Global controllability and stabilization of nonlinear systems

Global Controllability and Stabilization of Nonlinear Systems

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Series on Advances in Mathematics for Applied Sciences - Vol. 20

Global Controllability and Stabilization of Nonlinear Systems

S. Nikitin Centre for Systems Science and Applied Mathematics Arizona State University Tempe, Arizona

World Scientific Singapore • New Jersey • London • Hong Kong

Published by World Scientific Publishing Co. Pte. Ltd. P O Box 128, Farrer Road, Singapore 9128 USA office: Suite IB, 1060 Main Street, River Edge, NJ 07661 UK office: 73 Lynton Mead, Totteridge, London N20 8DH

GLOBAL CONTROLLABILITY AND STABILIZATION OF NONLINEAR SYSTEMS Copyright © 1994 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof may not be reproduced in any form orbyany means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 27 Congress Street, Salem, MA 01970, USA.

ISBN: 981-02-1779-X

Printed in Singapore by Utopia Press.

V

Preface This book is based on courses of lectures on nonlinear systems given by author at the Moscow State University (1989 - 1990), at the Moscow Institute of Steel and Alloy (1990 1991) and at the Kaiserslautern University (summer 1990, 1991 1992). The main purpose of this book is to provide a self-contained, complete and geometrically clear presentation of recent results on global controllability and stabilization. A good deal of effort was put into the development of geometrical control intuition and inspiring the reader to think independently. That is why, the book contains many pictures and exercises. The material presented is organized in such a way that, for Part I, it is only assumed that the reader has mastered the very basic notions of ordinary differential equations theory, general topology and analytic geometry. For Part II the reader is assumed to be familiar with the elements of differential geometry. The present book can serve as a two-semesters course for those students and post-graduates specializing in automatic control theory and mathematical systems theory in applied mathematics departments. The author wish to express his deep gratitude to Dr.Marina Nikitina who has helped to prepare this book for publication and made many valuable remarks which allowed the author to improve the material of this book. The author takes occasion here to acknowledge his indebtedness to Professor A.Krener and Professor H.Sussmann, whose works were original stimulus of the present investigations. The bibliography at the end of the book is not intended to be complete, but merely to list papers and books used by the author, or papers and books which may be regarded as related to this work. The author is very thankful to his friends Professor W.Demtroder and H.Demtroder for their support during the work over this book. The idea to write this book was suggested to the author by Professor W.Demtroder. The author acknowledges the generous aid furnished him by Alexander von Humboldt Foundation and Kaiserslautern University for the preparation of the manuscript. The author extends his thanks to Michael Marquard who has been so kind to read parts of the text and to offer valuable suggestions. The author is grateful to Professor D.Pratzel-Wolters who has greatly assisted in preparation of the text of the first part of this book. Arizona State University 22.02.94. Sergey Nikitin

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vii

Contents Preface

v

I

T w o - d i m e n s i o n a l affine n o n l i n e a r s y s t e m s

1

1

C o n t r o l l a b i l i t y of cart 1.1 Prisons and mysteries of a plain domain 1.2 Controllability criteria for linear systems under phase constraints 1.3 Controllability analysis in a convex domain 1.4 How do control constraints change life? 1.5 Exercises

3 3

2

N o n l i n e a r afflne s y s t e m o n a plane as a cart g a r l a n d 2.1 About one nonlinear system which is equivalent to a cart 2.2 Controllability of a cart garland 2.3 Affine nonlinear system with control constraints 2.4 Dynamic of populations under external influence 2.5 Exercises

3

S e m i g l o b a l s t a b i l i z a t i o n of cart garland 3.1 Stabilization of a system being equivalent to a cart 3.2 Stable covering and semiglobal stabilization 3.3 Some examples of adaptive stabilizers 3.3.1 Sliding stabilizer 3.3.2 High-gain adaptive stabilizer 3.4 A simplified auto-pilot system and tracking control problem of robotic manipulators 3.4.1 A simplified auto-pilot system 3.4.2 Tracking control problem of robotic manipulators 3.4.3 Exercises

4

7 21 25 32 35 . . 35 41 50 58 65 69 69 74 80 81 83 87 87 . . 90 93

T w o - d i m e n s i o n a l s y s t e m s w i t h singularities 4.1 Normal forms of systems with singularities 4.2 Controllability and stabilization of systems with singularities 4.2.1 Controllability of systems with singularities . . . . 4.2.2 Local stabilization at a singularity of type I . . . .

