existellce of a elf is equivalent to global asymptotic st,abilizability. For systems affine in the control, :i:

=

J(x) + g{a:)1./ t

1(0)

= 0,

(2.17)

the df inequality (2.15) becomes

av

-8 f(·7:) x

all" + -8 g(:,;)o'(:1;) ,I:

~ -Hi(X).

(2.18)

If ll"{x} is a elf for (2.17), t.hen a pal't.iculal' stabilizillg control law o(x), smootb for all :I: =f. 0, is given by Sontag's formula [171J

It

= a,{x}

=

I

O\'f + IJ.%

8V'

, ax g =f 0 o

oll" g

, 8x =O.

(2.19)

2.1

27

STABILI11f

It should be noted that (2.18) can be sat.isfied only if

oV

ng(x} = 0 => uX

avo

! l I(.}:) u.l'

< O.

(2.20)

and that in this case (2.19) results in

H' (.1:) =

ax I + (aV)4 a.}: 9 > 0 (DF)2

t

V.1.: =F

o.

(2.21)

A furt;her characterization of a stabilizing cont.l'ollaw 0'(.&) for (2.17) with a given df V is that 0' {.}; ) is continuous at. x = 0 if illld ouly if t.he elf satisfies the .'J111.all con/.ral property: For each e: > 0 there is a a{e) > 0 such that. if J: #- 0 satisfies 13:1 < 6, then there is Rome 1I with rill < e: snch that

av 03' rI{·};) + g(:r)u]

< O.

(2.22)

The mail1 deficiency of the ('If concept as a design tool is that' for most. nonlinear systems a elf is not knOWll. The task of finding au appl'opri.1.te elf lllay be as complex as that of designing a stubilizing feedback law. For several important classes of Ilonlinear systems, we will solve these two tasks simultaneously using a backslepping proeedure. To initiate this pro('edure we need t:o be able to find 1/(3:) and Q·(.l;) at least for scalar systems. Fort.unntcly. for scalar systems, lI'(:r) = !x:! is always a reasonable elf ~tnd the inequality (2.18) is easy to satisfy. This is illustrated by all example which also issues a warning that some designs Illay lead to t1 waste of cont:rol effort. Example 2.5 For the scalar system showll ill Figure 2.1,

:i'

=

3 COS.r - .r

+U

t

(2.23)

our task is to design a feedback cOlltrollaw which creates and globally stabilizes the c(luilibrium at :r: = O. We will ('ompare three different designs. In a feedback lineari:w.tion design, the control law ·U

-

cos.:r

+ .1:3 -

:&

(2.24)

cancels both llonlineal'ities (COS3' and _.r 3 ) and replaces them by -:t so t.hat the resulting feedback system is linear: .f = -.1:. T~l.killg (2.25) a elf for (2.23). we see tiu'tt t:l1e control law (2.24) satisfies the requirement. (2.18) with TV(:r) = .1,2, that. is, "(~r) ~ _~:2. However, there is an obvious irrationaJity of this control law: It cancels not only cos ,t:, but also -.:ra. For

115

28

DESIGN TOOLS FOR SrrADILIZATION

f

11.

•

cos ( .)

.r I--------r+-

1 + - - - -.......

Figure 2.1: The block diagram of system (2.23).

stabilizat.ion at x = 0, the negative feedback term _.t:3 is helpful, especially for h·rrge values of .1:. On t.he other hand, t.he presence of :r:i in the cont.rollaw (2.24) is harmful: It leads to large magnit.udes of u and IIl8Y cause Ilonl'obustness. A more reasonable design is not to canccl-.v3. \Vit.h "'(.1:) = ~:z~:! as before, we take lV(x} = .1: 2 + x·l , so that the cont;rollaw satisfying (2.18) becomes 'lL

= - cos:r. - x

ll. =

0:(:£').

(2.26)

In this case, the magnit,ude of 11 grows only linearly wit.h 1.1:1. Finally, as our third cont.rol law we employ Sontag's formula (2.19). Since t.his formula is based on the assumption that f(O) = 0, we first; cancel cos.?: by introducing'U = - cos x +u&. 'Ve again use V'(:t') = ~x2 as onr elf and evaluate 0'5(:£') from (2.19) with f = -3.,.3 and 9 = 1: (2.27)

A remm·lm.blc propert.y of (2.27) is that O:s(:r:) -+ 0 as 1:1'1 -+ 00, which means that for large 13:1 t.he ('ont.rollaw for ·u. reduces t.o t.he term - ('OS:l' requil'cd to place t.he equilibrium at :,; = O. The rationale is clear: Exccpt for t.he cancellation of cos .'t', the control is inactjv~ for large I:r.j because then the intc1'ual nonlinear feedback _x 3 takes over and forces :r towards zcro. In this way the control effort is not wasted to achieve a propelty alrpady present in the system. On the ot.her hand, for smal1 1.1:1 we have 0:8 R::: -.1.., which is t.he same as ill t;he pl'evions two control laws. It is casy t.o check t.hat u = - cos:l:+a"s(x) sat,isfies (2.18) with l-F(:r) = :r2 .j:c·1 + 1. This control law is superior because 0 it requires less control effort. than the other two. Example 2.6 The scalar syst.em shown in Figurc 2.2, .

X

=

X

3

+ :c-u, '1

(2.28)

is of interest; because it is smoothly stabilizable in spite of the singularit.y at :r = O. 'Ve proceed with l"(.t) = ~X2 and, because of t.he term :['3. we choose

2.2

29

BACI<S'l'EPPING

·U

Figure 2.2: The block dio.gl·am of system (2.28).

H'(x} =

CIX'"

where

CJ

> O. Solving

V(:r, u)

for

'I}.,

we obtain the control 11

=

a' [~l·:i + x 2u]

=

-cl.r4

(2.29)

ltl.W

= -(1

+ c.):!'

t::.

= a:(.r) I

wbich yields the globally asympt.otically stable closed-loop syst.em In this case, Sontag's fOl'lllUla yields

(2.30) j.

= -Cl.1:3 • (2.31)

which satisfies (2.18) with Hl(X) = X'IV! quires more control effort th~),n (2.30).

+ :1;4. Clearly, t.his control Jaw re0

For scalar systems t.he choice of a quadratic elf is obvious and the nonlinear design is straightforward: Tbe harmful nonlinoarities are cancelled by the control.

2.2 2.2.1

Backstepping Integrator backstepping

The simplicity of scalar designs motivates us to use them as start.ing points of recursive designs for higher-order systems. Let us first COllst:ruct dr's for second-order systems. Vve begin by augmenting the system (2.23) with an integnttor:

.r

=

cOS:l'-X +~

~

=

11..

3

(2.32a) (2.32b)

Let the desigll objective he the reguhl.tioll of ;z;(I.) , that iS t .1:(t) ~ 0 8S t -- OOt for all x(O), ~(O). Of course t ~(t) must remcun bounded. From (2.32a), the only equilibrium with :r = 0 is at (x,,;) = (0, -1). Vve will meet our design

30

DESIGN TOOLS FOR. STABILIZATION

/I

.r

./

./

- ( . ):.1

cos ( . )

Figure 2.3: The block diagram of system (2.;32).

objective by rendering this equilibrium GAS. In the block diagram ill Figure 2.3 the scai+++----f+}----to+t+---.]

/

.7:

- (. ):1 -0(:1:) cos ( . )

f4-------'

n (.)

Figure 2.4: Tntroduciug a(;/,') as the desired value for

~.

vVe now need to select n elf '~l for the system (2.:t2). Let: us try to constrllcl', it by Cluguleni.illg ''"(:I:) with a. quadratic j,enn in the PHClL' variable z: fj-") _.,)1 ( 'J The derivative of \~l along the solutions of (2.35) is comput',ed as :1: [-cI:r -CI:1:'1 -

.l

'U

:1::-1

+ ::] + ::

[II

+ «(',

+ :: [:Z: + 11. + (C1 -

sin .r) ( -('I:r - :r:3

sin :1:) ( -Cl::r

::

- :r:i

- (.

+ :::)].

(2.38)

:r

./

-ci'

+ :: ) ]

/

-c,

Figure 2.5: Closing the feedback loop in the dashed box with ping" -0: through the int.egra.tor.

+a

and "baclcstcp-

32

DESIGN TOOLS FOR STABILIZATION

As always, we let l~ he an expJicit function DC 11 and design u to sat.isfy t.he elf inequality (2.18). For this reason, the cross-terlll x=, which is due t.o the preseJl('e of =ill (2.35a), is grouped t.ogether with 'U. This is possible because u is also muJtiplied by z due t.o the chosen for111 of \~. This is the second key feat.ul'e of bacl{...,t.epping. NolV we choose the control 'U to make l~... npgativc deIinit.e in .1' aud z. The simplest. way t.o achieve this is to make the bracket.ed term in (2.38) equal to -C2=2, where C2 > 0: 11

=

-C2= -

=

-c!d~ + CtJ; + COS.T)

.1; -

(Cl - Sill.l:) (

-cl:r - .1':"

-;1; -

(CI -

+ =)

sinx) ({ + cos:r -

.1·:1).

(2,39)

Wit.h t.his control, the elf derivative is (2.40) which pl'Oves that. ill t,he (x. =) coordinates the equiIibrimn (0,0) is GAS. In view of (2.34), the equilibrium (0, -1) in the (.1:,~) coordinates has the same property. The resulting closed-loop system in the (.1:, =)-coordinates is (2.41) Although writt.en in a linear-like form, this system js nonlinear. An important. structural property of this system is t.hat it,s nonlinear "system matrix" is the sum of a negative dia.grmal aud n skew-symmelric matrix function of x. This is the third key feature of backstepping, which will he (lxtremely useful in other designs.

Avoiding cancellations. The above control law is not the hest way I () achieve negativit.y of \~, because jt invohres at least one unnecessary canceJlation. A closer examination of (2.38) re,rerus th at the terllJ _.: 2 si n.1: need not be cancelled because it can be dominated by -C2z2. A control Jaw which avoids t.his caneellatioll is 'U

=

-C2= - X -

(c) - sjnx)

(-CI'T. -

x 3 ),

C2

> Cl + 1.

(2.42)

\\lith t.his control, the elf derivative is

(2.43) Although more complicated than (2.40), this fUJlction is el:lSily rendered negative definite by the choice C2 > CI + 1. The resulting system in the (x, z) coordinates preserves it,s skew-synunetric form

(2.44)

I j ~.

I

t .,

I

2.2

33

BACI{STEPPING

Tile simplified control law (2.42) is an illustrat.ion of design HexibiHties in satisfying the elf inequality l~"l :5 0 and at the same time avoiding uunccessal:V cancellations. In fact~ more detailed calculations show that the control law ca.n be further simplified to (2.45)

with (2.46) Using this control wc obtain

1.,

--CIX· -

2

.,

c.,:;~

-

,

Intcgrtttor backsteppillg as a general design tool is based assumption:

(2.47)

the following

011

Assumption 2.7 COIu;ide7' th.e system .1: rIJhere

;1:

=

J(:I:}

+ g(:v)u., 1(0) = 0,

lL E lR is the control input. feedback control law

E lRn is Ute state and

conli1~tJ.ousl1J diffe7Y~ntiable

11.

= O'(:r) ,

(2.48)

Then? exist a

0'(0) = 0,

(2.49)

and a smooth positive definite, radially unboundelJ junction V : JR.'I that (x) [f(.l:) + g(x)a:(:r:)] :5 _HT(X) ::; 0, 'r/x E 1R", j

all -a X

when; llT : JR."

-+

lR is

po.rliti1Je

~

1R. sltch (2.50)

semidefinite.

Under this assumption, the ('ontrol (2.49), applied 1:0 the system (2.48), guarantees global boundeuness of x( t), and, via the LaSalle-Yoshizawa theorem (Theorem 2.1}t the regulatioll of H'(:r:(t)): lim t-1'(x{t)

'-00

=

o.

(2.51)

A stronger cOllvergellce result is obtained using LaSalle's theorem (Theorem 2.2) with n = lRll: x(t) converges to the largest invariant set J\1 contained ill tho set E = {x E m." I H'(x) OJ. Clearly, if IF(x) is positive defiuite, the control (2.49) renders :r = 0 the GAS equilibriulll of (2.48).

=

Lemma 2.8 (Integrator Backstepping) Let the h'lI.dem {2.48} be a1J.gmented by an intc,qrato7-: (2.52a) j= = I(x} + n(x)~

e=

(2.52b)

u,

and suppose that {2.52a} sa.tisfies Assumption 2. 71lJif}" { E R

D·S

its control.

34

DESIGN TOOLS FOR STABIL1ZATlON

(i) If 1-1'(:r.) is positilJe definite. Ihen (2.53)

is a elf for the full ,t;1Istem. (2.52), thn.t is. lhet'f! C:l:ists a Jeedba.ck control = O'n(:l',~) 1lJhich 1'e'TIders x == 0, = 0 the GAS equililn7.u7lJ of {2.52}. One S1J.(!h. control is

e

'lL

8n

8V'

11 = -c(~ - a{;r» + 8.1' (x) [J(.1:) + y{.1:)el - al' (,r}J](J~). (' > o. (2.54) (ii) If l,r(a') is onlll positive sem:idefirtite, then the'l'C e:-cists a fcedlJad~ control 'which 1Y~nde7'8 l~"l ~ - B'o.('}:,~) ~ O. such that H/o(a:. e) > 0 'whenelJer 1-1"(.1") > 0 01' ~ =F It(:l'). This gua.mntees glabal bouncleclness and con-

lIergcnee of [ B. = {[

~~!~ J to /.he ia.yest im",,.;ant sd AI.

~ J E IR"+!

I

W(.,)

contained in "he set

= 0, (= o(x) },

Proof. Int.roducing the crror varia.ble

==

~ - a(:l') ,

(2.55)

and diffcrent.iating"' with respect to time, (2.52) is rewritten as

:r. = I{:r:) + g(.1:) [a:(x) + =1 .: = u - Baa (:t) (/(x) ,t'

(2.56a)

+ g(x) (a(.1:) + :;)1.

(2.56b)

Using (2.50), t.he derivative of (2.53) along the solutions of (2.56) is

all". (J + ga + g=) + =[ ao. -a -ax (f + g(O' + =»J avo (I + gal + z [Da: a1/ ] -a 'U. - -a {J + D (a: + =)) + -a,g ~

=

.r

II. -

:r:

< -W(x) +: [It - :: (f + 9 (a + :)) +

J.:

:~ oJ '

(2.57)

where the tcrms conta.inillg = as a factor have been grouped together. By tbe LaSalle-Yos]lizlnva theorem (Theorem 2.1), ~tlly cllOice of the control'll which renders V;, $ -H~(.l:,~) ~ -H'(J~), wit.h H~"l positive definite in z = ~ - a(.l:}, guarantees global boundedncss of :t, =, and E.. = ;; + o:(x), and regulation of lV(a:(t)) and =(t). Fm·t.hermol'e\ LaSalle's theorem (Theorem 2.2) .IOnce again, 11Ot.e thai tbe t.ime derivative ti ill (2.56b) is implelI1ented analyt,icnlly l\'ithout the need for a diffcl'entiator.

2.2

35

BACI{STEPPING

guarantees convergence of [ the set {[ :

~~g

]

to the largest invariant set. contained in

1E lR.,,+1 I I-V (x) = 0, == o}_ Again, t.he simplest- way t.o make

l~ negative definite in : is to choose the control (2.5,1), which rencl~rs t.he bracket.cd term in (2.57) equal t.o -c,: find yields (2.58)

Clea1'ly, if 1-1'(.1") is posit.ive definite, Theorem 2.1 guarantees the global asymptotic stability of .r = 0,.: = O. which in turn impHes t.hat. \-;(.r,~) is a elf and :I' = 0, ~ = 0 is the GAS equilibrium of (2.52). 0 While the choice of ('outrol (2.54) is simple. t.his control may not be ucsirable because it involves cancellation of Ilonlinearities, some of which may be useful. As illustrated by (2.39) and (2AOL the requirement t.hat ,~'l in (2.57) be made negat.ive by u allows C'onsiderable freedom in the ('hoice of control law u = a·a(.J:,~) such that

.

V~"L

DV" ] :5 -HT(X) +.: [DO' (La(.1', €) - o:t' (l + 9 (0' + z)) + D:r fl .

= -1-1'~l{.1·.~)

:5 O. (2.59)

We stress that t.he main result. of bacl\:st.epping is not the specific form of t.he control law (2.54), but rat.her Ule construction of a Lyapunov function whose derivative can be made negat.ive by n wide variet.y of cont.ro] laws. III t.his way, the design of a stabilizing st,at:c-feedback cont.roller is effectively rt'du('ed to satisfying t.he scalar inequality (2.59). Example 2.9 As a design tool, bac1 0 which makes the> polynomial q(s) = 8 2 + k2 s + ~:1 Hurwitz, and dellote p(s) = hI + h'!,s. 'Ve can choose /':J = 0 2 and k'2 = 2a, with a > ~7 so t.lmt the transfer fUllction Z(s) = p(.r;)/q(.r;) is posit.ive rcaJ. \Ve eRJl t.hen write (2.1068) as

.r

.r = + gu: f = g = _.r~i. Clearly. J' = 0 is a GAS equilibrium for :i' = I, so all the conditions of Lemma 2.13 are satisfied. Using V(:l.') = ;r'l ill (2.D7), WE' obtain t:he ('"olltrollo.w

(2.10i) The situation is quite difl·crent. when II .11'2 < O! t.hat is, when t.he transfer fl1m:{:ion Z(s) = p(s)/q(s) is llonminimmu phase. Then, Lemma 2.13 does 1101. apply. In fact l a detailed (.'aknlation givon in [85, EXi:lmple 3A] shows that, in this cuse the syst.em cannot be globallr st.abilized. 0 The FPR propelt.y is a passivity property. Its nouliuear counterpart. will he employed in the stabilization of t.he nonliuear ('ascade j:

=

€ =

+ 09(.1:, ~)u I(O:~) = 0: "I ~ Em'l: J: E ffi.n, /1 E R + B(€)u, y = h(~), h(O) = 0, € E JIlq, ·u Ern..

1(:1',~) m(~)

I

(2.108a) (2.108b)

Our key assumption is that (2.1 08b) can be rendered paSSi1Je or strictly plI.ssive (d. ApprmdL"{ D) via. fI, feedback trnnsformat.ioJ] lJ. = ~-:(~) + 7'(~)l"

Definition 2.16 The s1l,t;tem

{= m(O + ,B(€)'lL, y = h({), h(O) = 0, ~ is said to be feedback passive (FP)

E IR.q,

U E lR

(2.109)

iJ t.hp.Te e:t."'ist.r; a jeetlback t7Ylnsjo'T7n.atio11. (2.110)

snc/" tha.t.

the 7'eS'lJ.iting s1jstem. t = m{E)

+ .B(€)kUJ + /3(€)1'(Oll,

!}

= liCE)

f.r; pa.ssil 1c with a· s/.omge fwu~tion U{!) 1IJhich i.r; posif.iYJe definite and 1udiallu 1/ nbo1Jndetl:

lui ;lj(a)v(u)du

~

U(E(t)) -

U(~(O)).

(2.111)

Th.e sust.em (2.1 09) is said to be feed hack st.rictly passive (FSP) if the feedbad~ (2.11 ()) rentlc7's it stl'lct/U passi1.ll~:

f' y(u)ll(a)da ~ U(~(t)

~l

- U(€(O»

+ [' 1/:(~((1))rl(1, ~)

(2.112)

where 1/J(·) i.r; the posit.ive definite ilissipation m.te. As in the linear case, FP systcms of I;he form (2.109) must ha\"e relative degree one.

2.2

4i

BACJ{Sl'EPPJNG

Lemma 2.17 (Stabilization with Passivity) Let V·(.r) be n 7ndialll1 ?J.nbounded Ll1apu.nov juncii01J. J01'.1: = f(.l~, e) sat·it;Jying

av'

tJ;t~ (.1:) 1(·1:, {)

:5

-IV(:t):$

o.

V.I: E

n,t. v~ Em..,

(2.113)

and let (2.108b) be FP as in Definit'ion 2.16. Then. a LlIllpu7lo'l} Junction 101' th.e cascade. 81Jstem. (2.108) 'is

and

tI/.f!

COl'1'espoll.di'Ttg contl"Ol

JfJW

(2.115)

guarantees that (

~g?]

is globally b01lnded and Wllve'1I'. to the lamest in-

",,,'ian! set !II. cof!lained

jn

Ihe sct E.

= ([ ~ ] E R h+q

1

HI(x)

=0 }.

If

(a.JOB1}) is FSP, then (2.115) gnanm(ecs cO'f}weryence to Ihe lmyest illVari.'ll1t

= {[ ~ ] E m. +,/ h

set M. contained if! lite sct E.

111'(.,,) = 0, € =

o}.

Fi-

na1l1}, if (2.108b) is FSP ancllV(x) is positi11e dejirlite. tJml. is, if.i: = f(:l"~) has a GA.S eqnilib'lium. at ,l! = 0 1I.17:ifminly if} {, Ihen tll.e equilihrium :1: .:::: 0, t: = 0 of (2.108) i.fJ ailla GAS. Proof. We lil'st recall Theorem DA ill AppendL"{ D, WIlich states t.hat the negative feedback intercOIlllec:tion of two (st.l'ictly) passive systems is (stridly) passive, and of Lemma D.3, which s1.at.($ that. n (strictly) llfl.Ssive l'iystem with a positive definite and l'arHaJly unbounded storage fUllction has n. G{A)S e{luiHbl'iulll at x = 0 when it.a ext(,l'llal input. is set to zero. The closed-Joop system (2.108) with the control (2.115) is J(:l\ {) + g(:c 1 {)y

.1;

=

~

= m(~)

11

=

+ i3({)A'(~) + f3(~)r(~)tJ

h({),

(2.1Hi)

= - ~~ (3')g(J', {) .

l'

Let us now express (2.116) as the feedba('k illterCOJlllet'tioll of hvo pn.~sivc systems ~ 1 and ~2: ~l ~:.!

{~ = {; ===

h{~)

=

-I}.

=

L'

+ Y(,l:, {).IJ ~~ (3:)0(·l',~)

f(x,~)

m({)

+ p(~)k{~) + p(~)r(~),t

(2.11in)

(2.117b) (2.117c)

48

DESIGN TOOLS FOR STABILIZA'l'(ON

Vole already know that E!! is passive, since (2.111) is satisfied. To sho\v that E t is passive wit;h storage function V(x), we use (2.113):

. all (f + gy) :5 -t"'(X} + -gy all ax ax = -H'(x) +1111.

l' = -

(2.118)

tl we obtain

Integrating (2.118) on [0,

fat 17(u)11(u)da ~

l/(x(t» - l/(.r(O» +

10' Hi{:t(a»du,

(2.119)

which shows that El is passive since Hi(X) ~ O. From Theorem D.4. we conclude tbat (2.117) is passive with the positive definite and radially Ullbounded storage function Vn{x,~) = "(x) + U(~). Lemma D.3 then states that 3! 0, ~ 0 is a globally stable equilibrium of (2.116). To see tllat Hr(x} ~ a as t -!o 00, we difFerelltiate (2.111) and {~omhine the result with (2.113):

=

=

.

v,"l = :5

.. l' + U

811

ax (J + gy) + lI 811 -H!(x) + -a. 911 + yv = -t·V(x) , 3, :5

V

(2.120)

Then, LaSalle's theOl'em (Theorem 2.2) guarautees convergence to the set l\'7[n' If (2.108b) is FSP, we replace (2.111) by (2.112). Then (2.120) becomes (2.121) wh.ich, since ?/J(E,) is positive definite, guarantees convergence to the set Af."l' Finally, if H'(x) is also positive definite, we conclude from (2.121) and Theorem 2.1 that x = 0, ~ = 0 is GAS. 0

Example 2.18 Consider the cascade system:

= t =

x

-a:(1+~)+x3e

(2. 122a)

{u.

(2. 122b)

=

The choice of output y {2 satisfies all the conditions of Lemma 2.17. First, (2.122b) is FSP: The feedback U

e=

_~3 + ~11, 11 results in U(e) = !e~\ since

= -(+11

(2.123)

= f', which is strictly passive with storage function (2.124)

implies that

10' y(a)v(u)da ~

U({(t» - U({(O)

+ lot fl(a)da.

(2.125)

2.2

49

BACKS'l'EPPING

Furthermore, (2.122a) can be represent.ed jn the form (2.108a) with (2.126) and (2.113) is sat.isfied with V'(:c} = ~:r2, lV(;.,.) = _.1,2. Applying Lemma 2.17, we conclude that the cont.wl

=

u

=

-e -

I

(2.127)

."C.

=

guarantees GAS of :1' 0, f. O. Indeed, thf' derivative of thc df ,~,(x,~) = ~(X2 + ~2) is negative definite: •

l~

=

"(

-:1:- 1 +

e"e ) + .1:'I" ~-

-

fI-

I


O.

(2.148)

AloreOtJer) if Vex) a,nil H/(x) a7'e posif.ive definite, then the eq'lJ.ilib7'iuffl. x = 0, ~ = 0 is GA S.

Proof. Since the relative degree of tb~ subsystem (2.147b) is globally defined and equal. to one uniformly in x, there exists a globa17 chauge of coordinates of the form (2.90). in particular (Yt () = (Y, t/J(x,~» wit.h ~:f3 == 0, which trfUlsforms (2.147b) into

7This change of coordimltes is il global dilfeolUorphislIl under additional conditions of nod complct eness [13].

t~ollnectedness

2.2

55

BACKSTEPPING

We now consider t.he cascade system consisting of (2.147a) and (2.149a). If we lineru'ize (2.149a) with t.he feedback given by (2.80).

'll

= ( a~ t3

81 )

-1 (

'l' -

a;

tJl) m ,

(2.150)

we obtain iJ = v. Then WE' can apply LCJ11ma 2.8, wit.h 'v as the new control input, to guarant.ee global bOllndedncss of .1' aud .lJ and regulation of nr(:r(1)) and y(t) - o·(x(t)). From (2.149b) and the ISS assumption on t.he zero dynamics, ( is also bounded, and thus ~ and u are bounded. Since all solut,ions of (2.147) are bounded, we can apply LaSalle's t.heol'em (Theorem 2.2) with n = lR"+ 1, t.o conclude convergence to t.he set. J1(... Combining (2.150) with (2.54), we sec that a particular choice of control is given by (2.V18). From Lemma 2.8 we also know that if "(:c) and 1T'(.1:) am positive definit.e, then the equilibriulll.l: = 0, IJ = 0 of (2.147a) and (2.149a), which is completeh' decoupled fr01I1 (2.149b), is GAS. The fact that in this case t.he equilibrium ~r = 0, ~ = 0 of the cas(:ade system (2.1,17) is also GAS follows from Lemma C ...J by noting that the st.ate (:r, y) of the GAS systcm (2.147a) and (2.149a) is t.he input; of the ISS system (2.149b). 0

Lemma 2.25 relaxes the global stabilit.y assumption of Lelllma 2.17 to global stabilizability of :1: = 0 through 11. As in the case of Lemmas 2.13 and 2.23, however, the price paid for this generalization is t.he st.l'cngthcning of t.he FP assumption of Lemma 2.17 t.o t.he ISS asslImption of Lemma 2.25.

The following example illustrates t.he usc of block backst.epping as a design tool.

2.2.6

Example: active suspension (series)

VVe return to the active 511Spf"nRion e.xample, but in {'outrast to the parallel configuration of Figure 2.6, we now work with t.he quarter-car model of Figurc 2.7, where t.he hydraulic act.uator is connected in series wit:h t.he spring/damper system. s III t.his configuration, the piston position .l·a is determined from the equation . .ra

=

1 Q

A

.

(2.151)

Again, the control objective is t.o stiffen the snspension near its travel limits. Negleci;iug the wheel a.cceleration xw , t.he accelerat.ion of the car body is equal t.o the suspension acceleration .1.\; and is given by (2.152) BBoth of t.hese configurntions nre currently llsed iu 8('I.jve suspension research and design.

56

DESIGN TOOJ"S FOR STABlLIZATION

Figure 2.7: Quartel'-car model with .lie.nee connection of hydranlic actuntor wit11 plL.'tsi\"e spring/damper.

Combining (2.151) and (2.152), we obt.ain the following suspension equat.ion:

t

f

(2.153)

I

For the flow Q, we consider again the linear equation (2.74):

Q=

-CrQ + kr7v •

j

The system composed of (2.151), (2.153), ~uld (2.154) is a linear system which ran be rewritten in the form of (2.135) with XI = x s , :Z:2 = '*81 ~I = .ha, ~2 Q, U. lv:

=

=

:i:1 =

:1:2

=

--Xl -

~l

= =

A6

~2 ;1/

=

ell

I(a

j:2

Alb

-:1:'1

JHb -

1

I(a

Alb

+'". (2.155)

-Cr~2

€ 1

+ J.:£,u. Ca

+ Jl1b A

f

1

~ 2·

CJ earl y, the assumptions of Lemma 2.23 are satisfied, Sill{'e the (~1' ~2)­ subsyst.em is minimum phase and its relat.ive degree is one. It should he noted t.hat the assumpt.ions of Lemma 2.13 are also satisfied, since the (x 1, .h'J)subsystem is GAS whell1J = 0, and the (~11~2)-subsystem is FPR. Thus, if the objective were just stabilization, Lemma 2.13 would be applicable.

\

2.2

57

BACI(STEPPJNG

Vvo first. rewrite the syst.em (2.155) in the form (2.139) with ( =

= ,1:2 [\n en J~] - T1 :1'2 + JJ = -11 1 h 1 I.J

.1'1

.;'2

iJ =

(J\a Cal:r Cakr ---- ) ~',+--11 _~lbA l\fbA ~ A/hA

(2.156)

Ah - 0, which yields H'h = 1F"_1 + c,.{~,. - c\',·_d 2 • \V 0 or e2 =F al(X I ), Once again, if wo want "'2(.\11';2) = H't(x,e,) + C2 [e2 - O'J(X1 )f:!, we need

:~:(,Xlte!l,e3)

-j; 0,

'V)[l E

m. +t , n

'Ve2 E JR, 'Ve3 E JR.

(2.191)

In most situations we would avoid this requirement: by direct1y finding a'2(-~2) to saj;isfy the inequality (2.190). Proceeding in the same fashion, in the kth step we arrive at the actual control u. in Fk - l (Xk-It ~I..)

Jk(Xk -

1,

(2.19~)

~~., ll},

lvbere

,-

.o'\.k-l

[

= X'~'-2] J: ':.It-l

'

1':" ('''' I' ~'-1 ~'\~'-l,

1:) = [Fk-2( 1 r ('-..Yk-2,f.k-l) 1: t:)' JIr-l -\.k-:h 1o:>1.·-b ':.1.'

':.1.'

(2.193)

For the system (2.192) we use the Lyapullov function

\Jj..(x,elt .. '1~k) =

Vi-l("'\k-l)

+ ~ [f.I.' -

1 k

=

V(:r:)

-df

O'k-l(Xk

+ "I: rei - a'i-l(Xi-1)f . .... i=l

(2.194)

64

DESIGN TOOLS FOR STABILIZATION

The design is completed by finding a rontrollaw 'U

= a·,J.l', €1 •. -. I f.1.')

whirh makeR l'k :::; -H'k :::; 0, wit.h Hl'k

l~- = '~'_I + (€k -

(2.195)

> 0 when TT'I.·-l > 001' €I.' #- Q"'-1:

Ok-I)

[ii.. - aa~l!--l FI.'-l] ·\.1.'-1

~/'. ( ,,... ) D\'I.-_1 1.- (C < - T'1"'-1 ·\.1.·-2. ~/"-I + -a-- /,,-1 ",I.' -

01..'-1

Ck-l

+(~I.' =

0'1.'-1)

[fk(XI.--I,Ck, /1.) -

)

aa~~I.'_1 Fi.'-I] "'\.1.'-1

-l·\Jk(.\""I.-_},~"'} ::; O.

(2.196)

alice again, under the condit.ioll (2.197) we would be able to find u. = 0:1.- t.o yield I·V". = HI''''_I +Ck [~k - a:k-If. Howevcr, not only ('an (2.196) oftcn be satisfied even if (2.197) is violat.ed, but. even when this condition is satisfied, we may prefer a different choice of Hrl...

2.3e3

Block-strict-feedback systems

Lemma 2.25 eall also be ~tpplied rE'peateuly to dcsign cont.rollcl's for nonlinear systems which can be transformed, by a change of coordinates, into tlle blac/.:-

stl-ict-fccdba.ck form:

x

= /(.1:) + .(}(xhJI \1 = 11 (x, XI) + iii (.r, \1 )Y2 ill = hI (:\:1) :\2 = 12(''}:, "I, :\2) + li2(·l~, \'1. \'2)/13 112

;;::;;;

"2(\2)

Xi

=

Ji(:t,'X.J, ... , Xi)

lli

=

hdxi)

= =

!P-l(:l', \11···, XI.--d hk - 1 tn-I) l"·(:l~, x) + 9k(X, X)u hll\'J..·) ,

\k-I YI.·-I

:\} = y". =

+ iii (.1:, Xl, ... ,:\i )Y;;.l

+ 9"'-1(.7:, \h'"

,:\'I.·-l)Yp

(2.198)

2.3

65

RECURSIVE DESIGN PROCEDURES

where each of the k subsystems wit.h stt1.te Xi E RRi, output 1li E nt, and input. Yi+l (for COlWOllicnce wo denote 1/1.:+1 == u) saUsfics t.he following conditions:

(BSF-l) its l'elative degl'ee is one uniformly in :1:, Xl!' " t \'i-l, and (BSF-2) its zero dynamics subsystem is ISS with respect to x, Xl,'· .• '\i-I. Yj' Under condit.ions (BSP-l) a.nd (BSF-2), the syst.em (2.198) can be t.ransformed into a form reminiscent of the strict-feedhac1( form (2.165). In particular, (BSF-l) is equivalent to

Bhi 9j - ..J. -8 r 0 , \JV:\:I Xi

E

lRu I, . , ... , \J v Xi E

]Rill

1

= 1, ... , h'I •

(2.199)

This means that for eacb Xi-subsystem ill (2.198) there exists a global chaugp of coordinates hi;, (i) = (hi(,d, q);(:c, Xl:' - .• \i»), with g~:Oi == 0, which transfor111s it into the normal form (2.149):

A

(i

=

fi(x, y], (1, .. ,y;, (j) + 0;(·1',/11, (1 , .. dj;, (i );IJi+l i-I

8q,'

j=1

Al

(2.200a)

_

1: 8 .'.(x,A l!"',Xi)[fJ(J:,A.1, ... ,lj)+liA.}:,xt",.,:\:j}11i+1] fJt/Ji +a(Xt,"\:ll"" xi)fi(:r:, tll·,·, Xi) Ai

~

ii(.X,XI"",Xi-hYi,(;)

A

il1:,llb (II""

Yi.-h (i-I, Ytl (i).

(2,200b)

\¥ith this change of coordinates, (2.198) is transformed into :i;

YI

= =

il2 =

J(:c:)

+ g(X)1I1

11 (~l", 111, (I) + 91 (.1', Yl, (1 )y:! h(x, Ytt (I, l/2t (2) + 02(3:, Yl I Cit Y2~ (2)113 (2.201)

Jik-l

= iJ.--l (Xdlt, (t"

ilk

=

(1

=

.. , Yl'-b ("-1)

+ 91.:-1 (:£,:111, (11' , , ,1Jk-h (k-dm·

.f".(x, YI (1,' .. ,y"., (l') + 91;(·1:, 111, (II' .. ,Yk, (,,-}-u I

~l(X, VI, (1)

66

DESlON TOOLS FOR STABILlZATION

If the zero dynamics variables (1, ... , (Ii \vere not present, (2.201) would be identical to the strict.-feedback form (2.165) with t;i replaced by Yi. Helice, the design procedure of Sect:ion 2.3.1 can be applied Tnutlltis mu.tl1,ndis to (2.201). The presence of (i genenl.tes a few lIew t.erms and requires a modification of the proof of boundedlless as well. First t.he bounded ness of :r:(t),111 (t), . .. , IJIt(t) and t.he regulation of TV(.r(t)). Yl (t) o.(x(l)), .. , ,Y1.'(t.) - Q'Ii-I(:z:(t),.IJt(t},(l(t), ... ,1I1.. (t),(".(t)) is esl.ablisllf.=-d via t1. LytlPUllOV-like argument llsing the fUnct;ioll Vi. (:l:, Yh ... , Uk) = F(:1:) + tE~=1 [11i- Oi-t(:t:,Yl,(l, ... ,Yi-l,(i-l)f· This implies that 111 is boundcd. Then, the houndedness of 112,"" JJk. (l,"" (I.' and u. (and, consequelltly, t.he boundedness of Xl, ... , Xk) is est.ablished via. ~1.n induction argument fol' i 2, ... , I,~ + 1 (with 11k+1 == '1/.); If Yl, ••. , lli-l and (I, ... I (i-2 are bOllnded, t:he ISS stabilit;y assllmptioll on the (i_l-subsystem guarantees tlUl1. (i-J is bounded. Tbis implies tllat. o'l'C t 111 t (It ... ,Yi-l, (i-I) is bounded, w]lidl implies that IIi is also bounded. This completes the indurt.ioll argument and shows that Xl, ... , :n·, 'U are bounded. since they can be expressed as smooth functions of x, 111 , ••. ,11k. (I, ... , (,..

=

2.4

Design Flexibility: Jet Engine Example

\Ve ha\'e presented several backstopping, cascade, and block-baekstepping design t,ools and procedures which make the design of nonlinear controllers systematic. 'Ve st.ress, however, that 'systematic' means neither rigid nor dogmatic. Following the same principles, various modifications DC the tools and procedures are possible. The recursive construction of control LY~lpl1nOV fUl1ctions is flexible, and so is the choice of stabilizing functions. At present there are no specific opt:imalUy criteria to help us select the best member of the bacicstepping controller family. However t there are cert.ain applic~1.tiol1s-orjented gUidelines which in most cases will lead to tt simpler and more robust controller. It. is clear from the design procedures that the complexity of the controller increases with the number of recursive steps. Much can be gailled if the number of steps ean be reduced. It is also desirable to satisfy the V-inequalities with as few cam:ellations as possiblE'. Exact cancellations can rarely be implemented and cancellation errors may lead to llonrobustness. Additional analysis lllay be required to identify useful nonlinearities and avoid their call('ellatioll. For this purpose, a more flexible construction of control Lyapunov functions can be cmplo)red. Some of these guidelines nre now illust.rated 011 a design example of major practical interest. 9 9Tbis section is based

011

I(rsLic and Kokoto\'ic [105J.

2.4

DESIGN FLEXIBILl'fV: JET ENGINE EXAMPLE

2.4.1

67

Jet engine stall and surge

Jet f'ngine compression systf.'ll1S (Figure 2.8) have recently become t.he subject of intensive control studies aimed at. understanding and preventing two types of instability: mt(lting stall and surge. Rotat.ing stall manife8t.5 it.self as a region of severely reduced How that rotates at a fraction of tho rot.or speed. Surge is an a..xisymmet.ric pumping oscillation which can cause flameout. and engine damage. The simplest model]!) tha.t describes t.hese instabilities is a threc-st:at.e Galerkin approximation of the nonlinear PDE model by IvIoore and Grt:'itzer [137]. This model exhibits bifurcations analyzed by ;McCaughan [128], and was used by Linw and Abed [113] for a nonlinenr feedback control design. Cont.rol designs with experiment.al verifications nre reported in Paduano et aI. [149] and Evoker and Nett [34]. We will design a ff'edbnck contl'oller for the three-state model ~

= -w + wc(cp) - 3ellR 1 ql = {32 ( O. \¥e can now use V; = ~R2 + ~¢~ and proceed to t.he second b1;op of backstepping. This design, whic1i is Iori to the reader, is completed in three steps.

2.4.2

A two-step design

'Vheu at the first step of a three-step design the stabilizing fUllction a-I is zero, this suggests thnt a. simpler two-step design is possible. vVc therefore considel' that the init.ial subsystem consists of the fit'st two equations (2.208) and (2.~09). This subsystem is ci1.Scaded \vith the integnttor (2.210) and the whole system cnn be viewed as being ill the ct1.Scade fOlm (2.147) where (R, c/J) is x and l/J is~. Following Lemma 2.25, we need to satisfy Assumptioll 2.21 using 1/, to stabilize t.Iu-" (R, tP)-snbsystel11. This can be done using the semidefinit.e \'1 = ¢'l because the R-equation is ISS as call be seen from (2.208)~

R :S -uR'J + 2uRI4tI- 0'4>2 R ~ _~R2 - ~R(R - 4/¢1) ,

(2.211)

2.4

69

DESICN FLEXIBILITY: JET ENGINE EXAMPLE

which implies that when R(t) > 414J(t)I, R(1.) rlecays fastpr than t.he solution of lb = -~UJ:.!. Hence, an upper bound for R(I) is (2.212) CIC81'ly: R is hounded if lP is bounded. and R 0 if t/J O. For ·tt, as the virtual ('antral of 1.he ¢)-cqualioll we choose -jo

-jo

(2.213) \Vith this choice we havE' avoided canc('llatioll of the useful Ilonlinearities _~tPa and -3Rl/J. Substit.uting·J, 1/.' - l.\'(rjJ, R) in (2.209), we get.

=

.

¢ = -CllfJ At the scconu st.ep we differentiate obtain

~=

-lL -

(CI - 3lfJ)

(-if) - ~q)2 -

21tP''i -

3Rf/J -

J, = .,j; -

~q)3 -

-

~}

ci:(r/J, R)

3tjJR -

.

