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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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