97 97 103 103 113

viii

4.3

4.4

II 1

2

Classification of singularities, controllability criteria of bilinear systems on a plane 4.3.1 Classification of bilinear systems 4.3.2 Controllability criteria Exercises

Multi-dimensional nonlinear s y s t e m s

120 121 127 143 147

G l o b a l c o n t r o l l a b i l i t y analysis 1.1 A brief review of basic concepts of topology and differential geometry 1.1.1 Topological spaces 1.1.2 Smooth manifolds 1.1.3 Tangent bundle and vector fields 1.2 Groups and monoids 1.3 Approximative groups and systems on foliations 1.3.1 Approximative groups 1.3.2 System on foliations 1.4 Necessary and sufficient conditions of global controllability 1.4.1 When does the existence of transitive approximative group imply controllability? 1.4.2 Full rank conditions of global controllability . . . . 1.4.3 Systems on foliations and controllability of nonlinear systems 1.5 Hypersurface systems 1.6 Exercises

149

Local s t a b i l i z a t i o n of n o n l i n e a r s y s t e m s 2.1 T h e simplest necessary and sufficient conditions of local stabilization 2.2 Liapunov's direct method 2.2.1 Sontag's formula of almost smooth stabilizer . . . . 2.2.2 Jurdjevic-Quinn approach 2.2.3 Stabilization of homogeneous systems 2.3 Decoupling normalizing transformations and local stabilization of nonlinear systems 2.3.1 Existence of decoupling normalizing transformations

213

149 150 153 157 164 166 166 181 186 186 190 197 203 210

213 220 221 222 223 226 226

ix

2.3.2

2.4 3

Local stabilization of nonlinear systems and approximation of decoupling normalizing transformations Exercises

Semiglobal stabilization 3.1 Necessary conditions of smooth stabilization in the large 3.1.1 Some facts about degree of function 3.1.2 Multi-input systems 3.1.3 Single-input systems 3.2 T h e relationship between controllability and stabilization 3.3 Exercises

References

239 247 251 251 252 256 257 . 259 265

.

267

3

Chapter 1 Controllability of cart In this chapter the most simple type of a system is considered. T h a t is a cart going along straight rails. Due to its simplicity the qualitative control behavior of the system can be relatively easy investigated in a transparent geometrical way. T h e controllability is analyzed under phase and control constraints. As we shall see later (Chapter II), every affine nonlinear system without singularities can be represented as a garland of such simple carts going along the rails that are not straight but in some sense twisted.

1.1

Prisons and mysteries of a plain domain

Assume t h a t we are driving a very simple cart (Fig. 1.1) along straight rails without any friction. T h e dynamics of the cart, in accordance with Newton's second law, is described by the system X\

=

i2

— u

X2

(1.1)

where i , ( i ) denotes the derivative of Xi(t) with respect to time, i.e. i ; ( i ) = jnXi, the coordinate Xi measures a position of the cart on the rail road, the coordinate x% is a velocity of the cart and u is a control force. The state of the cart is a pair x = (xi,X2). The set of all possible cart states constitutes t h e state space which is R2 two-dimensional real space or, in other words, the state space is a plane. Suppose it is permitted only to use controls from a class A given in advance. For example, 1) if controls continuously depend on time and act only when 0 < t < T, then A = C ( 0 , T ) , where C ( 0 , T ) is the set of real continuous functions denned on the interval [0,T]. 2) if controls are supposed to have a finite number of discontinuities such as switches, then A = PC(0,T). u(t) £ PC{0,T) if u{t) is bounded

4

Controllability of a cart

Figure 1.1: The cart is going along straight rails. x\ is a position of the cart on the rails, xi is a velocity of the cart. for all t e [0, T]; there exists a finite set {i, : i = 1 , . . . , N} C [0, T] such that *i = 0 , tn = T, i,- < ti+i (i = 1 , . . . , N — 1) and u(t) is continuous for all t,• < t < i, + i (i = 1 , . . . , iV — 2), tjv-i < i < *N (the finite set {i ; i = 1 , . . . , N} may depend on u(t) G PC(0, T)). 3) if admissible controls have to be piecewise-constant, then

A=

PC^t^T),

where PCcon8t{Q,T) C PC(0,T). u(t) e PCconSi(0,T) if the derivative ^u(t) exists and is equal to zero for all t g [0, T] except a finite subset in [0,T]. The finite subset may depend on u(t) e PC^^Q^). 4) Throughout this book the following classes of controls will be mostly used: C = D C(0,T), T>0

P C = f| PC(0,T) T>0

PC c o n s ( = f) PC c o n s ( (0,r). T>0

Assume that it is not permitted that our cart goes out of some domain D c R 2 , It is quite a natural assumption which can be faced in many applications of system theory. Then it is important to know the conditions under which the cart is controllable on D by an admissible class of controls, say A. Throughout the first part of this book unless the contrary is explicitly stated, D is an open connected subset of R2.