=

(2.214) -'U -

~~~ - g~R

and

3R) + 3uR (-2riJ - (p2 - R) .

(2.215) The control law is now rhosen to render the deorivntive of l~ = tjJ2 +.Jt2 llC'gati\'e definite:

11.

=

('';},;fi -

q) - (el - 3rb) ( -1/' -

+3t1'R (-2fjJ

_l/J2 -

~tP2 - ~rP3 -

R).

3¢R -

3R) (2.216)

Not.e that; in the (R: 1/>, -Jr)-spt.u:e the Lyapllnov function V; is only posit.ive scmidcfinit.e, which is allowed by ASSlllllpt.ion 2.21 and Lemma 2.25. In ('011trost., for the threoE:'-step design out.lined ill the preceding subse!'tiol1: the final elf would be 1"2 = R2+ljJ2 +·lip. and it.s derivat.ive would involve terms from the R-eqllat.ion. This would make the cont.rollaw morc complicat.ed than (2.210). Wit.h the control (2.216) i,he result.ing feedback system is

R

=

-uR2 - crR(2¢ + tjJ2)

~~ =

- (CI

~,

(jJ -

=

+ 4rp2 + 3R) cP _.J,

(2.217)

('2.J. .

The equilibrium (cp. J,) = 0 of t.he (li>,0)-subsystelll is GAS fol' an R ;::: O. In addit.ion, R(t) 0 berause of the ISS-property (2.212). This means thnt. surge and stall are suppressed within t.he region of validity of t.his jet. engine model. -jo

70

2.4.3

DESICN TOOLS FOR STABILIZATION

Avoiding cancellations

Alt.hough we have already avoided cRncel1ation of several usefulllonlinearities, a further systematic simplification of the above controller is possible by a bet.ter choice of o' and a more flexible construction of the control Lyapunov function \'2. We illustrate this possibility on the no-stall (R = 0) part of the jet engine model (2.208)-(2.210), rewl'itten here as (2.218) (2.219)

To design a stabilizing function 0:($) for .t/J in (2.218) with respect to Vi (q;) = ~ ¢2, we examine the inequality lij, < Ot that is,

3 - 2tP 1 3] < 0 , "2t/J-

¢ [-0: () c/J -

I)

't/ ¢

#- o.

(2.220)

\'Ale already noticed that -!t/J3 enhances this inequality and we did not cancel it in (2.213). But if we go iurthel' and rewrite (2.220) as (2.221) we recognize that (tP + ~) t/> is also useful and should not be cBl1celled. Therefore, 0:( rP) is choBen to be linear

-4

2

Cl

The derivative of Vi(t/»

> O.

(2.222)

= 4rJ>'l is then (2.223)

where {b = 1/, - a:(¢). In the second step we denote Co = we get

Cl

+1, and by differentiating ¢ = 1j;- co4> (2.224)

Proceeding as usual, our choice for the Lyapunov fUllction 1'2 would be quadratic, V2(rJ>, '¢I) Vj(q;) + ~.,jj2 = !t/J2 + ~tfo2t resulting ill ti2 $ -ClCP'J. +

=

2.4

71

DESICN FLEXIBILITY: JET ENCINE EXAl\IPLE

;p [-11. + Co (1/) + ~qJ2 + 4rf>3) -

t/J]. To satisfy the inequality ,~ < O. the

COll-

trollaw would lutve to cancel t.he llolllineru·it.ies CO~t/>2 and co~tb:l. ''''Ie will now illustrate the construct.ion of a more flexible Lyaptmov function (2.225)

where F(·) is yet to be selected as a continnously differentiable, llonnegat:ive, and increasing function, d~~;'d ~ O. III view of (2.223) and (2.224), the derivat.ive of (2.225) satisfies -Cl fIJ'2 -

l/J(P + dF( l/I}

(-CI rf>2

dl'l 3co.,

- ifJ-J;)

+1/'- ( -u + cO'if' + TCP- + Co2 tP3) . After collecting all the terms with ,,/,,2 -CI y~ -

(2.226)

,¢, we get

,/,'2 dF(Vi) CI If' - - -

dVi dF(l'd

3co Co +I/J- ( -II. - qJ - --q) + Co1P + -l/J~ +-l/J d''i 2 2 I)

3)

(2.227)

In addition to the choice of a control law for 'u, we now have the freedom t.o choose d~~~;I). With t.his choice we will avoid the cRncellat.ion of t.he cubic: t.erm

,r/J3

by

'lJ..

\Ve simply select d~~;'I) to eliminate 'fl/J:~: (2.228)

This yields

F(V) = 1

Co l/2 2 1

=

Co

8

4>"

(2.229)

'

and the resulting Lyapul10v function (2.225) is nonquadratic:

11') co ... 1'2(¢,1/J ) = '2tjr + SrP

1

+ 2' (1" -

'l

col/Jt .

(2.230)

Substituting (2.228) into (2.227), we arrive at •

l'2 ~

3co 'l) -ClrP2 -CI CD2 rP'I +1/J- ( -u-4>+cor/J+Tl/J.

(2.231)

The design could now be finished by selecting a control u which cancels -"'l/J2 • However, even this cancellation can be avoided because of the strong stabilizing in (2.231). Completing squares, 3Cn l/J2.;fi::; Cl~c/J·I + gel ?fD..~2, we get. t.erm CI!:fiq,4 2 ~ (2.232)

72

DESIGN TOOLS FOR STABILIZATION

Hence our cOlltroll}lW can be selected to bt' lineal', £'2

> 0,

(2.23a)

Clnd yield (2.234) This proves that. tht" equilibrium t/J

Denoting ~'l

= 1+

C:'!Co

= 0, l/J = 0 is globttlly alWmptot.icnJly stable.

9~ 8rt

+-

t

(2.235)

we rewrite (2.233) ill t.hc mare ('ompact form (2.236) and obt.ain the rIosed-loop nyRtem

(2.237) (2.338) For comparison, we also derive a feedback linearizing controller,

(2.23f)) which makes the SystPlll {2.218}, (2.219) appear lillt"ar in i:he coordillat.es :\1 = The cout.roller simplificat.ioll achieved wit.h backst.epping is impressive: \Vhile the liueal'izillg cont.rol (2.239) grows aR q,s ttud 'ij14J2, the backstepping controHer {2.23G} is linear. The improvement over the ('ont.rol lnw {2.21G} in Section2A is also signific-ant: (2.216) grows as cP'· and 1/J(P because the quadratic nonlinearity wa.c.:; cttllcelled at the first st.ept so tbe cancellat:ion could not be avoided at. t.he second step. III t.he remainder of the hook we will not assullIe t.he presence of usefnl nonlillearities. Ho\Vever~ it should always be ulldorstood that. whenever sueh additional information is available, backstepping designs should incorporat:o it.

13"'

("~~!.)

=?

> II~~;;.. Combining

Ii < 0,

(2.274)

This llleans that if l.r{O)1 .$ ''iiI (jl~!;"'), then

V{x{t)) ::;

')':10,3"'

("~!!.)

,

(2.275)

which in t.urn implies tlu1t (2.2i6)

If, on the other band, 1.1:(0)1 implies

> 13]

(!I7.ll?s )

I

then 1/(:r(t» :::; \l{:r(O))! which

-

(" ....,,--) (( Combining (2.270) and (2.277) leadR t,o the global uniform boU)ulcc1ness of

x{t);

11:"11", ::;

max {,,' 0 'Yo 0 1'3"

("~!;'

),

1'j"' 0 12 (I:r{O)I) } ,

(2.278)

while (2.274) and (2.271) prove the convergence of .r(t) to t.he residual set defined ill (2.270). Finally, the ISS property of the closed-loop system with respect to the dis~urballce input ll.(J:, lJ, t) follows from Theorem C.2. 0

80

2.5.2

DEs[GN TOOLS FOR STABILIZATION

Backstepping with uncertainty

Lemma 2.26 deals with the case where the unrerblinty is in the span of the control '1/., i.e, t.he matching condition is satisfied. Combining Lemma 2.2B with Lemma 2.8 allows liS to go heyond t.he mat.rhing rase, as t.he following example ill us t, l"t1t es.

Example 2.27 Consider the system j'

~

= ~ + :l'~ arctan e6()(/.) = (1 + ~2)U + e:re~D(t) ,

(2.279a) (2.279b)

where 6.0(1.) is a bounded tim('~varyillg disturbance. Clearly, the Ullcert·ain terms in (2.279) arc not in the span of the control 'lI.. Therefore, we will design n static nonlinear controller in two steps, combining nonlinear damping and badcsteppjng.

Step 1. The staltillg point is equation (2.279a) and the choice of a virtual ('ont.rol vRriJ:luJc. Clearly, { is tlle only choice. The lact that E. is also present in the U1u'ert.aill term docs not. present a problem, since it enters that term through the bounded fundion aJ'ctan(·). In the notation of (2.268), we 11avc (2.280) The uncert.ain nonlinearity /);.1 (~,/.) is bounded: (2.281) Hence, Lemma. 2.26 can bo lIsed to design a stabilizing function for~. The unperturbed syst.em in this case would be the integrator ± = ~, for which 1:t elf is given by "(x) = !x2 and the conesponding control is o(x) -CI'T.. From (2.269) we have (2.282)

=

which results in (2.283) with the error variable

z defined as in Lemma 2.8: (2.284)

The derivative of l,l(.:z') along (2.283) is

,i-

=

by (2.280) by (2.254)

(2.285)

2.5

81

STABILIZA"rrON WITH UNCERTAIN'l.'Y

which confirms that if :; == 0, that is~ if ~ were the actual control. tben (2.282) would guarantee global uniform bOl1ndeduE'ss of x.

Step 2. Using the error variable

=from (2.284), the system (2,279) is l'cwritt.eu

as :i:

=

-

::::

(2,286a)

=

(2.286b)

where the partial ~ is comput,ed from (2.280) and (2.282): 80') 8:.1.'1

= -c) - 1i1..i!-.(J:J:

[;l'tpi(:~')] ::::

-Ct -

5Iil:c1.

(2.287)

If the ~o(t)-terII1 "Were not present in (2.286b), then LeIllllltl 2.8 would dictate the Lyapunov [ullction

I"

112 ( x,) e = 2' .1;" +

1., '2:;-

=

(2.288)

and the following choice of control:

1:

" = li("" {) =

£;2

[-C2= + ';:' { -:r] .

(2.289)

To compensate for the presence of the ~o(t)-term in (2.28Gb), Lemma 2.26 is used again. From (2.269) we obtain (2.290)

which renders the derivative of 1l2(x, e) negative outside a compact set., thus gnaranteeing bounded ness of x(t) and ~(t):

V2 by (2.285)

by (2.286b) and (2.290)

=

Ii" + zz

::; "'x _ £.yr.2 ..,

=

-Ct X2

..

+ 1I~11l~ + --:.... 4lbl

+ II~I fI~ 4lil

+= { -c!!z + [e.( -

h~2:: [ e:re -

;::£2

arctan

a.r x 2 arctan e]2

80'1

e] ~(t)}

82

DESIGN TOOLS FOR STABILIZATION

(2.291)

by (2.254)

o The combination of Lemmas 2.8 and 2.26, illustrated in t.he above example, is now formulated a..., a.nother design tool.

Lemma 2.28 (Boundedness via Backstepping) C07uJidcT Ihe system :i' = 1(.1:)

+ g(J.')u. + F(J;)~I (:1', Il, t) ,

(2.292)

where :,. E IRn. II E JR., F{:r) is an. (n x q) ma.tri:r: oj known smooth nonlinecu' Junctions, c1.1l·d ~1 (:1.', II, t) is a (q x 1) vectol' oj uncertain 1wnlineu7i.tie.i which 11.7'f. uniformly bounded J07' nil vallle.~ oj x, tt, t. Su.ppose lha.t i.here C:L'ists a. Jcedback contl'Dl 'lL = o{l') that 7-enders x(/.) globall1J uniJ01'111.111 bounded, a.nd that this is est(J,blisheti 'uia positive definite and 'f'IJ.diallll U7l.bounded junclions ll(:l:),IF(x) and (]. constant h, such that all

D.'C (.1.') [f(:&)

+ g(.1·)(1(3:) + F(:r)d l (3:,11, t)]

~ -''I'(x)

+ b.

(2.293)

Now con.r;ider the aUflmentcci .'1y.r;tem .1: = E;.

=

j(.r} + g(x)~ + F(x).6 t (.1", u, i) 'U + cp(:z:, ~)1'~2(X, €, u, I.) ,

(2.294a) (2.294b)

when! 0

(3.53)

T(x, iJ)

(3.54)

J = -r [:: (x, 11)F(Xf (t; - a (.r., 11)),

(3-55)

where {j is a neul cstima.te oj fJ, r = r 1' > 0 is the adaptation gain maida:. Under Assumption 3.1, this adll.ptive cont1TJller g1l.c1.mntees global bo1t7l.dcdness oj x( t), ~(t), D(t), 0(1) a.nd 7"C!J'ull1.tion of fI'(J:( l), t?(t)) and ~(t) -ll'(~l:(t), iJ( t»). These ProlJe7'tie..r; can be e.r;tablished 1JJith lile Lyapunov ju.nction 1 1 -" 1 l~(:l:, ~t iJ, 17) = "(x, t~) + 2' [{ - O'(:l~, ,a)]- + 2(8 - '0) r- (8 -19) . (3.56) I)

e-

Proof. vVith t.lm error variable == 0:(;1:,19), (3.52) is rewritten as x = J(:I:) + F(x)() + g(x) [0:(.1;,11) + ~] (3.57a)

z

=

u - : : (x, iJ) [J(.'l:)

+ F(;r)O + g(.1:) (O'(x, ·O) + :;)] - :~'T{:r, .[)).

(3.57b)

Note that in (3.Sib) the derh'ative of {) was replaced by t.he update law (3.54). Introducing a new parameter estimate 11, we augment the Lyapullov function:

=

F(a:,·tJ) + _?1:;2 + !(8 -- ii)Tr- 1 (8 - 11) . ... 2 Using (3.51), it is easy to Rbow that the deriva.tive of (3.58) satisfies l'o.(X'';1 0,19)

all

.

(3.58)

all'

Va = a3.~ (J + FIJ+ go + g:;) + {J.t) T +: [II -

=

81'" 8J.:

00' (J

8x

lY (x I 19)

+ cH) T

+ F8 + g(a: + z}) - a~'T + av' 9] _,nTr- 1 (fJ af) 8:r

+ z [u. -

- [:: F: + jT

:;T] - ilr-'(9 - ;i)

811

(f + FB + go')

+:: [u 5 -

~: {f + F9 + 9(0' + =)) -

r-']

i9)

oa (f + F';9 + g( a + .:») _ 80: T + at' gJ o.T at) ua' (9 - 0).

(3.59)

Tlle (8 - 79}-term is now eliminated with the update law (d. (3.55))

~

=

-r (ao~F)TZ' u:J.

(3.60)

98

ADAPTIVE BAC1{STEPPING DESIGN

and the control (3.53) is cbosen to make the bracketed term multiplying :- in (3.59) equal to -cz (ef. (2.54)): IJ

=

aa ( -

-c=+ 8J.' f+FtI+g(a+=)

) + 80' al' 81?T- a~.g·

(3.61)

This results in the desired l1on]>ositivity of l~:

\~ :::; -rV(:L',1?) - c:2

:::;

o.

(3.62)

From (3.56) and (3.62) we conclude that "'(3:,0), i9 and z are bounded. By Assumption 3.t this means that. x(t) and l'(t) are bounded. I-lel1t'e,'; = z + a(.l:, ·0) and It are bounded. By Thorem 2.1, the boundedness of all the signals combined wit.h (3.62) proves the reglllat:ion of H'(x(t), t9(t)) and =(1).

o

3.2.2

Adaptive block backstepping

We now extend the Adaptive Baclc.stepping Lemma (Lemma 3.2) byaugmenting the initial system with a relative-degree-one nonlineaT system whose zero dynamics subsyst.em is ISS, just-like we did in Chapter 2, Lemmas 2.8 and 2.25. The adaptive counterpart of Assumpt.ion 2.7 was Assumption 3.1. "Ve now formulate the adaptive counterpart of Assumption 2,21, with analogolls changes in the properties of l/(x, t?) from Assumption 3.1,

Assumption 3.3 Suppose Assumpti.on 3.1 is 'Volid, b,J.t l'{x,19) is onl,} positi'Ve semidefinite. and the closed-loop system {3.49} 'U1ith the atJ.apf.ilJC controller (3.50) h{t.s the lJropert1} that x{t) a.nd 11(t) are bounded if l!(x(t), tI(t)) is bounded. 0 Under this assumption, the control (3.50), applied to the system (3.49), guw'antees global bouudedness of .t·(t), t9(t) and, by Lemma A.6, regulation of

IV(x(t) , O{t)).

Lemma 3.4 (Adaptive Block Backstepping) Let the sJ}stem {3.49} be augmcnted by a 11.o71.linear system llJhich ill linear in the un/""1lo'Uln param.ete7' 'Vector 0, (3.63a) :1: = I(:,;) + F(x)8 + g(x)y (3.63h) € = 1n(:l:, {) + l\1(x, {)O + ,8(x, e)u, y = h(eL

e

where E :IRq, and suppose that {3.69b} has rclati11e degree 071·e 1.t.niforrnly in x and tllat its ze1TJ dynamics SUbsystem is ISS 1lJith. respect to y and x. Under' Assumption 3.3, the feedback C07l.trol tJ.

8h = [ a{ (';),8(:z:, e)

]-1 {

ah -C(l} - o'(x, 19)) - 8e (e) [m.(x, e)

+ l\l(x, {)i9]

] + aCt + 80: ax (x, 'D) [J(x) + F(x)tJ+ g(x)y af) T(x, ·0) - av ax (x, 11)g(x) } (3.64) I

3.3

99

RECURSIVE DESIGN PROCEDURES

with c

> 0 and 19 a ne'lIJ estimate of (J, along with I.he 1J.1Jda.te laws

iJ = T(:r,19) !..

f)

=

r

(3.65)

[8h a~ (~)}.J(X,~) -

with the adaptation go.in matrix r =

1 (1) -

80: u:r (.r, {})F(.r)

T

0:(:1:, 17)) ,

(3.G6)

r T > 0,

guam.ntees global bO'lJ.ndednes.fl oj .r(t), ~(t), 19(t), lj(t) and regulation oj l'F(:r(t), ·tI{t)) cwd ~(t) - a:(:r(t), {}(t)).

Proof. As in Lemma 2.25, we employ the change of coordinates (u, () = (h(~), 4>(:r:, ~)), with ~:/3 == 0, to transform (3.63b) into the normal form ;lj

= ah D~ U) [m.(:r:,~) + 1&1(:r:, ~)O + f3(x, ~)1I] a~

( = D:r (J',~) [1(.1') + F(x)O + g(:l:)'lJ) + l::,.

<po(x, y, ()

(3.67a)

a~ a~ (:t,~) [m(:r,~)

+ cJl(x, y, ()tJ .

Introducing a new parameter estimate " =

to rewrite (3.63a) and (3.67a) as x = f(x)

1j

(3.67b)

ii, we lise the feedback t.ransformat.jon

r {II - ~~

(~~p =

'U

[m +MD]}

(3.68)

+ F(.1·)9 + g(:I:)1/ fJh

+ AI(x, ~)B]

+ a~ (~)l'f(:r:, ~)(tJ -

(3.69a)

-

(3.69b)

17) .

We now apply Lemma 3.1 to (3.69). The only difference between (3.69) and (3.52) is the presence of the additional parameter error term ~~ 1&/(0 - 0) in (3.69b). This term can be eliminated in ,~'l by adding the term -r(~~llJP'(y­ ex) to the update law (3.55). Combining this modifi('ation with (3.68), we see that the resulting adaptive controller is given by (3.64)-(3.66). This guarantees the boulldedness of x, of), 19, z and the regulation of H'(:t,19) and z. Hence, 1J = Z + Q·(x, '19) is bounded. Then, from (3.67b) and the ISS property of the zero dynamics, ( is also bounded, and thus ~ and 1I are bounded. 0

3.3 3.3.1

Recursive Design Procedures Parametric strict-feedback systems

Through repeated application of Lemma 3.2, the bacl\stepping design procedure is now generalized to nonlinear systems which can be transformed! into IThe coordinate-free charncterizat.ion of I.bcse syslems in t.crms conditions is givcn in Appendix G, Corollary 0.15.

or differential geometric

100

ADAPTIVE BACKSTEPPINC DESIGN

Figure 3.4: Block diagram of it. third-order parametric strict-feedback systcm with

P(x) = 1. The nonlinearitics depend only on varia.bles which llre "fed back." the paramel1"i.c .drict-feedback j07m

:h =

3:2

.1:2 =

:t'3

+ V'I(xJ)O + I.{)I (x 1, x2)6 (3.70)

i:n-l

=

:1:'1

=

Xu + tp~-l (Xb" • ,xtl-dO ,8(x)u. + V'~'(X)O,

where j3 (x) -:F 0 for all $ E 1Rn. The reason for the name "parametric strictfeedback" can be deducoo from tIle block diagram jn FigtlJ'e 3. t 1, whel'e, except for the integrators, there are only feedback paths. For systems in the fonn (3.70)1 the number of design steps required is equal to the degree 11 of the system. At each step, an error variable ::i: a stabilizing fUllction ll';, and a parameter estimate fJ, are generated. As a result, if a system contains p unknown parameters, the overparametrlzed adaptive cont:l'OIler may employ as many as p 11 parameter estimates. A schematic representation of this design procedure is given in Figure 3.5, and the resulting expressions axe summarized in the fonowing theorem:

Theorem 3.5 (Parametric Strict-Feedback Systems) For the sy.'Jtem (3.70) with. {3(:c) :F 0 f07' all x E IRn , consider the adapU.1Je controller' (3.71)

i = 11 " when; iJ i E lRl' lIte mu.ltiple estimates

of () r

rT >

n1

'l'

(3.72)

0 is the adaptation gain m.at7ir., and the varia.bles ::; and the stabilizing fun.ctions Gi, i = 1, ... , 1l t 1

=

3.3

REcuRsIve DESIGN PROCEDURES

----r -=.

,

I

I

I I

I

101

I

_.J

0'1

'- _____

~

___

Zq

I

-~.J.

ll'::!

~a:'

-

11-

U

Figure 3.5: Thc design procedure for overpa.rnmetl'i~ed schcmc.c:;. Each step generntes an error variable =i, a stabilizing fUllctioll Q;. nnd n. new cstimate 1?i of the unlmmvll pILT8metel' vedOl' 8.

defined by the following recu1'si'ue e:r.p'ressions (with C; > 0 being design constanl.s, and 4:0 :: ao == 0 '1I.S(1.(1 f01' notat.iDnal c01wenience):

aJ-e

This overparamct'li.=ed adaptive cO'1l,t1'(Jller ,Q1£lI1'llnlees global bO'Ulldetlness oj x(t), 'l?(t), ... ,'l?n(t), and regulation oj:t'.{t) ancl:r'i(1)-,l:Y, i = 2, ... ,n. where ':1'1 = -OT tpi-l (0, J;~t·· . , a:l_I)' Proof. Using the definitions (3.73), (3.74) nnd denoting Xo == O:u :: 0, P(.l:)ll~ the derivative of the error variable =j, i = 1, , .. ,TI, becomes

:1'11+1

==

102

ADAPTIVE BACI{STEPPINO DrnSJGN

The choice of control (3.71) guarantees that system can therefore be expressed 88

':,1-1

=

Zn

=

fJ i

=

Zn+l

== O.

The closed-loop error

or, equivalently, in the matrix form =1

Z2

d

==

dt

-Cl

1

0

-1

-C2

1

.,.

0 -1 -Cn-l 0 -1

0 0 [ WI

+ ~

0

rlt

: tJ n

=n-1

1

Zn

-C n

T

- f)2 [ 8 9d,

W2

1

(3.77)

0

0

~ [:~ 1 = [1

:!:1 %2

0

Zn-1 "'71

0

0

WI!

8 - On 0

0

r 0

11

[I

0

1 [ -, 1 ~ft :

W2

0

z..,

;,,'

3.3

103

RECURSIVE DESIGN PROCEDURES

where we have used the convenient notation i-l

WI

=

'PI ,

'Wi

= 'Pi -

L

00'i_l

-a--'Pj

I

i = 2, ... , n .

(3.78)

Xj

j;::;;;1

Tbis system has two important properties: (i) The z-system matrix in (3.77) has negative diagonal and skew-symmetric off-diagonal t.erms, and (ii) t.he transpose of the matrix that mu1tiplies the parameter errors in the z-equat.ioll appears ill the update law. This structure is a result of the design procedure, and it allows us to use the simple quadratic Lyapullov function 1 '1 lIn (.::1, ... ,Zn, Dh .. , ,t?,,) = ry + (8 - OJ )'I'r-] (8 - iI;)] (3.79)

L [=;

... i==1

t.o prove stability and regulation. Its derivative along the solutions of (3.77) is

,i"n =

=T

z-

n

~,t9lr-I(8

-

t9 j )

1=1 n

= =

L

[-CiZ~ + Zjw;(8 - iJ j )

;=1

,.

-L:CjZ;.

-

zi w ?,(8

-19;)] (3.80)

i=I

The LaSalle-Yoshizawa theorem (Theorem 2.1) now guarantees t.he g10bal uniform boundedness of z(t), ill (t) •... , '!9 n (t), as well as the regulation of =(t). Since Zl = X], we see that :l'1 is also bounded and regulated. The hOl.lndedness of X2, •• • , Xn then follows from the boundedness of O'j (defined in (3.74)) and the fact that Xj = Zi + ai-I- Since x is bounded, {3(.-.:) is bounded away from zero. Combining this with (3.71) we conclude that the control 11 is a1so bounded. Finally, the regulation of .'Vi - xf is concluded as follows: Since Zj(t), i = 1, .. _,11 converge to zero, Jdt), i = 1, ... ,n also converge to zero and =i(t) is integrable over [0,00). Furthermore, the boundedness of all the signals and their derivatives guarantees the boundedness of =i(t) and hence the uniform continuity of Zj(t). From Lemma A.G, we conclude that lim/_ oo Zj(t) = 0, i = 1" .. , rI. Since x can be expressed as a smooth vector function of =1, • , • '=11 and '011' • , , {)n , we can express X as a linear comhination of Zj and 17i wit.h coefficients which are bounded because they are smooth vector functions of the bounded signals ZI," . ,Zn and 1)1,.' . I {In' Hence, the convergence of Z and tJ j to zero implies that :i: converges to zero. Combining this with (3.70) and the regulation of XI leads to the desired result. 0

3.3.2

Multi-input systems

The adaptive backstepping design procedure of Theorem 3.5 can be easily e}..'i;ended to nonlinear systems which have been transformed into the multiinput pam,metric strict-feedback 101m

104

ADAPTIVE BACKS1.'EPPING DESIGN

j.'I,1

';'1,2

= =

Xl,!!

+ cprl (3,'1,11'1"2,., ••• ·1"2,P2-PI+h"

·1'1.3

+ cpI:ll.·1.1,J~I':!1.l~2.Jl'" •• , !·T.'rn.J,·"

J" l,p,-]

., :L'm.lI ••• , ,l:m ,Pn,-PJ+J)O

.1;2.p'.J-PI+21

,X rn •p,,,-Pl+2)9

= ·&.l.P1' ", + cp1'l.Pl-1 (...."1.1,···, "l,Pl-h· :/. .... ,., 2.h··· "'2,P2-h L

, •• I

·'I'm,], • , • 1 ,t m ,pm-l)6

m

i'l,111

I: f31J (.x) lJ.j + 'PT./)1 (:1:)0

=

j=l

(3.81) !i:i.)

=

Xi,j+]

+ cpl,j(xl,h' .. , 3.~l'P!-P.+jl' •• ,3:;,1, ••. ,l'i,j, •.. , :t'lJI.l, ..• , X m ,p".-PI+i}8

;i'm,l

,i'm,2

= =

+ It'!.,1 (.t'1.1,' .. ,.1'I,Pl-Prn+b .:r2,h ' , • X!!,p:t-p",+b •.. .1'm.3 + fP;'.2(XI.lt., • ,3:1,PI-P,"+21 ;]=2.1, , •• X2.P'.!-Pm+2, .1: 111 ,2

• • • 1 :l'""l t

:i:m'PPI,-J

=

Xm,Pm

:r:m,I)O

:l:m ,2)0

+ tp~'PfIl-1 (.J;],., ... - •• I

1

1

1'l,PI-]' X'J,l, ' •••1:2,{I2-J 1

Xm.l1 ••• ,Xm,p,.._I)O

711

:i'm,p".

=

2: f3".J(:V)-Ui + lP~'Pl (X)O

1

j=J

wberp 'Uh'"

, Um

are the inputs, and tbe input matrix is llollsingulru.' 'V:I:

e m,n

(ll=PJ+"'+Prn): clet B(l:) ::F 0, 'V ;1: E

m.n

(3.82)

I

The design procedure for this class of syst.ems cOllsists of applying the design proeedure of Theorem 3.5 to the fil'st Pi - 1 equations of each of the m subs)'stems of (3.81), to obttUll the system -CiJZi,j -

=iJ-l

+ ZiJ+J + tl1i ,)X, 191 " , , ,t),.-l )({I- llt) '('

;-1

e= L:(Pk -1) + j ,I$. j S Pi - 1, k=. = r Wi.j(X, 1111 , ,. , tlc)z;J' 1 $. e$. In 1]. -

1 S i $. m (3.83)

3.3

105

RECURSIVE DESJGN PROCEDURES

where t.ho functions Wi,j, rI> and HJ:II _ m +1 are defined appropria t.cly. Now let -o"-JII+I be a new estimate or 0 and define the cont:rol1J. as

lL

=

CI,PI ZI.Pl

B- 1 (.1:)

{

[ Cm'PrrI-7R,Pm

-

+ ':J ,PI-l

_:

]

_

- ID(.1:, #1, ...• ,0 11 - " , )

+ -m,p... -l

W~~m+l (x. 19" ...• U

n -.n jU,,-m+ 1 } •

(3.84)

and t.he updat.e law for 01l-m+l as

(3.85)

The stability properties of the resulting closed-loop system are analogous to those listed in Theorem 3.5, and can be similarly established Ilsing the Lyapunov fuuct.ion

{3.86}

3.3.3

Parametric block-strict-feedback systems

Lemma 3.4 can be applied repeatedly to design adaptive controllers for nonlinear systems which can be transformed, after a change of coordinates, into the pa7·ameL1i.c block-strid-feedback form

Xl =

11 ttl) + FI (XI)6 + iit (Xl )Y2 11.1 (:\:1)

111

=

:\,'1

= h6:'1, :(2) + F':!(XIt X2)8 + 02 (X.I ,X:!)Ya

Y'l.

=

h 2 (X2)

X·

=

Yi

=

/;("Xl, ... ,Xi) + Fi(Xll ... ,Xi)6 + 9;(YI, ... ,Xi)Yi+l hi(Xi)

.

I

Xp-l Yp-l

:\,p YP

(3.87)

= j,l-l (Xb ... ,Xp-t} + Fp- J (Xb ... , Xp-l)9 + 9p-l h'l' ... ,Xp-l )yp = hp _ 1(Xp- J) = jp(xJ + Fp(x,lO + 9p(X)U = hp(Xp) \

Ir06

ADAPTIVE BACKs'rEPPING DESIGN

lvhere each of the p subsystems with state Xi E m.nl , output IIi E lR, and input HI (for convenience we denote Yp+l == u) satisfies conditions (BSF-1) and BSF-2) (see Chapter 2, equation (2.198)), t.hat is, it has relative degree one niformly ill All' .. ,Xi-h and its zero dynamics subsystem is ISS with respect oX}'···, Xi-I! Yj·

Using the change of coordinates which transformed (2.198) into (2.201) in ection 2.3.3, we can now transform the system (3.87) into IiI

=

Yp-l

=

11 (yt, (I) + CPI'1' (Yll (1)0 + Yl(Yh (1)Y2 Y2 = hhJl,(ldJ21 (2) + cpI (111 , (11 Y2,(2)(J + 92(1111 (ltlJ':!, (2)1/:-1 /p-l (y., (1, ... 1 IIp-b (p-I)

=

(p

=

I

1/p-l. (p_I)8

+gp-1(Yl, (h' .. , Up-II (,,-I )yP (3.BB) 1p(/l1, (1, ... , Yp, (,,) + qJ~ (Vll (I, ... , 1Jp, (p)O + 9p(YI , (1, ... 1 Up! (II)?/'

YP = (1

-1' + 'Pp-l (VI, (II ...

cI> 1,O(Yl, (1)

+i

l (Yl, (1)9

Then we employ anotber change of coordinates which replaces Vi by

t/JdUh (h'"

1

Yi-I! (i-lt

=

YI ~ 1/'1 (Y1)

·'t·2

=

-;;;-(/1 (.lUI

:ti+l

=

+ 91112) =

II(y.,(I) + Ol(11I,(t}~J2

i-I 8~J,

i-I

=A '~'2(11I'(1'Y~)

tPi+ I (Yl t

(3.89)

81/Ji _

L ~(fj + gjYj+l) + L ar.~J WitO + 91 ... Oi-I!i + 91'" i=:1 ul/j

l:l

=

Hi), where

Xl

a~tl

Xi

9iYHI

j=1

'It ... ,

YEt (i I YiH)

i = 2, ... ,p - 1 .

f

FinallYl we use the feedback transformation p- 1

v =

L

j=J

8VJp

-0. (h Y.1

p-I

8t/Jp -

j=1

J

+ gjYi+l) + L

8(. tI>j,O + 91'" gll-l!p + gl'" 9p U. (3.90)

CondHion (BSF-l) guarantees tbat UI,'" ,gp :F Q everywhere. Hence, the change or coordinates (3.89) relating [YI! ... dIp, (l t • • • , (JJT to [:CIt- .. ,.1:/1< cT (J]T is a global diffeOJnorp11iSnl, and the feedback transformation (3.90) relat.ing II to u is llonsingular. It is now straighLfonvard La verify that (3.S9) and (3.90) transform (3.88) int:o a form l'Cmilliscent of tJle I ' •• t

3.3

107

RECURSIVE DESIGN PROCEDUR.ES

parametric strict-feedback form (3.70): !i'l

X2

=

+ 'PT(Xl1 (1)0 == Xa + \O~'(.Th X2t (it (2) X2

X,,-l == XI' + cp~_] (:rl1 .•• , ·2:,,_11 (" ... t (P-l)O .i;p == v + 'P~'(XI ()6 (1 = Wl.0(l!I,(1)+ ID 1(Xlt(dB (" = y

=

tPp,O(a:h'"

(3.91)

,.l:p , (I," . ,(p-lt (,,) + tP p(:l!I •• •• toVPI (It ... ,(p-ll (,,)0

$1·

In (3.91) each (i-subsystem is ISS with respect to 3'" ... , Xi, (1, ... , (;-1 ~i,O, eDit i = 1, ... ,p are defined as

ru;

its

inputs, and IPi,

a'Pi L a.(YII (Jf .•. i

11,

j=l

I

Yi-I, (;-1, lIi)

~J(Ylt'l" .. ,Yj,(j)

i-Io,p.

+L

j=1

0(' bJh(j. .. ·,Yi-lt(i-h1Ji) J

~AY1t (1, .... Hjt (j)

(3.D2) (3.93)

(3.94)

It is now clear t11at UlC class of parametric block-stric1.-feedback nonlinear syst.ems strictly contains the dass of paramet:l'ic strict-feedback nonlinear systems, sill('e (3.70) can be obtained by setting ni 1, p 71, and v p(x)1J in (3.91). We now state and prove t.Ile generalization of Theorem 3.5 to block-strictfeedback systems of the form (3.91).

=

=

=

Theorem 3.6 (Parametric Bloc1c.-Strict-Feedback Systems)

F07'

the

slJstem (3.91), consider U."e adaplive controller (3.95)

i

= 1"",p,

(3.06)

108

ADAPTIVE BACKSTEPPING DESIGN

whc7'e tJ j E 1RP a.re multiple e.r;tima,tes of (), r = r1' > 0 is t.he a,daptation .qain matri.:c, and the vmi.ables Zj anti fhe sta.bili:.iug jlJ.nctions O'j, i = 1, ... , P. are defined by the following 7'Ccursil1f C3.']J1'fJ8,1iions (with. Ci > 0 being design constants, and Zn :::;;: 0'0 == 0 1/-sed fo'" 1J.ot(].tional convenience): (3.97)

D:j

=

-Cj=j -

+

=i-l -

8a i - 1 ..T), 8(, ':iJ,O

[

_

J

~ (DO;i-l T 80';-1 )] 0; + ~ {80'i-J - - ! . p j + --Wj L- --J.'i+l 8x 8(j j=1 tJ.1:j

IfJi'1' - L-

I

i=l

fJO'i-1

8 11 • /'J

r

j

'1 ?' _ ~

[ 'PJ

L-

k=l

(80;j-l a

0 is the adapt~tt:ioll gain.

(3.106)

3.4

111

EXTENDED MATCHING DESIGN

Vie postpone the choice of update law for () until the next st.ep. The first error su bsystt"m becomes

(3.107)

Step 2. The derivative of =2

= =

To design the ('ontro]

Z2

=

X2 -

ao: I 'U -

l)XI (X2

0'1

is aO'l :. .

+ OIP) - 00 0

-ao

au) 1 -80'1 DOl ':.. u- -:r..., -8-1.{) - (J-I.{) - - . B. ax! 0.1'1 OJ:. {)e 'U t

(3.108)

we consider t·he augmented Lyapullov function 1'11,)

-:j

2

1-...

+ -.:; + -(J- . 2 -

2"',

(3.109)

The only difference bet.ween (3.109) and (3A2) is the absel1ce of t,be new parameter error (8 - O2 ) in (3.109). In view of (3.106) amI (3.108), the derivative

of l'2 is

(3.110) In the last equation, all the terms containing 0 ba.ve been grouped f·ogether. To eliminate them, the update law is chosen as

B= " (tpZl -

80:1

ox)

CP':2) .

(3.111)

Then. the last. bracketed term in (3.110) will be rendered equal 1.0 -C2':~ wif:h tl1e control lL

=

(3.112)

where for 9 we usc the analyti('al expression of tbe update law (3.111). Substitutlng the expressions (3.111) and (3.112) illto (3.110) we obtain (3.113)

112

ADAP1.'IVE BACKSTEPPINO DESIGN

(_lu lP ]

'1'

U.r:1

J

1+---------'

Figure 3.6: The closed-loop adaptive system (3.114).

and tile error system bec-omes (see block diagram in Figure 3.6)

~. [ ~ 1 = [ __~l _~] ~ [

iI =

1

]

+ ( _~~ ] 8

[~ - ~~ 1[ ~~ ] .

(3.114)

Comparing (3.114) with (3.47), we see that the system matrbc in (3.114) has preserved the important structural properties it had in (3A7): Its diagonal terms arc negative and its off-diagonal terms are skew-symmetric. Furthermore, we see that, as in (3A7), the matrix that mUltiplies the parameter error 8 in the z-equation is usecl (in its transposed form) in the update law for the parameter estimates. It is ruso instructive to compare tbe ex.pressions for the parametel' update laws ill (3.114) and (3.47): Even though the update law for iJ appears in the form of the sum of the update laws for t}. and il2 , the expressions (3.34) and (3.104) for 0'] depend on different parameter estimates (tJ 1 and 8, respectively), fUld thus Z2 and the partial derivative ~ will have different values in (3...:14) and (3.111). Due to the structure of the errol' system (3.114), its stability and convergence properties are derivE'd in a manner almost identical to those of (3.47) and are therefore omitted here. In the extended matching case we avoided the overparametrizati?D by postponing the choice of the update Jaw until the second step_ When 6 appeared in the second step, it was replaced by its known analytical ex.press~.on. Beyond the extended matching case we need more than two steps, so that iJ and bigher derivatives of 6 appear. Instead of the simple idea of postponing the choice

3.4

113

EXTENDED lVIATCHING DESIGN

of the update law, we need the more intricat.e tunin.q juncf;ions method, t.o be developed in Chapter 4.

3.4.2

Example: biochemical process

Extended matc:hing design is also applicable to pure-feedback systemH, int.radured in Section 2.3.2, provided that the unknown parameters appenr linearly. 'iVhile the general case of these parmn.etric pu.re-feedback s1}stc'rns is presented in Section 4.5.3, t.he extended matching design will be illustrat,ed on u simplified model of a biotechnological process which goes as far back as IvIonod [135]. In spite of its simplicity and somewhat unrealist,ic assumptions, this example is representative of several successful applications of adaptive nonlinear control to more complex processes described by Bast,in [7]. In a model of no fed-bat.ch process. S is the concent.ration of the growt.h limitiug substrate, X is the concentration of the growing microbial population, ". is the yield const.ant, D is the dilutiollrate: and the control u is the substrate feed rate. In a batch process, that is, when both D = 0 and u = 0: the rate of microbial growth i; is modeled as i = p(5).¥, where p.(S) is t.he "spedfic growth rate." The nonlinear function p,(S) is usually poorly known. and for our illustrat.ive purpose we parametrize it using unknown parametel's: (3.115) Note that .X, 5, and J.L(5) are nonnegat.ive quantities. \Vith this panllnetrization the fed-batch process operating at constant temperature is modeled by the following two rnass-balance equa.tions:

..:t

=

S =

[IPo(S) + ()IIP1(S) + ()2IP:!(S)].Y - D ..Y -k [IPo(S) + 81IP1(S) + 1J21P2(S)].\ - DS + 'U.

(3.116a) (3.116b)

The control objective is regulation of X to Ute set point X r • To furt.her simplify the system (3.116), we use the change of coordinates Xl = In ...\'", X2 = S, which is well-defined and invertible since ~y > O. Then (3.116) becomes Xl = IPO(:C2) + ()lIPl(X2) + 82 IP2(X2) - D .1:2 = -I;: [lPO(X2) + 814'1 (X2) + 82CP2(.l:2)] e:r'1

-

D.l:':]. + 'll.

(3.117a) (3.117b)

This system is clearly not in t.he paramet.ric strict.-feedback form (3.70), since the nOl1linearities in (3.117a) depend on the second state variable :r:2. However, we can still apply the design procedure illustrated in Section 3.4.1 to this system. In Section 2.3 we saw that our recursive design procedures can be applied not only to strict-feedback systems (2.165), but also to pure-feedback systems (2.180), whose nonlinearities are allowed to depend on one more state variable. The price to be paid is that the stability properties are no longer

114

ADAPTIVE BACI{S'l'EPPING DESIGN

Substrate feed (control -u)

Heating/ cooling

Figure 3.7: Fed-ba.tch stirred tank rendar.

global, bu t regional: They are guaranteed only for a compad set of iuitial condit,ions. The same is possible for our adaptive designs. For t.he system (3.117), our design pl'Oceeds by choosing tpO(3:2) as tht:' virtual ('ontrol variable ill (3.117a) and dcsigning for it the stabilizing rnlle-tioll

(3.118) where Zl

=

=1

=

Xl

= IPO(X2) -

-In X r • \'Vit.h Z2

-C1Z1

at,

the error system becomes

+ =2 + (0 - 81)IPl + (fJ - 92 )V'2

(3.119)

(aa~O + 0/a:,1 + 82 8aV(2) { - ~~ [V'o + OIIPl + lJ~tp2J eaJ

=2 =

X2

,1:2

X2

I

-

D:V2 + u}

+ct [-Cl=1 + =2 + (0 - 6d'P1 + (0 - 82 )V'2] + B1V'] + 02V'!! .

(3.120)

Following the development. of Sectioll 3.4.1, we choose the update law

~

r [ : }{ =, + [c,

=

= [Ol 021.

wherc 01' u

!

-

+

(3.121)

Tbe corresponding control law

= (!!!!! 8 ~ -I- ~,~) O:r:l

- k ( : : + 8, :~; + 8, ::) ] =, } ,

I 8:1::1

{ -

C2=2 - =1 - Cl [-C1Zl

+ z!! - D1V'l - 92V'2]

- DJ:'J

[~ .. v>ol r [ ;. ] {., + [ct - k ( : : + 8, :~; + 8

2 ::.: )

+k [cpo + 81lP! + 62 V'2] e + DX2, Z

'

1o.}} (3.122)

3.4

115

EXTENDED MATCHING DESIGN

is feasibile only in the region in which ~ + 01 ~ choicps, the derivative of the Lyapunov function

+ O:!~ =/: O.

'Vith thcsp

(3.123) is nonposit.ivp: (3.12-!)

" =

As we will see in Section 4.5.3, stability is guarant.eed for all initial conditiolls inside the largest level set of the Lyapunov function 3.123 (·ont.ained in the feasibility region.

3.4.3

Transient performance improvement

The llonlillem' damping with h:-terms introduced in Section 2.5 can easily bE' incorporat.ed int.o the adaptive design procedures we have discnssed so far. The resulting adaptive controllers guarantee boundedness even when the adapt.ation is switched oIT, and their transient performance can be improved in a systemat:ic way t.hrough trajectory) initiuli:;ation and the choke of design paramelers. To illustrate the design with Ii.-terms and t.1m process of trajedory init.ializatioll, we consider again the system {3.29} with the out.put /I = :1'1:

.1:1 = :1:2

:r2

=

+ 8cp( .1' J ) (3.125)

1I

Y =

.1'1'