5 Global controllability

and

stabilization

Figure 1.2: We cannot leave the subsets i \ , P2 without leaving the domain D. T h a t is why, P\ and P2 turn out to be prisons for the cart. D e f i n i t i o n 1.1 The cart is called controllable control class A iff V x,

x £ D

3

T £ R, T > 0

and

on a domain D C R 2 by a

u(t) : [0, T)

-+

R,

u(t) £ A,

such that xu(T,x)

= x

and

xu(t,x)

where xu(t,x) is a solution of the system and is generated by the initial condition xu{t,x)\t=0

£ D

V i e [0,T],

(1.1) governed by the control

u(t)

= x.

R e m a r k . We will say that a subset G C D is reachable (and/or accessible ) from V C D if there exist two points x £ G, x £ V and a control u(t) £ A, which steers t h e cart from x to x in a finite time T > 0 and xu(t,x) £ D for a l l i e [Q,T]. Controllability property of the cart crucially depends on the geometry of the boundary dD of the domain D. To this respect, it is convenient to discuss some examples. E x a m p l e 1.1. Consider controllability of t h e cart on the domain D shown in Fig.1.2, i.e. 22 D5 = = {{x {{x1},x x2)£R ; D {(xi,x 1 2)€R 2 ) G; K*;

\xx\ ,

(1.3) < 6, x > the system (1.2) has the form 2i =

< bx,Ab± > _ _ j-r-jj z\ + det(b,Ab) ■ z2, (1.4)

Z2

=

< b, Abx > •2l 16 I4



+

IM2

■ z2

+

u

2

where I b | = < b,b >, < *,* > denotes the scalar product < x,x > = 2\_i Zi^i on R2; det(vi,vi) is the determinant of the matrix composed by two columns vj., v2 G R2; *_L G R-2 such that < i x , b >= 0, and det{b,bi_)=\b\2 2

.

The system (1.2) is controllable on R iff the system (1.4) is controllable. In turn, the controllability of (1.4) implies det(b,Ab) ^ 0. Indeed, it is easy

8 Controllability of a cart to see that unless det(b, Ab) ^ 0 we cannot influence the dynamic of z t in (1.4). Moreover the inequality det(b, Ab) ^ 0 means that the system (1.2) is equivalent to the cart in the sense of the next definition.

Definition 1.3 Two linear systems £ : x = A- x + b- u E :

£ = A- x + b- u

are said to be equivalent iff one can find a linear transformation x = Tx, u=

T e R2X2,

det(T) / 0

q € R2,

+ p-ii,

p € R\0,

under which the system £ becomes £. Taking the linear coordinate transformation z\

=

*2 = u —

z\,

()2

« =

_ _ z\ + det(b, Ab) ■ z2, + det(b,Ab)-

r^F +

_

det{b,Ab)-

±

< b,Ab>

|6|2

.

z\x +

"

|6|4

■ z\,

((1.5) )

+ det(b, Ab) ■u,

we see that the system (1.4) is equivalent to the cart. Therefore we come to the next proposition. Proposition 1.2.1 The system (1.2) is controllable on R2 by C(0,T) T > 0 iff it is equivalent to the cart.

with

Proof. If the system (1.2) is controllable, then det(b, Ab) ^ 0 and under the transformations (1.3), (1.5) the system (1.2) becomes the cart. The relation introduced in Definition 1.3 preserves the controllability property. Consequently, to prove the other implication of this proposition we have to consider controllability of the cart on R2 by C(0, T) with T > 0. Given x i£ R2 we now construct the control u(t) = 6oi + 26 with

a, b e R

9 Global controllability

and

stabilization

steering the cart from x to x in finite time T > 0. Having solved the equations (1.1) describing the dynamic of the cart we obtain from xu(T,x) = x the following linear equations for a, b £ R

( where £,-,

xi - xt ■ T - £x \ £2 -x2 )

( T3 \ 3T2

x,- (i = 1,2) are entries of x, x

T2 \ ( a \ 2T j ' { b ) '

£ R 2 , respectively.

Since

det

(3?

Ti

IT ) = ~

and T > 0, we can solve the equations for a, b Thus, for every pair x, x £ R 2 , there exists a continuous control steering the system from x to x in t i m e T. Q.E.D. As an exercise we leave for the reader to prove the extension of the proposition to t h e situation where the class A of admissible controls is PCoonst(Q,T). For the hints see Exercises 1.1, 1.2, 1.3. If det(b, Ab) = 0, then the system (1.4) is not controllable on any domain D C R 2 and so is the system (1.2). Thus, as long as we are interested in controllability of the system (1.2) under phase constraints, we have to focus our attention on analysis of controllability of the cart (1.1) on a domain D. Let us consider the cart under controls from PC^n^. Taking u(t) = k for all t 6 [0, T] (where k > 0, k £ R), we can go along the parabola (Fig. 1.4). P u t t i n g u(t) = —k for all t £ [0, T] we obtain the cart going along the parabola drawn in Fig. 1.5. After applying the control u(t) = 0 our cart has to go to t h e right whenever £2(0) > 0 and to the left whenever x2(0) < 0 (Fig.1.6). If u(t) — 0 and x2(0) = 0, then the cart will stay in equilibrium (x a (0),0). It is easy to see that the parabolas (in Fig.1.4 , Fig.1.5) tend to a vertical line denned by xi = c as k goes to infinity. In what follows, we can drive our cart along any vertical line X\ = c (more precisely, we can move up and down in some neighborhood of the vertical line) in both directions and the cart drifts to the right (left) on the half-plane defined by the inequality x2 > 0 (x2 < 0). Thus t h e controllability analysis of the cart on a domain D c R i s reduced to the investigation: when can every two points x, x £ D be connected by an oriented p a t h which is in D and consists of vertical and horizontal segments of straight lines (Fig.1.7). Following the orientation of the path one can go from t h e point x to x without leaving the domain D. The orientation of the p a t h should obey t h e following rules: a) along horizontal lines with x 2 > 0 . (x 2 < 0) it is allowed to go only to t h e right (to the left);

10 Controllability of a cart

Figure 1.4: The arrow shows the direction of the cart drifting under the control u(t) = k, k>0.

Figure 1.5: The control u(t) = -k pushes the cart along the parabola in the direction marked by the arrow.

11 Global contTollability

and

stabilization

Figure 1.6: For x 2 (0) > 0 ( i 2 ( 0 ) < 0) the cart under the control u(t) = 0 goes to t h e right (to t h e left) with the velocity equal to x2(0). If x2(0) = 0 and u(t) = 0, then the cart remains stationary.

Figure 1.7: T h e cart is controllable on D because every two points can be connected by the p a t h with the appropriate orientation.

12 Controllability of a cart

Figure 1.8: The darkened region is surrounded by the curve x"1. b) along vertical lines one can go in both directions, i.e. up and down. We recall that an image of the segment [0,1] C R is referred to as a smooth curve 7; under a continuously differentiable mapping x"1 : [0,1] —> R2, such that | xdy |> 0 for any r e [0,1]. The mapping ^ ( r ) is called parametrization of the curve 7. Orientation of the curve 7 is called the direction of the motion of point s 7 (r) as the parameter r increases from 0 to 1. A curve with an orientation fixed is called oriented. To formulate the first criterion of con­ trollability of the cart we need the following topological property of a plane domain. Definition 1.4 A domain D C R2 is said to be simply connected iff, for any continuous curve x7(£) : [0,1]->D with clockwise orientation such that s 7 ( n ) + X 7 (T 2 ) for all rx + r 2 ,

ru

T2 e [0,1) and x 7 (0) = x 7 (l),

the region surrounded by the curve (i.e., the region that, going along the curve, you have on the right side (Fig.1.8)) is in D. In other words, a domain D is called simply connected iff it has no "holes" inside. Theorem 1.1 Given a simply connected domain D C R2 the cart is control­ lable on D by PCconut iff x = (xx.x2) £ D

13 Global controllability

and

stabilization

implies (x1,rx2) for attr Proof. implies x, x 6 t h e cart

GD

G [0,1]. Sufficiency. Since D is an open subset in R2 and x = (x1,x2) G D (X1,TX2) G D for all r G [0,1], we can connect any two points D by the p a t h like t h a t in Fig.1.7. That means the controllability of on D.

Necessity. If t h e cart is controllable on D, then for each x = ( x i , x 2 ) e D there exist T G R, T > 0 and a. control v(t) G P C c o n s t , such that x „ ( T , x ) = x, xv(tux) ^ x „ ( t 2 , x ) for a l l * ! ^ R, p(xt, xs) = xt. It is clear from geometrical point of view (Fig. 1.10), that prisons and mysteries arise in the points of D \ {x2 = 0} where the function p(x) has local maxima and minima. Let arg

extrf,(p)

denote t h e set of points where the function p has local extremums. More precisely, a connected segment rfcflof vertical straight line is in arg extrg iff there exists an open connected neighborhood 0 ( $ ) of i9, such that either p(x) < p(tf) for all x G 0(i?) n D or p(x) > p(rf) for all x C 0(d) n D, where p(#) is a value taken by the function p on d. Evidently, p is constant on i9. T h e following theorem gives necessary conditions under which the situations shown in Fig.1.10 are impossible. T h e o r e m 1.2 If the cart is controllable on a domain D c R ! , sup p = 8D

sup

then

p,

Dn{ij=0}

(1.6) inf p = 9D

inf Dn{x2=0}

p

14

Controllability of a cast

Figure 1.9: The cart is controllable on these domains.