The control objective is to asymptotically t.racl\: a reference out.put Yr(l) with the out.put JJ of the system (3.125). 'Ve assullle that not. ouly Yr. but also it.s first t.wo derivatives Yr, jjr are known and uniformly bounded. and, in addition, Yr is piecewise cont.inuous.

Step 1. The first errol' variable is now the t7'lJch71.g =1

= Y - Yr =

e7'1'07'

I1r ,

Xl -

(3.126)

whose derivative is ':-1 = :1:2 + 0'1'91 (.t·l) -

Viewing

:1:2 ~lS

.llr·

(3.127)

t.he virtual control we define the st.abiliziug fUlldion 0'1

= -c} =1

-

n.1 =1 cp2 -

Ocp + Yr .

(3.128)

Comparing {3.128} with (3.104) we note two new t.erms in (3.128). The term which is intended to cancel t.he corresponding t.erm in (3.127), is due to the t.racking objective. The nonlinear damping tE'l'lll -11:1 =1 cp2 is motivated

Yn

116

ADAPTIVE BACI(STEPPJNG DESIGN

by Lemma 2.26. It contains the square of the term ( 1012/4noco, where 8 = 0 - O{O) is constant. since adaptation is turned off.

Transient performance improvement with trajectory initialization. Let ns now investigate the transient performance of the adaptive dosed-loop system (3.137). The derivative of the nonnegative fUllction \1(.::) defined in (3.140) along t.he solutions of (3.137) satisfies the same inequality as in (3.141):

(1, dt 2

2

-d -:; I") - ::; -co Iz 1'-1 + -0

4n:o

.

(3.143)

8 has already been established from (3.132) and (3.136), we can strengthen the inequality in (3.143) by replacing ij'J with its

Since the boundedlless of

3.4

119

EXTENDED lvIATCHING DESIGN

bound 11811~. This bound is est,imated from (3.132) using t.he fact that ~ is nonincreasing:

:5

1 ., 1 - '1 2Iz(t)l- + 21,8(t)- = ''2(l)

:5 ~(O) = ~1=(O)12 + ')1 0(0)2. ...

(3.144)

....1'

This implies

IIOII~ $ 71=(0)1 2 + 0(0)2.

(3.145)

Combining (3.143) and (3,145) we obt.ain

d

at

(1;;12) :5 - 2col=l:! + _1 ["'11=(0)1 2 + 8(0)2] , 2nn

Multiplying both sides of (3.V!6) by results in

e'lcol

(3.l46)

and integrating over the int.erval [0, I]

(3.VJ7) The bound (3.l tl7) suggests that the transient beha.vior of the error system can be influenced through the choice of design constants Cn, n.o and 1', ~'hat is not clear, however, is that an increase of h~OCO alone may not reduce the ma..ximulll value of 1=(t)1 and \Vill certainly not reduce the comput.able .coo-bound of =. In fact, it may even increase this bound by increasing the initial value 1=(0)1. To clarify this point, let us recall the definitions of =. and =2: Zl

=

·r-l - llr

Z2

=

X2 -

0'1

=

;1:2

+ CI Zl + /i'IZ1cp2 + Otp -

lir·

Suppose now that ZI(O) is different than zero. In that case, all increase of and may increase the value of :~(O) and thus also the value of 1=(0)1. IvIOl'cover, this increase may more than offset the dccreasing efrect of the term 1/4n.ocn in (3.147), since Iz(O)1 2 will increase in proportion to ci and It would seem that the dependence of =(0) 011 the design constants Cl, C:Z, 11:1, n.:z eliminates any possibility of systematically improving the t.ransient performance of the error system through the choice of Co and 11:0' Fortunately, it is not so. The remedy for this problem is to use tmjector!J initialization to render =(0) = 0 independently of the choice of these design constants. The initialization procedure, present.ed for the general case in Section 4.3.2, is straightforward and is dictated by thc definitions of the =-variables: Cl

li,

h-i-

• Stal'ting with

z},

set.

=1 (0) =

0 by choosing (3.148)

120

ADAPTIVE BACKSTEPPING DESIGN

• Since :'1(0) = 0, (3.128) shows that

where we use t.he notation !p(O) = CP(:t:I(O)). From (3.149) it is ("lear t.hat we can set. .::AO) = 0 with the c:hoice

(3.150) With the t.rajcct.ory initialization defined by (3.148) and (3.150), we have set. .:(0) = O. III t.he case of model reference control, this is achieved by adjusting the init.ial conditions of the reference model. If, on t.he other hand, the reference trajectory is given as a prccomputed function of t.ime, t.hen it can be iuit.ialized t.hrough the addition of exponentially decaying t.erms whicll define the refcl'Cnce l1Yl1t.rlients. Vve note t.hat. (3.148) and (3.150) are independent of tht" deRign ('onstnnts c., C:!. h'lt 1i2' This means t.hat. different choices of Co and KO will stilll'esult in =(0) = 0 with the same values of Yr(O) and 1jr(0). Ret.urning to (3.147), we substitute .:(0) = 0 to obtain 2 1=(t)1

::;

1

-

'l

-8(0)~lli:oco

(3.151) j

which imp1ies

(3.152) Hence, the .coo-bound on the transient performance of the error system is direct.ly proportional t.o t.he initial parametric uncertainty and can be reduced arbitrarily by increasing the values of Co and 1\:0. In particular, this implies that the transients of the tracking error Zl = Y - Yr are directly influenced by the design const.ants Cj and Iii' This possibility of arbitrary reduction may seem peculiar, since it can be achieved for all initial conditions. vVe must. remember, however, that this elTor is defined with respect to the reference signaJs which have in tUrn been initialized to set =(0) = O. Hence, the effect of the plant iuitial conditions has been "absorbed" into the reference t.ransients. To provide some further insight into the process of trajectory initialization, let liS return to the Lyapunov function (3.132). When .:::(0) = 0, the initial value of this function is reduced to the initial value of the parametric uncertainty. 1£ we interpret the value of this function as a distance between the actual system trajectory and the reference trajectory, we see that trajectory initialization places I.he initial point of the reference trajeclory as close as possible to thc initial point of the system trajectory. If the parametric ullcertainty were zero, t.rajectory ini tiaJization would have placed the reference output and its derh'at.ives at the true values of the plant output and its derivatives. Tilis

NOTES AND REFERENCES

121

is easily seen if B{O) is replaced by () in (3.148) and (3.150):

Yr(O) = .1:1 (0) = !I(D) Yt(O) = l'2(0) + 81;'(0) = 1j(0) , However, since the paral11etpr () is unknown, trajectory initia.lizat.ion placed only the reference output at: the t.rut:' value of the plant output,. while ii,s derivo,th'es wore placed at the estimated values of the plant ont.put: derivat.iveF;. Through this process, the iniUI:11 value of the Lyapullov function is (3.153) \Vhich~

in the presence of parametric ullt"el't.ainty, is its smallest possihl(' valuo,

Notes and References Adaptive backstepping (Kallellakopoulos, Kokotovic\ and rvlorse [60]), first present.ed in a Grainger lecture [87], was a culminat;ion of an intellsivp eIrod of severol groups of ~1.uthors. The path to adaptive backst;cpping was not, as direct as it may appear from t.his chapter. It led ~,brough the makhed ease in Taylor, Kokotovic, rvIarino, a.nd f\:anel1akopoulos [186J, itnd then thp case of extellded mat,ching ill Kanellakopoulos, Kokot:ovic, and :Marino [65J, and Bastin and Campion [8]. Even t.hough nonadaptive backstcpping was ftvaHablc from Saberi, Kokotovic, and Sussmann [163}. the st.eps beyond the ext(:'uded matching case wore delayed lUld ruterlmtivo npproaehes were explored. The focus was on estimat:ion-based designs sllllullarizec1 in Pral)" Bast-in, Pomet'l and .Jiang [157]" The rIass of paramet.ric strict:-feedbal'k systems was characterizerl via coordinat.E"-fl'ee geometric conditions by Kanellakopo ulm; ct 0.1. [63, 69), and the class of pru"ametl"ic }lure-feedback systelIlR by Akhrif tlud Blankenship The multi-input design £01' parametric pUl'L.1..feedback systems was prcs('l1t,ed

rlJ.

in [B7]. Teel {lBB] increased the feasibility region for parametric pure-feedhack syst.ems by cast.ing the scheme [69J in an observer-baserl setting. Severa] extensions of [69] were proposed by Seto, AUllaswamy, and Brumettl [3, 167J.

Chapter 4 Tuning Functions Design

The adaptive baclcst.epping solution to the problem of nonlinea.r stabilization and tracking in the presence of unknown parametel's is a starting point for more elaborate adaptive designs which lead to new properties of the designed controller and the resulting feedback system. One of t.he improvcmcnts t.o be a.chieved with t.he tuning functions design in this chapter is the redudioll of t.he dynamic order of the adaptive controller to it.s minimum: The number of parameter estimates is equal to the number of unknowll pammet.ers. This minimum-m'der design is advant,ageous not only for implement.ation, but also because it gunrantees the strongest achievable stability and convergence properties. In the tuning functions procedure the parameter updat.e law is designed recursively. At each consecut.ive step we design a tuning function as a potential update law. In contrast to adaptive ba("kstepping in Chapter 3, t.hcse intermediate update laws are not implemented. Instead, the controller uses them to compensate for the effect of parameter estimation t.ransients. Only the final tuning function is used as the parameter update law. We start this chapter with Section ~1.1, which introduces a g(meral framework for Lyapunov-bnsed adaptive design via adaptive control L;l/ap U7/. 0 II functions (adf). VVe depart from the certainty equivalence principle and approach the problem of adaptive stabilization of the original nonlinear syst.em as a problem of nonadaptive stabilization of a modified system. In this setting, the tuning functions design is a method for recursively genera.ting aclf's. The design procedure is presented in Sections 4.2 and 4.3, which are independent from Section 4.1 and can be read first. In Sect.ion 4.4 we derive transient performance bounds on the error state of the adaptive system. An essential part. of t.he technique for improving the t.ransients is t.he trajectory initialization presented in Section 4.3.2. Several e..xtensions of the design are presented in Section 4.5. In Sect.ion 4.6 the tuning functions design is applied to suppress the wing rock jnst.ability in aircraft flying at high angle-of-attack

124

TUNING FUNC'rIONS DESIGN

4.1

Adaptive Control Lyapunov Functions

The basic: iclea of t:he Lyapunov approach to adapt.ive c:ollt.rol is t:o design a control law and a paramet.er update law to gnal'ant.ec that. t:lu' derivative of a suit.able Lyapunov function is nonposit.ive. \¥e are therefore sent. to search for a triple: Lyapunov function, ('ont.rol law. and update law. For a class of nonHnetu· systems caIled paramctric'-st.ric~t.-fceclback syst.ems we will be able to make til is search systel1la tic. To hegin with. let. liS investigate the possibility of adaptive design for the syst.("lll (4.1) i: = f(:l') + F(.l:)O + g(:t}u, :[' E JR.". 'U E 1R,

n

where E JR.]) is a vector or unknown constant. parameters, and ,((:1:), F(3;) and g(.1") are smooth. For simplicity let j(O) o. F(O) = 0, so that. x 0 is an equilibriulll of the uncontrolled plant..

4.1.1

=

=

Departure from certainty equivalence

Ivluch of the traditional adaptive cont.l'ol employs some form of "certa.inty equivalence" thinking. FoJlowiJlg t.his path one first performs a design fol' t.he C8.Cie when the exact. value of 0 is known. Suppose that. t.his noutrivial ta.."k is compJeted and that its result is a fecdb~lck contra} 'U = (JA~I:, 0) wliich stabilizes t.he equilibrium x 0 with respect to a known Lyapnnov function \~(x, 0). The suhscript 'r:' st.ands for "certaintyequiva]cllre.'· \Ve 1mow that V';,.(;v,lJ) is posit.ive definite and radially unbounded in .1' for all 0, and thnt there exists a funetion l.f!(x, 0), which is also positive definite in .1' fol' all 0, such that l

=

a,~

ax' [f(,T) + F(x)B + g(x)Q:r.(:c: 8)] ~ -TV(:z:, H).

(4.2)

Hmv can we c>..1Jloit the knowledge of ll:c(:l', 8) and 1~(.1·1 0) for adaptive design when 0 is 110t. known? The certainty equivalence idea is to replace () by an estimate 9(1) obtained from a parameter Ilpual.e law (4.3)

r

where t.he adapt.ation gain matrb:: is posit.ive clefillit.e. \Ve want. to select 11and T 1.0 guarantee that the derivative of a Lyapullov fUllction is nonpositivc. For t.he syst.em (4.1), (4.3), a Lyapunov function candidate is

(4.4) 1 Throllghont. the chapter, we will drop t.he arguments in lJ\~.8) wId O\I/~~·/j) , nod write shortly ~~ and ~~. However, we will keep the flrgument.s in f(x), F(X)l g(x), and a(x. 0).

4.1

125

ADAPTIVE CONTROL LYAPUNOV FUNCTIONS

where the "certainty oquh7a]el1ce" form of \~ is augmented by a term quadratic ill the parameter estiIllation error

(4.5) Upon tbe substitution of F(:t)(J = F(x)9 along thE' solutions of ("1.1), (4.3) is

.

l'

+ F(.l')O,

the derh'at.ive of l'(.J:,8)

D1~ ( J(:I') + F(:z·)(J~ + g(x)u ) + -~'rT D1{, -. (D'~. )T_ = -0' + Or ~F(.r) - liT T. ~ 00 ~

(...t.6)

To eliminate t;be indefinite c1epclldell(~e of ,:r on the unknowl1 parameter error 0, we select r to cancel the last ';wo terms ill (4.6):

)'1'

8V r(x,O) = ( a.l~ F(;.. )

(4.7)

With this choice of T, the expression (4.6) is reduced t.o

81~ (f{3.) . 1'F -_ -8 X

•

...

. . al~ r (8\~ + F(.l)O + y(.t)lI) + -. ~F(.l) )'T . 88 v.l'

(4.8)

=

Our nc..xt task is to select a control law u a'(x,8) to make 1> llonpositive. The "certaintyequiwtlenceH controll1. a·c (.t',8) fails to acllieve i.bis hecause then (4.2) and (4.8) yie1d

=

.

•

\f S -1'li(:c,6)

O\~ (D\~

+ D9 r

lJ.r F(:r;)

),r •

Clearly, l' is not nonposith'e because a. sigll-illclelinit;e term is added to - (.t!(:r, 8). In search of a bettel' control law 0'(.1',8). we augment aA,'~, iJ) by 0-.,. (x, 0), (4.10) The substitution of (·tl0) into (4.8) shows that the desired nonpositivit~t -IV(x,8) will be achieved if aT CRn be found to saUsfy

al~

. + -~ aVe an r

~g(:t)o'l'(X' 0) vX

(81' c ) -0 F(:.:)

T

.1:

= O.

if s

(4.11)

This condition (or a'.,.. demonstrates tbe difficulty of adapt.ive designs for a general nonlinear system (4.1). It is casy to S(,E' that a r satisfying (4.11) is unlikely to c..xist: The scalar quantity ~g(:c) may be zel'O at: a set of points. StiU, the condition (4.11) is of iuterest because of an hUJJOl'f.ilUt. special caso, which will be the starting point of our recursive design. This special elISe is

126

TUNINC FUNC1'(ONS DESIGN

the "extended matching" studied in Section 3.4. In this case, a smooth vectorvalued fUllction cp : R n +1J --+ JRP is known such that ~ can be fadored as follows: a\~ aVe (}.'1: () 'P(.?", (j~)T . (~1.12)

ao = -ax'

-A

Then, irr~pective of the zeros of t~~.t}(X), an A

A

07"(:1',8) = - I) The proof of this part is based 011 Sontng's cOllstructive proof [171J of Artstein's theorem [4J. "Ve assume tbat 1~1 is an adf for (4.28), tha.t is, a elf for (4.31). Sontag's formula (2.19) appJied t.o (4.31) gives a rOllt.rollaw smooth all (JR" \ {O}} X lRP :

.I

a:(x,6}

+ = - !!Y.a.! 8:1'

(4.34)

o

!

Qiag(x Ox • t 6)

= 0I

130

TUNING FUNCTIONS DESIGN

where

(4.35) 'Vitb the c110ice (4,34), inequality (4.33) is satisfied with t:he ('Old.inuous function

IF(:Z',8) =

_ )2 + (tn~ ).a ax 1(3:,8) Dxg(J.·,O) , ( al~

(4.36)

which is positive definite ill x for each 0, because (4.32) implies that ~ j(:r, 8) < 0 whenever IlJ:g(~v, 8) = 0 alld 3' i= O. We note that tbe COlltrollaw o'(:L',8) will be continuous at .1' = 0 if and only if the aeIf \fa satisfies the following property, called the small controi,}roPC7"ty [171]; For each () E B" and for any E: > 0 there is a 6 > 0 such that, if x =1= 0 satisfies 13:1 S 0, then there is some 'u, witb luI ~ E: such that

all axu [J(:r:) + F(:r.} ((all 0 + r ao )T) + g(~l:)ll ] < o.

(4.37)

Assuming the existence of an aclf we now show that (4.28) is globally adaptively stabilizable. Since (2 => 1), there exists a t~rjple (a, Va, f) and a function ll' such that (4.33) is satisfied, that is,

~~ (f(x) + F(·l:l8+ g(xla{x, 8)1 +

:"r (~

F{''I:'f

~ -W{x, II) .

r- 1(0 -

8) .

(4.38)

Consider the Lyapullov function candidate 1

~

4T

lI{.T.,O) = l~(:r:, 0) + 2(6 - 0) A

~

(4.39)

'Vitb the help of (4.38), the derivative of V along tbe solutions of (4.28), (4.29), (4.30), is .

V

= 81~ ax =

l)8'~ x

[

cJl~

-]

-;;J'

,.

f + FO + .qa:(x, 8) + 80 rr{x,8) - 0 r(x, 8)

[J + PO + ga(x, 8)] + 81~ rT(X, 0) + ~~ F8 88

eTr(x, 8)

uX

,. al~ :::; -H'(;r,O) - -,. f

80

"

+0m' (al~) ax F -

(aVa )T Dl~ ~ -;;-F + - . fT(X,O) 80

uX

° T{:C,O). nT

-

(4.40)

Choosing

r(x, 0) =

al~ (

T

ax (x,6)F{x) ) A

,

(4.41)

4.1

131

ADAPTIVE CON'rnOL LVAPUNOV FUNCTIONS

we get

11 ::; -1"'(.1',8) ,

VB E ffiP.

(4.42)

Thus, the equilibrium x = 0,0 = 0 of (4.28), (4.20), (4.30) is globally stable, and by the LaSalle-Yoshizawa theorem (Theorem 2.1), :c(t) ~ 0, that is, (4.28) is globally adaptively st.abilizahle. 0 The adaptive controller const;l'ueted ill the proof of Theorem .4.3 consists

=

of a control law lJ. = a(:r,8) given by (4.3 l1), and an update law iJ fr(:1.',8) with (4.,U). It. is of int.crest to inl;erpret this controllcr a.1Oj a certaint.y equivalence COIltroller. The control law 0:(:1:,8) given by (4.34) is stabilizing for the modified syst.em (4.31) but may not; stabilizing for the original system (4.28). However, as t.he proof of Theorem 4.3 shows, its cel'tainty equivalencc form 0:(:1:,8) is an adaptive globally stabilizing control law for the original system (4.28). Hence, if ~t certainty equivalence approach is to be applied t,o a nonlinear system, the system is to be modified to require a cont.rollaw whi{'h anticipates the parameter estimation transients. In the proof o[ Theorcm 4.3, this is achieved by incorporating thc t'uning JUTlction T in t.he control law 0', Indeed, the formula (4.34) for a: depends on r via

ue

av;. -. _ aVa .. T a,T. f(:,;, 0) - D."C J + r(.l, 8)

avo (0 + f ( 80 )T)

'

which is obtained by combining (4.35) and (4.41). Using (4A1) 1,0 rewrite the inequality (4.38) as a~

ax

[j(x)

+ F(x)8 + g(x)o'(x, 0)] + 8~ ao fr(x. 8) ~ -H'(,l.', 8)

I

it is not. difficult. to see that the control law (4.34) containing (4.43) prevent.s

r fl'om destroying the nonpositivity of the Lyapunov derivativE'. Remark 4.4 A relevant question remains unanswel'ed: If there exists an aclf for (4.28), is this system globally asymptotically st,abilizable for each f) (and vice vel'sa)? In other wOl'ds, does the existence of a pair (a, l/~) satisfying (~J.33) for some r > 0 imply the existence of it pair (0:°, l~O) satisfying (4.33) [or r 0 (and vice versa)? Adaptive Lyapunov designs available in the literature [59, 65, 69, 94, 156, 157, 186] are all for systems whirh are not. only globally adapt.ively stabilizable, but. also globally asymptot.ically stabiliznble for each O. 0

=

As is always the case in adaptive control, in the proof of Theorem 4.3 we used a Lyapunov function li(.1\ 8) given by (4.39). which is quadratic in the parameter errol' (J - O. The quadratic form is suggested by the linear

132

TUNING FUNCTIONS DESIGN

dependence of (4.28) on 8, and the fact that (J cannot be used for feedback. VVe will now show that the quadratic form of (4.39) is both necessary and sufficient for the existence of an adf. Vve say that systcm (4.28) is globally adaptively quadratically stabilizable if it is globally adaptivellJ .litabili::able and, in addition, there exist a smooth function 1'1I(.1·,9) positive definit.e and radially unbounded in x for each 9~ and It continuous function lV(x,O) posit.ive definite in .T for each 0, such t.hat fat' all (x(O),8(0)) E lR"+P tl.lld all () E JRP, the derivat.ive of (4.39) along the solutions of (4.28), (4.29), (.t.30) is given by (4.42).

Corollary 4.5 The 51JS tern (4.28) is gio bally lI.dap ti1Jei1J ljualiratically stabilizCLble if and only if there exists an adf Va (x, 8). Proof. The 'if' part is contained in t.he proof of Theol'em 4.3 where the Lyapullov function F(x,O) is in the form (4.39). To prove the 'only if part, we start. by assuming global adaptive quadratic sta.bilizability of (4.28), and first show that 1"(:1:,0) must be given by (4.41). The derivatjve of l' along the solutions of (4.28), {4.29}, (4.30)1 given by (4.40), is rewrit.t.en ru;

This expression has to be nonpositive to sat.isfy (4042). Since it is affine in 0, it can be nonpositive for all (x,6) E lRn +p and ~Il (J E lll.P ouly if Ule last term is zero, that. is, only if T is defined as in (4.41). Then l it is straightforward to verify that

a;;

[J(Xl + F(x)

(0 + r (a;) T) + Y(X}l>{X,Ol]

=V+(6T-~ir) (T-(~:Fr) ~ -lV(x,8) for all (~'I 0) E IR"+p. By (1

(4.46)

=> 2) in Theorem 4.3, Vo(:r, 0) is an aclf for (4.28). o

The above analysis applies also to t.he ('ase where t.he unknown parameters entcr the control vector field:

.i· = f(x)

+ F(x)8 + [g(x) + G(:r)O]u..

(4.47)

4.1

133

ADAPTIVE CON1'ROL LVAPUNOV FUNCTIONS

In this case the existence of an adf Va is equivalent to the existel1cc of a elf for the syst.cm

±=

/(x)

+ F(,,')

(OH (lJ~y) +

[9(X)

+ G(x)

(6+ r (~ f)]

II.

(4.48) Tl1e ext.ension 1:0 t.he multi-input case is also st.raightforward.

It. is of intercst. to examine the itl}lllt-output Pfopclties of I.he system resulting fro111 tbe application of the adaptive control Jaw Q·(.r.9) to the plant (4.1):

± = I(x) + F(x)9 + O(J:)O'(:I', 0) + F{a:)6.

(4.49)

In early Ly~tpUllDV designs for linear systems of relative dcgree DIU', an important pl"opert:y was the strict positive realness of t.he transfer I'ullctioll between the parameter error and tbe out;put error [142J. For an analogous passivity pl'operty of the nonlinear system (4.49), let us consider that it.R input is 0.

Corollary 4.6 (Passivity) Suppo.'Je a j'll1lcLion l~,(.r: 0) is hU:J1Im. to be an ad! 1lritll. (1.11. associated control la'lJJ o·{ X t 8). Then the By/stern

'llnth

+ F(:J:)O +g(:t)o'{J:, 0) + F(.")ii

j:

=

f(:L')

T

=

81'0 ~ )T ( 03: (:1:,O)F(:d

0 lIS the input and T

as the D1Jlp'Ut is

(4.50)

l;#'riCtly

passizlc.

Proof. Along the solutions of (4A9) we have .

1~1

= ~

01~

-8

x

(

~

o\~

.

a\'~I-

f + FO + go·(:r, 8) + -. rT(:!:, 6) + -8 FO A

,

]

W

- T-

-ll"(:r:,O)+T(X,8) 0,

7

(4.51)

wbich, upon integration, yields

10' TT(J=(S), O(s)}6(s)ds ~ V:1(.t'{f), 6(t)) - \~I(X{O), B(O)} + 10' l·{l(.r(s) , O(.~»d8. (4.52) Using V;,(.l~, 0) as a storage functioll and TV(x, 8) as fL dissipation rate, and r noting that. l~ and al'e positive definite in 3: [or each B~ tlle inequality (4.52) estnblishes strict passivity by Definition D.2. 0

n

Hence, our closed-loop ada,pt.ivc system represents a llega.1.ivc feedback connection of the strictly passive system (4.50) and tbe int.egl'ator

- -T, r

- 9=

S

(4.53)

134

TUNING FUNC'fIONS DeSIGN

which is passive (positive real) because

!!.. (!orr- 1o) = dt 2

_8'r T

(4.5~1)

implies that (4.55)

For snch a feedback connection, Theorem DA establishes that the equilibrium x = 0,9 = 0 is globally stable, and x{t) -+ 0 as t --I> o. Thus, t,he problem of adapt.ive stabilization can be approached as the problem of finding an output T with respect to which it is possible to achieve strict passivity from 8 as the input.

4.1.3

Adaptive backstepping via aclf

\Vith Theorem 4.3, the problem of adaptive stabilization is reduced to the probJcm of findhlg an adf. We }lOW address the problem of systematic COllstruction of an adf. Our aim is a recursive approach because we already know how to find ac1f's for systems with the extended matching property, and expect to recursively enlarge this initial class of syst.ems with repeated use of backstepping. So, we assume that an aclf is known for an initial system, and construct a new aclf for the initial system augmented by all integrator.

Lemma 4.7 If the system

± = f(x)

+ F(l')6 + g(x)u,

(4.56)

is g{oball1J adaptivcl1J q1l.adratica1l1J stabilizable with a: E Cl, then the augmented system :i; = /(:1') + F(x)6 + g(x)~

{ =

(4.57)

'I/.,

is also globally a.daptively quadra.tically stabilizable. Proof. Since system (4.56) is globally adaptively stabilizable, then by Corolhuy 4.5 there exists an aclf {~(x,O), and by Theorem 4.3 it satisfies (4.33) with a control law 'U = 0:(.'&,6). Vve will now show that

l'i (:I:,~, 8) = V'a(.'&, 6) + ! (~ - a:(x, 8))2

(4.58)

2

is an ac1f for the augmented system (4.57) by showing that it satisfies aVI

8(x,~)

[f + (8 + r (~)T) + F

0:)

(.1:, ~ t 6)

lJf. ] $ _ HI _ ({ _ a

f

(4.59)

4.1

135

ADAPTIVE CONTROL LVAPUNOV FUNCTIONS

with the contra] law

=

aa (/ + Fe + g~) ax

oVa - - 0 - (~- 0') + -

ax

+ 80'r (8Vi F)T + av;,r (BIl'F)T olJ

OJ'

08

o:r

Let us start by introducing for brevity:; = t -0:(3:,8). With (4.58) we compute

alii [ J + FO + gf. ]

o{x, f.)

01 (x,

{, 8}

alii

=

81't 8x (/ + FIJ + g{)

=

(av0; - =ax ao:) (J+ FO+gf,)

=

8\'0 8d~ (1 + FO + go:)

+

o{

0:1 {x,

{, iJ}

+ZO:l

81~

Oet

+ ax gz - ;; aa: (J + FO + .Qf,) + =°1

80: ) = 01'a ax (/+F8+go:)+z ( 0]+ tn! 8:9- ox (J+F8+gf,)

. (4.61)

On tbe other hand, ill view of (4.58), we have

8111 O(x, {)

--

[Fr (!:!!i)T ] = -Fr alii (alii) 80 -8e l' 0 ox

(en;;. _:; au) Fr (8\!a _; an) T

=

ax

ax

ae

89

a)T

8V'", Fr ( 8V

=

ax

-z

ao

(oar (8Viox. F)T + ollar (OO·F)"). 8S

80

83.:

(4.62)

Adding (4.61) and (4.62), with (4.33) and (4.60) we get

alii

[f+

F(0 + r (~ )T) + DC; ] 0'1 (x,

8(x f.) t

=

{,O)

aVa (/ + Fe + go:) + 81'", Ff

ox

a.r:

+Z

(0:

1

(aVn)T {)f}

+ al'a g _ 80' (/ + FO + gf.) ox 8x

_ Bar ( 81"i 00 ax ::; _HT(X,O)

F) T_tH~, r (00' F) T)

_.::;2.

01J

ax

(4.63)

136

TUNING FUNC'l'IONS DESIGN

This pI'oves by Theorem 4.3 that \'1 (x. {, 0) is an aclf for syst.em (4.57), and by Corollary 4.5 this system is globally adaptivcly qnadrat:ically stabilizablc.

o The new tuning function for system (4.57) is determincd by the new aclf

Vi and given by T] (:1', ~~O) =

[F ])T = (~Vj F)'f = [(aVa _(e _ a,}aa:) F]T ( B(~ .1.:, 0 o:r B.T. a3: ~)

(4.64) \Xle note that the new tuning function

I)

is obt.ained by augmenting the initial T

tuning IUllct:ion T with the term - (~~F) (~- 0') which accounts for the faet t hat the ac1f 1~, is augmented by ! (~ - a(.1: I 8))2. The form of the controllJ:l.w Cl:l~(3:1 {, £J) in (4.60) is of part.icu1ar interest. It consist.s of two parts~ 0:1 = O'ltC + O'I.T' TIll" first part., ltl,c(Xt(,IJ)

a\~ = --a' 9:r;

(~- 0')

ao

+~ cr + FIJ + g~) v.1.

,

{£l.GS}

would becomc the "certaintyequivalencc" controlla,w for the augmcnted system (4.57) if we were to set r = 0. 2 The set'olld part consists of t;wo terms,

aJ

,,(X,e,8)=:r(:Ff +~;r(~;Fr

Their role is to produce ~Fr (~)

'f

(4.66)

in t.he aclf inequality (4.59). Observe

that the first. term ill (4.66) incorporates TI = (~F) T. The controlln.w 0'1 (x, €, 0) in (4.60) is only one out of many possible control given by (4.58) is an ac1f for (4.57), we can laws. Once we have shown that uset for example, the CO control law 0'1 given by Sontag's forllluia (4.34) witl1 ...illi.. g - - "'lld O(.:E'.~I 1 - '" u

1'.

~

I

I

It can he shown that the following function, used as a. elf ill [1581, is a 1110re general aclf than (4.58): {tj (x,~, 6) :!Note, however, I.hat are also fUllctions of

r.

al.e

(-a(.:E'IO)

= "11('1', 6} + .L n

is not. obtained by setting r

l1{s)ds,

(4.68)

= 0 in 01 since o'(x, O} and Va(xlO)

I

}

4.1

137

ADAPTIVE CONTROL LYAPUNOV FUNCTIONS

where TJ is a CO function sneh that 811(8) 7J f/. £1 « -00,0]) U £1 ([0, +(0)).

> 0 whenever s

The following example illustrat.es t.he use of Lemma

=f:. 0, 7]/(0)

> 0,

and

~L7.

Example 4.8 Let us consider the system Xl X2 ·'Va

=

x:!

=

X:i

=

+ cp(:r:} )T (1 (4.G9)

'/I.

"Ve will treat the state 3'a os an integrator added to the (:1:1, .r~)-subsystem from Example 4.1. In that example, we have already designed an adaptive control law for the syst.em

Xl ·1:2

= =

.l'2

+ cp( X1 ) 'I'fJ

(4.70)

Xa,

considering X:i as a control input. With (4.18), (4.10), (4.20), (4.22), it ean bc shown t.hat

=

which means that l'a{xt, :1:2, 0) = '''=(XI, 3:2, 0) ~(zr + =i) is an aclf for the system (4.70) considering X:i as a control input. Therefore, Lcmma 4.7 is directly applicable. \Ve define z = X3 - a(x, 8). By Lemma 4.7, the function

Vi (:r; fJ ) = '12 ( =j + =2 + '::3 'I

'J

.')

I

(4.72)

is an aclf for the system (4.69). With (~L60) and (4.64) we obtain 0'1

(.1',0)

= (4.i3) (4.74)

'\Tith the following adaptive cOlltrollaw and the paramet.er upclat.e law:

(x, 0)

(4.75)

B = T}(x,9),

(4.76)

·u. =

0'1

138

TUNING FUNCTIONS DESIGN

it is straightforward to verify that the closed-loop adaptive system is

where ;;1, -=2,'::3 are used with (j as all argument. The global stability of this system is established using the Lyapunov fuudion l'{x,O) = lIt(r,O) + !(JTO. ~

0

While in Lelllma "1.7 the initial system is augmented only by an integrator, a minot' modification is sufficient t.o obt.ain an ~Ulalogous result for the more general system

x =

f(x) + F(x)8 + g(.T,)c;

t =

u + F1 (x,c;)8.

(4.79)

Corollary 4.9 The Junction Vi (x, ,;,8) defined in {4.58} is an acl! Jor the system. (4.79) wit}, the control law and the tuning function gi'uen as (4.80)

(4.81) A repeated applica.tion of Corollary 4.9 will furtber extend t.he class of nonlinear systems for this type of adaptive design. With the knowledge of Va, i, alld 0: for the system (4.79), it is llot bard to see that by applying and 0:2 for the system Corollary 4.9 twice we can find 1'2,

'2.

.1:

{l {!!

= = =

f(:v) ~2 'U

+ F(x}8 + g(X)(l

+ Fl (3:, €l)8 + F2(X, elt ~!!)8.

{4.82}

4.2

139

SET-POINT REGULATION

In fact, it is clear that an n-fold appJication of Corollary 4.9 ",ill provide us with \;;' , Tn, and Q' n for the system :i;

=

1(:1.') + F{.1:)6

~i

=

~2

+ g{J:)~l

+ FI (J:,~,)6 (4.83)

~r'-l

=

~11

\Ve

WillllOW

4.2

€" + F,,-l {x: {b' .. , {1I-l)6 1/. + Fn(~t:,~1., ... '~II)8.

develop a detailed design procedure for such systems.

Set-Point Regulation

\Vith repeated use of Corollary 4.9, we can design an adaptive cOllj:!'ollel' to globally stnbilize a desired equilibrium .-,;C of the paramel1i.c sf.1'icl-Jeedback system (3.70): :i:) = ·1:2 + !PJ (xd T 8

x!! =

X3

+ Y'2(X1, l':SJ'e

;r:n

+
Our task is to stabilize the (z\, Z2t Z3)-system with respect to

1 'l V:... = If..- + --:: 2 ....... ' wbose derivative along (4.105)

V:i. =

Rlld

').,

-c'l:j - c2 zi +'::.1

(4.109)

(4..107) is

lJa:t (

:.)

8iJ rT,2-8 BCt'!

8Q''1

Va .. :.)

+=:i [ .":2 + .:., + a':~ - ax,-,1'2 - aX2-:l'!i + wJ8 - ---;;:.(} ao +OT (1;) + W:~Z:i - r-1o) . Vva can ciimi.nate

efrom

~a with the update hnv

A

B = rT:",

\Vbel"e

(4.110) T3

is our

tuning fund.ion

If x:-\ were OUf actual contl'ol, we would have ':., == 0 and achieve l~ = -Cl =r C3Z~ hy designing 0:3 to make the hra.elected term multiplying =3 equal to -ClZ3, Dft.nlcly

C:!=5 -

(4.112) where "'J is a eorrectioll term yet to be chosen. 5 Substitutiug (4.112) into (4.110). and noting t.hat

jj -

1'T2

=

0-

rT3

+ rT3 -

==

e-

rT3

+ rUJa=a,

rT;z

(-!.113)

(4.110) is rewritten as

Ii;,

= - 0, and 1·(t) is bounded and piecewise continuous, A realization of (4.1Si) which is of particular interest is

(4.187)

because, in this case, t.he derivatives of Yr arc available as the states of the reference model: y~i) = .l:1II 1i+l, i = 0, ... , n - 1.

4.3

157

TRACJ{ING

1

-c) [ -1

-C2

1

r s Figure 4.3: The feedback connec.'tion of the strictly pnsRive passive update law.

4.3.1

(=., =2)-system with a

Design procedure

The design for tracking is only tl minor modification of t.he set-point design procedure. As before, the first =-variable is the tracking error, =1 = :rl Yr' However, because the reference signal Yr(t) is not constant., its derivative y~i-I)(t) appears in the definition of the i-t.ll error state :i, i = 1, ... ,71. The only change this creates in the design is the addition of the sum L~:A u(::~)y~~.) OYr in the definition of ai. As we showed in Section 3.'1.3, nonlinear dampirlll can be used to guarantee global bouncledness in the absence of adaptat.ion, as well as to enhall('e performance. Therefore, the general design also in('orporat.es t·he nonlinear damping terms (4.188)

in the definition of a/so It is sufficient that we now only give a complet.e set. of recursive expressions for the stabilizing functions ll:i and the tuning fllnctio,ns Tj leading to the final adaptive control Jaw 11 and the final npdate law for O. These expressions, organized in Table 4.1. give a succinct summary of the t.uning funct.ions design.6 It can be checked that the resulting errOl" system has the following form similar t.o t.he set-point regulation case: : E UFor notational convenience we define =0 ~ O. ao ~ o. TIl ~

o.

lR"

(4.195)

158

TUNING FUNCTIONS DEStaN

Table 4.1: Tuning Functions Design for Tracking (4.189)

(4.190) (4.191) (,1.192)

i = 1, ... ,n ,-(i) _

.Ilr -

Un Ilr, ... , Yr(i»

(.'

Adaptive coutrol1aw: 1L

= {3(x) _1_ [a

(x

JI"

Parameter update

0"'/r 'ii(n-l» + y(I.)] r

(4.193)

law~

(4.U)4)

rWk,

the matri'l: ;1:; (z, iJ, t) has the form where, with the definition O'iI.. = - 8';;0-1 of (4.V!8) with the addition of t.he nonlinear damping terms -liiI1ll;j2Zj: -Cl - Ii.llwl1 2 -1

A:;

=

0

o

-C2 -

1 0 1i21w212 1 + 0'2:"

-1 -

0 0'21.

0'2:"

1 + Un-J.n -0'2n

-1 -

O',,-l,n

-C" - Ii ll lwn

l2

(4.196) alld 1-1/(.:,8, t) has the same form as ill (4.139). Although the functions O'iI•• and 'Wi lllay appear t.o be the same as ill the set-poillt regulation case, tbiR is not so, because uow they include y~i)(t) through the partial derivatives of ai-J, wbich is reflected in the dependence of A:;(.:, 8, t) and H'(.:, 6, t) on t.

4.3

159

TRACKING

The change of coordinates (4.189)-(4.192), which we compact1l' write as

(4.19i)

is smooth in 3: and jj and bounded ill I. Note also that the invprse transrormation

x is smooth ill

= 4>(=,6,1)

(J.108)

=and jj and bounded in f.

Tbeorem 4.14 The closed-loop (1.dClpti'l./c systcm. c01)"r;iBting oj the 1,lant {4.185}, I.lIe controller (4..193), and th.c 'U.pdatc law (..I. 194}· h.as a glol,ally ufl,ijo;'nly stable equ.ililnium at (:,0) = 0, and lim :(1) = O~ which meanB, in 1-00 particular, t/f.{I,t global asymptotic trrJ.cking is achic'ucd: (4.199)

lim fy(t) - Yr(1)] = O.

1-00

lIf()re.o1Jcr~ if lim y~i) (t)

'-00

Proof. Denote

Co

Ol i = 0, ... ,11 -1, and P(O) = 0, the1' ,lim x{t) = O.

-00

= mi1l1SiSn Ci. 1l'n

The uerivative of tbe Lyapunov function

1 T = 2"= =+ 'l-T 2 8 r-1o-

(4.200)

along t.he solut;ions of (4.105) aud (4.104) is (4.201)

=

which proves that the equilibrium (.:,6) 0 is globally unifOl'mly stable. Fl'om the LaSalle-Yoshizawa theorem (Theorem 2.1), it further follows that, as t ~ 00, all the solutions converge to the manifold = = O. From the definitions in (4.189)-(4.192) we conclude that, if lim y~t)(1) = 0, i = 0, ... ,71 - 1, and

F(O) = 0, then a:(I)

~

0 as f

~ 00.

'-00

0

The proof of Theorem .l1.V! reveals the st.abilization mechanism employed in the tuning functions desigIl. The update law is chosen so as to mal\:e t.he delivative of the Lyttpu110V fUllctioll llol1positive. The update law is fast because it does not use any fOl'm of normalization common ill t.raditional certaillty equivalence adi1.ptive C:Olltrol. The speed of adaptation is dictat.ed by the speed of the nonlinear behavior capt.ured by t.he Lynpunov fUllction. The tuning functions controller incorporates the knowledge of t.he update law and eliminates the disturbing effect of the parameter estimation transients on the error system. The controller and the update law designs are interlaced. The nai. theorem shows t,hat in the absence of adaptation the nonlinear damping terms guarantee boundedness. In addit.ion, global asymptotic stability is achieved for sufficiently small parml1eter error ii.

160

TUNING FUNCTTONS DESIGN

Theorem 4.15 (Boundedness and Stability Without Adaptation) Consider the closed-loop adaptitw system con.sisting oj the plant (4.185), the contmlle1' (4.193), and the ·upda.te la'llJ (,1.19.0, with r = 0 and Iii > 0, i = 1, ... ,71. All the solu.tions are globalll1 lJ,'niforrnly bounded. Fu,'the.7inoTe, if F(O) = 0 and Yr(t) == 0, then there exist Rn > 0 sytch that /01' each constant iJ E ]R!), 10 - 01 :5 Ro, the equilib7'ittm:c = 0 is globl/·1l11 asymptotically stable.

(Ei~J

t)-l.

r

= 0, the paramet.er est.imate 0 is consttUlt. Denote For the nonadaptive system (4.195) we have

Proof. Since

From Lemma C.5(i), taking

t1

IiO

=

= ;:2 and p = Fn 101, it follmvs that (4.203)

This proves that the solution .:(t) is globally uniformly bounded and, sin('e 3.' = q, ( :; I ij It) is smooth in :; and fJ and bounded ill t, this also proves that x( t) is globally uniformly bounded.

Figure 4.4: The solutioll enters the larger ball (solid) in finite time, Then, the equilibrium z 0, wbich is exponentially stable for 5ufficiently small 9, tal(es over and attracts the solution.

=

4.3

161

TRACKING

=

Now, suppose that F(O) = 0 and 1Jr(t) O. Since x = Dwhenever == 0, in view of (4.166), we have TV(D, 0, t} == O. Thus = 0 is nn equilibrium of (4.195). To prove asymptotic stability of == 0, we only need to consider t,ile case 0 "# O. The situation is depicted in Figure 4.4. From (4.203) we know

=

that =(t) converges exponentially t;o the ball of radius

4ink around == o.

This means that =(t) enters a larger ball, say the ball B of radius finite time t :::; T = rna..x {O l enters B it decays

~ In :!~I:(O)I}.

A, in

Note that before the solution

e~-ponentially:

I $. T.

(4.204)

VVe now examine tho trajectories inside 8. Because H'(:;, 8, I.) is locally Lipsrhitz and vanishes at 0, there exists a finit;e positive number L sllch that. fol' 0.11 : E B we have jlV{:::, 0, t.}j :::; Lj=j, and therefore

==

!!.(! jzj:!) dt 2

:::;

:::; - (co -

LIB!) 1=12,

t?. T.

=

The equilibrium z 0 is (locally) m.-ponelltially . st.able provided j81 z(O) E B we have T = 0 and

I=(t)j :::; 1=(0)je-(co-1J1fil)t . 'Vhen z(O)

rt B then T > 0 and, for t ;?: T 1=(1.)1
0, the. following inequality holds (4.231) Proof. For the adaptive system (4.195) inequality (4.202) gives

d (~lzI2) d.t _

::; -colzf + ~1912. 4#\'0

(4.232)

From Lemma C.5(7.), taking v = z2 and p = Fo191, it follows that (4.233) A bound on IIOII/Xj is obtained from (4.211) as 11911~ ::; 19(0)1 2 + 'Ylz(O)f, By substituting the latt.er e.xpressioll into (4.233), we complete the proof. 0

Remark 4.20 All elegant way of removing the term 'Ylz(O) 12 from t.he bound (4.231) wit.hout I;ntjectory initialization, is using t.he "observer"

z(O)

= z(O).

(4.234)

T As we shall see later, a similnr bonnd can be derived for linear system.! even without these t.erms.

4.4

167

TRANSIENT PERFORMANCE

VVe introduce the observer error

-

~ ==Z-Z',

(4.235)

and instead of (4.194) we employ the updat.e law driven by the observer error: (4.236) Combining (4.234) and (4.235) with (4.105) we obtaiu the observer error system ~ A:(.:,O, t)z + HT(=, 6, t)TO.

=

Now, along the solutiollS of {4.236)-(4.237} we have (~1.238)

which implies that (4.239) because =(0)

= =(O) -

=(O) = O. By substituting (4.239) into (4.233) we get (4.240)

instead of (4.231). Using this method, the £2 bound (4.226) and the £00 bound 0 (4.230) remain unchonged.

With our stal1dard initializat,ion ::;{O) = 0, combining t,hc bound (L1.231) with the bound (4.230), we get 1 1 } Iz{t)\ ~ min { ?~'" r::; \8(0)1· -

COliC

v'Y

(4.241)

It follows t,bat the £00 bound for the tuning functions scheme can be systematically reduced by increasing anyone of the design parametel's: Co, lio, Dr 1'. The tuning fUllctions scheme is uniqne ill its capabiHty to use the adaptatiol1 gain "y for improvement of transient performance. In Theorem 4.15 we proved that: nonlinear damping guarantees boundedness without adapta,tion for any (unknown) amount of parametric ul1(·ertaint.y. For the nonadaptive system we also estabUshed an Lee performance bound (4.203), which with =(0) 0 becomes

=

(4.242)

168

TUNING FUNC'l'IONS DESIGN

where 8 is a constant parameter error. Let us suppose that. tIle paramet.er estimate initial condition 8(0) in the adaptive design is the same as the const,ant parameter est.imate iJ in the nonadaptive design. (This a..'isumption is reasonable because the a priori lmowledge is the same.) Comparing the adaptive bound (4.241) wUh t.he nonadaptive bound (4.242) we come to an em;y but. fundamental conclusion: ''YIth adaptation gain "'Y > -kalio! the adaptive bound (4.241) is lower than the nonadaptive bound {4.242}.

4.5

Extensions

For t.he sake of clarity, the adaptive design in this chapter \Vus presentcd for the class of panullct;ric strict-feedbad: systems. We now givc three extcnsions of the tuning functions design. Vve first consider strict~feedbRck systems with unlmown virtual cont.rol coefficients. Second, we explain how the design is modified for parametric block-strict-fccdhack systems, Third, we extcnd thc design to parametric pllre~fcedback systems.

4.5.1

Unknown virtual control coefficients

\Ve consider systems of the form .1;j

i' n

== bi:Z: i + 1 + 0 Po + p'r (0 - t 1-

1

(t1.297)

Xn

k=2

r

(OD:k::l)T =k) 88

Let us firsf'. analyze condition (4.297). From (4.290)

WE.'

> O. compute

(4.2!J8)

178

TUNINO FUNCTIONS DESIGN

Thus, condition (4.297) becomes (4.300) By repeating the computations as in (4,209), condition (~1.297) is brought to the f01'm EJ'PiT ( 0'" )PI' )] '"'" r (0. £l:k-l -L...., IT 1 +-a:r:i+l 1.'=2 ao- .-'k, 'j

n-l [

i

;=1

>0

.

(4.301)

vVith (4.301) and (4.298), we have the following proposit:ion. Proposition 4.24 Let B z em." and Bo C lR" be open sets such that (.1:,8) = (0.0) E B.r x Bo, and f01' all (.1:,6) E B.r. X Bo,

)'f) >, L r (J:l- ...- )T) >

~ ( (J~ _ L...., . 8xi+ 1 '-=2

1 + lJ'PiT

f30 + f3

or

-

(

0-

Then :F = Bz Table

X

r ( ao:/.-_~ 1 lJ(J

U

UO:k_)

k=2

()f)

0

..

-k

Zk

O.

i=1, .. &,11-1

(4.303)

Bs is a subset of the sel in which the design. p1vcedu1'e oj

4.2 is feasible.

Let us now examine the clulllge of coordinates in Table 4,2. For all. i

=

1t •• , ,n we have

(4.304) 'Ve showed above that (4.305)

4.5

179

EXTENSIONS

By subst.ituting (4.305) into (4.304), and inductively applying the implicit function theorem, we cOIu'lude that the change of coordinates

(.1·,0).- (:,0)

(4.306)

is one-to-one, onto, smooth, and has a smoot.h inverse on:F. !Vroreover, since IPo(O) 0, CPl (0) = .,. = 'P,. (0) = 0, by examining (,1.304), one can show

=

inductively that. x=O



(4.307)

==0.

Because of (4.307) it is not hard to show that. a feasibility condit.ion, equivalent. 1.0 Proposition 4.24, is that there exists a set :F' = B~ x B~ cont.aining (;1:,8) = (O,B) such thaI. for all (:c,O) E :F', 1 + a"'i(x)T 01 O:Ci+l

> 0,

; = 1", . ,n - 1

1

IJ3o(x)

+ ,8(.1.),1'01 > O.

(4.308) (4.309)

It is then possible to find a subset. of :F' which is anot.her est.imate of the feasibility region. In general, the feasibility region is not global. However, t.his is not due to the adaptive scheme. Even when t.he paranl('t.cl's 0 al'e known, the feedbac1\: linearizat.ion of the system (~1.285) ran only be guarmlteed for lJ E Bo clAP, an open set such that for all (.1;, B) E Br x Bo, 1+

1

a;i(~·)T lJl > 0,

l,8o(:r)

i

:rj+l

= 1, ... , n -

+ ,8(x) TO I > O.

1

(L1.310)

(4.311)

Let us now return to the closed-loop adaptive system (4.29..1), (,L287),

(4.312)

The derivative of t.he Lyapunov function

1

" = -:; 2

'1'

I-T =+ -lJ r-1o2

along the solutions of (lL312) and (4.288) is •

JI

"I

V ---~c·-­ ,...;. ~

i==1

(4.313)

180

TUNING FUNCTIONS DESIGN

=

This proves that the equilibrium z 0,0= 0 is stable. Since we showed above that the coordinate change (4.306) is a diffeomorphism which preserves the origin, we condude t.hat the equilibrium :£ = 0, {j = 0 is also stabJe. ""Ve HOW give an est.imate c :F of the region of attract.ion of t.his equilibrium. Let. n(c) be the inval'iant set defined by l"(.1:,6) < c, and Jet. c· be the largest constant c such t.hat O(c) c:F. Then! an ostimate 0 of t.he rogion of at.t.nu.·tion is

n

c* = arg sup {c}.

( 4.315)

O(c)CF

Finally, hy applying [81, Theorem 4.8] (a local version of the LaSalleVos hizawa t.heorem (T heorem A. 8») to (4.314.), it foHows that z (I.) ~ 0 as t - 00. In view of (4.307), this means that x(t) - o.

Theol"mn 4.25 Suppose Ihat sllstem, (4.285) sati.sfies Propo.r;itioTi. 4.24. Then the closed-loop adn~)tive sysfem CD7}"f~i8tirr.lJ oj the plant (4,285), the controllu'w (4.293), (J:ltd the u.pdatc ia.1IJ (4.294) has a stablc cquiiib7'ill.m.1: = 0,6 = 0, and its 1'Cgion of attra.ction in.cludes the sct fA defined in (4.315). F1J.7·the'lmo7'e, for all (:£(0),0(0)) E 0, we hn.IJC lim :l'(t) = O.

1-00

4.6

(4.316)

Example: Aircraft Wing Rocl
2,

(5.13)

the solution (5.12) escapes t.o infinity in finite time, that. is 1

:I:(t)

-j.

00

as

:c(O)O(O)

(5.14)

t -+ 2111-X(-o....:....)O=""(-O)"--_-2

This cat.astrophic instability is due to cp(x) = ~.:2. If, instead. we had i:t nonliuef! rit.y 'P(.1:) with a linearly bounded growth, Icp(:I:) I $ 1..:lxl, the above instability would not have Dccuned. A far-reaching conclusion to be drawn from the above example is that a modular design, with a certainty equivalence controller and a standard identifier as it.s modules, is not applicable 1;0 systenls with nonllnearities whose growth is higher than linear. The mechanism of instability is clear: The ident.ifier, whose speed of convergence is at. best. exponent.ial, is not rast enough to cope with potentially e..xplosive instabilities of nonlinear syst.ems. This simple example shows that; t.o achieve stability we need either a much fast.er identifier, as in our tuning functions design, or a stronger controller with a biggf'r stability margin for dist.urbances such os 0 in (5.3). \Ve consider the parameter estimation enol' and its derivative as two independent disturbance inputs and design stronger cOllt.rol1ers t.hat guarantee boulldednesA of all the states of the closed-loop syst.em whenever 1. t:he parameter estimation error 2. its derivative

8 = -0 is

8 = 0 - 0 is bounded,

and

either bounded or square-integrable.

These new controllers create a possibility for a complete identifier·cont.rol1er modularity. Such (.'omplete cont.roller-identiliel' separation has been an unachieved goal of adnptive cont-rol, even for linear systems.

5.2

5.2

189

ISS-CONTROL LYAPUNOV FUNCTIONS

ISS-Control Lyapunov Functions

In Section 4.1 we introduced a framc\Vork for Lyapunov design of adaptive stabilizers where thc central role was played by adaptive ("ont.rol L)'apunov functions (aclf's). In this section we develop a framework for modular design of adaptive l1011lillC:'al' controllers. Let us consider t.be system

:i: = /(:I:)

+ F(:r}(J + g(:I~)11

(5.15)

where /(a:), F(3.·), and gel:} are smooth, and 1(0) = 0, F(O) = O. \Ve say that. syRtelll (5.15) is globally adaptively stabilizable l if tlu1J'e exist a function a(:l:, 8) cont.jnuons on (Hl" \ {o}) x IV with Cl (0,0) == 0, continuous functions T(a~, 0, 71} amI fl(:r:, 0, ',), and a positive dpfinitc symmetrie p x p matrbc r, such that the dynamic controller 'lL

=

6 =

O:(:l~, 0)

(5.16)

rT(X, 8, 17) fl(x. 8, 1])

(5.17)

= " guarantees tbat the solution (:c(i), OCt}, .,,(t» is global1y boundcd, and .t(t}

(5.18)

as t -;. 00, for all (J E W. We I'cler to (5.17)-(5.18) as an identifier, wUh (5.17) being it.s update "Vc st. art by rewriting (5.15) with (S.lG) as .1; = j(.r)

+ F(:r)O + g(.1:)a:(X, {}) + F(;,:)8.

-+

0

IRW.

(5.19)

Suppose we know how to find Ii ('ontl'olla\V a:(x, 8) UUIt fitabilizes this syst.em with 0 = O. As we saw in Section 5.1, ill tbe presence of t.)le dist'.tubance inpnt 0, this cOlltrollaw, in gencral, does not preserve st.ability evcn if 8 is bounded and exponentially decaying. To preserve stabilit.y, we need a strollgel' conj~rol1er. Sincc the standard parameter estimators guarantee thn t. jj is bou llded, \VC are int.erest.ed in designing controllers which call guarantee input-to-stRt:C stabiHt:y of (5.19) with respect to 0 as input, in the sense of Dpfinition C.l. Howe\'er, ~he time-varying character of the paramet.er estimate OCt) forces us t.o consider

8(t) as another disturbance input, even though it. is not explicitl)' present. ill (5.19). Our goal is to find a controlla\v 0'(x,8) continuous on (R" \ {O}) x mp with 0:(0,8) == 0, such that the following inp1tlrlo-st.nte stabilitIJ (ISS) propert.y is satisfied:

Ix(t)\ ::; P(\J'(O)I, t) + "I (sup

O::;T::;I

[~(T) J) . O(r)

(5.20)

J'fhjs definition is more general than the one from Section 4.1.2 to suit the rnodulnr approach.

190

MODULAR. DESIGN WITH PASSIVE IDENTIFIERS

where {3 is a class JC£ fUllction and 7 is a class JC function (see Appendix C for detaHs). Lin, Sontag, and Wang [115, Theorem 3] recently proved that a necessary and sufficient condition for (5.20) is the exist.ence of an ISS-LlIapunotJ function. (The proof of sufficiency is given in Theorem C.2.) 'Ve say that a smooth funct.ion l' : lRJ1 X lR.I) -+ 114, positive definite and radially unbounded in :c for each () is an ISS-Lyapullov function for (5.15) if there exist.s a class /Coo ~unction p such t.hat the following implication holds for all :z: =F 0 and all

0,8, ii E lRP :

(5.21)

JJ.

aV' [

-8 f(:c) x

avo

- av' 8;. < O.

+ F(:c)8 + g(a:){l'(x, 8) + -a F(x)O + A

A

]

x

W

-A

If there exists such a triple (0, V',p) we say that system (5.15) is input-to-state stabilizable with respect t.o

(0, 0) .

\Ve have t.lms chosen 1:0 st.udy the problem of modular adaptive stabilization as a problem of input.-to-state sta.bilization. Similar to Section_ 4.1 where we cast Lyapunov adaptive stabilization in the framework of control Lyapunov functions, in this section we develop a clf framework for modular adapt.ive stabilization.

Definition 5.1 A smooth function v· : [ttl x JRP ~ 1R.+. positive definite and radially 'imbounrled in x jor each 0, is called an ISS-control Lyapunov function (ISS-elf) jor (5.15) if there exists a class /Coo .function p such tlla t the following im.plication holds for all x # 0 and all 0,6, fJ E nlP :

1·1: 1

~p( [ ~] ) ./J

inf

uEIR

(5.22)

all' [j(x) + F(x)8] + 8V' av' ~} < o. + g(x)'U -8 F(x)8- + -.8 { -a. ~ x ao

We now show that. the existence of an ISS-c1f is a necessary and sufficient condition [or input-to-st.ate st.abilizahilit.y. The proof of sufficiency is construct.ive--we design a control law starting from a given ISS-elf.

Lemma 5.2 (Input-to-State Stabilization) System, (5.15) is input-tostate stabilizable with 7'Cspect to if and only if there e.-vists an ISS-elf.

(8,9)

5.2

191

ISS-CONTROL LYAPUNOV FUNCTIONS

Proof. The 'only if' part is obvious because (5.21) implies that there exists a particular cOlltrollaw 'It = a'(x,9) which satisfies (5.22). \Vc \yillllOW proye the 'if' part by showing that the following control law achieves input.-to-state stabilization:

~~O(x,8) -# 0 l:J\'

(5.23)

•

o;:O(:r:,8) = 0 t where

.

w(a:,8) =

elf [f(x) + F(3:)8]. + [81' " 8\']1'_1 -a -0 F, - . p (lxl) . .1' an

(5.24)

:1'

\¥e first sbow that (5.23) if; continuous on (lRfl \ {O}) x 1Rl'. In [1(1), Sontag provC'd that the fll11ct~ion (5.23) is slllooth provided that. its argUlllents wand ~~ 9 are such thitt

all"

-g=O => w 0 there is a 6 > 0 s\lch that, if ;1: " 0 satisfies p

then there is SOlUe

avo [

11

with

-0 f(x) .1'

lu.I

~ E:

(I [: ] ) : ; 1.'1 ::;

6,

such that

. ] DF F(x)O- + -~ at-' :. + F(.J·)8 + g(x)1J. + -8 0 < O. :1' ao

(5.30)

Now we show that. t.he controlla,w (5,23) achieves input-tcrstat.e st.~tbiliza­ t.ion, Along t.he solutions of (5.19) and (5.23), the derivative of l' is

fJlI]T all F, -~ [-a :z: aB

-1.

p (Ixi)

av F(.t)B . - + av:./J + -a ~. ao -a


, ~~r P-'(I.vl)r + (~~'g)' (5.31) In view of (5.29) this proves that 1i' < 0, 'rI.v " 0, whenever

1·r.1 ?: P ( [

~ ] I) ,

that is, V' is an ISS-Lyapunov func:tioD, which by Theol'em C.2 establishes that (5.15) is input.-to-state stable with respect to

(0,0).

0

This lemma shows that, if an ISS-elf is available, it is easy to design ~ cont.rol law which guarantees bOllndedness of the state x whenever 0 and ~ M'C bounded. Therefore, we neE-d identifiers which can guarantee that, 0 nnd iJ are bounded. As.we shall see in Cha,pter 6, the swapping identifiers guarantee t.hat both 8 and iJ are boundcd. The passive identifiers that we present in this chapter can guarantee only.the bounded ness of O. Fortunate1y, even though they cannot guarantee that 0 is bounded, they can guarantee that it i~ squareint.egrable. The boundedness of 6 and the square-integrability of jj will be enough to establish the boundedness of x because we will design controlJers

5.2

193

ISS-CONTROL LVAPUNOV FUNCTIONS

which guarantee tbe following linear-like relationship;

i\ > O. Wit1~ inequality (5.32) we will show t.hat

.1:

(5.32)

is bounded whenever ij is bounded

and {} is square-integrable. In contrast to the Lya.punov design, where it is not dear if i.he global asymptotic stabilizabilit.y for each (J is a necessary condition for tlu:> existence of an aclf (sce Remark 4.4), t.he global asymptotic stabilizabiJity for eat'll (; is a necessary condit.ion for the existence.of an ISS-df. This becomes ohvious by set.ting B(t) Ne~i;

== 0,

whi(~h implies 0(1) = B(t}

== 0, into

(5.20).

we give sufficient conditions under which a:(t)

-+

0 as t -+

00:

Duf"

to the ISS pI'operty (5.20), one sufficicnt condition is thnt both 6 and {} tend to zero. However, ill general, identifiers cannot guarantee that 8 goes to zero, so the lle>...i; lelIlma. gives a less demanding condit.jon. Both the passive (see Remark 5.15) al1d the s,vappil1g (see RClIuuk 6.8) identifiers will be able to guarantee that these conditions arc satisfied.

Lemma 5.3 (Regulation) Suppose f.he contro/lull1 U = Q·(.v,6) gUll1fl.nlees that system (5.1,5) is ISS with 1'eslJcct to (OtO). If a:(t) is bounded, and F(x(t»O(t) and 6(t) converge Lo zef"O a.s I ~

00,

then lim,_oo x(t} = O.

Proof. Since the system :i: = J(x} + F(x)O + g(x)a:(x, 0) is ISS with respect to

(5.33)

(0,9), i;hen the same system with 6 = 0= 0, name]y

the system

.1! = I(x) + F(a')O + g(:t:)a:{x, 8)

(5.34)

is globally asymptotically stable. Therefore by [174, Theorem 2] there exist f3 E K.£, "y E IC, al1d a continuous fUllction 0' : R+ -+ R+, O'(s) > 0 for" > 0 such that for each continuous and bounded input wet)

6. [ =

F(:r:(i))8(t) ~

OCt)

1, for

each x(to} E JR." t and for aU 1 ~ to 2: 0, t.he following implication holds;

(5.35)

-lL

Ix{t}1

:s; ,B{lx(lo)l, t - to) + 'Y (suP~J$"':91'W(T)1)

.

194

l\1:0DULAR DESIGN WITH PASSIVE IOEN'fIFIERS

=

Let itl be sucb that /.l:(t)/ :5 1"[ for all t ~ O. Let e mil1{ u(r) I", :5 ill} > 0, and let. T 2: 0 be such that Iw{t)f ::; e for aU t ~ T. Then from (5.35) we obtain

Ix(t)l ::; ,8(I:r{to)l, t - to) + '1 ( sup

t~I$'T9

IW(T)I)

(5.36)

for all 1 2: to ~ T. TllUs, from time T onward, the system satisfies the ISS inequality (5.36). To complete the proof, we have to show that. tbe ISS property implies thl:1.t :ret) -+ 0 as t -+ 00. Our computations folloW' those in the proof of [173, Proposition 7.2]. First, we 11ot:C that there e.'1i.:ists 8. lllonoLonicaHy decreasing to zero I'unction "1 continuous all [T t (0) such thil.t

Iw(t)1 :5 T/ {t - t·o} ,

'tit ~ to

~

T.

(5.37)

Then we have

Ix{t}1

~ P (Ix C~ T)I, t ~ T) +-y c;~s, IW{T}I)

~ P (p (Ix (T)I, t ~ T) + -y (TS~~~ IIII{T)I) , t ~ T) +y

(¥~:9 '1 (T - T))

.

Not.iug that for any class 1C function 0, 6(a + b) nonnegative a ll.nd b, we llroceed from (5.38) with

which converges to zero as t

-+ 00.

{5.38}

:5 b(2a) + 6(2b)

for any

0

As we shall see later in this chapter, as well as in Chapter 6, ~otb the passive and the swapping identifiers guarantee that F(x(t))ii{t) and O(t) converge to zero whenever x(t) and u(t) are bounded.

5.2

195

ISS-CONTROL LVAPUNOV FUNCTIONS

The two lemmas in this section outline a framework for modular adaptive design. Lemma 5.2 shows how to design a control law once an ISS-elf is known. While Lemma 5.2 gives sufficient conditions that the identifier has to saHsfy to gum'untee boundedness of the plant state x, Lemma 5.3 gives sufficient. identifier conditions for regulation of .1'. Identifiers satisfying t:hese conditions will be designed in this chapter and Chapt.er G. Therefore the main task is t.o find an ISS-elf for a given system. For t.he simple scalar system

:?

i'

±=

I(x)

+ F(a;)B + g(.l:)l1 ,

where g(x) ¥= 0, a valid ISS-elf is Vex) control law .

U

1

= x'J..

= 0-(3:,0) = () (- J(x) g :c

(5AO)

.1: E 1R I

This is easy to see be('ause the

(5,41)

F(:I:) -:r -IF(x)I.1;)

yields

all'" [J(.1:) + F(.r.)O- + g(x)a(.'l!lB). ] + -F(a:)8 av - + Otl -A ·0 a.... 0:1: [)fJ = _2x2 - 2IF(x)lx 2 + 2:I:F(:v)O $ _2:1.,2 - 2IxF(x)I(lxl -191) < -2:c 2 whenever IJ:I ~ 16L which implies that system (5.40)-(5.41) is ISS with respect to

(5.42)

(0 0). 1

(To make the control law 0;(.1', ti) smoot.h, we replace IF(x)1 in (5A1) by JF(x)F(x)T + 1.) Since we know how t.o design ISS-clf's for scalar systems, our approach is recursive: We a.':isume that. an ISS-elf is knowll for an initial system, and construct a new ISS-elf for the initial system augmented by an integrator using backstepping.

Lemma 5.4 (ISS-Backstepping) If the system

.r. = J{:I:) + F(:c)8 + g(x)u is input-to-state stabilizable with respect to b01mded, then the augmenteci system .i'

=

I(x)

~

=

u

(0,0)

(5.43)

'using

+ F(.1:)O + g(.1:)~

is also inpu.t-ta-state stabilizable with 7'fSpect to ( 6,

B) .

o' E

C l J and

9 is

(5.44)

196

l\JI00ULAR DESIGN WITH PASSIVE IDENTIFIERS

Proof. Since (5.43) is in pli t- t.o-state stabilizable with respect to

(0, 8), there

exist·s a. triple (0', V', p) and a class K. function 1" stich that

Ixl

~ p ([

r)

(5,45)

.lJ. •]

avo [ • aV' - aV' :. -a f(:r) + F('-l:)8 + g(.r)a.(x, 0) + -a F(.l·)O + -. (J :5 -p(l:r/) . x x 00 In fact, without loss of generality we assume that 11· is class /Coo' It. was shown ill [173] t.hat if Jl is only in class /C, t.he given Lyapunov function V' can be modified so t.hat the new I' be in class /Coc . For p. E /Coo, it wa..1:j shown in [177] t.hat (5.45) is equivalent to the following 'dissipa.tion' t.ype of characterization:

aV' [ /(x) + F(:r.)O + g(:I:)a(.T 0) -a x A

-]

t

of - aF:.() ::; -jl·(I:rl}+7f +'7lF(x)O+-.

ao

uX

(I [0 ] ) :.

8

.

(5.4G) where 7r is a class IC Cunct.ion. Since the proof of the affine case considered here is simple, we give it for eompleteness. It is clear that (5.46) implies (5.45). To see the converse, one only needs to cOllsider the case

1."1 ::; p

(I [~ ] ).

Since

ois bounded, with Young's inequality one obtains a1' [ . - ] DV' - aV' :. -a /(:r) + F(.l·)O + g(:t)a(x, 0) + -a F(x)O + -. (J + JL(lxl) x x 00

av- [

::::;; a.1.: f(x)

+ F(x)1J. + g(x)a'(x,lJ). ] + 11·(/.1..'1)

+~ [~~ F(.r), ~~r' + [: ::; Mlxl) + [~ll' fi

:;

0

r

P ([:: )

+ [:

~ rr ([ : ] ) ,

r (5.47)

where p. is 8 dass /Coo fUllction. This completes the proof of (5,46). We will now use (5.46) to show t.lmt

.

Vi (x, ~, 0)

• = l/(x, 0)

1

+ 2(~ -

- '1 a(x, 0))-

(5.48)

5.2

197

ISS-CONTROL LVAPUNOV FUNCTIONS

is an ISS-clf for (5.·U). \¥e do tbis by showing that, the controllA\\, 11

= a:1 (:r, e, 0)

=

811

- - g - (~- 0')

a.l'

-

Do: ( +a.t f + F8 + .O~ 4

oa]T 2(e -

80: -.. [-F(;}:), a'-I: 88

)

(SAg)

0:)

achieves input-to-stat,e stnhilizut.ion of (5.44). Towards this cnd, consider

l~

=

~V [/(:1.') + F(:l,)jj + g(:r)o·(x. 8)] + av F(:r)9 + D\~ 0+ a\r !}(e -

ox

11.1;

+(€ -

~

a) (u - :: (f + FH g€) -

- JI(lxil + w

of}

a:l'

Q.)

~~8)

(I [~ ]I)

+«(-0) (u+ ~:g- :: (J+FIi+g~))

ao: -

00' ~

-(e - 0') a.~ Fe - (e - a) ao B

~

-p·(I,eil + 7r (

[ : ]

I) -(( - aV

_ [aa: F, D~]T 2 (e _ O,}2 _ (~_ Q.) faa: P, a~]']' [ ~ ] a.J;

~

a.l'

Df)

-JL(lxi) + 1r

(I [: ]I) -(~ -

of +

~

ao

[:]

l

0

•

=

(5.50)

Denoting 'iil{") 11'(1') + ~1·2 and picking a class /Coo function /II{l') min{JI.{1·), r!.!}, be(~ause of t.he boundedllcss of 0, we gct:

:5

where Ji'1 is a cla..IOiS }Coo fUllction. Thus, \Ii is an ISS-Lyapullov function. Bl' applying Theorem C.2 to (5.51) witll P = pi] 0 211"11 we prove that s)lstem

(5A4) with control law (5049) is ISS with respect, to

e,

(8,0).

0

The control law al(.r. 0) in (5.49) is ouly one out of many possible control laws. Once we have shown that Vi given by (5.48) is an ISS-elf for (5.44) (with

198

lI!fODULAR DESIGN WITH PASSIVE IDENTIFIERS

=

P J.i2'J 0 27rd, we can use, (or example, the CO control lawa'l given by the fonnula (5.23). While in Lemmo 5.4 the initial system is augmented only by an int.egrat.or, minor modificat.ioll is sufiicient to obtain an analogous result for the more general system .f = fr:c) + F(x)f) + g(.t)( (5.52) ~ = 11 + FI(x, ~)(J.

1:1

Corollary 5.5 The Junction Vi (x,~, 6) defined i1l, {5.48} 7.s a.n ISS·clj Jor S1)Stern (5.52) 1l1i.lh I.he control law

QI (x, ~, 6)

= 0:1 (X,~, 6)(5.49) -

Fl (X, {}O -IF1(:l", ~)12(~ - a(x, 6)) .

(5.53)

An n-fold application of Corollary 4. 9 will provide us with \~ and ar! for the system :& f(x} + F(x)fJ + D(X)~l

=

{I

=

{2 + PI (:1:, ~l)B

(5.54) ~n-l ~n

= =

€n + F,,-l(X, ~lt··· ,~,,-d8 U

+ Fn(x, ~lt ••• '~fI)6.

We will now develop a detailed design procedure for such systems.

5.3

ISS-Controller Design

Our goal is to develop a modular adaptive design for nonlinear systems in the pal'amet7;'c strict-feedback form

XI

=

X2

+ CPI (xd T 8

2;2

=

.'1:3

+ CP2(Xl! X2}T() (5.55)

:&11-1

Xu

:Z:1l

=

+ ¥'n-1(Xl1"

• 1.7:n _I)T()

j3(:c)'U. + 0 U = o(x, t} - '\Ip(l:, t}l- ax o(x, t) ,

guarantees that: 1. // eithe'"

of the

(a) dE £00

condition.,

01'

(b) dE £'}, and U(x, 1.)

~

cl/(x, t), c> 0

is satisfied, then x E £00' ~. If d E ~

n £00

and U{x, t) ~

clxr!,

then limt_oc x(t)

= O.

Proof. 1. Due to (5.57), the derivative of" along (5.56), (5.58) is

if =

~~ [I + ga: + 9 (-,\p'fp:~g + pTrJ)] + :

i 0;':

< -U _ A p of 9 -

~

-U + 2:..ldl 2 • ..:1,\

2

d1 + 2.l d l2 2. 2'\ 4,\ (5.59)

(a.) Since U is positive definite and radially unbounded, there exists a class A.oo function l' such that U(x, t) ~ 'Y(lxl), and tberefore {5.60}

200

MODULAR DESIGN WITH PASSIVE IDENTIFIERS

It follows that. if I:vl ~ I-I C!~\ IdI2 ), thon V ::; -~/(lxl). By Theorem C.2, system (5.56) with ('antral (5.58) is ISS with respect. to d. Hence, if d E £0::;, t.hen .1: E £0::;' (b) In the case when U(l.', 1) ~ cV(.t', t), from (5.59) we get .

1

~

V" ::; -(:1' + 4. .\ Idl- .

(5.61)

\Vit.h d E £'!.t by Lemma B.5, .1" E Coo. 2. Integrating (5.59) over la, t], we obtain

c 10' I·r(r)fdr < ::;

\ (' Id(r)1 2 dr + 11(0) Jot U(.t'(r), r)dr ~ 4A.k 4~\ IIdll~ + v"(0) ,

V"(t) (5.62)

which implies that. .J' E £2' By part 1 of this lClluna, x E £00' and therefore 'U E £rxJ' Hence:i: E L. oo • By Barbalat's lennml, (Corollary A.7), l;(t) ~ 0 as t

~ 00.

0

Our primary inl,erest. is in prut 1 of Lemma 5.6, which states that. :1' j~ bounded if d is either bounded or square-integrable. In our design, 8 and ij play t,he role a! d. The ident.iIiers that we shall use will gnarantee that 8 is bounded, and [) is eit.her bounded or square-integrable. Part 2 of Lemma 5.6 is also useful. l~. will help us to achieve tracking because our identifiers will guarantee t,hat. iJ is square-integrable. Before we develop a general design of a cont.rollaw for the parametric strictfeedback systems (5.55) we illustrate the use of Lemma 5.6 on a secolld-ordpr example. Exrunple 5.7 Let us consider the system ·1:1 =

·1:2

x:! =

IJ.

+ 'P(.1:d'l'8

(5.63)

Viewing .1:2 as n cont.ro] input, we first, design a cont.ro] law at (Xl, 0) t.o guarantee that the state :l'1 in .i:, = .1:2 + 1p(:rd T Ois bounded whenever 0 is bounded. Following Lemma 5.6, we design (5.64)

Then we define the elTor vcuiable =2 = =1 = Xl. The first equation is now

J.·2-0'1(Xl,

8). and for uniformity denote (5.65)

If Z2 were zero, the Lyapunov fUllction l'I = ~zi would have the derivative

(5.66)

5.3

201

ISS-CONTROLLER DESIGN

which would mean that ZI is bounded whenever this is no longer clear because then

8 is bounded.

\Vith

::2

:F

0

(5.67) The second equation in (5.63) yields •

•• 0:1

=2 = X2 -

80') (

=

l/. -

lJx]

The derivative of the Lyapullov function

Tn)

+ cp

-

t12 = Vi +

~=i

!l':!

ae n:' .

80'1

(5.(8)

= ~I=I~ is

(5.69) Qur design is now led by Lemma 5.6. To make the bracketed term equal to ')

-C2Z2 -

U

=

wbere

#b2

'~CP\"" Z2

'1

g21~ T," =21 we design t.he control law

-

ea.}

-4;1 -

C.,Z", -

....

> O.

C2, 1i.'J"g'l •

V'2:S

-

'1

- 1ax}"" 1 Z"- - g.)-

Ii...,

-(I')

TI'1 Z., -

1DOl"

-.

80

-

ao']

(X'1 8.'1.:1 -

+ .,..

T·

,I')

n) ,

(5.70)

Thus, using completion of the squares as ill (5.59), we get ')

(1 ~'1 1 ~'1 + + - 1 ) 181+ -191-, 4Ii.1 41i2 492 'J

-CtZi - C!!z:;

(5.71)

which means that the state of the error system

(5.72)

:I 'R

l

is hounded whenever the disturbance inputs

8 Rnd iJ are bounded.



'Ve now consider the parametric strict.-feedback systems (5.55). Tbe recursive design procedure is given in Table 5.1. 2 :!For Dotational convenience we define :0 ~ 0,

00

~ o.

202

l\'10DULAR DESIGN WITH PASSIVE IDENTIFIERS

Table 5.1: ISS-Controller

(5.75 )

(5.76) i = 1: ... : n 'y' -.vi

('1'"1,···

,-(i) _ ( .

Yr

'j' ) , l'-i

-

,

(i»)

?/r,!ln···)!lr

Adaptive cont-rollaw: ,II _ -

1

-(.)

/3

,r/-

[. ( . B~ -(11-1») 011 ,1., ']}r

+ ]}r(11)]

(5.77)

By l'Olnparing the expression for the stabilizing function (5.74) in the modular design with the expression (4.190) for the tuning functions design we see that the difference is in the second lines. \Vhile the stabilization in the tuning functions design is achieved using the terms iJ~Jit [Ti + 2:t:~ a(~;;l fWiZk: the stabilization in the modular design is achieved with the nonlinear clamping tenn -Si:':j.

Clahn. The dosed-loop system obtained by.' applying the design procedure (5.73)-(5.77) to system (5.55) is (5.78) where }-l:, Hl, and Q are matrix-valued functions of z,

A.:(z, 0,1)

-C1 -.51

1

0

-1

-c:! - 82

1

0

-1

0,

and t:

0

0

1 0

0

-1

-ell -.5 11

5.3

203

ISS-CONTROLLER DESIGN

o _ili:u. Of}

_

E Jff!>;p. (5.7D)

{h~'I_-l i:J(j

Proof. For i = 1 we have

':2+0:1 + -(('I + sd.:) +':2 + For i 2, ... , n - 1, since (. '"» fJ, !iI" ...• yr'-- 1 wc have ~

O'i"-l

is a function

wTo.

(5.80)

or only the variables :1'11 . . . :

( .1: /,._!_}

.rj_]

1

+ i.p,.T 0)

(5.81)

For i

n ,,,e have -11

(5.82)

in vedaI' form yields (5.

o

System 78) will be referred to as the el'ro1' Notc that the first. component of its 2:1 = :CJ - Yr = Y - ~Jr, represents tlH:~ tracking errOl', The ebange of coordinates (5.73)-(5,75), which we compactly write as (5.83)

204

MODULAR DESIGN WrTH PASSIVE IDENTIFIERS

is smooth in x and iJ and is hounded in f. N ot.e also that the inverse transrormation x=iI>(z,6,t) (5.84) is smooth ill z and iJ and is houndp.d in t. Except for the term Q( z, iJ, t)T 0, tbe error system (5.78) is similar to the error syst.em (4.195) in which the t.erm

Q,(=, 8, trrO was accounted COl' by using

tuninlJ fund·ions. Here we let both ii and iJ appear as disturb!U1ce inputs. In the modu1al' design their bound~dl1ess will be gum'auteed by parameter identifiers. We now establish the basic inpllt-to-shtte properties of the error system (5.76), (5.78), (5.79), lllalullg use of the following cOllstil.nts:

Co

= min Cit

"·0

1:$i:$11

=

1) -I L --: ;=1 Ii; rI

(

t

90 =

(rlL-1)-1 ;=1

9r

(5.85)

Lemma 5.8 In the error system (5.78), (5.7!J), (5.76), the following inputto-state pmpc1'ties hold:

1

,=(t>l ~ 2 ~ (~IIOII~ + ~1I611~) 2 + Iz(O)lc-CUL • veo

no

Yo

(ii) If 0 E £00[0, tf ) and 0 E £2[0, if), then

Iz{f.)'

~

(5.86)

=, x E £00[0, tf)' and I

(-4 1_1I01/~ + -!-IIOII~)'!i + Iz(O)le-~t. Colio ....go

(5.87)

Proof. Diffel'Cntiating ~1:::12 along the so1utions of (5.78) we compute

and arrive at

1(1,-," 1:.

-dtcI (I?) -'2:::1- ~ -Co I::: 1'1 + -4 -Ko (J .. + 90 -181-'1) . *

(5.89)

5.3

205

ISS-CONTROLLER DESIGN _

(i) From Lemma C.S(i), hiking 'IJ

~

1/2

= =2 and p = (:',1912 + ;;1(12) , it follows

that (5.90) which proves.: E £00[0, i/) and (5.86), and

(ii) By Lemma C.5, taking PI = 0 and P2 =

br (5.84), :t~ E £00[0, tJ). ii,

from (5.89) we get

1=(1)1 2 ~ 1.:(0)12e-2Cl)t + 4c~no IIOII~ + 2~o 11911i, which proves

=E .coo [0, tl) and (5.87), and by {5.8'1}, 3; E .c00[0, 'I}'

(5.91)

o

As we can see fro111 Lemma 5.8, with 110nlinear damping we achieve not only input-to-state stability (5.86) in the sense of ~efinjtion C.1. but also the input-to-state propcrt:y (5.87). With respect to 0, this property ean be understood. as ".c2-illPllt -+ .coo-output" stability, but it. can also be seen as ISS with

1I{jlb

considered as input.. \Vhile this property is not important in schemes, it is crucial in passive schemes whel'e boundedlless

sw~pping-based

of ii C~tll be independcntly guaranteed by the identifier only in the .c2 sense. The quadratic form of the llo11lille~U' damping functions is only one out of many possible forms. Any power greater than one would yield an ISS property. but the proof with quadratic nonlillem' damping is by far thc simpJest. A consequence of Lemma. 5.8 is that, even when the ad!lptatioll is switched

off, tha.t is, when the parameter estiuul.te iJ is constant (8 = O) tlnd the only disturbance inpnt is 0, the state =of tbe elTor system (5.78), (5.79). (5.76) remains bounded and converges exponentially to a posit.ively invariant co Ill-

TI2 =i are not needed when i1. = 0.) I Moreovel" when the adaptation is switched off, this boundedness result holds pact set. (Note that the terms -OJ fJ~i'

even when the unknown parameter is time-varying.

Corollary 5.9 (Boundedness Without Adaptation) If 0 ; lR+ -+ R,f is piecewi,se continuo'us and bounded and iJ i.9 C01U;tn.ut, then .:, x E .ccr.t and 1 1=(t}1 ~ ?JCOiiO supIO(r} ...

co~o

T'~O

81 + 1=(O)le-ent .

Proof. Since B(i} == 0 1 (5.89) balds with 8(t) = 9(t}

-8.

(5.92)

o

Thus, the controller module alone guarantees bounrledlless, and the task of adaptation is to achieve tracking. In fact, a stronger result given by Theorem 4.15 for the tuning functions scheme also holds: if tldaptatiol1 is discollnected and the parameter error is sufficiently small, not only will the global boundedness be achieved, but also the global asymptotic stability.

206

A10DULAR DESIGN WITH PASS1VE IDENTIFIERS

5.4

Observers for Strict Passivity

.

.

Having designed a controller module which achieves input-to-state stability with respect t.o

8 and iJ,

we turn our at-tention to ~he design of an ident.ifier

module which guarantees t.he boul1dedness of 6 and 8. This chapt.er deals on]:y with those modular schemes which use passive identifiers. Swapping schemes will be the subject of the next cbapter. In order to design identifiers which guarant.ee boundedness of 8, let us consider the negative feedback connection in Figure 5.1. It consists of a transfer

mat.l'ix

!:., 8 r = r"

> 0, which is passive, and ll.llOlllinear dynamical system E _

whose input is the parameter error lJ. If we can design the system E and select. ~).n output T so that E is strictly passive from 8 to T, then by Theorem D.4 the equilibrium at i;he origin of t.he interconnected system ill Figure 5.1 is globally uniformly stable (and, ill addition, the state of system E cOllverges to zero). Thus, ill order to guat'allte~ the ~oundedncss of 8, it suffices to desigul:t strictly

=

= -rT.

pasRive system E and let 6 -8 Now we present. the design of observers whose errol' systems ~tre st.rictly passive from ii as the input, to a judiciously selected output T. These error systems will play the role of E ill OUI' identifier design.

The parmnetric .:-model Let

llS

first discuss t.he parametric model (5.93)

If the term Q{=, 8, t)Tij were not prcseut, we would have strict. passivity from t.he input 0 to the outpnt H/(Z, 6, t):. To see this, let, us consider the system

T

r s Figure 5.1: Negative feedback interconnection of the possive system rIB, r = We have to design tile system E nnd select an output T suell that 'E is strictly passive with input O.

rT > 0, lvith 8. dynamic nonlillear system E.

5.4

207

OBSERVEBS FOR STRICT PASSIVITY

which is a copy of (5.93) without Q(z, iJ, t)T8. In view of (5.79), system (5.94) satisfies d pr' ~ T(5.95) dt 21zl~ :5 -cl=l~ +.: Hi(=, fJ, t) 8.

(1

'1)

.,

Integrating over [0\ t] we obtain

- ~1=(0)12 + C 10It 1=(T)j2clr, 10f' (HT=)TOdT ~ ~1=(t)12 _ _

(5.96)

which by Definition D.2 proves that (5.94) possesses a strict passivity property from 1:he input jj to the out:put HT(Z,8 f i)=, or ill ot.her words the nonlinear operatol' (5.97)

is strictly passive. To eliminate the term Q(=, 0, t)TiJ from (5.93), we jnt.roduce the observer ~ = A:(=, 6,/)= + Q(=, 8, t}TO (5.98) and define the observer error ==z-':.

(5.99)

It is readily verified tbat. .: is gm'erned by (5.94). Heuce t with the addit.iou of an observer, we lutve generat:cd a strictly passive error system wboFie sj:ate is available.

The parametric x-model Our goal is to design a parameter identifier for nonlinear systems in the parametric strict-feedback form (5.55): .i:i =

x"

=

+ IPi(Xlt ... , Xi)T6, 1:5 i :5 TI f3(x)u + IPn(:v)TIJ, Xi+l

1

(5.100)

wbere IJ E RP is the vector of unknown constant parameters, and the complete state x is assumed to be available for measurement. System (5.100) is a special case of the general affine parametric model: ;1:;

where vector

f

E JR.",

(5.l0l)

and tile "regressor" matrix F are defined by 3

X"

J(x,·u) =

:

[

1 ,

(5.102)

.7,1

{30(.1:)U though F ill (5.102) does not. depend on II, we llllow t.bis dependence in (5.101) because our identifier design will be applicable to general linearly parametrized nonlillenr syst.cms.

208

rVIODULAR DESIGN WITH PASSIVE IDENTIFIERS

It was easy t.o achieve strict passivity with the parametric z-model because the undl'iven system was eA'"Ponentially stable. How can we bring t.he parametric x-model (5.101) into a form similar to (5.94)1 First, we need the presence of 9 instead of 8; second, we need an e}..llonentially stable homogeneous part.; and, third, we must remove f(:r;, ·u.). Namely, we would prefer to have the model

.t =

A{.~, t):1: + F(x, u.)T6.

(5.103)

whose homogeneous part is exponentially stable:

PA(.r, t) + A(x, t) P:5 -I, ')'

P= pT > o.

(5.104)

To obtain (5.103), we introduce the observer

i.: =

A(x, t)(x - :z:)

+ /(:1:, u) + F(x, ·u)T{).

(5.105)

It.s error state (5.106) is governed by the error system (5.103). This error syst.em satisfies (5.107) which upon int.egration turns into (5.108)

By Definition D.2, this establishes the strict passivity from the input 0 to the output F{x, u)Px, that is, the strict passivity of the nonlinear operator

Ex : 9 H F(:I:, u)Px.

(5.109)

It is important to note that ,wit.h passivity we can only claim the boundedness of O. The bounded ness of 0 is yet to be dealt ~ith. It turns out that with passive identifiers we can achieve boundedness of {) in the £2 sense but not in the Ceo sense, which means that we will depend on part (ii) of Lemma 5.B, We present two passive schemes: the z-passive scheme and the x-passive scheme. The z-passive identifier is based on the parametric z-model, whereas the x-passive identifier is based on the paramet.ric x-model.

5.5

209

=-PASSIVB SCHEME

Figure 5.2: The =-passive identifier.

5.5

z-Passive Scheme

'Ve consider the parametric .:-model

(5.110) and the observer

(5.111) which is a copy of the system (5.110) with the term I'F'(.:, 0, t)TO omitted. The observer error (5.112) is governed by an equation driven by . E

8:

= A::(z, 8,... t)€ + 111(z, 8,... t) T-8.

(5.113)

As we have explained in Section 5.4, the observer error system possesses a strict passivity property from the input 8 to the output l'V(z, 9, t}f, that is, ~be operator E: defined in (5.97) is strictly passive. Therefore, we choose 6 = -rE::{8}, that is,

iJ = rHf(z, 0, t)£,

r = rT > o.

(5.114)

The basic properties of the z-passive identifier (Figure 5.2) are as follows.

Lemma 5.10 Let the ma:r.imum inte'l'1Ja.l of existence of solutions of (5.110), (5.113) and (5.114) be [0, If). Then the following identifie'" properties hoM: (i) (ii) (ii.;)

8 E £00[0, tf} €

E

£2[0, tf ) n £00[0, tf}

9 E £2[0, tf) .

(5.115) (5.116) (5.117)

210

A10DULAR DESIGN WITH PASSIVE IDENTIFIERS

Proof. Let us introduce a Lyapullov-like function 1 -..,

1'" = -IOIF-' 2

1

')

+ ;;Ifl. ...

(5.118)

Its derivative a.1ong the solutiolls of {5.113)-(5.114} is

,i"

=

-oTr-

=

-cleF! -

6+ ~fT (A:: + A~) f + f'J'HiTO

1

t (liilWil2 +.Qi lI:Ja';-:1 T12) f; + 8 r- (rUiF. - 9) T

,=1

1

f)8

/I

< -clff~ -

L n:ilwd2f~ .

(5.119)

.=1

The nonposi t.ivity of Ii" proves that; 0 o,nd E nrc in £00 [0, t I ). Intflgrating (5.11 g)

\Vege!: c

,Ifl!!dr ~ - £1.Vd, ~ V(O) - l"{t) ~ V{O)
..F(:,;)T F(:l')P) +Q(=, 8,~ t) T r F(x)P ]

(5.157)

f,

which, in view of (5.148), can be expressed as .

( =

Ad"

€,

-

0, t)(

T + H',(, €, 0, t) f,

(5.158)

where lYe is smooth in (, € and 9, and bounded in t, and the homogeneous part of the system (5.158) is e.xponentially stable. Starting from (5.158), one CRn prove that the equilibrium ( = 0, f = 0, B= 0 is globally uniformly stable by Definition A.4, which along with (5.148) implies that the equilibrium z = 0, f = 0, 9 = a is globally uniformly stable. The difference from Theorems 5.11 and 4.1~J is that the stabiHty proof is not based on a quadratic Lyapunov function encompassing the complete state of the closed-loop system. 0

218

MODULAR DESIGN WITH PASSIVE IDENTIFIERS

Remark 5.17 At the first glance, one may thinl,: that a passive identifier can be designed using only one of the state equations in (5.55), for example, (5.159)

(or a combination of several state equations). The observer (5.160)

would yield the observer error system (5.161)

where e =

Xl -

;i':l.

The update law would be (5.162)

A simple a.nalysis with a Lyapunov funct.ion V = e 2 + 161!:! would show that 8 is bounded, iJ is square-integrable, and e is both bounded and square.-integrablp. Therefore all the states are bounded. Unfortunately, we cannot prove that :;: converges to zero. If we apply the sa.me approach as in the proof of Theorem 5.13, namely, if we define ( = -=1 - e and subtract (5.161) from t;he Zl-Cquation (5.163)

then we get (5.164)

Since e can easily be shown to converge to zero, in Ol'der to show that =1 does so too, it, suffices to show that ( converges to zero. However, z!:! in (5.164), which we only know is bounded, keeps us from concluding that ( goes to zero.

o

5.7

Transient Performance

Transient performance of our adaptive system will be estimated by £.2 and £00 bounds for the error state z. Since ZI = Y - Yrl these bounds also bound the tracking error Y - Yr' Vle analyze first the z-passive scheme and then the J:-passive scheme. To simplify the derivations, without loss of generality, we assume that £(0) = z(O) (in the z-passive scheme) and .i:(0) = x(O) (in the x-passive scheme), which implies f(O) = O. Such observer initializatiolls should be performed in practice to eliminate the disturbing effect of the initial observer error. For simplicity, we also let r = "(I.

5.7

219

TRANSIENT PERFORMANCE

Theorem 5.18 (z-Passive) In the adaptive system (5.55), (5.77), (5.111), (5.114), the following inequalities h.old:

(i) (-ii)

(1 + - (+ 10(0)1

11=112 :::; 19(0)1

wy:.! ) 1/2

JC?i

Iz(t)1

~

+ -1-1::{O)l ,

2goli,m

1

2v'

con.o

2n-r~)I~ 90 lim

Fa

+ 1=(O)le-cnl .

(5.165) (5.166)

Proof. (i) Since £(0) = 0, (5.119) implies that (5.167) Substituted into (5.122), E(O)

= 0 gives 1

-

IIElb:5 . ~18(0)1·

(5.168)

V"Co"Y

Now, for po
0 t

11

~

o.

Again, by aUowing II = 0, we encompass unuormalized gradient: and lenstsquares. The complete a:-swapping identifier is shown in Figure 6.2. Lemma 6.5 Let the maxim.al intenlal of c:L-islence oj solutions of (6.76)! {6.87)-(6.88} with cithc7' (6.90) OT (6.91) be (0, tJ). Then f01* lJ ~ D. the following identifie7' pmpe'rlies hold: (i) (ii)

Uri)

6 e £00[0, tJ)

(6.92)

E £2[0, LJ) n .coo [0, t,}

(6.93)

iJ e £1[0, tJ} n £00[0, tj) .

(6.94)

f

Proof (Outline). Along the solutions of (6.81) we have

c1t

(npn")

=

n(pAo+A~P)OT -2AOPpTFPOT +nPF1t + FPOT

=

-noT - 2A ( F PO T -

1 II' )T ( F pnT 2.,\

-

1 IS) ) 2,,\

1 II) • + 2,\

{6.95}

6.4

251

X-SWAPPING SCHEME

Using the Frobenius norm we obtain

(6.96) In view of the fact. that d(P)IO,} $ tr {flPO'l'}, (6.96) proves t·hat 0 E £00[0, t J). From (6.82) and (G.85) it. follows that

~ < dt (,-,2) f p _ _1-12 E,

(6.97)

which implies f E £2[0, tf) n t:.~[O, if). Gmclienl, update la:w (6.90). ''''e consider the posit.ive definitc fune'Hon \f =

" + '_f1'1P "21,-,' {} r-

(6.98)

I

whose derivative is readily shoWIl to satisfy

3 IEI2 41 + 1)101} .

(6.99)

The nonposith'ity of li proves that 9 E £oc[O, tf). Due to f = nTo + land the boundedness or n it fo]]ows that. f E £~ [0, f f) I which, in tnrn proves that :. Of f B= r 1 Inl:! E £00[0, t f)· Integrating (6.99) wc get E £2[0, t'I)'

+V

H

J1 + I)IOI}.

:F

Since n is hounded, then

f

E £2[0,1 f). The boundedness of n and the squa.re-

integrability of f prove that Least-sl]1J.U7"eS

8= r 1+11fllfI!'j: E c,,[0, t f).

update law (6.91). \Vc consider the function (6.100)

which is posit.ivc definit.e becanse f(t)-J is positive definite for cal'h I. After routine calculations we get. (6.101)

1 + vtr{OTrO} , which, due to the positive definiteness of r(1)-I, proves that Integration of ineqnality (6.101) yields

-

J1 + vtr{OTrn} f

E

8E

C~(O, f f).

.c, [0, tf).

Now

using t.he boundedness of rand 0, followipg the same line of argument as for the gntdient update law, we prove that

f,O E £2(0, tf) n £00[0, tf).

0

252

l\!fODULAR DESIGN WITH SWAPPING IDENTIFIERS

Our x-swapping identifier, unlike the standard lill~fl.r parameter estimators, guar811j,ees boundedness and square-integrability of iJ even with ul1Ilormalized update laws. This is achieved by including nonlinear damping into filters (6.87)-(6.88) to slow down t,he adapt.ation. Remark 6.6 As we noted in Remark 6.2 for the =-swapping scheme, there is R. passivity interpretation of the x-swapping idelltifier. The signal x = + OT8 is driven by the 'observer'

no

j. =

(A.n -

;\F{3:, 'lt)T F(x, u)p) (x

~ :1') +1(.1:, '11) + F(x, 'u)T8 + 0'1'0

l

(6.102)

which differs fr 0, which means that we allow only normalized update laws. This is in contrast to the z-swapping identifier from Section 6.3, where the normalization is not necessary because the nonlinear damping substitutes the normalization in the task of slowing down the adaptation. From the proof of Lemma 6.1 we recall that the nonlinear damping slows down the adaptation by guaranteeing the boundedness of 11 for any input Hr. It is important to understand why we have to use normalization here. To this end, let us note from (6.241)-(6.244) that the regressor HI'17 is given by i

= 1, ... ,m-l (6.253)

8a'j_l

---Xm+1 8x m

,

wJT

j

= m+2, ... ,TI

6.8

281

UNKNOWN VIRTUAL CONTROL COEFFICIENTS

We recall from (5.223}-(5.226) that the relevant terms in the nonlinear dnmp~ ing functions are 81

8m

Sm+l

Sj

= Kil w il2 + ... =

J

r b+am ((y<m

li':m

m

= I<m+l

((

(

= I<j (

r ")

+ l1lJm l- + ...

8X

m Xm+l)

(6.255)

+ Iwm+d" ) + ... + IUI;!- + .... ")

8o'm

Zm - 83'm Xm+l)

aO'i-l

(6.254)

-

-

1 + k. This implies that irrespective of bow fast ~(t) converges to 7..ero or how small its initia.l condition ~(O) is, there always exist initia.l conditions x(O) from which the system escapes to infinity in finite time.

u

Figure 7.3: Replacing

eby eas tile virtual control in (7.9).

7.1

289

OBSERVER BACI{STEPPING

Nonlinear damping. The remedy for this problem is, again, nonlinear damping. Starting with (7.9a) and using i, as the virtual control, we modify the stabilizing function O'I(X) by adding to it a nonlinear damping term -s(x).r.: (7.12)

Follmving the developll1cnt ill Section 2.5, equations (2.248)-(2.252), we design sex) using the function F(:r) = ~x::!, whose derivative is

,i"

=

_3.;2

+ 3:3 .:; -

x 2 s(x)

+ :1:(x!!)t.

(7.13)

The choice of nonlinear damping

s(.t:) = el j (x 2 ):!, d1 > 0

(7.Vl)

yields the stabilizing fUllction a I (J:) = _:r 2

d 1.1: 5 ,

-

(7.15)

the error variable (7.16)

and the closed-loop mqnession foJ' (7.9a)

-x - d1x

:i: =

5

+ ;r= +x-~. '1

")-

(7.17)

The derivative of \i becomes

V = =

_x2

-

.) -.V-

+ x 3 :;; -

:!

:5

-:r;

d}x6

+ X:i:; + x 3e dl

("

.1:" -

1 -) '). 1-'1 2d ~ + 4d ';t

.3_

+.1 ... +

1

1-2 /old ~ .

(7.18)

l

The last inequality illlplies that if z == 0, .1: will remain bounded if ~ is bounded. Here, bowever, wc can achieve more than that by exploiting tile fact that €(t) is the error of an exponentiaJly converging observer. To this end, we augment tIle function ll"(x) with a quadl'atic term ill

e:

1") 1 '"2 _ 1.2 1 Z2 1 (x +?d k'; - ?,.:I +?d I .. ~ • -

1

(7.19)

. . . . . . . Ill.

Using (7.18), we see that the derivative of 'Ii satisfies .

Vi

=

. 1-... Vdt


0,

Y - Yr Xi - 0';-1 (y, .f}, ... I Xi-l,llr, . .. ,y!i-:l}) - y~i-l}, i -CIZI -

= 2, ... , P

d1=1 - "'1{Y)

(7.35) (7.36) (7.37)

"l

-CjZi -

Zi-l -

80'i-I

+-~- [X2 UY

A

d.; (

i-I

j=1

OYr

-

J

I~i

(

Y-

8ai-l _

-

Xl

)

-

'Pi ( Y)

+ 'PI (Y)] + L -8.... [Xj+l + It'j{1I -

OO'i_l. (j+l) L." u) Yr +~ j=l

T

OO:i-l) -Zi

1

• 1

?

XJ

= -, ... , p •

..

Xl) + 'Pj(1/)] (7.38)

7.1

293

OBSERVER BACI<STEPPING

Proof. From (7.27), (7.31), and the defillitions (7.34)-(7.38), we can eAllress the derivatives of the error variables J • • • J ::p as follows:

=.

Zl

.

Zi

= iJ - fir = X2 + 'PI (y) -tir = :1;2 + :1:2 + tpl (Y) - Jir = =2 + at + c,ol(Y) + x!! = -CI Z I + =2 - d1 z 1 + X2 =

•

xi+1

~ ) + c,oi () + k4i (Y - 3:1 Y -

~ 8a:i-l [ftXj+l + J..111 ( 3

oai-l [..r2 + !PI () 11

• ) Xl

- {- Oi. 3=1

--au

zp

-CiZi -

= Xp+l

Zi-l

+ Zi+l -

+ bmP(y)-u. + ~~p(y -

-E°0a:~~1 j=l

[.Tj+l

:1:J

80;_.) lJ]J

Zj -

J=l

A

-

Yr

oO:;_t - . {- . a y X 2 ' 'I = 2, ... ,p - 1, u:lO}

:i"l) + !Pphl) - 8a'p-I [&2 + !PI (Y) 8 11

+ ~~(y -

5: 1)

+ tpj(U)] -

Eaa~~)ty~j+l) j=l

= bm f3() Y U + Xp+l =

di (

-] + :r::!

~ Oa:i-l (J'+t) -1J (i) + tpj ()] Y - ~ lJ U) Yr r 'I

=

(7.30)

O:p - Cp"'P -

-

"'p-l -

dp

Ollr

+ .V2] -

y~p)

.,

(oo'p_t)" OOp_1 -:. ay :'p - ay3.'2 - Ilr(p)

-CpZ, - =,_1 - d 8i~-1 fzp - 8';;;1 ft.. p (

(7.41)

The resulting error system is

(7A2)

Due to the piecewise continuity of y~p}(t) and the smootlmess of the nonlinearities, the solution of the closed-1oop system (7...:12) exists. Let its ma."{imulll illtel"V1u of existence be [0, tf). On this interval, the nOllnegative function V'p defined by V;J(=,X) =

t [~zJ +

j=l -

dl.XTPOX] ,

(7.43)

J

wbere Po is the positive definite symmetric solution of the Lyapunov equation PoAo + A6'Po -1, is llonincreasing, since its derivative along the solutions

=

294

OUTPUT-FEEDBACK DESIGN TOOLS

of (7.42) satisfies

.

Ifp

0 for aU t 2:: 0, we can guarantee that (7.66) achieves the control objectivcs from a set of initial conditions that can be determincd a priori. To see this, we first combine (7.,,19), (7.58), (7.64), and (7.66) to compose

300

OUTPUT-FEEDBACK DESIGN TOOLS

the system il Z2

+ at (X:4 X.1 - X2 XS) - d1ai(x; + X5)=1 + =2 = -C2=2 + W:!-I.T..1 + W2S XS - d2 (wi.. + W~5)~ - Zl =

2:3 =

=. . =

-Cl=l

-CaZa + z.. -C4="'Ponentially stable transfer function of relative degree greater than or equal to i. By (7.141), this in turn implies that the vectors v~~}), 0 ~ j ~ 111, are bounded, where _(i)

Vj

=

[ Vi,l,'"

]

I

V;,i •

(7.143)

7.3

313

AOAPTIVE OBSERVER BACl{STEPPING

In particular, by (7.122), this implies that vC!) is bounded. Combining (7.119) and (7.116) we conclude that WI and 0:1 are bounded, which ill turn implies that 'v m.:! = ;;2 + a1 + iJ1.1iJr is bounded. Hence, by (7.141), H p- l (s)[,8h/)U], and thus 'jj~::;}, a ::; j :5 m, are bounded. This in turn implies that W!!, a'!! and 'Umt3 are bounded. Continuing in the same fashion, we use (7.115), and (7.J.cll) to show that Hi(s)[P(Y)u], p - 2 ?:: i ~ 1, are bounded, wbich impJies that t' is bounded. Since {3(y) is hounded away from zero, we conclude from (7.110) that u is bonnded. Furthermore, from (7.108) we see that x is bounded. "ve have thus shown that all of the signals in the closed-loop adaptive system are bounded on [0, tr) by constants depending only on initial conditions. Hence, tr = 00. The convergence of the tracking error to zero can now be deduced from the LaSalle-Yoshizawa theorem (Theorem 2.1), since =h"" =P and e converge to zero as t - t 00. 0

7.3 . 3

Example: single-link flexible robot

As all example, we consider a single-link robotic manipulator coupled to a de motor with a nOll rigid joint. When tile joint is modeJed as a linear torsional spring, the dynamic equations of the system are J1iil

+ F1cil + I\,(ql - ~) + mgdcosq1

=

a

q2)

=

/\Li

J" F!' /(( 2q2 + 2q2 - N 91 - N

(7.144)

LDi + Ri + /\-bq:! = 'u, where ql and q2 are the angular positions of the link and the motor shaft, i is the armature current, and 'U is the armature voltage. The inertias .It, J!!, the viscons friction constants FIt F2t tile spring constant 1(, the t.orque constant J(Lt the back-emf constant J(b, the armature resistance R and inductance L, the link mass wI, the position of the lin] 0, Ci > a, i = 1, ... , p)

Zi)

pl > 0, PoAo + A~ Po = =

=1

Y - Yr (7.1GG) Xi - O:i_I(Y,Xt, •.. , Xi-I, Yrt ... , y~i-2}) - y~i-l), i = 2, .. , 1 P (7.1G7)

-. =

...., 0:1

=

ai

=

-CIZI -

-c,,,, i-I

'PJ{,/)

(7.1GB)

erdo', + 8:~' [X2 + 'PI(Y)]

k,(y - XI) - 'Pi(Y) -

Zi-I -

lJ "

i-::! {) '.

+ E {)O~~l [Xj+l + kAy - Xl) + epAy) + eJdol'i] + L: a~;)l Vp+l) x,

j=l

+dOPD- 1C2

j=l

E(e&. - E{)aa:~~l

k=2

=

1,1

ia =

j=I

OYr

i = 2, ... , P

Cj) T 80/:J'i-l =kt

(7.169)

Y

''C J

(7.170)

PO- Ie2=1

8a'i-l Ra 1e2 -ay=i'

"i-l -

.?

I = .... , ...

(7.171)

,p

guarantees global boundedness of x(t), x{t), and regu.lat.ion of the tracking er1'01':

Jim [yet) - 1Jr(t)] = O.

(7.172)

I-co

Proof. From (7.27)1 (7.165), and the definitions (7.164)-(7.171), we call express the derivatives of the error variables ZI

%1, ••• , =p

as follows:

= iJ - ilr = X2 + epl{Y) - ilr = X2 + X2 + lPl{Y) - ilr =

Zi• =

Z2

+ itl + 'Pl ClJ) +:1:2 =

A

i-l

-L

+ Z2 + X2

(7.173)

T

A) + eTdotp + t.fJ( () y - aa:i-J [A3:2 i

r ( + h:j ;rJ -

Xi+I

-CIZI

Xl

-] + ~l () Y + 3:2

a

~'~~1 [Xj+l + kj(y - Xl) + qJj{Y) + eJdoLp]

i=1

uX J

i-2 a

U+1)

' " UO:i-l

- .t- ---u>11r j=I

=

81ft

-c-z. - z· •

1

1

1-

(i)

- Yr

(e. _~ EJO:8 -. . c.)

+ ""+1 - 80:;-1 aY XII- + do •

i-I (

d R- l ' " +OOe2L."

k==2

-1

L."

j=l

i 1 _

XJ

J

T

"-1 n ) T 8 O'i-l _ '"" UOk-l ' _ ? Ck-L.,,••.a Cj -{)-';'k,1.--, ... j=I''CJ Y

(t p _ t·) I

,p

(7.174)

7.4

319

EXTENSIONS

The resulting error system is

(7.1i6)

For t.his error system, consider the Lyapunov function

~ 2 V( z,fi ) = ?1 ~Zj + ?d1 a:-T Pox- . ... j=l

-

0

(7.177)

320

OUTPUT-FEEDBACK DESIGN TOOLS

Its derivative along the solutions of (7.176) is llonpositive:

(7.178) Thus, Zt, ... I zP' i are bounded and converge to zero. The remainder of the 0 proof is similar to that of Theorem 7.1.

7.4.2

Design with partial-state feedback

All t.he nonadaptive and adaptive schemes we have presented so far as well as those to be presented in the remainder of the book, assume either that the full state is measured, or that only one scalar output is available for feedback. In this section we present a natural generalization of these results which bl'idges the g~tP between full-state and single-output feedback. Combining the design tools we have developed for these two cas~., we call easily incorporate information available from additional sensors and accommodate systems whose nonlinearities depend on measured state variables only. Such designs are applicahle to systems in partia.l-state-feedback form which have k groups of measured state variables, denoted by XlI'" I XmJ and Xn .+1, . .. ,xmi+I' i = 1, ... ,Ii: - I, and k groups of unmeasured variables, denoted by xTlli+lt .•• ,xn • , i = 1, ... k, where 11 k = nand mIt ~ p. For notational convenience, we adopt the following definitions of the vectors of measured variables for 1 ~ i ~ k : I

As we can see, the nonlinearities in (7.180) are strict-feedback nOlllinearities, since they depend only on measured state variables which are fed back: In the

7.4

321

EXTENSIONS

xi-equation, the nonlineal'ities depend only on measured state variables up to Xi- An example of such a system is3 P

Xl =

X2

+ 'Po•.(xa) + l:lJj'Pj,l(:t'I) j:::::1

:C2

=

,1:3 + 'Po.:;! (Xl I :1"2) +

1l

L 8j rpj,2(J;t, :J::!)

j=l

,1

Xml

=

Xm1+l

=

:1"n 1 +1

=

,1:'n,+2

+ 'PO,ml (Xfllt) + L

OJ'Pj,mt (x'" I )

j=1

p

Xnt

+ lPO,n

(xm I )

I

+L

0jl.pi,n I (:e" I )

i==l p

X,. 1 +1

+ O.

(8.55)

\Ve pause to clarify tlle ru.'gllment.s of the fUllction 0'1- By examining (8.50) along with (8.39) and (8.40), we see that 0'1 is a fUllction of 7J,~,S,6,e,.Vo.2, ... ,'LIm-).2 and Yr' For brevit.y, we denotr- ,x = (y,~,::.,8,B). From (8.31) one can show that l'iJ can be expmssed as

(8.56)

334

TUNING FUNCTIONS DESIGNS

where Ak ~ 0 for k > n, and * denotes entries that can have any values. 'Vith (8.56) we conclude tbat III is a function of y,~, ~t 0, il, Ah"" Am+l' Yr- Deuoting Xi = ("'\1,' •• ,Ai), we write O:I("~t Xm+b Yr). In the backstepping procedure, ai is a fUllction of Xm+il y~i-l»). To make the expositions shorter, this fact is postulated here, and then verified once the a:i'S have been selected.

(X,

Step i =2, .•• , p. Differentiating (8.46) for i = 2, ... t P -1, with the help of (8.43) we obtain

~~1~

- L

~(-kj'\I+Aj+l)~1 a~

Noting from (8.46) that

Zi =

Zi+l

'Um,i+l - By~i) = Zj+! aOi-l ( ay \WO

+ O:i - k"i1Jl'lr,l -

_ 8ai-l (A

a~t

C

o~

+ k·Y + .J..) _ rp

_mfl a;~~1 j=l

l+

(-kj A

~

~

•

'i::ljj_~B. 00 ~

(8.57)

+ O:i, we get

+ W T (J + e2 )

8O:i - 1 (A -= 6.) _ ~ 80'i-l (i) 8= 0 .... + '.I:' ~ U-l)Yr ......

Aj+l) _

j=18Yr

(y~i-l) + a~i-:l) B. (8.58)

80;-;:10 _ a(J

J

(l

From (8.44) it follows that the final step i = P can be encompassed in these calculations if we define O:p = u(y)u + 'urn,pH - ey~p) and ZP+l = O. To make the choice of a stabilizing function easier, we add and subtract _D':;;1WT9 lJ~i;l rTj ill (8.58) and get

Zi

=

OJ -

~":iVm,l

_ 80:;-1 (A

at ~

-

rn+i-l

-

t'

o~

OCti-l ( Wo + ay

+

k

.J..) _ y + If'

80:i - 1

'~ " -OA' - ( -k-Al J

J=l

80:i-l

J

8a:i-l

aOi-l

a-=.....

-=

(A

0-

+

tP) _ ~

.LJ

j=l

aO: i - 1 + AJ'+1) - --fT.' ali

-aye ayw 2 -

'r ..)

W (J

Tf}-

I

11

al1:i-l

80

(0':.

-

r) 1i

(i

aO: i - 1 (j) (j-I) 'Yr

8 1

Yr

aai- 1 ):"(! an

Y(r - } + - t:

_

+ ""i+l •

(a.59) ,

The stabilizing fUllction 0:; will be chosen to cancel all the terms except those e2 in the last line of (8.59). The potentially destabilizing disturbance -

87;1

8.1

335

DESIGN WITH K-FILTERS

(8';;;1) 2Zi

is counteracted by includillg the nonlinear dampillg term -d j in G:j. Dealing with the remaining three terms in the last line of (8.59) is no different from the way we dealt with similar terms in Section 4.2.1:

• The term _O';II-lwT8 determines the ith tuning function ? 1' -~, •••

• The term -

0';;0-1

(8.(0)

,po

(6 - rTi) represents the mismatch between the actual

update law and the tUlling fUllction, and it appears because at step i only Ti can be cancelled in (8.59). As u~ual in the tuning functions design, our choice of the update law will be we ha.ve •

8-

0=

rTp.

p

rTi

=- L r j=i+l

Therefore, in view of (8.60),

a a:j-JWZjf ay

(8.61)

which yie1ds Oa:i-l

ao

---

(0:' -

r)

=

1'.' I

(8.62) The stabilizing function Q:i will include the term - L~:~ O'jiZj to achieve skew-symmetry in the error system. • The 'above-dia.gonal' term diagonal' 2i-l-

is compensated by adding the 'below-

Zi+l

To summarize, our choice of the stabilizing fUllctions for i

., G:i

=

-Zi-l - Ci 2 i -

di

ay

aO'i - 1 ) -

(

a

Zi

= 3, ...

I

P is

( T A) + kj'Una,l + Oa:i-l By Wo + W (}

a

i-I

a

a:i-l (A '=' 4» '" Qi-l (j) + lti_1 ac"e (A Dc"t' + k11 + If'.I.) + a= 0 ..... + + j=l L.J (i-I) Va..... BYr

ao-

k \ \ ) +-i-I r + L -OA'i-I - ( -"'1\]+1\'+1 . 'J :J 00' T,'+ J=1 3 m+i-l

!:la:

U

I

a ac:ti-l.,.. _ ' " a:j-I r L;=2 ao 8"'" y

(

(i I)

Y r

a....)

"'~i-l ~ +-an (} r::

i-J

A

(8.63)

336

TUNING FUNCTIONS DESIGNS

For i = 2 the stabilizing fUllction differs because of the term compensate for brn32 in (8.53):

-bm 3]

needed to

(8,G4)

At the end of t.he recursive procedure, the last stabiJizing function used in the actual control law:

O:p

is

(8.65)

and the last tuuing function

Tp

is used in the updittc law:

By sUbstituting (8.63) and (8.64) along with (8.62) int.o (8.59)1 the error system becolnes d1 z1 + bm z2 + e2 + {w - U(Iir + 0'1) et )"

-Cl =1 -

-bm (lir + 0:) ) 0 Z2

=

d'},

-C2Z2 -

(~~'l) '}, ':2 - btnZ1+ Za + vY

aCXl_

t

0 (8.67)

O';}..jZj

j=3

T ii

aCX1

(8.68)

--e"l--W u

ay -

Xi

=

-C;=i -

ay

di

(aao'~-l) '},

By

zp+J

M

By

E

lTjiZj -

Z;_I

+ =i+l +

j=2

OO'i-l,,-

OCXi-l

---C'l---W

where

Zi -

1J

8,

t

lTijZj

j=.+1

i =3, ... ,p,

= O. This system is compactly written as

(8.69)

8.1

337

DESICN WITH K-FILTERS

\vhere the system matri.'1t A;(z, t) is given by

-CJ

bm

-d]

-brn

o

-1-

o

0

d.,. ( ~ )!.! 1+0'23

-C2 -

0'24

0'23

1 + Up-l,P

o

-1-Up _I,p _ Cp -dP

-U'l,P

(OaiJJJe- )2 1

(8.71) and H'E"(.z, t) and H'o{z, t) are defined as

(8.72)

To prepare for the sta.bility analysis in the lla.-t subsection, along with the elTor system (8.70) we consider the erl'or equations for parameter estimators (8.66) and (8.55), as well as the state estimation error (8.18):

8

=

-rTJVo(z, t)z

(8.74) (8.75) (8.76)

g = 1'sgn(bm) (ilr + 0:1) eI ;;

e

=

Aoe.

The candidate Lyapullov function for system (8.70), (8.7.!!), {8.75}, (8.76) is 1 tr 2

1~ 2

Ihml_'J

~ 1

1" = -z z + _9' r- 8 + - r [ + L..J - e Pe. 1-

2')'

i=l

T

(8.77)

4di

Recalling {8.12}, the derivative of 1" is

l:r

=

zT

(A; + A;) ;; + .;:TvVeE:2 + z'l'Hilb -

-8-T H'oz + Bbm (ilr + {l.t> el'1 z z T ( A:::

.f. 4d.1 E:" E I

~ Zi--E2 aO:i - 1 - ~ 1 I 1"1 + A:T) z - L..J '- -. e - , , i=l

ay

+ 0:1) elB

L..J i=1

=

Z Tbm(Yr

i=l

4d

(8.78)

338

TUNING FUNCTIONS DES10NS

where for notational convenience we have introduced ~ ~ -1. The skewsymmetry in (8.71) gives us

C.-., + d.,.. (!!!ll)2 Uu

(8.79)

CP+ dP(80.,'By_1)2 which substituted in (8.78) yields

" = p

:s; -

L CjZ;.

(8.80)

i=l

From this inequality we can conclude that z, 8, g, and e are bounded. In the next subsection we use tllis along with the minimum phase Assumption 8.2 to establish the boundedness of all signals and asymptotic tracking. To guarantee bounded ness witbout adaptation we add (strong) nonlinear damping terms which counteract the parameter estimation error. To motivate the choice of these terms, we first rewrite the error system (8.70). After replacing (B.67) by (8.51) and adding ±bmz1 in (8.68), we arrive at

(8.81) where the only difference between A:: and A; is that (2,1) is replaced by bmt

bm at positions (It 2) and

A;(z, t) =

o

bm 0 -cl-d1 -cl)-d~(~ )~ -bm 1 + 0'23 8u -

0

M

-1-0'23 -U24

O'p-'J.,p 1+O'p-l,p

0

-(J2,p

-O'p-'J,p

-1-Up _l,p

-Cp -dp

(B8"y-l)2

8.1

339

DESIGN WITH K·FYLTERS

and TV; is given by

(8.83) _lJo.,,_1 WT

Ou

While the system (8.70) is more ade(i)(Y)o.

= 1, ... ,11 •

]}

,

(8.109)

The boundedncss of y, t:he smoothness of r/J(y) and ~(y), Assumption 8.2, and (B.109) imply that "\1, .. , ''''\m+1 are bounded. We now return to the coordinate change (8.94), which gives vtII, ;

_

_

-

':'j

• (i-I) - (JA + (}Y + lti-l (& y, ~ ,=-, r

I

~ '\ -O-::l») tl, "'m+i-l! 'Yr ,

.-

')

l . - ... , ...

,p.

(8.110) Let i = 2. The boulldedlless of Xm+1! along with the boundedness of Z2 and y, {,.:=, 0, B, ;IJr' Yn proves t.hat Vm,::l is bounded. Then from (8.56) it fol1ows that Am+2 is bounded. Continuing in the same fashion, (B. 110) and (8.56) recursively establish that. ,\ is bounded. Finally, in view of (8.19), (8.31), (8.32), and the bOllndedness of ~,=:,,,,\, and E, we conclude t,hat l' is bounded. Since u(y) is bounded away from zel'O, 'u. is bounded. VVe have thus shown that an of the signals of the closed-loop adaptive system are bounded on [0, t f) by const,ants depending only on the initial conditions, design gains, and the e)..i;crnal signals Yr(t), ... , y~n)(t.), but not on tf. This proves t.hat t I = 00. Hence, all signals al'e globally uniformly bounded for aU t ~ o. By applying the LaSalle-Yoshizawa theorem (Theorem 2.1) to (8.106), it further follows t,bat =(t) ~ 0 as t - t 00, which impJies that limt-oc [yet) - Yr(t)] = o. 2. The boundedness of all the signals without adaptation follows from (8.91), by repeating the argument from point 1. 0 TlleOl'C1ll B.5 established global uniform boundedness of all signals but not g10bal uniform st.ability of individual solutions. To refer sucb a stability property to the origin, we now determine an error system such that all of its states

8.1

343

DESIGN WITH K-FILTERS

except the parameter error cOllverge to zero. 'Ve start with the subsystem (z, E, 8, 0) whose 2n + q + 2 states are encompassed by the Lyapunov fUIlction (8.77). Then we derive additional equations to complete the error system. For filt.er states we introduce the reference signals f/ and ;::r defined by

~r

sr

= =

Aoer + k.lJr + q,(Yr} Ao:::' + lP(IJr) •

This allows us to define the error states { = t;. by

~

=

Aol + li'::1

(8.111) (8.112)

for and 3; = :::i - =f governcd

+ ¢(.:::ltYr)ZI

(8.113)

..... = AoE + (;(ZI, 1Jr)=J ,

(8.IV1)

wbere ~ and i are SIJlooth functions defined by the mean-value theorem. The system (::, E,~, 2,8,0) hus (q + 3)11 + q + 2 states~ while the original (.7:, f., S, A, 6, 0) system has (q + 3)n + q +Tn + 2 states. Vve recover the missing m erl'or states ill the inverse dynamics of (B.3). Let. us consider the similarity transformation

°PX1h ] .".

T

(B.115)

''',

where

T = [Atel," . ,Abel, In.] Ab =

[

-bna-tlbm

-bo/hm

(8.116) lm-l

0

(8.117)

o] .

Tbe following two identities are readily verified:

(8.118) With these identities the inverse dynamics of (8.3) arc expressed as

(8.119) Introducing the reference signal (r as

"

= A.C' + T (AP [

ny, +

¢(Yr) + el) Zl 8ai-1

y

(8.227)

i

= 2, ...

t

P

(8.230)

is defined as (8.231)

The followillg t\VO identities are straigbtforward to verify: Tl=O,

(8.232)

B.2

359

DESIGN WITH MT-FrLTERS

Table 8.5: Tuning Functions Design wI MT-Filters (cont'd from Tab. 8.4) Adaptive control law: (8.233) Parameter update laws:

iJ =

rTp

(8.234)

~ =

-')'sgn(bm)(Yr + Q'd Z1

(8.235)

Observer: (8.236)

(the latter is immediat.e froln (8.172)). Using (8.231)-(8.232), we compute

=

[Aie},

AI] = A, [A,e},

In-I]

= A,T.

(8.237)

Combining (8.217) and (8.220) and using (8.237) we have

Ti = T Aoc - ~olwl2TleT e + Tlw'l'8 = TAoE' = A,Tc I

i] =

(8.238)

which because of (8.220) gives (8.239) This system represents the e,'1lponelltially stable inverse dynamics of (8.217). Since they are not controllable by Cl, they are also the zero dynam.ics of (8.217). Now we examine the cl-equation: (8.240) The second identity in (8.232) gives eTI\Q = EI = - (co + li:olwl2) el

Co

+ ll'

Therefore

+wTO + £2 -[lel.

(8.241)

Subst.ituting (8.231) into (8.220) we see that (8.242)

360

TUNING FUNCTIONS DESIGNS

and obtain (8.243) Now we al'e ready for a Lyapullov stability analysis for the closed-Ioo}> system consisting of the error systiem (8.205)\ the error equations for parameter estimators (8.20Ll) and (8.195), and the observer error system (8.243), (8.239)~ :;

=

A=(z, t)= + l'l'e;(=, t)C2

+ H'o(.:, t)T8 -

bin (Yr

+ lh) elU

-r(B'o(z, t)z + '.lWEI) g = "( sgn( bm) (Yr + iiI) eI.:

8 =

E"t

iJ

(8.244) (8.245) (8.246)

= - (Co + n.olw\-9) el + WT-(J + 711

(8.247)

= Am·

(8.248)

(The system matrix A;: in (8.2M) is oftbe C01'Dl (8.71), but it also incorporates the lIonlinear damping terms (S.84)-(8.87).) A candidate Lyapunov function for t.his system is

V'

1 T 1-T 1 ~ Ibml lJ = -;; Z + -f) r- IJ + -U- + -£i + 1117 2 2 2"( 2 t)

t)

"

where J1 is a positive constant to be chosen later, and solution to the Lynpunov equation

P'17

/1

'

= Pl

/1Ar + All1 = -I.

(8.249)

> 0 is the (8.250)

vVith calculations similar to (8.78)-(8.80), we a.rrive at (8.25] )

By applying Young's inequality to the term . li :5

-co/zl-.., + -4(11E2 IJ

0

Noting that: (8.242) implies E~

V. :5 -co IZ 12

-

(-vC

o

2

-

o

Ile .., -Ei -

2

IJEttl17

(

we obtain

po - -

1.1) 117I"- •

2eo

(8.252)

:5 21i£I + 21Jr, we finally get -

Ii) El -

2do

IJ

(

p. - - IJ - - 1 ) 1111-'1 • 2co 2do

(8.253)

By choosing v as in (8.218), and f.L = 2:.,. + ~fI' we prove that " :5 O. This implies that z, 8, U, E1 t T/ are hounded. We now use this to establish the boundedllesS of all other signals ill the adaptive system. In view of the similarity transformation (8.220), the bounded ness oC El and 11 establish that £ is bounded. The boundedness of Yr and ':::1 implies that y is

8.2

361

DESIGN WITH MT-FILTERS

bounded. Therefore ~ and first re\Vrite (8.236) as

:e: are bounded.

To prove bounded ness of :\:, let us

(8.254) and note that the boulldedlless of Cl and y implies that Xl is bounded. To prove that. the remaining componcnts of Xare bounded, we employ the similarity transformation

[ Xl ',p

1~ [ Tx. X: 1= [ eTT ] ,.

(8.255)

.\.1

and, from the observer equation (8.254), obtain the system (8.256) which shows that tP is independent of the input Ii'D Iwl:! (1/ - XJ) + Wo + w T fJ. To arrive at the last equation, we have used the identities (8.232) and (8.237). Because of the boundedness of y and the Hurwitzness of AI, (8.256) proves that 'l/J is bounded. By (8.255), the boundcdness of Xl and l/J establisbes that X is bounded. We have yet to prove the bounded ness of A and ;t'. Our main concern is ,,\ because the boundedness of .r will follow from the boundedlless of E I X, ~,3, and A. The proof of boundedness of ,,\ is similar to the corresponding part of the proof of Theorem 8.5. From (8.168) it follows th~tt ,A.

= .'Ji-l + It,si-2 + ... + li-t [ L(5}

I

J

(, )

a II

U

i

,

= 1, .. 'In -1.

(8.257)

Substituting (8.108) into (8.257) we get \

I

,\,: =

+ llsi- 2 + ... + 'i-l

Si-l

L(.'1) B( s)

{d

RlI

dt,n -

d t; dt"-; r'Po.iCl1) + n

n- i

} Ii given by (8.218), not. found in the design with K-filters, is needed in the design with IvIT-:6lters because the parameter error 8 appears not only ill the z-system (8.205) but also ill the observer error system {8.217}. The need for each of the factors to be large CRn be explained as follows: should be large because the term weI needs to be given a sufficient weight relative to the other terms in the update law (8.204). In other 'Words, t.he adaptation with respect to the e-system needs to be sufficiently fast. because £2 appears as a disturbance in the .:-systcm.

• 1-'

• do should be large to prevent the destabilization of the :;-system by £2 if the adaptation with respect to the e-system is slow.

• Co should be large to maIm the part of the £-system controllable by w T6 fast enough if the adaptation with respect to the £-system is slow. Hence, if codo is small, l.I should be large enough t.o satisfy condition (8.218). However, v must not be too large because, as our performance analysis in Section 8.2.4 shows, an increase in 1-' may cause a deterioration of pel'fol'mance of z.

1-

As in Corollary 8.6 for t.he design with K-:61ters, we can show that the error system ."

= A.:(z, t)z + IVE(z, t)e2 + H'o(z, t)"l'6 - bm (fir

E =

;p

(Ao - olwl leT) £ + lw 8 2

fL

-

2

= A,'l/J + A,elzl

T

+ (j'I) fIB

(8.261)

(8.262) (8.263)

.~-

~ ~

8.2

363

DBSIGN WITH MT-FILTERS

,

= Ab( + 0 and j:3 > O. The Lyapul10v function (8.306) WillllOW he used to estimate the region of attraction within:F. Let O( c) be the invariant set defined by l'{.'J:l,X2,X3,~,(,Bl,{?at61,62) < c. Theil an estimate n of the stability region is

n = {. Y.. = (xt, X2, ,1:3, e-", ~1t Ba, OJ, 62 ) Il'(...Y) < arg ll(c)C.r sup {c}}.

(B.30B)

36D

NOTES AND REFERENCES

From (8.307) we also conclude that all the states except possibly the parameter estimates converge to values given by t.he equilibrium (8.304). Using LaSalle's invariance theorem one can prove from (8.299) t,hat Q1 and 81 convel'ge to zero, which means t.hat t.he estimates of ill and 1/1'1 converge t,o their true values. This is due to the gravity 9 which causes t.he force of the elect.romagnet to be nonzero even when Yr = 0, thus introducing in the regressor tenns which do not vanish at the equilibrium. AB we explained in Example 4.13, nOllvanishing elements in the regressor contribute to convergence of parameter estimates. From LaSalle's invariance theorem it also follows that. a linear combination of U3 and 82 converges to zero. Since V' converges to a constant, we conclude that and O2 con.verge t.o constant values.

ea

Notes and References Both the output-feedback scheme of Marino and Tomei [122, 123) and the modi.fied scheme of Kanellakopoulos, Kokotovic, and Morse [72} inherited overparametrization from the original adaptive backstepping design 169]. The designs presented in this chapter avoid Dverparametl'ization using the tuning functions technique which WiiS first employed for output-feedback adaptive designs in Krstic, Kanellalmpoulos, and Kokotovic (95] and in Krstic and Kokotovic [99]. \Vhile the I(-filters employed in this chaptel' arc the same as those in [72] patt.erned after Kreisselmeier [91], the !vrT-filters in this chapter are simpler than those proposed for liltered transformations in [122, 123]. The use of nonlinear damping as a tool to improve performance and guarantee boundedness without adaptation was suggested in Kanellakopoulos [64] and in K811ellakopoulos, Krsti6, and Kokotovic [80]. Teel [189J proposed an output-feedback design where high gain is employed to allow the regressor to depend only on the reference signals, which means that the adaptation is active only in the case of tracking. Khalil [83] and Jankovic [57J developed semiglobal adaptive designs for a class of nonlinear systems which includes some systems not transformable into t.he output feedback form. Their designs employ high-gain observers and control saturation. While Khalil's identifier is Lyapunov-type, Jankovic employs a passive identifier. Tao and Kokotovic developed adaptive designs for systems with unknowll backlash [184] and dead-zone [185].

Chapter 9 Modular Designs In this chapter we ext.end the modular approach of Chapters 5 and 6 to the case of output feedback. The output-feedback modular approach results in separation of three design modules: the control law, the identifier, and the state estimator. With the K-filters we design output-feedback control laws which guarantee input-to-state stability with respect to the parameter error, its detivative, and the state estimation enor as the inputs. At the end of t.his chapter we briefly present some schemes with NIT-filt.ers. Following the ideas introduced in Chapters 5 and 6, we develop outputfeedback forms of passive and swapping identifiers. The schemes in this chapte.r are simpler than the tuning functions schemes because they do not eliminate iJ from the error system, but inst.ead use stronger ISS-controUers. As in Chapter 8 we consider the output-feedback systems :i;

=

Y =

A,,' + ",(y) + 1I we compute 1;'111 -

9.1

379

ISS-CONTROLLER. DESIGN

where for notational convenience we have defined ~ ~ -1. Now, with (9.52)(9.54) we get

(9.59)

By completing the squares in (9.59), we obtain 1 oai-l 1 P P P 1 -dtd (-Izl-) ::; - L Cjz;2 - E di ( - Z j + ;;-E.'l )" + L: -£2 2 i=l i=l fJy ... ~ i=l 4d. I)

I)

380

MODULAR DESIGNS

which with (9.55) becomes dt' d

(-211=12)

$;

-col~f +!4 (2-c~ + ~1812 + .qo ~IOI2) do Ii.Q

(0.61)

By applying Lemma C.5 to (9.61) we est.ablish the following two inequalities:

1=(t)1 2 Iz(t)12 whic11, in view of the boundedl1ess of c, prove that z E £:00[0, tl) and (9.56)(9.57). It. remains to prove that the boundedlless of = and 8 implies that x,~, and A are also bounded. A proof of this implication was already given for the tuning functions design in Theorem 8.5 (cf. (8.107)-(8.110». The same argnment is applicable here with (8.110) replaced by

=,

.VnI,i = ..._j + ~Yr 1 (i-I) +O:i-l bnl

The bounds on

(& - 0" '\ -(i-2)) y,~,.=., '''m+i-l, Yr

,

=, x,~, =:, and A are independent of tl'

i = 2, ... , p. (9.64)

o

A consequence of Lemma 9.2 is that even in the absence of adaptation 011 the closed-loop signals remain bounded, as we stated in Corollary 5.9 for the state feedback case. "Vith Lemma 9.2 at, haud, our next task i~ to design t.he identifier module which guarantees t.hat 0 is bounded, and iJ is either bounded or squareintegrable.

9.2

y-Passive Scheme

'''ith the K-filters in Table 9.1, we have obtained the parametric y-model (9.10): (9.65)

9.2

381

V-PASSIVE SCHEME

where Wo and ware measured and defined in (9.14) and (9.12), respect.ively, Except for the state estimation error ;2, the parametrk y-model displays little difference from the parametric :t'-model (5.128) we used in the st.ate-ff'edback design. What malres (9,65) desirable is the relative-degree-onc pl'operty between (J as the input and y as the output. \-Ve introduce t.be setuar observer

ii = - (co + n.olwI2) (;Q where

Co

and

lin

are as

ill

y) +wo +wTiJ.

(9.66)

(9.55). The observer error

{9.G7}

f=y-iJ is governed by

e= -

(co + IiO\W\2) f + wTO + e2' (9.68) It call be shown tlmt. this system and t11e system g = AoE form a syst.em wit.h a strict passivH.y property from the input 8 to the output WE:. This determines our choice of the paramet.cr update law: fJ = Proj

{rWf} ,

(9.69)

h". where the projection operator is employed to guarant.ee that fbm(t)1 ~ 0, \:It 2: 0. (For a detailed tl'eatment of parameter projection, see Appendi.'\: E.)

Lemma 9.3 Let the ma:r:imal interval of existence oj solutions oj (9.65), (9.66) and {9.69} be (0, t J). Then the following identifier prope7'ties hoM: (o) (i)

Ihm(t)\ ~ r=rT>o, ,,;:::0

m

(9.95)

or tIle least-squares:

iJ = Proj &,,.

{r wi r}' 1

+IJwY.

wtv"

t = -r1 + lItv I rr, f

(9.96)

r(Q) = r(O)T > O!

II

2:: 0

'III

where the projection operator is employed to guarantee that Ibm(t)1 ;::: > 0, Vf ;::: 0, and by allowing II = 0 we encompass unnormalized gradient and least-squares. (For details of parameter projection, sec Appendi.x E.) Lemma 9.5 Let the maximal interva.l 0/ existence of ,90l7J.ti0l1S of (9. 89}, (9. 90}-{9. 91 ) with either' {9.95} or {9.96} be [0, i/)' Then f07" IJ ;::: 0, the following identifie1' properties h.old:

IbmO.)1 ;::: ~nl > 0, Vi 6 E £oc[O,t/)

(0) (i) (if)

(9.98)

E £2[0, t / ) n £00[0, tJ)

(9.99)

{} E £!![O, tl) n £00[0, tl) .

(9.100)

f

(iii)

~

(9.97)

E [0, tl)

Proof. First, from LemmaE.1 we conc1ude that Now we consider

=

-co Iw 1'1- - lio

Ibm(t)l2;: C;m > 0, Vt

(T

E [O,t/)'

1)2 + - 1 2n:o 4n~0

tV W -

-

~ -colml!! + -.!:.... ,

(9.101)

41\.0

~.

which proves that

tr1

E £00[0, t I)' \Vith (9.4), along the solutions of (9.94) and

(9.3) we have

I'))

d (-., 11 1E-+-ep

2dt

Co

= ~

.J} -COE-

- - - 1 IeI"lio IW I"""" -f- + ee:! 2co

Co _') Co --f---

2

2

(_

1

f--e2

Co

)

:3

1 1 I I') +-e2--te'l

2cQ

2co

(9.102)

386

MODULAR DESIGNS

which shows that f E £2[0, t,) n £00[0, tf). Gmdient update law (9.95). We consider the positive definite function 1 (-2 = -21 1()-,'1r-1 + -2co E + -1 I€ 12) P • Co

V

(9.103)

Using (9.102), the derivative of V is

,i' $ _il'r-J~ - ~t! .

(9.104)

By Lemma E.l t \Ve have

_ iJI'r-le = -iJI'r-J Pro' {r WE 2 } < -rrr 1 + IJIWl

J

-

(9.105)

mf

1 + vlml:!

1

so (9.104) becomes i' \

(}nT

:s; -

wf

1 + vlm/:!

1_,) - 2c .

(9.106)

In view of (9.93), we have

,i' ::; _OT m(m

~ =

T

9+i) _ !f2 = (wT9)2 _ eTrol _ 1 + IJlml:! 2 1 + vlml 2 1 + vlml:! 2 1 (W'l'iJ)2 1 (mTO) 8Twi _ ~l2 21 + IJlwl!! 2 (1 + vlwl2)2 1 + v\wPz 2 1

!l2 2

(wT O)2 _! ( mTe +f)2

21+vlm1 2

(9.107)

1+v\mI2

2

and alTive at (9.108) which implies

ii E £..[0, tf)

J1 +evlwl:! E £2[0, tf)·

and

Using this and the

boundedness of wand e, we readily show that f,B E ~[Ottf) n C,o::r[Ottf). Lea8t-Bqua1Y~B update law (9.96). We consider the function -2

V = 19tr(t)_1

1_,) 1/') + -e" + -;; (Ii>, I

Co

CO

(9.109)

wllOse derivative is readily shown to be

v< _ -

which proves that

ii E £ ..[0, tf)

and

2 f

(9.110)

1 + vlmlr. t

J

e

• E L2[O, tf ). Using this and

1 + vltrJlf

the boundedness of m and f t we show that

f,

BE £2 [0, tJ) n Coo [0, tJ ).

0

9.3

387

y-SWAPPING SCHEME

Theorem 9.6 (y-Swapping) All the signals in the closed-loop adaptive system consisting of the plant (9.1), the control law in Table 9.2, the filters in Table 9.1, and the filters (9.90)-(9.91), 1lJiUI, either lhe gradient (9.95) 07' the least-squares upda1.e law (9.96), are globally uniform.ly bounded, and global asymptotic tracl.:i.ng is achieved: lim [y(t) - Yr(t)]

1-00

= O.

(9.111)

Proof. The projection operator in Appendix E is locally Lipsrhitz, as stated in Lemma E.1. Therefore, as argued in the proof of Theorem 8.5, the solution of the closed-loop adaptive system exists and is uniqne on its ma.'ti.mum interval of existence [0, t f). From Lemma 9.5 we have 0, iJ E £00[0, tl), which in view of Lemma 9.2(i) implies that z, :c, {, 3,'\ E £00[0, t f). Equations (9.90) and (9.91) imply that to'o and 1AJ are in L.oa [0, t f). Since bm(t) is bounded away fronl zero, the control 11. is also bounded. By the same argument as in the proof of Theorem 8.5 we conclude that tf = 00. To prove the tracking, let us consider the error system (9.84),

(0.112) along with the error equation governing

i = - (CD + Kolwl'.!)

£:

+wTO + c2 - mTO.

f

(9.113)

We could finish the proof of convergence of z to zero by an argument similar to (9.86)-(9.88) in the proof of Theorem 9.4. Instead, we present an alternative proof based on the idea in Remark.6.B. As in Remarl~ 6.8, we show that i(t)

:-+ o.

it follows that B(t)

--+

Since,

O.

by

therefore, H'E(=(t), t)T (w(t)1'6(t)

Q(z(t), t)T8(t)

Lemma 9.5, {) E £DCJ

o.

+ C2(t))

Thus the input H'E (9.112) converges to zero. In view of --+

:t (~lzI2)

n

£2 and

From (9.113) we conclude w(t)T8(t) --+

O.

Since O(t)

--+

",

£001 and

°

0, we have

(wTO + C2) + QT8to the error system

:::; -colzl 2 + ZT (I'T'e (wTO + C2) + QTO) :::; - ~Izl' + ~ IIV, (wT8+£2) + QT01'

m

8E --+

by applying Lemma B.B, we arrive at the conclusion that z(t) = Y - Yn this proves the asymptotic tracking.

Zl

(9.114) -+

O. Since 0

Remark 9.7 In Theorem 6.4 we showed that the state feedback ISS-controller can be simplified by setting 9i 0, i 1, ... ,n, provided that a particular

=

=

3BB

MODULAR DESIGNS

form of normalizat.ion is introduced in the parameter update law. 'The output feedback case is no different. It is possible to show that the modified gradient

o= ProJ' {--:...'_r__

wE

1 + v'IQI.F 1 + IJIWl 2

iJ",

}

1/

>0

(9.115)

'

and the modified least-squa,res {)

=

Proj bm

t

=

{r + I)'IQIF 1

1

1 + J)'IQI.F

r

me} , 1 + vlwlr.

(9.116)

l'

ww r 1 + vlmlr~

update Jaws, which guarantee the properties (0)

(i) (ii)

(iii)

Ibm(t)1 ~ ~m > 0, 'Vt E [0, if) 8 E £00[0, tl) f

E .coo[D,tl), '1':'

E

/(1 + zlIQIF) :..

(9.117)

(9.11B) E

Q 0 E .coo [0, tf)' 0 E £2[0, t/),

.c2 [0,t/)

(9.119) (9.120)

allow one to set gj = 0, i = 1, ... ,71 in the ISS-controller in Table 9.2, while 0 retaining the result. of Theorem 9.6. 'Vhile the 1}-passive identifier uses an e~-tra integrator to generate fj, the y-swapping identifier employs p + 1 integrat.ors for wand woo Therefore, the price paid fol' having flexibility in the selection of the update law is the increase of the dynamic order by p.

9.4

x-Swapping Scheme

In this section we design a swapping identifier which uses the K-filters alrea.dy employed for state estimation. Instead of the parametric y-model we consider the parametric J'-model (9.2): (9.121)

This parametric model is already in the static form due to the use of the K-filtet's from Table 9.1. However, this parametric model may seem unimplementable because .7: is not measured. Only ~ n and the first component of x (the output y = XI) are available. Fortunately, it suffices to consider only the first row of (9.121) where all the signals are measured except for Ct: I

(9.122)

9.4

389

x-SWAPPING SCHEME

It is crucial that el is bounded and exponentially converging to zero. We introduce the "prediction" of y as 'r~

A

Y=

so that, the "prediction enor"

f

f.l

+ 0 1 (J ,

(9,123)

~ Y - iJ is implemented as (9.124)

and satisfies the following eqnation lincar ill the parameter error: f

'1'= !l18 + Cl'

(9.]25)

fJ is either the gradie11t:

The update law for

bill (0) sgn bm > 'III r= r T > 0, v> 0

(9,126)

or the least-squares:

iJ

=

t

= -f

P.roj 11",

{r 1

~;r fl

+1.1

I

0

Ot~I

1

} , (9.127)

f~

1 + vf11foO I

r(o) = f(O)T > 0,

II>

0

lvltere the projection operator is employed to guarantee t.hat Ibm{t)1 ~ \m > 0, Vt 2:: 0, MatrL'\: ro in the least-squares update la\v only indicates t:hat we = + 1'0/' 1'0 > 0, 1.0 lwep the either use covariance resetting for or let normalization positive definite, The above update laws are normali~ed. '~Titb ullnonnalized update laws we are not able to guarantee bonndedlless of n governed by (8.16),

r

ro r

o'T = AoO"I' + F(y,u) T

!

(9.128)

independently of the boundedness of F{y, 1I).

Lemma 9.8 Let the maximal inten,"l oj existence of solutions of (9.1), (9.5)(9.7) with either (9.126) 07' (9.127) be [0, tf). Then the following ide71.tifie7· propertie/i h.old:

(0) (i) (ii)

(iii)

Ibm {t)l2::

'm > 0

1

(9.129) (9.130)

Vt E [O,tf)

9 E £o;,[O,tf) ..;

E

. ' ..;

1 + "10 1 1-

f

1 + I.lOIfoO l

8 E £2[0, tf} n .coo [0, tf)·

E

4[O,tjl n£~[O,t,l

(9.131) (9.132)

390

MODULAR DESIGNS

Proof. First t from Lemma E.1 we conclude that Ibm(t)1 ~ C;m > 0, 'It E [0, tJ). Gradient update lauJ {9.1B6}. We consider the positive definite function

V

= 2"11-1" () r- 1 + 2"11 e 12p.

By virtue of Lemma E.l we have -ffI'r- 19~ show that

.

V ~

1 21

(9.133)

_{jT 1+e1hll::l' which enables us to

13.2

+ IJln11!l .

(9.134)

The nOllpositivity of li proves that jj E £00[0, tJ). Integrating (9.134) we get f

-r==== E £2[Ottj). From

JI + vln l l

2

(9.135)

in view of the bOllndedness of jj and

g,

we establish..; E E ,c",,[0, tJ). 1 + vlnll!!

With Lemma E.l we have

iJ E £2[0, tJ) n £00[0, tj). Least·squa1-es update law {9.127}. We consider the function

which proves that

(9.137) " = l{jl~(t)-I + lel~ which is positive definite because r- 1 (t) is positive definite for each t. Using Lemma E.l and the fact that 1, (r-1o) = 1+11~1f~onl' it is straigbtfonvard t

to arrive at

(9.138) In view of the positive definiteness of r-l(t) this proves that jj E Coo[Ot tj). It also proves that

J1 +vnlron .. f

E C2 [0, tj). From 1

(9.139)

9.4

391

x-SWAPPING SCHEME

in view of the boundedness of

esta.blish

8 a.nd

E

and positive definiteness of

J1 + IIlnllrn E .c",,[O,tl)· With Lemma E.1 we ha.ve f

r 0,

we

•

o Combining tbe x-swapping identifier with the ISS-controller with I(-filters, we obtain the following result.

Theorem 9.9 (:v-Swapping) All the signals in the closed-loop adaptive system consisting of the plant (9.1), the control law in Table 9.2, and the filters in Table 9.1, with either the gradient (9.126) or the least-squares u.pdate law (9.127), arc. globall]l uniformly bounded, and global asymptotic t"acking is achie'lled: (9.141) lim [yet) - Yr(t)] = 0. i-co Proof. Tile projection operator is locally Lipschitz, so the solution of the closed-loop adaptive system exists and is uniq}le all its ma.~imum interval of existence [0, 1,f). From Lemma 9.8 we bave 8, 8 e £00 [0, t f), whicll in view of Lemma 9.2(i) implies that z,X,e-,S,A e £oo[O,tf)' Hence, tf = 00. To prove tbe tracking, let us consider the ell'or equation (9.125): f

=

"-

(0.142)

+ O2 + FJ(y,u).

(9.143)

{llO+EI&

First, from (9.128) we note that

fh

= -kin!

=

Recalling from (9.9) that f22 [Um .2, Vm -l,2, .•• , Vo,:!, =(2)]'1', and from (8.9) that F1(y,u) = [0, ... ,0, l)(l)]T, we conclude that O2 + F1{y,u) = w, so (9.144) With (9.142), (9.144) and (9.3), we now get (9.145) The rest of the proof follows the lines of (9.112}-{9.114) in tile proof of Theorem 9.6. The centraJ part. of the argument is to show that (9.145) implies that w T 8(t) --. 0. 0

392

l\10DtlLAR DESIGNS

Remark 9.10 As in Remark 9.7 for tile y-swapping identifier, we can modify t.he update laws (9.1.26) and (9.127) by a normalization with 1 + v'/QIF, which guarantees that QTfJ E L oo , and allows us to set gi = 0 in the ISS-controller in Table 9.2. 0 The x-swapping scheme is the only modular scheme whose dynamic order is as low as that of the tuning fUllctions scheme. The x-swapping scheme is simpler than the tuning functions scheme, but, as we shall soo, its performance propert;ies are a little less strong.

9.5

Schemes with Parametric z-Model

The lllod1l181' schemes we designed in the last three chapters were based on the parametric IJ-ll1odel (9.10) and tIle parametric x-model (9.122) rather tban 011 the parametric z-model

(9.V16) with A:, l·F" and Q defined in (9.B5), (9.52), and (9.54), respectively, and defined in (9.47}-(9.49). (Note that (9.39) cannot be a parametric model because 11m in A~ in (9.39) is not known.) Even though the parametric :;model was central in the state-feedback design, our attention was devoted to other parametric models because of the lower dynamic order of the resulting adaptive schemes. For example, while the v-passive scheme employs a scalar observer that generates ii, the :;-passive Bchenle would use an observer of order p for :. The differt:llce is more drastic between the x-swapping scheme which does not use any extra fiIt.el'S, while the z-swapping scbeme uses additional filters of total dimension P(1l + 1). For the sa,lre of completeness and continuity with our state foedback designs, we briefly llre.C3ent t'VQ schemes based on the parametric z-model: tIle =-passive scheme and the .;;-swapllil1g scheme. We omit tbe stability proofs. Sj

z-Passive scheme In analogy with the state-feedback ::;-passive design in Section 5.5, starting from the paramet.ric model (9.146), we consider the identifier

z = A:(z,t)z + Q(=,'l)T8 f

iJ

= =

(9.147)

z-z

~roj {rwH'7 E} ,

(9.148)

bm(O) sgn bm > -;m 1

r=rT>o.

(9.149)

brA

If bm were known, in which case we would not have the estimate bm (and would not use projection), then one could prove the same result as in Theorem 9.4

9.5

393

SCHEMES WITH PARAMETRlC z-IvloDEL

(cr. [103]). With unknown bm , the identifier (9.147}-(9.149) is not directly applicable. To see this, recaU that t~e crucial property for establishing global boundedness in passive schemes is 0 E £2. This property was in the statefeedback z-passive design guaranteed automat.ically by t.he nonlinear damping terms built into A=. When bm is unknowll, the nonlinear damping terms (9.47)-(9.49) are capable of guaranteeing input-to-stat.e stability with respect. to 8 in the system (9.39), but in the observer error system f

= A=(z, t)f + H/£(z, t) (w T 6 + C2)

(0.150)

with the paral~eter update (9.149), they canllot guarantee the squareintegrability of {J as they would if bm were known (cf. Lemma 5.10). To remove this difficulty we strengthen t.he observer error system (9.150) by including the additional nonlinear damping term -diag { li:llwj:!, "~21~wI2 10, ...

,o} (£ - z) in the observer (9.147):

t,= A:(z, t)z-diag {"dWI" "'" 1"

,0, ...

,o}

(z-.o)+Q(z,t)TO. (9.151)

It is possible to show that the adaptive scheme consisting of the control law in Table 9.2, the filters in Table 9.1, the observer (9.151) and t.he updatE' law (9.149) achieves global uniform boundedness, as well as asymptol:ic tracking.

z-Swapping scheme The difficulty with inadequate nonlinear damping terms due to unknown bm in the z-passive scheme is much easier to deal with in the z-swapping sche!ne. A key property of swapping identifiers is that they guarantee that iJ is bounded. VVe often achieve this with nonlinear damping terms which guarantee that the filtered regressor is bounded. However, when bm is not knowll the nonHnear damping terms built into A= are not capable of guaranteeing the boulldedness of the filtered regressor (cf. Section 6.8). Fortunately, in the swappin~ design we can use normalizat.ion which can guarantee the bounded ness of 9 even when the filtered regressor is growing unbounded (cf. Lemma 6.26). With this observation one can show that the following z-swapping identifier guarantees the global ulliform boundedness and asYlllptotic tracking:

UT = A=(=, t)U T + H'E(Z, t)wT , UO = A:;(z, t)Uo + 1,Vf:(z, i)w'r(j - Q(z, t)TiJ, f = z+Uo-U 8,

lJ E IRP>:P (9.152)

U E HlP

T~

(9.153) (9.154)

with either the normalized gradient:

{Uf}

~ = Pl'oj r lUI'" b l+v 7F

(J

m

bm (0) sgn bill > C;m

r=rT>o,

1.1>0

(9.155)

394

MODULAR DESIGNS

or the normalized least-squares update law:

0

r 9.6

=

=

p[~j {r 1 +~~UI}} , -f

b,.,. (0) sgn bm > C;m (9.156)

T

tRJ r 1 + llIUI}

f(O) = r(o)T > 0 I

II> O.

I

Transient Performance

In this section we derive £2 and £00 transient performance bounds for the error state z, which include bounds for the tracking error Zl = Y - Yr' We first consider the passive schemes (f}-passive and z-passive), and then the swapping schemes (y-swappillg, .~-swapping, and z-swapping). For simplicity, we Jet r=;I.

9.6.1

Passive schemes

First we derive an £00 performance bound for the 1}-passive scheme. To eliminate the effect of the initial condition of the estimation error f = Y - fJ, we initialize the observer with y(O) = y(O), which sets c(O) = o.

Theorem 9.11 (y-Passive Scheme) In th.e adaptive system {9.1}, (9.5), (9.6), (9.7), {9.50}1 {9.66}, and {9.69}, the following inequalit1J holds:

Iz(t)1 ~ . ~ (AlI8(0)12 + Nle(O)I~) 1/'!. + Iz(0)le-COI / 2 I v Co

(9.157)

111here

l t l =1 -(l +"(2 -) 2lio li.oDo

N

=

(9.158)

"(3 ) +-1 (1 +"(2) -"(- (1+- , 2coli.o

li.ogo

2.a( P)

do

Con-ago

(9.159)

Proof. To obtain an £fXl bound on z, it would s~em that inequality (9.56) could be used along with £00 ,bounds on C2, 8 and 0. However, it is not clear how to obtain a bound on 1191100 depending only on design p81'ameters and initial cOllditi~ns. Therefore, we apply a different approach which eliminates the need for 1181)00' First, we note from (9.68) that =

-coe:! - li.o/wI 2 e2 + f

~

--fa< -

Co 2

I)

(w T 8+ e2 )

lio 12 + -c2 1 2 + -1 -IWE 2

2co

2li.o

I-I" f) -,

(9,160)

9.6

395

TRANSIBNT PERFORMANCE

From Lemma E.1(H) and (9.69) with substituted into (9.160) yields

r

= ,It ,ve have

161 ::; l'lwel,

which

(9.161) In this inequality 161 2 appears with an opposite sign of that ill (9.61). Theref~re! by adding these t,vo inequalities with an appropriate scaling t \ve eliminate

1812:

By applying Lemma C.5, we get

Iz(t}I' + 2~~:(t}2

~ 2~ [Uo + Co~:Yo) 11£.11;' + :0 (1+ ":110) 11811!'] + (lz(0)1 2 + i

2~o!1o

e(O)2) e-cot .

(9.163)

Since f(O) = 0, we have 2 ) 2 ) Iz{t)1 :s .~ [ ( d1 + _7_ lIe211~ + 2:.. ( 1 +.:::L 11611~] 1/2+lz(O)le- Col/ 2 •

v 2co

0

COliD9o

From (9.3) and (9.4) we have 1ilel~

lio

Ko90

::; -lef2, which gives

(9.164)

(9.165)

396

MODULAR DESIGNS

To obtain a bound on 116!1~ we recall (9.77) and (9.74), which give

11611~ :5 10(0)12 + 1.Ie(0)1~.

(9.166)

Co

Substituting (9.165) and (9.166) int.o (9.164), we obtain (9.157) with (D.158)(9.159). 0 The initial condition z(O) in the bound (9.157) is, in general, dependent ou the design parameters CO, dO,lio, go. However, as explained in Section 4.3.2 for state feedback, with trajectory initialization we can set z(O) = O. Following (9.40) and (9.41), :(0) is set to zero by selecting Yr(O) =

y~i)(O) =

y(O)

(9.167)

bna (0) [Vrn'i+l (0) -

O'i

(y(O), e(O), 3(0),0(0), Xna+i(O), ii~i-I)(O))] , i

= 1, ... ,p -

1.

(9.168)

Upon setting z(O) to zero, the bound (D.157) can be systematically reduced by increasing Co. By examining (9.158)-{9.159) one can see t,hat the bound (9.157) can also be systematically reduced by simultaneously increasing liD and

do. A careful comparison of (9.157) with (8.139) reveals that the bound for the tuning functions schcmc is lower. Also, an advantage of the tuning functions scheme is t.hat an L.2 bound like (8.138) is not available for the y-observer scheme. We now give performance bounds for the z-passive scheme. The observer initial condition is set to z(O) = z(O). For comparison with the y-passive 6. scheme, we select ~1 = ... = n.p = ~o.

Theorem 9.12 {z-Passive Scheme} In the adaptive system (9.1), (9.5), (9.6), (9.7), (9.50), (9.1,/9), and (9.151), the following inequality holds:

Iz(t)1

~ ~ (M",,19(0)1 2 + NooIE(0)I~f/2 + Iz(0)1.-",'/2,

(9.169)

where

M"" = N"" =

2~o (1+ ~~~) 8~u [(1+ ,,~~) (:. + ~(~)) + ~(~)l·

(9.170) (9.171)

AforeDver, if bm is kno'Wn, then the z-passive design with the observer (9.147) results in the L.2 bound 1 ( 1\1 10-(" IIzlb:5 . r;:: 0)1- + N'J. IE(O)I:p., ) 1/2 + 2

vCo

1 ~lz(O)I, v.:.co

(9.172)

9.6

397

TRANSIENT PERFORMANCE

M2

=

H1+ 4::Uo)

(9.173)

N2

=

~ (1+ 4~Uo)·

(9.174)

While the Coo boullds (9.157) and (9.169) are similar, the £2 bound (9.172) is available only for the z-pnsshre scheme.

9.6.2

Swapping schemes

First we derive an £00 performance bound for tbe v-swapping scheme. For simplicity, we consider only the gradient. update law. To eliminate the effect of the initial condition of the estimat.ion enol' f = y + ron - ro TO, we initialize it with ro(O) = 0, wo(O) = --y(O), w1Iich sets l(O) = O.

Theorem 9.13 (y-Swapping Scheme) In the auapti1Je slJstem (f).l), (9.5), (9.6), (9.7), (9.50), (9.90), (9.91), and (9.95), the Jollo'wing ineqlJ.fJ.lily holds 1=(t)I =s;

~ (1\118(0)12 + lVle(O)I~) J/'2 + 1=(O)le-cllt •

(9.1 (5)

yeO

whe1"e

ill

= -1

N =

(1 +

.))

'1-

411:0

Bc5noDo

-

1+

l[-2- (

4 cfi~o

"'r'1

8CB~oDo

(9.176)

1]

+2.. ) +_...,-4{Jo

doLl( P) .

(9.177)

Proof. 'Ve derive an £00 bou!ld all :; using (9.56). It rcmltillS '.0 det.ermine bounds on

lIe:zlloo, nOnce itnd lI'ince'

First, from (9.3) and (9.4) we have (9.178)

In view of (9.108) and (9.103), using e(O) = 0, we get (9.179) 'Vith the help of (9.95) we write

e2 "11 II" f2 '111 III) II 11'1 1B::"1"- =s; '1-"IW\2 (1 + vlwl 2 )2 ~,.- to' ;;.;, (1 + vlrol 2)2 ~ ')'- ro ;;.;, e ;;.;,

(9.180)

398

MODULAR DESIGNS

and by substituting (9.93) we obtain (9.181) From (9.101), using m(O) = 0, it follows that (9.182)

To obtain a bound on -d dt

11;IICXl, along the solutions of (9.94)

(11-1') - f - + -1 Ie I')) p 2

4eo

~ -CaE-2

+ f~2 -

-

1

~1ca

and (9.3) we have

IE 12 ~ 0

(9.183)

which yields

IIfll~ ~ -2 Ic(O)I~· Co 1

(9.184)

By substituting (9.179), (9.182), and (9.184) into (9.181), and then, along with (9.178) and (9.179), into (9.56), we arrive at (9.175) with (9.176) and (9.177).

o The initial condition z(O) ill the bound (9.175) can be set to zero by the trajectory initialization procedure (9.167)-(9.168). Upon setting z(O) to zero, the bound (9.175) Crul be systematically reduced by increasing Ca. By examining (9.176)-( 9.177) one can see that the bound (9.175) can also be systematically reduced by simultaneously increasing Ka and dll . The £00 bound (9.175) for the y-swappillg scheme is lower that the bound (9.157) for the V-passive scheme but higher than the bound (8.139) for the tuning functions scheme. Nmv we derive an £00 performance bound for the x-swapping scheme.

Theorem 9.14 (x-Swapping Scheme) In the adaptive 81/stem (9.1), (9.5), (9.6), (9.7), (9.50), (9.126), the Jollo'wing inequality Il.olds: (9.185)

where (9.186) (9.187)

9.6

399

TRANSIENT PERFORMANCE

Proof. We derive an £a;J bound on z using (9.56). The b01:1nd on in (9.178). It remains to determine bounds on and (9.133) we get

1\81100 and 110\100'

1Ie-211co is as

From (9.134) (9.188)

By sUbstituting (9.135) into (9.136) we get

18\2

~ 21'2 (.!.IBI2 + le-d2) . 1.1 1/

(9.189)

Since (9.190)

then (9.189) yields (9.191)

By substituting (9.188), (9.178), and (9.191) into (9.56), we arrive at (9.185) with (9.186) and (9.187). 0 The initial condition z(O) in the bound (9.185) call be set to zero by the trajectory initialization procedure (9.167)-(9.168). Upon setting =(0) to zero, the bound (9.185) can be systematically reduced by increasing Co. By examining (9.186)-(9.187) olle can see that the bound (9.185) can also be systematicaHy reduced by simultaneously increasing "0, go, and do. We now give performance bounds for the z-swapping scheme. The filter initial conditions are selected as U(O) = 0 and Uo{O) = -zeOl, to set leO) z(O) + Uo{O) - UT(O)8(0) o.

=

=

Theorem 9.15 (z-Swapping Scheme) In U"e adapti1Je system (9.1). (9.5), (9.6), (9.7), (9.50), (f).152), (9.159), (9.155), the jollo'llJing inequality holds Iz(t)1

~ ~ (J1/0018(0)12 + NoaIE{O)I~) 1/2 + Iz{O)\e-COi , veo

(9.192)

where {9.193} (9.194.)

Moreover, if bm is I.."nown, then the z-swapping scheme results in the £'}, bound

a--

1 ( 1112 19(O)I- ") + N2 1e-(0)lp 2 ) 1/2 1 IIzlb::; t;:: + n;:lz(O)1 + -3 16(0)1, v~

v~

7

{9.195}

400

MODULAR DESIGNS

where

AI, N,

= =

H2;~n +;(~ +2~Ko)] 3~, [1+ ~~ + ~V (~ + 2~Ko)]'

(9.196) (9.197)

'Vhile the £00 bound for the x-swapping is similar to that for the :;swapping scheme, the £2 bound (9.195) is available only for the =-swapping scheme.

9.7

Swapping Schemes with Weak ISS-Controller

In Chapters 5 and 6 we discussed the possible undesirable effects of the strong nonlinear damping terms. In Sect.iDll 6.7 we introduced a weal L Tbe design for the case p = 1 can be easilY-.ded-nned from',th~ first. step of the' recursive procedure. Thanks to the minimum pbase Assumptirinl'l0~1, the design is· restricted to the first p equations in (10.2):

Xl Xp-l

Xp

=

= =

X2 -

an-lY!

xp -

am+I'Y."

(10.41) xp+1 - amY + brn u •

We will return to the behavior of the last m -equa.tions in the.stability proof. In the backstepping approach we view the. state variable Xi+l as a control· input to tbe subsystem consisting of the states Xl,'" ,Xi, and- we design a stabilizing function 0:, which would achieve the control objective·if Xi+l were available as a control input. The control law for the actual control input 'U is obtained at the pth step of the recursive-design. Because-only tbe system output y = .'1:1 is measured, we replace (10.41) with a new system·whose states are available. We start with (10.38), which is just an alternative form of tbe first equation ill (10.41). Equation (10.38) suggests that 'Vm ,2 is chosen instead of the unmeltSUl'ed X'2 to be tlte 'virtual control' in pu t for backstepping. The reason fOl" this choice is that both X2 and V m ,2 are separated by only p - 1

10.2 ''DnNI:Nc IFiuNCmICdNS 1l!)~SIGN

423

in~gv81hors !ham lUbe .aatmill contrulJlu, which

is clear {from' (10.17) for j

= m: (10.42)

A (dl(!)sEJr!exaniina1iiollICl{lthe i fi.ltersiain Table 10.1n'e,re8:ls that more integra.tors stand iin ithe .way ·of any· other\va.r.ia:ble. Therefore,! bhe design system chosen t'fl Tftplace{(10.f:.I!lJ)Js ,iJ -= :lb,rivm,2 + ~2 +:(iil)6 + E:2

"'Um ';2 -:::

'vm ,3 -

ki11rn,1

(10.43)

k j V na ,1 ··,vm ,p+1 -1t:'"Vm.li+ u.

'iJrn.p":"l -::: '":Vm,p -

'il""p .:::

~ll'

of its states are, avit1l8lble

for feedback. ' Our design tasle is to forooe the

ontpmt/lll' to 8&1'lllPttlltto8ll1y. track :the 'reference outlmt Yr while keeping all nhe

closedL.}(!lCllp signals, .bounHdd. . As in' the tuning:fu.nctions desigll'in Chapter 4, we employ the change of !

coorainates .E:l

-:= :,1/- Yr

.,., := ",,,.-. ~,l

ny(i-l) -

r:: r

a·.- 1 t

i

,(10J14) ,(10,t15)

= 2, ... "p,

\vhere '0 is an estimaltc\of (! = lIbRa' Our goal is to regulate:; = [=h .... , zp]T to.zero because by- regitlating z to:lzC!ro we will achieve asymptotic tl'a
+ (V

71l .2

-

T-

".

BYr - al) el (J + bmz2

e(fir- + 0:

')'-

1)

e1 (J

'I' -

+ bm =2 A

(w - b(Yr + Q'd el) 0 + bm=2' A

(10.52)

Substituting (10.52) into (10.51) we get =1 = -Cl=l -d1 z1 +E2+(W - iJ (ilr

+ 0'1) el)T O-bm (ilr + 0:1) O+bmz'J.

{10.53}

Tbis systcm along with (10.V!) is to be stabilized by selecting update laws for the paramet.er estimatcs 0 and fl. These update laws will be chosen to achieve stability with respect. t.o the Lynpunov fUllction

V:I

1 1 -"J' 1 mI " l I T = -=+ -0 r- 0+ -Ib2-y { ! - - + - E Pe 2 1 2 4d • '1

(10.54)

l

We examine the derivative of Vi:

,~

=

=2 [-CtZt - d1=1

ii1'r-l~f1 -

-(I

=

+ £2 + (w -

lbml_:.

-(!{! -

-y

-clzr + b.m =t=2

iJ (Ur + 0:1) edT 0 - bm {lir + al}

e+b z

m 2]

I_T

-e £ 4d1

-Ibmlo..!:. bsgn{bm ) (Yr + al)' /81l1Hwlloc:u and a.':iYlllptotic stability follows as in the proof of Theorem 10.8. Substituting (10.23,1) iuto (10.233) and I

solving for

IIzlb.1 = (J~ 1=(r)12dr)!i, we get

)1 2d ) ~ < 18l11Hwiloo (I II ( io('1-( ... r T - 2..Jcodo -19111 Hwlloo 1Jr 2,1'

(10.235)

JJ

and (10.226) follows because IIYrll~,1 = IlJr(r)1 2 dT 5 IIIJrll~t. From (10.209)(10.210) and (10.211)-(10.21'2), by Theorem B.2(ii) we have

l\ i111b :5

1/(,112

s

IttVij/lool/(Zl)tl/2 I/H',lIooII(=d,lb,

which 1 in view of (10.226), proves inequalities (10.227) aud (10.228).

(10.'236) (10.237)

o

Theorems 10.8 and 10.9 provide two different stability conditions, (10.188)

a.nd (10.225), of which (10.225) is directly computable [16] and less conservative because IIHwl/oo S IIhwllt (see Theorem B.2(iii)). Another way of expressing the performance properties is by comparing the detllned closed-loop transfer function (10.207) ,vit.b the desired transfer function YrII(CRR») 1.

=

10.4 TRANSIENT

453

PERFORMANCE WITH TUNING FUNCTIONS

Theorem 10.10 (Frequency Domain Performance) In the nonadaptive .'lysf.em (10.207), the design IJa1'O:mete7's Ci and di, 1::; i :s; p, can be chosen t.D satisj1J the following tracking pe7jonnance .'fpccification far any Dc > 0: Vw E lR. Proof. By setting t =

00

in (10.233) we sec that the induced £'.}, norm of ::~:~

is 110 grpatcr tluul ~. This, ill tllrl1~ means that ~""O(IO implies

B:(jW)

IlX::(jW) From (10.20i) we

G .

I c(Jw)

llOW

_ 1 _

1-

(10.238)

1
2V Co db > IOlllhwlh, the perfo11Twnce ratio Rc.:.:. is no greate7' than B.C,,,,,- < 1. Fl'om this corollary we ean deduce two furt.her advantages of the adaptive controller. • First, the adaptation gain l' provides an addit.ional degree of frel"dom wit.h lvhich the performance can be improved when thc adapt.ivE'" and the nonadaptive gains arc the same, c~d{} := c~d~ ~ codol and sat.is(y the stabilit.y condition 2Jeodo -IBII!hwlli > O. In this case the adaptive bound is lower than t.he nonadaptive bound provided t,hat. (10.290) and the bounds are the same when l' ::; tive plot of tbe quantity

BN

,*.

Figure 10J:i shows a qualita1

Qh') =In- = I n BA R.c. x

(10.291)

obtained using the bounds (10.283) and (10.287). ''''hile Corollary 10.15 demonst1'8,tes a performance improvement due to adapt-at.jon ouly fol' 'Y > ,'''' t.he simulations, some of which are shown in Example 10.16, exhibit a performance improvement for all 'Y > o. • Second, and more important, performance improvement can be achie,red even with c8c1~ sma.ller than c~d~A. In the presence of a, large parameter

462

LINEAR SVS'l'EMS

Q(-y) = In ~: = performance improvement

,,"

," ," ,,"

,,'"

__ -

..... ...........

, ,'"" simulations,' , "

,,"

, ,,' "

,/

,,

Corollary 10.15

"

Figure 10.6: Performance improvement due to adaptation.

uncert.ainty 0, the nonadaptive controller must use c~d~A sufficiently large to satisfy 2Jc~d~ -18'llIhw ll l > 0, thus increasing the bandwidth. From Corollary 10.15 it is clear that with the adaptive controller such an undesirable bandwidth increase can be avoided, because when both 8' and c~d~ are large, the condition 2Jc~d~ Rt.r:c > 2J~d~ -IOlllhw!1t can be satisfied with c~d~ much smaller than c~d~. This analytically confirms that adaptation is an efficient tool for reducing the effects of large parametric ullcertainty without unacceptable widening of system bandwidth. For small parametric uncertainty, the linear controller is effective.

Example 10.16 The improvement of performance due to adaptation is now briefly illustrated with the example introduced in Section 10.2.4. We consider the unstable relative-degree-three plant

y(s) =?( s- 8

1 -

a

) u(s) I

a

> 0 unknown.

(10.292)

The control objective is to asymptotically track the output of the reference model 1 (10.293) Yr(s) = (8 + 1)3 1'(8) . The tuning functions design for this problem was showll in detail in Section 10.5. To illustrate the parametric robustness (Theorems 10.8 and 10.9), we switch off the adaptation ('Y = 0) at a constant estimate O. = 1, when

10.4

463

TRANSIENT PERFORMANOE WITH TUNING FUNC'l.'IONS

Tracking error y - Yr 0.1

-0.1

o

10

20

30

o

40

10

20

30

40

I----.---r-----r-_

0

10

20

30

40

Control u 2

2

-2ii-----r---r--'T'-_ o 10 20 30 40

2

-2ii--__r-........- . . -...... o )0 20 30 40

-211----..---.---r----, o 10 20 30 40

Parameter estimate a 4 2

o

a fixed 10

20

30

40

o

10

20 'Y

30

40

o

10

20

30

40

= 0.3

Figure 10.7: Adaptation improves the tracking error transients without an increase in control effort. The plant is driven by r{t) = sin 1'1 and the plnnt parameter is a = 3.

the parameter error ii = 2 is significant, With Cl = C2 = Ca = 3 and d1 = d2 = da = 0.1, the resulting detuned linear system is unstable. \Vith an increase to Cl = C:l = Ca = 5, the system is stabilized. However, without adaptation, the tracking error, shown ill Figure 10.7, is about 12% of the reference input, \vhich is not acceptable in most applications. The adaptive controller is sbnulated with tbe same coefficients Cl = C!! = Ca = 5 and d1 = d2 = d3 = 0.1. The effectiveness of the adaptive scheme is demonstrated by the fact that even with slow adaptation h' = 0.3), tbe tracldng error is reduced to zero after a few periods of the reference input, as shown in Figure 10.7. It is remarl{able that even dill'iug the adaptation transients, the tracldng errol' is smaller than in the nonadaptive system, while the control effort is about tbe same. When the adaptation gain is increased to 'Y = 1, the tracking performance is further improved with about the same control effort,

464

LINEAR SYSTEMS

vVhile Coronary 10.15 shows the performance improvemellt only beyond a certain 1', the simulations indicate that the performance improvement is 0 present for any "y ~ o. As a conclusion to tllis section, we point out that the improvement of performance due to adaptation is the first such result in the literature. TheI'e al'e two reasons for this. First, the traditional certainty equivalenc!e adapt-ive controllel's do not possess the para.metric l'obustness property, so they do not have nonadaptive counterparts which can achieve stability, let alone a given level of performance. Second, even if they stabilize the plaut with some C011stant estimate, the adaptation is likely to make the performance worse during the transient because it is not based on tbe control objective (the identifier is not driven by the tracking error) I and the controller docs not account for the parameter estimation transients,

10.5

Comparison with a Traditional Sclleme

The tuning functions scheme is now compared using simulations with a standard certainty-equivalence scheme on the basis of tral1sient performance and control effort, The comparison is made for the relative-degree-threc unstable system from Section 10,2.4.

10.5.1

Choice of a traditional scheme

The comparison with a direct MRAC scheme is llOt pursued because such a. scheme updates at least tll,ree parameters. This is clear frOll1 its control la,v

(10.294) where .s2 + 'm18 + n12 is a Hurwitz polynomial. A calculation using the Bezollt identity gives 8

5

+ s4[ml -

83

-

a] + s3[m:! - 8.1 -

a(ml - 83 )] - s2[Oo + (n1.2 - 8.. )a] = (8 + 1)3(82 + mls + m'2) + IJJs + 62 I (10.295)

which shows that 60 ,O;:s, and 84 have to be updated, while 81 and 82 can be -711.) - 3m'!h 82 = -m2' Simulations showed that the upd~tte of three parameters results in transient performance infel'ior to indirect linear schemes which update only one parameter estimate. Therefore t we compare our new controller to a standard indirect scheme [43 t 129}, ill which the plant equation 8 2 (8 - a)y(s) = 1/.(s) is filtered by a Hurwit.z

fixed at 61

=

10.5

465

COMPARISON WITH A TRADITIONAL SCHEME

observer polynomial s!-a

+ kl s2 + k',!s + ka to obtain the estimation equation

4J

=

1jJa.

tb

=

1/J

=

+ k 1 s 2 + k'ls + ka y{.s) .., s---::-------:-----y( s) S!i + /0:18 2 + k:2 8 + ~~:i '

1

S3

S:i

S3

+ kJs2 + k2 s + ka u(,s)

(10.296)

and the parameter update law is a normalized gradient:!: •

1/Je

il = 'Y ---:Jil, 1 + .jJM

e

= l/J - 1/Jo, •

(10.297)

The control law (10.294) is implemented by replacing a with a. in (10.295) and then solving it for the controller parameters: 83 = -(3+&), (J.J = -(3+3'111.] + a(ml -8a )], 00 = -[1+ 3m l +3m2+(m.2-04)aJ, 01 = -ml-3m'2, ()2 = -7J}·2. The indirect. adaptive linear scheme and the tuning functions scheme were applied to the plant (10.158) with the trne parameter a = 3. In aU tests the initial parameter estimate was a(O) = 0, so that, with the adaptation switched oH, both closed-loop systems were unstable. The reference input was r(t) = sin t. For a fair comparison, our first taslc was to adjust the design parameters of the indirect scheme to achieve the best transient performance wit.h a prescribed control effort. This was done in detail in [95, Section VII]. The trade-off between transient performance and control effort was examined for various initial conditions. To reduce the transients due to the mismatch of initial conditions, the initial condition of the reference model output was set in all tests to be equal to the initial value of the plant output. The available design constants were the adaptation gain 'Y and the coefficients of the observer polynomial ,r;3 + k1s 2 + l-:28 + k3 and of the controller polynomial s2+mJs+m'2' All the roots of the observer polynomial were placed at s = -2 with kl = 6, k2 = 12, k3 = 8, while the roots of the controller polynomial were placed in a Butterworth configuration of radius 3 with ml = 4.2426, m2 = 9. These were judged to yield the best trade-off between transient performance and control effort for different initial conditions. On the basis of the simulation results shown in [95, Figures 2--,,:1] the best compromise between transient performance and control effort was judged to be for the value of the adaptation gain '1 = 1000.

10.5.2

Comparison of the schemes

For a comparison of transient. performaJlce, the tuning functions scheme was adjusted to employ about the same control effort as the indirect linear scheme. 3Thc simulation results \Vith a leastMsquares update law were virtually identical and are therefore omitted.

466

LINEAR SYSTEMS

Tracking error y - Yr

o

4

2

10

B

6

Parameter estimate

o

-a.

2

4

6

Control u 20

o

-~Ir---~~--~----~

2

4

Indirect linear

6

o

2

..

6

Tuning functions

Figure 10.8: Comparison for 11(0) = o. The tuning functions controller improves performance with about the some control effort by incorporating the update law

a.

10.5

467

COMPARISON WITH A TRADITIONAL SCHEME

Tracking error y - Yr

ludirect linear -10r-----r----.----r------r----,

o

6

4

10

8

P81'amcter estimate -0

2

-s 3

4

0

Control

2.

3

4

'U

200

lao

-100

-200 0

Indirect linear

-200 2.

3

4

0

2

3

-I

Tuning fUllctions

Figure 10.9: Comparison for lI{O) = 1. The parameter estimate ill the tuning functions scheme lS smootber because its update law is driven by the state of the error system z(t).

468

LINEAR SYSTEMS

Tracking error 11 - Yr I.S

1.5

1.5

0.5

0.5

0.5

-0.5 ;----r""-..,..----.--, o 2 3 4

0

2

3

-i).5 ;----r---r--~-.,

-i)j +----.--.....----,---.

0 1 2 4 Parameter estimate -ci

4

0

2

Control

1000,

4

3

0

2

3

4

0

2

3

4

0

2

3

4

1J.

0 -1

-1000· 0

d 1 = d2

2

3

4

0

= da = 0.001

2

3

4

d1 = d2 = d:t = 0.2

d. = d"].

= da = o.a

Figure 10.10: Nonlinear damping fol' 1J{O) = 1. The effect of state estimation error is attenuated.

=

=

=

Tbis was achieved with kl 6, "'2 = 12, h:3 = 8, Cl C2 = Ca 1, d. = d2 = da = 0.1, and the adaptation gain "'( = 0.5. The plots ill Figures 10.8 and 10.9 show that. the transient performance of t.he tuning functions scheme was far superior for both sets of illH:ial conditions. .Measured by any norm, the tracking error with the tuning fUllctions scheme is only a fraction of the indirect linear scheme error. The simu}i;ttiollS presented here confirm the strong transient performance properties of the tuning fUllctions design derived ill Section 10.4.2. As we explained, the distinctive feature of the tuning functions design is tbat the COlltroller incorporates the parameter update law = 'YT:'1t with which it accounts for parameter estimation transients. The effect of this additional information about ii. is tbat the settling time of the tracldng error is much shorter for the tuning functions scheme. Figures 10.8 and 10.9 show that the settling time of the tracking error is closely coupled to that of the parameter error. III contrast, tIle tracldllg elTor of t.he indirect linC!ar scheme continues to grow even aft.er the parameter estimate has converged to its true value.

a

10.5

469

COMPARlSON WITH A TRADITIONAL SCHEME

Tracking errol' 11 - Yr 1

1

-1

-I

0

1

:~ -soo[ 0

1

2

3 4 0 Paramet.er estimate -iJ.

2

3

1

0

3

4

2

3

4

2

3

4

Control u.

-1500 0

4

2

2

3

n011-initialized

.-It:

4

0

initialized

Figure 10.11: Reference model initialization fol' 1/(0) = 1. The large initial value of control is eliminnted, and the parameter transient is made wrnosL monotonic.

Two other most important factors which contributed to the superior performance of t,he tuning functio11s scheme are nonlinel:lI damping and referelu'e model initialization.

):!

Nonlinear damping. The nonlinear damping terms -di (ad;1 Zj contributed to a significant reduction of the effect. of initial conditions on the new adaptive system. Its attenuating effect is displayed ill Figure 10.10. If the damping is increased over an optimum rate, the tracking elTOI' continues to decrease, but the control effort increases. Re.ference model initialization. In contrast to the indirect scheme, the new tuning functions scheme provides clear guidelines for reference model initialization, which follow from the design objective of driving the z-val'iables to zero. According to (10.167), (10.170), and (10.173), the initial values of :;variables are set to zero by choosing 1'1 (0) = y(O), "2(0) = 112(0) - (tl (0), and T3(0) = va(O) - 0;2(0). In general, it is always possible to set z(O) = 0 by e."\:pressions (10.250)-(10.251). In all tests, the reference model initialization was

470

LINEAR SYSTEMS

found to significantly improve both the transient performance and the control effort. A typical e.."Cample is Figure 10.11.

10.6

Modular Designs

The tuning functions controller and update law are interlaced in an intricate fashion, which makes t11e design complex. In Chapter 9 we introduced modular output-IeedbaclL schemes with independently designed controllers and identifiers. Vle pursue here the same idea and design modular backstepping schemes for linear systems. However, for lineal' systems we do not use the strong ISScontrollers because their underlying nonadaptive controllers are nonlinear even for linear systems. Instead, we employ an SG-controllc7' which is a certainty equivaJence version of tIle llOllo.ch1.ptive linear controller in Section 10.3.1. This controller is different from traditional certainty equivalence controllers because its backstcpping st,ructure endows it with design coefficient.s which are useful in shaping the transient behavior. Because of their certainty equivalence nature, the modular schemes will serve nicely for a qualita.tive comp81ison between the traditional certainty equivalence sehemt's and the tuning functions sehelDt'. In addition to Assumptions 10.1-10..4, we make the following assumption about a lower bound on the high-Iff'C!llency gain, standard in 'indirect' adaptive control.

Asswnption 10.17 In addition to sgn{bmL a positive conodant fmch that Ibm I ~ C;m'

c;'m

is known

Assumptioll 10.17 stl'engthens Assumption 10.2 ill the tuning functions design. It allows the control law to contain a division by the estimate bill! wbich is kept away from zero by paraJneter projection. In the tuning functions designs it was possible to avoid this assumpt.ion by intl'oducing an additional estimate of !l == 1/b,rt. This section is organized as follows. In Section 10.6.1 we present the SGcontroller design using the knmvledge of the tuning funct.ions controller from Section 10.2.1. Then in Sections 10.6.2 and 10.6.3 we present two identifiers and stability analyses for corresponding closed-loop adaptive systems. Finally ill Section 10.6.4 we compare the modular designs wit.h the tuning functions design.

10.6.1

SG-controller

The SG-controller, given in Table 10.3, is a modification of the tuning functions controller ill Table 10.2. 'Ve briefly discuss the modifications leading to the controller in Table 10.3.

10.6

471

MODULAR DESIGNS

Table 10.3: BG-Controller for Lloear Systems (10.298)

i == 2, ... ,p

= a] = =

0:1

a'2

a, = r-ll fJ.

(10.299)

1

(lO.aOO)

-:-0:1

b",

- (Cl

+ d1) Zl -

" -b,n=l -Zi-l -

_TA

[ (n c2

(10.301)

{2 - W (J

+ d2

aat 811

[ + d; (8a;-1 ---a:y-r] Ci

Z2

Zj

+ /32

(10.302)

+ Pi ,

i = 3, ' .. P (10,303) I

8aj_l ( TO-) + -a- (A0'11 + cnY) + ~ ~ 8ai-l (j) = -ae2 + W --r=T)Yr Y 1] ;=18yrJ [JOi-l

(10.304)

Adaptive control law: (10.305)

• The adaptation gains r and 'Y are set to ~eJ:ob ,~bjch ~leans that we eliminate ~r'T2 from (lO.IOl), ~il Ej;1 '6;' r~;' Zj Il'om (10.102),

rT. -

aud (y~i-l)

+ 8';:i 1 )

~ from (10.I03) .

• With Assumption 10.17, 0 is replaced by crossing zero by parameter project.ion.

l/bm, and bm{t)

is l(ept from

Straigbtforward but lengtllY calculations show that the resulting error system is

(10.306)

472

LINEAR SYSTEMS

where

bm

-cl-d1

A;

0

-bm

-c.., -d..- (808u )?

0

-1

=

!!!ll -

~-

0

1 0

(10.307)

1

0

0

HIe

=

[

-~ 1

-1

-Cp-dp(88"u-1 )2

E 1R.P

_1:_.

(10.308)

l!1I

t-

+ (Yr + itt) e'f _~lwT +ZteT

(ijT

IV,...T 0

_fJs!:.1wT

=

E RPxp

8U

110.,._1

(10.309)

T

-lfiIW

Q'l' =

-~

[

+ ""21 iJr eT1

8IJ"1II

_~ + ~'fP-l)eT 0

80

b~·r

1

RPxp

(10.310)

E

1

By examining the expressions for ~ is a function of only 0,

Cit

ll:i in Table 10.3, it is not hard to see that and di, which means that A; and loVe depend all

0, but not on other states of the closed-loop system.

Thus, in contrast to the tuning functions design and all the other output-feedback designs so far (cr. Chapters 8 and 9), the matrix A~ and the vector I,VE are functions of fJ only. This is due not only to the SG-controller design, but also to the linearity of the plant. In view of (10.98) and (10.99) we have

(10.311)

10.6

473

MODULAR DESIGNS

On t.he other haud, - bmz 1 -

Bal a Y

TW (J

+ zleTt /J =

-

-bmzl

-

Bat Ta W (}.

(10.312)

Y

With (10.311) and (10.312), the errol' syst.em (10.306)-(10.310) is rewl'iUcn as (10.313)

where -Cl -

-bm AA8) =

0

d1 -c,:! -

bm

0

d 2 (~)2 lly

1

0

-1

0 1

o

0

-1 -c - d P

P

(OQP_I)2 Oy

(10.314) The error system (10.313) is similar in form to the error syst.em (9.84) in the modular nonlinear design. However, because of the absence of the Iii and ,qi nonlin~ar damping terms, (10.313) is not. inpnt.-to-state stable with respect to

8 and 8. .

For a brief comment on the underlying nonadaptive SG-controller, we let fJ == 0, so that the error system (10.313) becomes

iJ = ('onst.

(10.315)

Let, for simplicit.y, bm = bm = 1. The detuned error system (10.188) in the tuning functions design and the detuned error system (10.315) in the modular design are identical. Therefore, the parametric robustness and nonadaptive performance properties established in Theorems 10.8-10.10 for the tuning functions design hold also for the modular designs. These properties distinguish our modular designs from the traditional estimation-based certainty equivalence designs. The backstepping approach results in a nonadaptive controller which has parametric robustness and performance properties, and it can be made adaptive in two different ways-using either a Lyapunov methodology (tuning functions) or a modular input-output methodology with identifiers designed separately from the SG-controller.

10.6.2

y-Passive scheme

Consider the parametric y-model (10.37)

iJ =

€2

+ wT /J + c::! ,

(10.316)

where {2 and ware available and defined in (10.29) and (10.39), respectively. \Ve introduce the simple scalar observer (10.317) where error

KO

is a positive constant, and

as

defined in (10.213). The observer

f=y-:Q

(10.318)

Co

is

is governed by the system

.=

f

-

(Co + li'llIWI-'1) f + W T-8 + e2 .

(10.319)

The parameter update law is chosen to be

iJ = Proj {fWf} , bm

bm(O) sgnbm >

r=rT>o

'III

(10.320)

where the projection operator is employed to guarantee that Ibm(t)1 ~ <m > 0, \It 2: 0, (see Appendix E).

Lemma 10.18 Let the m.aximal inie71JaI of existence oj sol'utions oj (10.316), (10.317), and (10.320) be [0, tf). Then the following identifier properties hold:

(0)

Ibm (t)l2: (;m > 0, \It E [0, If)

(i) (ii)

9 E Loc[O, tf) f E '£:2[0, tf) n .coc[o, tf)

(10.321) (10.322) (10.323)

(iii)

wf,8 E £'2[0, tf) .

(10.324)

TIle above identifier is the same as t.he identifier used in Section 9.2. Therefore, the proof of Lemma 10.18 is identical to that of Lemma 9.3. Hmvever, the stability proof for the resulting error system is different from that of Theorem 9.4 because the SG-controller makes the proof of the following theorem considerably more involved.

Theorem 10.19 (y-Passive) All the signals in the closed-loop adaptive .'lystern consisting of the plant (10.1), the control law in Table 10.3, the filters in Table 10.1, the observer (10.317), and the update law (10.320) are globally uniformly bounded, and asymptotic tmcking is achieved:

t1!n! [lI(t) -

t/r(t)] = O.

(10.325)

Proof. The projection operator in Appendix E is locally Lipschitz, as stated in Lemma E.!. Therefore, as argued in the proof of Tlleorem 10.6, the solution of

10.6

475

MODULAR DESIGNS

the dosed-loop adaptive system exists and is unique on its ma.."\imum interval

of existence [0, t, ). Let us consider the systems (10.313) and (10.319), (10.326) (10.327) and define the signal

.

~

( = :; - IV£(8)f.

(10.328)

Equations (10.326), (10.327), and (10.328) yield the system

(. =

[. ] • 81l' (9) :. A::(8)( + A::(O) + col H'e;(O)e 8'0 Oe

+Q(z, t)'rO + 1l0 HT£(O)WT Wf • From Lemma 10.18,

9 E .coo [0, t,).

(10.329)

Since ~ are smooth functions of iJ when-

ever bm =F 0, and Lemma 10,18 guarantees t.I~at brn(t) 1= 0, then H'~, A: and

flJ/t

are bounded. Since, by Lelnma 10.18, we conclude that

f,O

E £2[0,t,) alld e E .coo[O,tJ),

• ] 8HT£(ti):. ~ 80 Oe = Ll E 4(0, t,). [A:(8) + col H'£(O)e -

(10.330)

Tlms, system (10.329) bccomes

( = A:(O)( + Ll + Q(z, t)T ~ +li.oHl~(O)WT ~ . E£2

f

(10.331)

E£:',!

Since A:(8) is e~llollential1y stable unifomlly in 0, we view (10.331) as a perturbed lineax time-varying system which cannot be destabilized by ~he square-integrable disturbance L 1 • So we focus our attention to Q(:;, t)'J'iJ + IiO HIE (O)w T we.

Claim 10.20 There e:r.istfYJ,nctionsp~ E £2[0,tl) andqt E L2[0,tf}, i,j = 1, ... , p, k = 0, ... , '11. + m, such that Q(z, t)T8 + li oHTE (8)w'l'we = B(t, s)[z]

+ G(t, s) [U!P-l)] ,

(10.332)

where the linea,' operators Hand G are n+nt

B(t, s) = {hij(t, s)} pxp ,

hij(t, s) =

{Oij(t,S)}pxp'

gij(t, s)

=

SN

(10.333)

r.; q~(t) 1\(S~B(8)

(10.334)

n+m

G(t, s) =

..

~ pf(t) ]\'(s)B(s) k

476

LINEAR SYSTEMS

The proof of t.hiR claim is leugt.hy but straightforward. It relies 011 tbe fact that: the O'i '5 are linear ill'll, 71, ,,\ and ti~p-l) and nonliuear only in 8. SubsLitut.ing (10.328) into (10.332) fmd then into (10.331), one obtains

O. The derivative of K/2/2 along the solutiolls of (10.337) with

; (~I(I~) ~ -col('!! + (TH(i, 8)[(1 + (T L2 ~

_ CO/(/2 2

+ ~/}I(l,8)[(]12 + .!..IL2 12 • Co

Co

(10.339)

Combining (10.330) with (10.338), we get

:, (1(12) ~

-(Co

-/1)Id. A proof of tllis implication \Va!; already given for the tuning functions design in Theorem 10.6 (cf. (10.124)(10.126)). The same argument is applieabJo ho1'O wit'.h (10.126) I"oplnccd by

Vm,i

=

_ 1 (i-I) ( - .- 0 because it can also guarantee sta.bility withont. adaptation.

10.6.3

x-Swapping scheme

Of thc two swapping ident.ifiers presented in Chapter 9, we choosc the :1'swapping ident.ifiel' beea.use it. uses only t.he K-filteJ.·s wir.h no additional swappillg Iill:ors. COllsider the parametriC' .1:-model obt.ained by substituting (10.12) into (HI.13): (10.3~lG)

In

HlP

first. row of (10.346),

(10.347)

all the signals are measul'ed c.xcept. for the bounded, c."{pollPutiaily deC-Dying e 1. The

(~predict.ioll

error" is hn plcment.ed as

(10.3-18)

478

LINEAR SYSTEMS

and its relationship with the parameter error is linear: (10.349)

The update law for

8 is either the gradient: bm (0) sgn bm > o, 11>0

(10.350)

or the least-squares:

(10.351)

f(O) = r(O)T > 0,

LJ

> 0,

where the projection operator is employed to guarantee that Ibm(t)1 ;::: C;m

>

0, Vt ~ O.

The proof of t.he following lemma is t,he same as that of Lemma 9.8.

Lemma 10.21 Let the ma:cima.l interval oj existence of solutions of (10.2) and (10.26)-(10.27) with either (10.350) OT (10.351) be [O,if)' Then the Jollowing identifier properties hold:

'm

(D) (i)

Ibm(t)1 ~ > 0, jj E £00[0, if)

('ii)

J1 + 1110 1-., E£2[O,tf)n£00[O,tf)

(10.354)

oE £2[0, if) n £00[0, tf).

(10,355)

vt E [O,tf)

f

(10.352) (10.353)

1

(iii)

In contrast t,o the y-passive identifier, the, normalization is employed here ill the update law. The normalization malees 8 not only square-integrable, but also bounded. It slows down the adaptation sufficiently so that the following stability result holds.

Theorem 10.22 (x-Swapping) All the signals in the dosed-loop adapti?le system consisting of the plant (10.1), the cont7'Dlla1u in Table 10.3, and the filters in Table 10.1, with either tile gradient (10.350) Dr the least-sq1J.o,res 1Jpdate law (10.(51), are globally u1J.ifonnly bounded, and asymptotic lra,eking is achieved:

lim [y(t) - Yr(f)] = O.

L-oo

(10.356)

10.6

479

MODULAR DES1GNS

Proof. The projection operator in Appendix E is locally Lipschitz, so the solution of the closed-loop adaptive system exists and is unique 011 its maximum interval of existence [0, t I)' In the proof of Theorem 9.9, we showed that

o} == -/;;}fh +W

(10.357)

I

which, along with (10.349) and (10.14), means that T-

i. = -kJf: + W

Let us

1l0W

(J

+ e2 -I"lel -

T:'

(10.358)

OJ ().

define the signal

, ~ z - HleUi)f: .

(10.359)

Combining (10.359) with (10.313) and (10.358), we get

, =

A:;(8)( + ~ + [Q(z,t)T + HlE

(9)nr].l. E~

A =LIE'c:I

+ {[A:(8} + col] Ttll£(D) - 8T1'".(8) •

EJ(J

e£CIC

o}.;·+ "10 ~

1

L

E£.:a

2 11

';1+ IIl nd' ,

J

... (10.360) where the £2 and £00 signal properties are established using Lenuna 10.2l.

Claim 10.23 There exist junctions p~ E £2[0, t/) and q~ E £2[0, tJ), 'i, j :: 1, ... , p, k 0, ... , n + m, 81tch that

=

(10.361) Inhere the linear operators Hand G are n+m

H(t,s)

{hij(t, 8)} pxp

,

G(t, s) = {gij(t, s)}pxp

,

::

"

sk

= k=O L p~(t) }.( )B( ) \8 S n+m .. it, 9i;(t, s) = E q~(t) J( )B( ) . k==O ('s S hij(t,s)

(10.362) (10.363)

The proof of this claim is omitted. With (10.361) and (10.359), syst.em (10.360) becomes ,

=

A=(8)( + Ll + B(t, s)[(J + R(t, s) [H/£(B)f]

+L2V1 + v\n1\2.

+ G(t, 8) [y~p-l)] (10.364)

480

LINEAR SYSTEMS

Since ii~p-l) is bounded and the coefficients in G(t: s} are square-integrable, theu Ll

( =

+ G(I,s) [y~P-l)] ~ L:t E £2[0,I'f)'

Thus (10.364) becomes

A~(O)(+L:~+H(t,s)[(l+H(l,s) [We(O).j1 +•vlfl •.jl + V1fl IP] 1\-

+L'l.Jl + v/!1d 2 • ~

\Ve not.e that lVe-(8}

{l

(10.365) E

E

.coo (0 tf), which implies that, there exists 1

1 + v(01/2 a function BI E .coo(O,t,) and functions r~ E 0, ... ,71

+ m, such that

H(tt s) [W.(Ii)

.c [0,tJ), 2

i,j

= 1, ... ,p, ~: =

J1 + 0 because it can also guarantee stability without adaptation. Our presentation in this subsection was with normalized update laws. It is also possible, however, to prove the result of Theorem 10.22 with the unnormalized least-squa7"es update law. T!Je proof exploits the properties of the unnormalized least-squares algorithm: iJ E £1, f E £2 (see, e.g., [157]).

10.6.4

Comparison: modular vs. tuning functions design

Even though the tuning functions design and the modular design both nse the backstepping approach with the same underlying nonadaptive controller, they result in fundamentally different adaptive schemes. The tuning functions scheme is designed using a single Lyapunov function, and results in a simple and direct stability analysis. The stability analysis of modular schemes is far more involved. Even though one can derive transient performance bounds for the modular schemes, they are neither as simple nor as insightful as the tuning functions performance bounds. 'Ve now illustrate the difference in performance behavior between the tuning functions and modular schemes on the example introduced in Section 10.2.4. Of the two modular schemes, we present simulations only for the x-swapping scheme. The responses with the y-p8SBive scheme are qualitatively

10.6

=1

483

MODULAR DESIGNS

O'sD:

ZIO.s~

-O.S

0

'lJ,

~J=

2

3

4

5

i

I

i

i

2

3

4

5

-t::: 1

0

2

,

i

i

4

S

tuning functions (,.

5

3

4

i

i

3

4

S

~jt i

0

ii

3

2

0

u

0

a

-0.5

2

-b 0

= 0.5)

1

2

3

x-swapping (g

I

i

4

5

= 500)

Figure 10.12: Comparison of responses with the tuning functions and the xswa.pping schemes. While the tuning functions update la\v (driven by the tracking error Zt) monotonically reduces the parameter error at the x-swapping update law generates an 'overshoot' in iit whicb results in a. considerably higher peak in the response of the control u. similar. The SO-controller used in the modular schemes is obtained by' setting "'( = 0 in the tuning functions controller (10.167)-(10.176). The x-swapping scheme employs tbe update law !

'::::1 11 f

a = -g-I-+-=-=-=2- , .... ].1

£

= y-

~1

+ a':: 1•1 -

'Vt •

(10.379)

For both schemes we use the design parameter values Cl = C2 = ('3 = 1, d 1 = d2 = da = 1, h:1 = 6, k'J. = 12t and k3 = 8. A comparison of responses with the tuning functions and the :z:-swapping schemes is given in Figure 10.12. While the tuning functions update la\v (driven by the tracking error %1) reduces tbe parameter error a almost monotonically, the x-swappillg update law generates an 'overshoot I in iit whicb results in a considerably higher peak in the control u. It is important to stress that the responses given in Figure 10.12 are not the best possible with eitber of the schemes. They are only chosen to illustrate a fundamental difference in transient properties between the tuning functions and modular scbemes. Even though the modular schemes may not have performance properties as strong as the tuning functions scheme, tbey are simpler to design and offer flexibility ill the selection of an update law.

LINEAR

SYSTElvlS

\Vhile tilE' rnodular schemes presentecl in this cbapter have a lot in common with the certaiut;y equivalence adaptive linear schelnes, the.y renlOve soniE' of tile shortcOlnings of the cf'rtainty equivalence sciIelnes. As shown in Section 10.3.21 t.heir underlying nonadapt.ive controller can achieve s1:(lbilir,y even without adaptation, and t.heir dcsign paramci.ers can bc uscd for systematic improvelllcnt. of llonadapt:ive performance.

10.7

S UITll1.1.ary

\Ve have presented two classes of adaptive designs for lincar systellls: t.uning functions and modular. These clesigns have the same underlying nonadaptive linear controller based on backstepping. This nonadaptivc con1Toller ca.n guarantee st.abilit.y without adaptation when a b01lnd on the pararnetric uncert.ainty is known 1 and in aclcli hon l achieve a prescribed level of tracking pcrfonllililce. This is an improvement over tradit.ionnl adaptive designs which cannol: gllarant.ee stability w}1 0;

>

• globally uniformly stable, if (A.:1) 1.8 satisfied with 'Y E Koo fOT any initial

state .'I:(to); • globally uniformly asympt.otically st.able, if {A ...I} is so.tisfted with j3 E K£oo for an1J initia.l state :c(l.o); and • globally exponentially stable, if (A ...1) is satisfied for any initi.al state .1:(to) and wilh {3(1", s) = k,'e- OB , k > 0, a: > O. The main Lyapul10v stability theorem is then formulated as follows (see, for example, [81, Theorem 4.1, Coronaries 4.1 and 4.2]):

Theorem A.5 (Uniform Stability) Let x = 0 be an equilibrium point of {A.1} and D = {x E lRn Ilxl < ·r. Let V : D X lRD --+ ~ be a continuously differentiable junction such that 'Vt ~ 0, 'Vx E D, (A.5) (A.6)

Then the equilibrium x = 0 is • uniformly stable, if 'Y] and 'Y2 a·re class IC functions on [0, r) and 'Y3 (,) ~ 0

on [0, r); • uniformly asymptotically stable, if 1'11 'Y2 and 'Y3 are class JC ju,nctions

on [O,T); • e.'\."Ponentially stable, if 'Yi(P) = kiPQ on [0,7"), ki

> 0,

ll'

> 0, i = 1,2,3;

• globally uniformly stable, if D = IRn, 'Yl and 'Y2 are class /Coo functions,

o.nd 'Ya(') ~

a on IR+;

491

LVAPUNOV STABILITY AND CONVERGENCE

• globally uniformly asymptotically stable, if D = n" t "il and 12 are class A.oo functions, and 1'3 is a etas,., JC function on 1R;..; and

= n"

• globally e>"''']lonentially stable, if D kj > 0, o· > 0, i 1,2,3.

=

and 7i(P)

= kiPQ

on lR+1

In general, 01.11' goal is to achieve convergence to a set. For time-invariant systems, the main convergence tool is LaSalle's Invariance Theorem (Theorem 2.2). For time-varying systems, a more refined tool was developed by LaSalle [110j and Yoshizawa [2011. For pedagogical reasons, we will introduce it via fl technical lemma due to Barbalat [155]. These key results and their proofs are of utmost importa.nce ill guaranteeing that all adaptive system wi1l fulfill its tracking task.

Lemma A.6 (Barbalat) Consi(ler the function tP : R+ --i' lR. If cP is 1/.nifO'lm11J continuous and lim,_oo f;o q,(T)dT exists and is finite, then lim q,(t) t-oo

= O.

(A.7)

Proof. Suppose that (A.7) does not hold; that is, either the limit does not e..-ust or it is not equal to zero. Then there exists e > 0 such that for every T > 0 one can find tt ~ T with 14t(tdl > e. Since q, is ulliformly continuous, there is a positive constant r5(e) such that Iq,(t) - ,p(t})1 < £/2 for a11 tl 2: 0 and all t such that It - tIl ~ b(e), Hence, for all t E (tll tl + 6(e)], we have 1q,(t)1 =

Iq,(t) - ¢(tl) + ,p(tl)1

2: f4>(tdl-\q,(t) - 4>{l1)\ > which ilDplies that

/1

'1+6'(£)

e

£

(A.B)

£-2=2'

I

q,(T)dT =

1"1+6(£)

'1

il

eo(c) 1q,(T)/dT>-, 2

(A.9)

where the first equality balds since 4>(t) does not change sign all [tlJ tl + bee)). Noting that fJl+ 8(S) q,(T)dT = f~l q,(T)dT + .r,'11+6(E) q,(T)dT, we conclude that t/J(T)dr cannot converge to a finite limit as t ~ 00, which contradicts the assumption of the lemma. Thus, Hmt-IX! ,pet) = o. 0

fJ

Corollary A.7 Conside7' the function q, E L.p for some. p E [1, (0), then

4> : lR.t

lim q,(t) =

t-oo

o.

--I-

JR. If,pl ~ E £00' and (A.I0)

492

ApPENDIX A

Theorem A.S (LaSalle-Yoshizawa) Let x = 0 be an equilibrium. point of (A.i) antll:11lppOSe J is locally Lipschitz in :r. 'lLnij'ormI1J in t. Let V : 1Rn x R+ -+ lR+ be a continuO!tsly diffe7Y:!nliable j·u11.ction such ULat 11 (Ixl) ~ V(:r, t) ~ "2(lxl) «

11' =

(A.II)

av 81' t) < -H'(1') < 0 -at + -j(x ax «, "-

(A.I2)

~ 0, T/:l: E JR.'I, 11Jherc 11 an.d 1'2 are class /Coo junction/; and 1-1' is a COfl.I.inuous Junction. Th,en, all solll.tions oj (A.l) a1'e globally unijo1.,nly bounded and sa.ti,r;jlJ lim Tl'(:r(t)) = 0 . (A.I3)

Vt

t-oo

In addition, ij T-l'(x) is posili1Je definile, then the equilibt'iu.m. J; = 0 is globull1J 1tnijormly asymptotically stahle, Proof. Since l:r ~ 0, V' is Ilollincreasing. Thus. ill view of the first inequality in (A,ll), we conclude that :r: is globally ulliformly bounded, that is, 1:1:(l) 1 ~ B, Vt ~ O. Since 11'(x(t,), t.) is nonincrellsillg and bounded from below by zero, we conclude t.lu·..t. it has n limit VIX) as t -+ 00. Integra.ting (A.12), WE' have

lim

1,

t-oo In

I-t'(x(r»dr

l' ll'(:r:(r}, .

~

- lim

=

L-·10

=

V(:r.(io}, to) - VIXl ,

1-00 to

r)dr

lim {'i(:z:(tO)' to) - V(x(t),

tn (A.I4)

which means that ftC: H'(x(r»dr e.'Cists and is finite. Now we snow that l-l'(x(t)} is also unifOl'lllly continuous. Since Ix(t)1 ~ Band j is locally Lipschitz in x uniformly in i, we sec that for allY t ~ to ;::: 0,

Ix{l) - x(to)1 =

11~ j(:r(r), r)drl ~ L

l:

Ix(r)ldr

~

LBlt - tol , where L is the Lipschitz constant of fall {Ix 1 ~ B}. Choosil1g tS(e)

(A,I5)

= tB' we

have

la:(t) - x(to)1
O. Note that. the ratio between the bounds (B.22) and (B.18), and the ratio between the bounds (B.23) and (B.19), are of order 11/1111 when IIl111t ~ 00. 0 because rr < (1

For cases where II and l2 are functions of time j:ho.t converge to zero but are not in Cp for any }J E [I, 00) we have the following lemma.

Lemnla B.B Con.sider the differentia" ineqllality v{O)

= 1'0 ~ 0,

(B.24)

c> 0 and TO ~ 0 a1"f constants, and /31 anll f3'J. are class IC£ functions. Then there e:r.ist.i a cla,ss ICC function f3v and a class IC Junction "I,. such that

whe7'i~

1J(t) ~ f3v(vo

+ "a, t) + l'v(P),

'tit ~ O.

ltforeo Vf7', if f3i(7', t) = 0'; (l')e-O',1 , i = 1,2, where O'j E K. and Uj the,.,~ exisls lXv E IC and U v > 0 such that f3v{1', t) = O:v(1")e-D'II/.

(B.25)

> 0, then

497

INPUT-OU'l'PUT S'rABILI·.ry

Proof. Vve start by introducing 'U = v -

~ and rewriting

(B.24) as

e

~ ::; -[c - PI {l'O, t»)ii + p1 (1'0, t) + fi:!{1'O, t} . C

(B.26)

It then follows that

'l'{t,) ::; '110tJ;rPJ(rOJ8)-clds +

t [!!./31 (1'0, 'T) + /32 (J'u, 'T}] c.f;18J(l'o,JI)-c],lSdT + !!.. . c r.

10

(B.27)

"Ve l'ecall the standard result (see, for example, [81, Lemma 4.6J):

(B.28) where ~: is a positive, continuous, increasing function. To get an est.imate of the overshoot coefficient k(ro), we provide a proof of (B.28). For each c there c."'{ists a class JC function Tc : R+ -+- lI4 such that

(D.29) Therefore, for 0 ::; 'T $ Tc(1'0) $ i, we have

so the ovm'shoot coefficient in (B.28) is given by

k(ro) ~ eTc(ro).t:JI(ru.O} •

(B.3l)

For the other two cases, t $ Tc(ro) and Tc(1'O) $ r, getting (B.28) with k(ro) as in (B.31) is immediate. Nmv substituting (B.28) into (B.27), we get

v{t) ::; Vok(1'o)e-;1

+ ~~(ro) lot [~/31 (1'0, T) + /32 (ro, 'T)] e-I(t-'T}dr + ~.

(B.32)

To complete the proof, we show that a class JC£ function /3 ('onvo]ved with an exponentially decaying kernel is bounded by another class IC£ fUllction:

498

ApPENDIX

B

Yl

Figure B.l: Feedback connection.

Thus, (B.32) becomes

vet)

~

k(1"o) {[vo + ~~,81 (7'010) + ~P2(1'O' 0)]

+ 2~ PI (ro C'"

l

t/2) +

e-~L

~8.!(1·OI t/2)} + e . C C

(B.34)

By applying Young's inequalit.y to the terms k(7'o)~,Bl(ro, O)e- 1' and ":(7'a)~PI (rol t/2), we obtain (B.25) with

=

k(r)

{[7' + k(:) pt(r, 0)2 + ~,82(7'I 0)] e-it c c~

k(r) ,81(1', t/2}2 + :'(j,.z(r, ? +-.-) t/2) } c,'" c

{B.35} (B.36)

The last statement of the lemma is immediate by substitution into (B.35). 0 All the above results describe input-output properties of individual syst.ems. To prepare for a basic result on feedback connections of systems, we first give a definition of L.p stability.

Definition B.9 A mapping H : £p,e exist finite positive numbe7's l' and u E £p,e and all t E [0, 00 ).

f3

~ L.p,e is said to be L.p stable if the7'e such that U(HU}Lll p :5 'YIiULli p + P Jor all

The following, theorem known as the small gain theorem [28] I gives sufficient conditions for L.p stability of the feedback connection in Figure B.l.

Theorem B .10 Consider the s;IJstem in FigU7'e B.1. Let H 11 H2 : L.p,c ~ C.P,£I P E (1,001, be two L.p,c slable operat07'S wit), finite gains 1'1, 'Y!! and asSDciated constants f31,{32 • Let the operator HIH2 be stlictly ca1Jsal. IJ (B.37)

499

INPUT-OUTPUT STABILITY

lI'uull" :::;

1-

lIu:ullp :::;

1

1 ')')')'2

1

- '11'2

(livull" + 121\ v2fllp + 1'2 + '/2{jl)

(B.3S)

(lI l12Jllp+"YJII Vull,.+,BJ

(B.go)

+1'1P2)

for all t E {O,oo). II, in addition, VhV2 E £p, th.en (B.38)-(B ..99) /told with all subscripl.s t dropped, which ifllplie.r; that lJ.1r Ilz! Yh and 1/2 have bounded L..

"

norms.

Now we give a version of Gronwall's lemma. For the proof, the readpl' is referred to lSI]. Similar lemmns can be found, among other references, ill [28, 81, 165].

Lemma B.II (Gronwall) Consider the contirlU01J.S functions A: lR+ -+ JR., --. lI4, and II : 1R+ --. ~, where J.l- and II are also nonnegat.ivf. If a co'n.tinflOUB function y : 1R+ --+ If{ sa.tis./ies the inequality

p. : 114

y(t) ::; A(t) + IJ{t)

t

ltD

(BAO)

11(.9)y(s)ds,

then 'TIt In particular,

if A(t) == A is

a constant and jJ.{t)

\.

J:

1J (t) • < _ ",e

0

V(-r)dT

•

== 0,

~

to

~

O.

(BAl)

then

'tit ?::. to ?::. o.

(B.42)

Appendix C Input-to-State Stability Input-to-state stability introduced by Sontag [173] plays a cruciaJ role in our modular adaptive nonlinear d~signs. We e~..t.end Sontag's definition to timevarying syst.em.c;:

Definition C.l The sy.IIlem .1: ::::: f(t, :J:, u)

(C.l)

I

1.IJhere f LIJ piecewise conlin1J.ou.'l in t and IDcally Lipschitz in .1: and 1I, i.s said to be input-to-state stabJe (ISS) if there e.a:ist a class IC£ f1J,Tlction t3 and (l clas,fj K:, function -y, such that, ID1' any :c(O) a.nd for any input u(.) cDl1tinnow; anti bounded on [0,00) I,he solu.lion exists fo'l' aU t ~ 0 a.nd sal,isji.es

l:v(t)1 ::; ,B(lx(ta)/, t - to) + I'

(sup "U(T)I)

(C.2)

fo5;T:st

for all to a.nd t such t.hnl. 0 ::; to ::; t. The following theorem estabHshes t.he connection betwf"ell the existence of a Ly8~)UnD\'-like function and the input-t.o-state stability.

Theorem C.2 [173, Claim on p. 441] Suppose that f07' flle slpdem (C.l) there ea:ist8 a C 1 Junction \1' : ~ x lR,R -0- 114 such I.hat jor all .1' E lRn and tl

E

m.m,

where 'j'l, i'!!, and p a71~ cla,,,,, K.o:J Junctions anrl1'3 is a cla,c;,'l1C function. Then the system (C. I) is ISS with l' = I'll 0 "(2. 0 p.

502

ApPENDIX C

Proof (Outline). If :r(to) is in the set (C.5)

then x(t) remains within the set

St" =

{x E JR" 1.,,1 :'0 1'1' 01',0 P (~~r.IIl(T)I) }

(C.6)

1

for an f. ~ to. Define B = [to! T) as the timc interval before :z:(t) enters Rto for the first t.ime. In vicw of the definition of Rio! we have

'Vt E B.

(C.7)

Thcll, by [173, Lemma G.1], there exists a class K£ function /3" such that

'Vt E B,

(C.B)

which implies

'tit E B.

(C.g)

On the ot.her hand, by (C.G), we conclude

Then, by (C.g) and (C.IO),

1·1:(t) I :5 /3(I."v(to)I, t. - to) + "( (sup 1u.(T)I)

'tit

I

1"~to

'2:. to '2:. O.

(C.II)

By causality, it follows that

Ix(t)1 $ P(I·l:(to)l, t - to)

+ I' (sup

'o$r9

IU(T)I)

I

'Vt ? to

~

o.

(C.12)

o A function l' satisfYing conditions of Theorem C.2 is called an ISSLyapunov function, Sontag, Wang, and Lin recently proved that the inverse of Theorem C.2 is also true. They also introduced an equivalent dissipativitytype characterization of ISS.

503

INPUT-TO-STATE STABILITY

Theorem C.3 [115, 177] For the system

x = f(x,u) , the Jollo'wing properties are equi'tJa.lent: 1. the system is ISS, 2. there exist,rJ a sm.ooth ISS-L,/apunov function, 9, tl&e1-e exists a smooth posiUve definite 'radially unbounded function 1; and class ICoo functions Pl and P'l such that the following dissipativity inequalit7/ is satisfied:

The following lemma establishes a useful property tbat a cascade of two

ISS systems is itself ISS.

Lemma C.4 Suppose that in the system Xl = !I(t,xl,X2,U)

X2

=

h(t, X2, 'u)

the Xl -subsystem is ISS with respect to with respect to lJ" that is,

3:2

(C.l3) (C.14)

and 11., and the X2-subsystC'ITI, is ISS

(~~~l {lX,(T) I + IU(TlI})

1",(t)1

$

.8,(lXI(8)1. t - s) + '11

IX2(t)1

~

.82(1x2(8)\, t - s} + 1'2 (sup IU(T)I) , Ii~T!;t

(C.15) (C.16)

where /31 and fJ2 are class ICC. junctions and 1'1 and 12 arc cla,9s JC fUTtdions. Then the complete x = (Xl,X2)-system is ISS with

Ix(t)\ :$ ,8(lx(s}I, t - s) + 1

(sup IU(T)I) ,

(C.17)

S'5T~t

where (j(r, t) =

"(7')

=

.8. (2,81 (r, t/2) + 271 (2{j2(r, +1'1 (2{j2(r, t/2)) + f3J(r, t)

a», t/2} 1

(C,18)

tJl(2"1(2')'2(7') + 21'), 0) + ')'1(2'Y2(~") + 21") + '"Y2(r), (C.19)

504

ApPENDIX

Proof. With (8, t)

= (l/2, t), (C.I5) is rewritten as

~ 131 (/.1!I(tj2)f, t/2) + "'It ( tJ'J$T$,t sup {lx!!(T)1 + /1/(T)/}) •

IXt(t)l

C

(C.20)

From (C.lo) we have

,ft.~~SI 1·:'(r) I




(D.9)

(VrlJlr/a

>

(D.10)

.hl/' Unv, Jl o

Adding inequalities (D.9) and (D.10), we obtain (D.ll) where the storage fUllctioll F and the dissipatiou rate :r-s~rstem are defined as

F(t, .1')

I/{J:)

'I' for

the C'LHnplE'te

+ V2(:C2:t) 1/ 1(;l'd + 1!':.!(·1:2).

(D.12)

Fi(:rj,t)

(D.l:3)

1

Since V is positive definite, radially unbounded and decreseent, and I/; is positive definite, this proves the strict passivity. If at. least; one of the systems EI and is passive but not. strictly passive, thell its dissipation rate 1/'1 is at best: positive selnidefinite but. lIot positive definite, and the ovt'rall s~rsl('nl is onl,v passive. Finally, when 'L't 0, if::S l is strictly passive and 2:::2 is passive, then 'I/!':!, is positivE' scmjrlefinitc, and b~T differentiating (D.ll) we get

(D.14) Thus, by Theorem 2.1, :r = () is globa.lly uniformly stable al1d lim,_c:c :/'j (t)

o.

o Now we turn our attention to linear time-invariant passive sysi:ellls,

Definition D.S A Hlt'iO'll.nl t"'(l1I.~rCl' IuncUon 0(.5) 'is sai.d to be positive real if 0(8) is Teal for all Tcal8, (/TIlPRe{G(.9)} 2: a for all!}ce{s} 2: o. fl, in addition., G(8 - f) is positive. real fO'l' some [ > 0 then G(8) is 8ahl to be sLridly positive 1

real. For eompldeness, we quote the celebrated Popov-Kalman-Yakubovich lelllma. A recent \'('rsion of its proof can be found ill Tao and Ioannoll (183].

Lenllna D.6 LeJ the strictly posit,iuc Tea.l iraTl.sjc'!' jllnci.io7/. G( s) have the slale-space l'epreseTdation (A, h, c, d), d 2: O. Then . fOT {InlJ given L L'1' > 0, til C1'C e:t:ists (J, scalar 1) > 0, a Vf;cf,l)'l' q, an.d {J, P = pT > 0 81/.ch til at ATp+PA = Pb-

-ql/I'-IJL

= qV2d.

(D,15) (D.16)

510

ApPENDIX

D

Witb this lemma a. Lyupunov function l' = x T Px can be constructed such that P satisfies 110t only the Lyapullov equation (0.15) but also the inputoutput condition (D.IS) from which the restriction to rela.tive degree zero (d > 0) or one (d = 0, cb > 0) is apparent. The maiu utility of tIns special Lyapunov function for adaptive and c8Bcade designs is that the indefinite term in its derivative l' depends on the output y and not 011 the whole state :1:.

Appendix E Parameter Projection The modular adaptive controllers have a point. of singularity bill = 0, where bm is the estimate of the high-frequcncy gain (virtual control coefficient) bnl" In order to prevcnt bm from taking the value zero, we use the parameter projection in our identifiers. For this, we need to know the sign of the actual highfrequency gain bm • We first give a treatment of projection for a general ('onvex parameter set and then specialize to the case where only the high-frequency gain is constrained. Let us define the following convex set

n=

I

{{) E RP P(8)

:5

o} ,

(E.1)

where by assuming that the convex function P : IRP --+ rn. is smooth, we assure o that the boundary an of n is smooth. Let. us denote the interior of n by n and observe that VoP repl'esents an outward normal vector at 8 E an. The standard projection operator is

I

T,

ProHT} =

(

VfJ PT ) 1- r \/.P1T'V _p

(E.2)

ViJP o

T,

0

where r belongs to the set g of all positive definite symmetric p x p matrices. Although Proj is a function of three arguments, T, {) and r, for compactness of notation we write only Proj {T}. The meaning of (E.2) is that, when {) js in the int.erior of n or at the boundary with T pointing inward, then Proj{T} = T. When {) is at the boundary with T pointing outward, then Proj projects T on the hyperplane tangent to 811 at 6. In general, the mapping (E.2) is discontinuous. This is undesirable for two reasons. First, the discontinuity repl'esents a difficulty for implementation in continuolls time. Second, since the Lipschitz continuity is violated, we cannot

512

ApPENDIX

E

usc standard theorems for e.\':istellce of solutions. Therefore, we need to smooth the projection operator. Let us consider the following COIlVe.\,: set

rr~ = {(j E

lR,P

I1'(B) ::;

(E.3)

E} ,

which is a union of the set. TI and an Q(e)-boundary layer around it. We now modify (E.2) to achieve continuity of the t.ransition from the vector field T on V·'p 't'-pT) r V;1'rr~Ii'P r

the boundary of IT to the vector field ( 1 -

on t.he boundary of

II!":

ProHT}

=

[(T'

_

V-PV-P'I") J - c(O)f V :p'I'r~ 07'

T,

c(O) = min { 1, P~iJ)} _

(E.5)

It is helpful to not.e that. c(8TI) = 0 and c(aTI E ) = 1. In the proofs of stability of identifiers we need the following technical properties of the projection operator (EA).

Lemma E.l (Projection Operator) The following are the p1"Ope1·ties oj the pl'Ojcction ol}eralo'f' (E.4): (i) The mapping Proj : lRP x lIe: x Q -+ JRP is locally Lipschitz in its a7'guments r, 0, r.

VO E TIE'

(ii) ProHr}'1'r- 1 proj{r}:5 rTr-1r,

(iii) Let r(t), ret) be continuously differentiable and

8=

Proj{r},

Then, on its domain oj definition, the solution (jet) remains in TI~.

(iv) -B'rr- I Proj{ r} ::; -o'rr-Ir,

VB E n~,f~ E TI.

Proof. (i) The proof o[ this point is lengthy but straightforward. The reader is referred to [157, Lemma (103)]. (ii) For {} or VfJp'r r ::; 0, we have Proj{.,.} = rand (ii) trivially holds

En

with equality, Othenvise, a direct computation gives

~ (v apTT)

Proj{r}Tr- I Pro.i{r} = rTr-Ir _ 2c(O)

0

vfJPTrviJp

IV-Pv-pTr I

2

2 A

+ C(O)2

0 0 r (Vo pTrVOp)2 2

=

r

T

r

-1

(VOpTr) r - c(O) 2 - c(O) VopTrvoP •

(

.)

(E.6)

513

PARAMETER PROJECTION •

~

a

where the last inequality (ollows by notillg that c(6) E [0,1] for 6 E TIE:\ n. (iii) Using the defillitioll of t,he Proj operator, we get

iJ

V,/pTT, VopTproj{r}

=

{

err

or V;/PTT:5 0

(E.7)

c(iJ») Vjj1'

(1 -

T

iJ

T,

E

IIE\ IT and ViJ1"J'T > 0,

which, in view of the fact that c(O} E [0, 1) for

ii, implies that 8 E an£ ,

iJ E TIe: \

V o1'T Proj{ T} :5 0 whenever

(E.B)

that. is, the vector Proj{ T} either points inside lls or is tangential to the hyperplane of ane: at Since O(O} E lli:' it follows that O(t) E TIt: as long as the solut.ion exists. 0 0 (ivJ For (J En, (iv) trivially holds with equality. For 8 E lli! \ sincE' 8 E n and P is a convex [unctiol1, we have

e.

A

n.

A

(8 - 8)TVoP :5 0 whenever

0 E II!\ Ii .

(E.g)

\¥ith (E.g) we now calculate

- 8Tr- 1 Pl'oj{r} =

-oTr-IT

! 0,

+

Q

4

c(8} (Ol'vIP),eVi/pT.,.)

vo'pJ rv,,"

(J E

,

TIE \ II

V;/pTT >

~tnd

a

(E,lO)

o

which completes the proof.

Since we intend to use the projection operator only to keep the estimate of the high-frequency gain bm from becomiIlg zero, we now specialize the projeetioll operat:or for t.his l'1:l.fie. We assume that Ibm l ; :;: C;m > 0, where sgn b", and C;m ate known. Recalling that bm is the first element of the parameter veetor 8, i.e.~ 8 [bRl I O!;!, ••• ) 8,l]T, we define P(6) C;1I1 - bm sgnb m and 11ol.e that VoP = -sgnbmei'. Let us denote the nominal vector field fOl' the parameter update la~v by T = [Tl' T:h ••• ,1j,]T and choose E E (0, C;m). The updatp law of

brn

=

tbe form

8=

=

Proj{'T} using the projection operat.or (EA)-(E.5) is given hy

! ! 1,

bm

=

Tl

a,X rnao

{a,

m }. E-c;m+b,:SgUb .. .

A

bm sgn bnI

_

(E.Il)

an d TI sgn I)m

bm sgn b", > C;m

Ti,

8i = . Ti -

< ~I!I

{l

. 7"1 ill r] I ll11n,

C;,JI-b",SgUb ... } E

b b < ' I l l sgll m _ 4

~m

or

Tt

T6] - T[H1T]B E £2 is a.lso referred t.o as Swapping Lemma. Our next lemma is a nonlinear time-varying genel'~uiz(Ltioll of this result.

Lemma F.4 Consider syste'ms (F.l}-(F.9) willi. the sam.e set of (lSs1l1!,ptions as in Lem.ma F.l. Jilm-the''71W1'e, assume that:; E £00 and e E £2- 11 jj E £2t then 111 - Y2 0 E £'! .

(F.14)

If 0 E £2 n £001 then

(F.15)

Proof. Since z E £001 then gH!T E £00 and Q E £00' Due 1:0 the eXpOllf.'lltial stability of ;.1{=,1), it fo]]ows t.hat E £00' \¥e need t.o prove that Ya E £2 and 11£ E £2. The solution of (F.3) is

n

(F.16)

518

ApPENDIX F

where (FA)-(F.5) guarantee that the state transition matrix «}): : :IJ.=4 x R+ --t is such that IIP:(t, r)/2 :S ke-o(f-T), k, a: > O. Since nand Q are bounded, then

m.nxn

I'!/J(t) I
ressed as 90

_

a

= {3{X}7j Xp

90=

I

a

8 Xp '

(G.35)

We now show by induction that conditions (iii) and (iv) imply that XmI+11 ••• , Xn can be replaced by Ilew coordinates x m1 +11 ... I.'1'n such that in the x-coordinates (with 3:1 = XIt ... , X'nlI = Xm I ' xr = Xr) t.he system (G.34) takes 011 the form {G.32} and, moreover, the vector fields ad~)go, 0 ~ i =::; p - mIl become

.

adinOo =

, a

(_1)1~ UXp_i

1

0~i

~

p - ntl.

(G.36)

• First induction step (i = p): In the coordinates of (G.34) the vector field adinDo is expressed as

(G.3T)

Then, the conditions [Yo, adlilDo]

= 0 and [go,90] =

0 imply that

(G.38)

(G.39) Hence, {1(.} is independent of Xp and the function 1lC\:) can be expressed as (GAO)

530

ApPENDIX

Let us then define tbe new coordinate Xp

= 'Xp -)0r

Xp - 1

G

X'p as

Jl.:1(Xl,"" Xp-21 s, ;{)ds

A == X" - P(XI, .. ' Xp-l, X'). (G.41) t

In the coordinates (Xh« .. , XP-l: XPt ){)~ the system (G.31) becomes

:\:i

=

Xi+1,

Xp-I

=

Xp + j2(Xl,« .. , Xp_I,X r)

:\p

=

.£1,1 (Xlt ... ,Xp-lt Xr)

it

=

tI>O(:Xll ' • Aq, x'}

Y

=

Xl,

1 $. i $. p - 2

+!3(Xll'

« •

,'Xp-l, xt)u

(GA2)

a

(G,43)

« ,

Do, ad/Duo become

and tIle vector fields

- = -_a axp

go

ad10Yo = - - - . aXp-l

I

:s

• Induction hypothesis ('i = k + 1, m 1 + 1 k ~ p - 1): Assume that we bave replaced Xk+I"" Xp by new coordinates ele+1,«'" p such that in the (Xli' .. ,Xk,';"+b'" '';Pl xr}-caardinates tbe system (G.31) is expressed as

=

Xj XL:

ek+l =

e

1$;j~k-1

Xi+ 1 ,

e"'-+l + VIt(;\:l~ , •. ,XJ.:, X') el.:+2

+ Vk+l (Xl, ...

I

Akl

X') (G.44)

fop =

l/p(X) , ' .. , Xk, A.r)

Xr

=

(lO(XI, ' .. I Xql X')

Y

=

Xl,

+ P('tJ, ... ,Xk, Xf)U

and, moreover, the vector fields ad}oUo, 0 $. j ~ p - k, are expressed as

fJ adi 9a == (_l)i_ _ 10

,

a~p_j

05 j

~ p-

ad P-

k -1,

k

/0

7'

= (-l)P-k~ 8Xk"

(G 45)

• Induction p'l'OoJ (i = k, ffll + 1 ::; k ::; p - 1): In the coordinates of (G.44), tbe vector field adi;'=+llio is expl'essed as

adj;k+llio

= [/0

~

i

ad'};kgo J

a

-l

E Xj+l 8 AJ,. + (~k+l + lit) '=1

+11 -

vAk

p-I

+

E

{)

({j+l

j=J.'+l

+ Vi)

£\t; .

V~J

a + L ()O'-, a (_l}P-k_ a] i=p+l J ax.j aXk

pa~p

=

a A.,

(_l)p-l-+l

n

(_8_ + 0"X.k-l

Bilk

~+

OAk 8Xk

t

j=k+l

8Vj~). 0XI: fJ{j

(G.46)

531

DIFFBRBNTIAL GEOMETRIC CONDITIONS

Then l the conditions [adj;kjjo 1 ad~;k+lgo] = 0 and rOOt that a'll)'

~ 8 XI;

= 0 :::} IIj(Xh .. ·

1

adj;kgoJ =

0 imply

Xk, XT) = lIi,J{:\:h ..• I AJ.·_l,Xr ) +lIj,2(Xl," . ~ XI:-l, Xr)Xk, k'5j '5P (G.47)

tJ{3 -8 (:\1, .. , I XI.·, :{) = XI.-

o.

(GAB)

Let us then define the new coordinates (1:1"" {p as /Xk-I

(k

:=

~

:=

Xk-10

~k-I

eJ-.10r

IIk,2(;\:1,. Vit 2(:\'I,

Of

.'Ic-'l, s, xT)ds

,) •

... 'Ak-!!I s, x')ds,

6 = XI: -

iik-l(Xh ... , Xl-It X')

(GA9)

I~ + 1 '5 j '5 n .

III the coordillates (XJ,' .. ,Xk-J I (kt' .. I {PI X'), the syst.em (G.31) is expressed as

Xi

=

:\:;+1,

1'5j'5~~-2

T :h'-l = {It + Vk-J (\:1, .. - , Xk-ll X ) (k = {1'+1 + Vk(XI, ... Xk-l, XT) I

(G.50)

ep = X'

vpCn, ... ,Xk-lt X + ,8h~ll ... f

)

Xl-I, Xf)71

= tPO(Xl \ •.. 1 XIJ1 X'}

Y =

Xl,

and, moreover. the vector fields ad}ooot 0 adi10 90

I

= (-lV--!} a!~ . ~P-J

I

0

'5 j '5 P - A: + I, become

< J. < __IP-k+J- = - P - k , tLU 10 go -

8_ (_1)P-k+l_

8 '

:\:1.'-1

. (051) .

o The necessity is again straightforward. The COllditions of Proposition G.3 are necessary and sufficien1: on1), fot" the local existence of a. diffeomorphism tral1sforming (G.31) into (G.32). At this time there are 110 necessary and sufficient conditions [or the global existence of such a diffeomorphism. Of course, the global validity of conditions (i)-(iv) of Proposition G.3 is necessal)" as are the completeness of the \'ector £elds adJp 90, 0 :::; i :5 p - 7111 - 1, and tile connectedness of the manifold ill = {( E lR": h«() =L/oh(C.) = ... = L,/;lh(() = O}, as proved in [13J. Therefore, to formulate the cOlluterpart of Theorem G.2, we make tbe following assumption: Assumption G.4 Tile system. (G.31) can be transformed via a global diffeomorphism x = 4>«() into (G.9S).

532

ApPENDIX G

Theorem G.5 Under Assumption G.4: the s1/.dem (G.l) can be f.ran-sJonn-ed via a global pmnmetc1'-'illdependenl diffeomorphism .1~ cjJ( () into the partialstatc-feerlbll.ck Jorm (G.30) if and oni!/ iJ the foliD'll/ing conditi()n.r; arc globallll satisfied:

=

(i)

[Ij ad}olio] 1

= 0, 0 $ i ;;;; p -

m,1 -

1, 1 $ j $ p,

(ii) t1 (LfJL}uh) E span {dll , .«. ,n (L~oh)}, 0 $ i $" q - 2, 1 $ j $ p, (iii)

[li, ad~.go] E (l = span {YOl adfnyo,., . 1$ j

~

f

acl}oyo}, 0 $ i $ p - q - 1.

p,

(itl) [9j' a dj.. 90] = 0, 0 $ i $" p -

1, 0 $ j ~ p, and

7111 -

1)

(v) 'L,6j Uj = (b m - 1)yo, i=l 1JJhe7'C

Yo is the vector field

defincd in (G. 93}.

Proof. Sufficiency. In the x-coordinates, which are defincd glohally by APrsumption GA t we have

ad}ooo

0 ~ i $ p - ml

(G.52)

f)

(G.53)

= (-1)i_ a , aTp-i

f) g'. = spall { -8

I • ,

'l'p

«,

}

-!:}-

,

vXp--i

0 '5:. i '5:. p - q.

Hcnce, condition (i) becomes (G.54) which implies that the ,'cctor fields

Ii are e..'\."}ll'cssed in the x-coordimttes as (G.55)

Since in the x-coordinates we have a;H.1

'Pi,i+l(.t'j" •• , X m1 ,

xr)

=

L}oh,

0 $; i '5:.

= LfJL}lIh,

0

1111 -

Si 5

1

711) -

1, 1 $ j ~ p,

(G.56)

condition (ii) becomes

d'Pj,; E spall {dx., .... dXi} , 1 '5:. i

5

q - 1, 1 5 j ~ p,

(G.57)

533

DIFFERENTIAL GEOMETRIC CONDITIONS

or, equivalently, 8CPj,i _ 0 8:Vk -

i t

+ 1 $ ~. $

11,

1 $ i $ q - I, 1

~j

$

]J.

(0.58)

From (G.53) ~lncl (G.55) we see that condition (iii) is equivalently expressed in the J'-coordinates as

[8 a.' IJl E span {a~.-c '''C _ 1 P

1""

p

a~ .}, 3.· P_ 1

0 $ i $ p - q, 1::; j $ p! (G.59)

which implies that

Dcpj.i

=

D.J:I.' 8tpj,; 8:r:/r

=

0, i + 1 $ II! $ p, q $ i ::; p - 1, 1::; j ::; P

(G.GO) 0, q + 1 ::; ~: $ p, p + 1 ::; i $

11,

1

~

.i $

p.

Combining (G.55), (G.58), and (G.GO), we sec that t.he vettor fields fj, 1 $ j ::; p, are expressed in tile .1:-coordinates as

Ij

=

Similarly, conditions (iv) and (v) imply that in the .1!-roordinatcs we have

(G.G2) From Assumption G.4, (G.Gl), and (G.62), we conclude t.hat: ill the .Tcoordinates t.he systcm (G.1) is expressed as (G.30) with CPo,; == 0, 1 $ i $ m'l - 1. The proof of necessity is straightforward. 0

G.2

Output-Feedback Forms

Setting 1111 = 1 in (G.2), in Proposition G.1, and in Theorem G.2, we obtain thc following corollary, which was proved in [122, 121]:

Corollary G.6 The system (G.3) can be tronsj01'7ncd via. (/ global param,ete1'independent diffeom.orphism x = !/J(() into the paramet1i.c outp'ut-jeedbackjo11n (7.101) if and only if the following conditions hold globally:

(i) rank {dh, d (LJoh) I ' " ,d (Lj;lh)} = n,

534

ApPENDIX G

(ii) [ad}or,

ad}!1,,] = a, a ~ i ~ n -

2,

0,

0 ~ i:5 n -

2,

(iv) [gi' ad}o"] = 0,

0:5 i ~ n -

2, 1 ~ j :5 p,

(iii)

[fj,ad}o1']

=

p

1:5 j:5 p,

m

(1Ji) 90 + L,(Jjgj = f3( ,)~bi( -l)iad}o", and j=l

(vii) the 1Jeclo7' fields

i=O 7',

ad/07', ... ,adji~Jl' are com.plete,

whe1'C f3 is a smooth nonlinear junction anti" is the 1Jector field defined by L Li h = { 0, ~ = 0, ... ,n - 2 r 10 1, 1 = 11 - 1 .

(G.G3)

Corollary G.G gives necessary wId sufficient conditions for (G.3) to be globally transformable into the parametric output-feedback canonical form (7.101) via a pU7umeter-independent diffeomorphism. However, this would unnecessarily exclude numy systems such as the robotic example of Section 7.3.3, for which a ]Jaramete1'-dependent diffeomorphism is needed to go from the physical coordinates into the output-feedback form. In the full-state feedback case, we need parameter-independent diffeomorphisms, because we want to be able to calculate the new state variables from the measurements of the original ones. \\Then only the output is measured, the dependence of the diffeomorphism 011 the unknown parameters is acceptable because t.he states do not appeal' ill the control law . Therefore, we now give necessary and sufficient conditions for the system ( = f((;0)+g((;8)11 (G.64) y = h((dJ) , where 8 is a vector of unlmowll parameters, to be globally transformable into (7.101) via diffeomorphism which is allowed to depend on the unknown parameters. The following result was first given in [72J:

Corollary G.7 The system (G.64) can be transformed via a global diffeomorphism x = ljJ((; Ii) into the output-feedback canonical jorm. (7.101) ij a.nd only if the following conditions a7'(~ satisfied f07' all ( E JR." and jor the true 1}o.luc of the pam.meter 'llcctD1' ij:

535

DIFFERENTIAL GEOMETRIC CONDITIONS

(iii) adjr

= n-l t; [ tp~.n-j(y) + ~P 6j fPj,n-i(Y)]

(-I)n-i adjr,

'whe7'e 'Pj,n-i(Y) == iou fPj,n_i(s)ds, 0 $; i $; n - 1, 0 $; .i ::; p,

(iv)

[u, adir] = 0,

O:S; i ~ n - 2,

fit

(11) 9 = {3(.) 'l)i(-I)iad}r, and i=O

(vi) the 'llector fields 'r, 2:tdfl', . . , ,adj-I r an~ com.plete, where {3 is a smooth nonlinear junction and r is the 1Ject07' fielll defin.ed by i I _ { 0, i == 0 , ... , n - 2 L r L f)'. 1, I=TJ-1.

G.3

(G.65)

Full-State-Feedback Forms

III this section, we consider the full-state feedback case and, hence, we require the diffeomorphisms to be parameter-independent. The result.s in this sectiol1 were first obtained in [69], except for Theorem G.9, which was first given in [1]. Setting k = I, ml = pin (G.30)-(G.32), in Proposition G.3, and in Theorem G.5, we obtain the following corollary: Corollary G.S There exists a parameter-independent diffeomorphism x = r/J( (), satisfied in a neighbol'hood U of a point (0, which tmnsfonns {G. 1) into the form

.1:1 = 2:2 = Xq-l Xq

X2

+ IPf(Xl)B

X:J

+ 'P2(JT (Xl, X2)(J

+ tp~_l(Xll'" = Xq+l + 'P:(Xl1'" =

Xq

,Xq-l)6

(G.66)

, Xql ;Jl)9

+ t.p~-1 (:'Cl' ••• Xp-l , xr)9 .ip = 'Po,p(X) + t.p;(x)6 + f3(x)u P if = q,o(Xlt ... ,xq , x') + L 8j lbj (x., ... ,3:1'/1,1:')

Xp-l

=

Xp

I

j=1

Y =

Xl,

if and only if the following

conditions are valid in a 11 eighb01'hood Ut ;2 U :

536

(i)

ApPENDIX

LgnL}oh == 0

I

0~i~

G

2, L!/n L'j;lh #- 0 I

P-

(ii) the distribution gP-q = span {gO, ad/ago, ... ,adj;qgo} i8 in'IJol·uf.ille and oj con.r;iant rank p - q + 1 , (iii) d (L/JL}/l) E span {dh,

(iv)

[h, ad)n90]

E

... ,d (L}ah)},

0

~ i:S;

q - 2, 1:S; j $ p,

gi = span {go! ad/uUol"" adJoYo}, O:S; i :s; p - q - 1,

1 $j:5 p,

(v) 9j

:E

0, 1 '5: j S; p.

For the diffeomorpbism of Corollary G.B to be globally valid, it is ll·ecessa1lJ that the above ('onditions (i)-(v) be globally valid and that the manifold A{ = {( E 1Rn : h«) = L/uh«() = ... = L1;1 h«} = o} be connected. As ca.n be shown nsing tbe results of [13], these conditions, together with the completeness of the vector fields 90, ad/uUo! ... , adj;l YOt where go is defined in (G.33) and = In - L 10 90, are sufficient for x = ,pee} to be a global diffeomorphism. In the case q = 1, these conditions are actually necesso,nJ and sufficient [13, Corollal'Y 5.7]. However, for q > 1, the completeness of ad/ugo, ... , adj;lyO is not necessaly. For example, consider the system

10

Xl

=

:1:2

3:2 =

(G.67)

'U-

3 -Xa -

X:i = y =

')

a

X1X

Xl'

This system is already in the form (G.66), but the vector field

(x~ + x~xi) aBXa a a = --+X38Xl 8xa

adiolio =

[X2 J::i8 UXt

-

1

!ol0

UX2

1

2

(G.68)

is not complete, since the solutions of the system .t! = -1 0 X2 ..,

=

.T3

=

(G.69)

.T5

starting from any point with X3(O) > 0 escape to infinity in finite time. "Ve now turn our attention to nonlinear systems of t.he form (G.70)

537

DIFFERENTIAL GEOMETRIC CONDITIONS

where h, gj1 0 :5. j ~ p, are smootll vect.or fields in a neighborhood of the origin ( 0 with h(O) = 0, 0 ::s; j ::; ]), 90(0) =F 0, and give necessary and sufficient conditions for (G.70) to be locally transformable via ~l. parameterindependent diffeomorphism x = q,( C) iuto the pll'l'ametric pU7'e-feedbuck j01'7n (4.285), which is repeated here for convenience:

=

XI = ~'2 ·1:2 = X3

+ ",T(x},x::!}8 + 'PI(Xh X 2,.J;3}8 (G.71)

= .tn + 'P~(.t'1"'" xn}tJ :i:n = 'Po.n{z) + ,/;(x)9 + [Po(x)

.l:n-I

+ pT(x)Olu,

where

lPo,n(O)

= 0,

'PI(O) = ...

= 'Pn(O) = 0,

PoCO) =F o.

(G.72)

Theorem G.9 A pammeter-independenl. diffeo1n01phism :1: = ¢J( (), with t/J(O} = 0, trnns!07'1ning (G.70) into (G. 71), exisf.s in a nei.qhb07'hoollB:r c U of the origin if and only if the following conditions. an'.! sal.isfied in U: (i) Feedback linearization condition. The distributions (G.73) a1"C

in'lJolutive and of constant rank i + 1.

(ii) Parametric pure-feedback condition. 9j E [~Y,

h]

E

go,

1:5. j ~ p

gi+l, \I ..1( E gi ,

0 =::; i ::; n - 3, 1 5: j

5: 11.

(G.74)

Proof. Sufficieftcy, As proved in [56], condition (i) is sufficient for the existence of a diffeomorphism x 4>(C) with 4>(0) 0 which transforms the system (G.75) fo(C) + 90«()'I1, fo(O) = o. no(O} -# 0

=

=

i: =

int.o the system

Xi

=

Xi+] ,

1 =::; i :5.

11 -

xn = 'PO,n(x) + .Bo(:c)u ,

1

(G.76)

with

«PO,n(O)

= 0,

.Bo(O) =F

o.

(G.77)

538

ApPENDIX

G

Hence! in the coordinates of (G.76) ,ve have

8

=

·'l:2 -

90

=

a f30(x} a:Vn

Qi =

aXl

span

a

8

+ ... + In - + 'PO,n{X)8:1: _1 aXn

1£1

(G.78)

n

{G.79}

{aa:r: ,... , aX"_' a .},

D,$i$n-L

(G.80)

,1

Because of (G.80), the parametric pure-feedback condition (G.74), expressed in tbe :twcoordinates, states that

gj

,Ii ] [ ao. X.

E spall!

E span

a~n },

1 $ j '$ P

!:J.~ UX

0- ~

, ••• , fl

:IJ-l

}

I

{G.81} 3 $ i S n 1 1 $ j 5: p.

But {0.81} can be equivalently rewritten as

Furthermore, since q;(O) = D and fAO) = 01 1 S j S p, we conclude from (0.82) that c,ol{O) = ... c,on{D) = O. (0.83)

=

Combining (G.78), (G.79), (G.82) and (G.83), we see that in the x-coordinates the system (G.7D) becomes (G.71). The necessity is straightforward. 0

Remark G.lD The design of Section 4.5.3 can be applied to the system (G.7D), after using the diffeomorphism of Tbeorem G.g to transform it into (G.71). Then, the feasibility region :F = B:t; x Bo of Proposition 4.24 must be a subset of t.he region on which the diffeomorphism exists. This can be ensured by selecting Bo;: to be a subset of Bz CU. Remark G.ll A special case of the parametric pure-feedback condition (G.74) is the extended-matching condition of [65]: (G.84)

539

DIFFEREN'rIAL GEOMETRIC CONDITIONS

This is clear from the proof of Theorem G.9: if (G.74) is replaced by (G.84), then (G.82) still holds, but \vith 'Pl == 0, ... t'PII-2 == O. Then, the system (G. 70) is expressed in the x-coordinates as

XI

·1:2

= =

Xn -2 =

XfI-l a!1~

= =

.t2 :1::3

(G.85)

·1:'I-J

+

In

'1' IPn-1 (.LIt .••

,xn)O

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