15 Global controllability

and

stabilization

Figure 1.10: p has a local minimum at x and a local maximum at x. and, for every connected component i? C arg

extrfi(p),

■dD{x2 = 0} jt$ holds.

R e m a r k . T h e domain D in the conditions of this theorem can be unbounded in R 2 T h e proposition of the theorem is true for any class A of admissible controls. Proof. It is clear t h a t sup p < sup p Dn(i!=0} SD

= sup p D

and inf

p > inf p

£)n{a:2=0}

= mi p.

3D

D

If sup p < supp, Dn{ij=o} D then there exists a point x = (xi,x2) p(x)

£ >

D such that

sup p Dn{x2=o)

16

x2

Figure 1.11: The cart is not controllable on D. and either x2 > O o r i j < 0. The inequality x2 > 0 (x2 < 0) implies that the set {x G D; p(x) > p(x)} contains a prison (a mystery). Thus it follows from the controllability of the cart on D that the equalities (1.6) hold. Consider now a connected component $ C arg extrp(p) which does not have any common points with the coordinate line {x2 = 0}. Without loss of gen­ erality, assume that i? C {x2 > 0} (the analysis of the other possibilities is completely analogous). Then there exists a positive real number e, such that the connected component of the set {x e D; p(tf) - e < p(x) < p(d)} containing i? has empty intersection with {12 = 0} and consequently there is a prison for the cart. Therefore controllability on D implies d n {i 2 = 0} 7^ 0 for every connected component 1? C arg extrr>(p). Q.E.D. The conditions given by Theorem 1.2 are necessary but not sufficient for the controllability on a plane domain. Consider, for example, the domain D shown in Fig.1.11. The cart is not controllable on D but all conditions of Theorem 1.2 are satisfied. As we shall see in the next section, Theorem 1.2 turns out to be a criterion for controllability of the cart on a convex domain. We now turn to the study of sufficient conditions under which the cart is

17 Global controllability

and

stabilization

Figure 1.12: T h e cart is controllable on D, which has no "holes" inside, because D is a foliation over D D {x2 = 0} with a leaf being a connected vertical line segment. controllable on a domain D with "holes". It is important to note that if D has no "holes", i.e., D is simply connected, then by Theorem 1.1 the cart is controllable on D iff D is a foliation over {x2 = 0} [~l D with a leaf being a vertical line segment. In other words, D is shadowed by connected vertical line segments having non empty intersection with D n {x2 = 0} (Fig.1.12). A more general concept of foliation will be discussed in Part II of this book. Here we use t h e following simplified version of it.

D e f i n i t i o n 1.5 Let XJ(T) : [0,T] -> R 2 be a differentiable mapping such that ^M ^ o for all T 6 [0,T]. Then a domain D C R is called a vertical foliation over the curve 7 = {^(T); T £ [ ° . r ] } iff there exist mappings R such that d_(T) < 0 < d+{r)

D=

U

{(*7(T),*a)i d-{r)

Vre(0,T)

< x» - * J ( r )
-1 (0) denotes t h e equilibrium set of E. Corollary. A linear two-dimensional system E is controllable by PCcan,t a convex domain D iff det(6, Ah) # 0, sup(det(6, x)) =

sup

(det(6, x)),

inf(det(6,x))=

inf

(det(6,x)).

on

(1.8)

Proof. If t h e system E is controllable in any domain, then det(6, Ab) ^ 0 and consequently E is equivalent to the cart. Therefore using (1.3), (1.5), we can rewrite the equalities (1.8) in the form (1.7). Thus an application of Theorem 1.4 completes the proof. Q.E.D. Using Theorem 1.4 and its corollary we can investigate controllability of a linear two-dimensional system on a convex domain with the help of convex optimization methods.

24 Controllability of a cart Example 1.3(linear programming for the controllability analysis). Let us inv estigate controllability of the system Let us investigate controllability of the system x — Ax+i \u (1 .9) x = Ax+(\\u (1.9) on the convex set D denned by the inequalities on the convex set D defined by the inequalities xj xj + + 2x 2x22 + + ll > > 0, 0, 2xi + + xx22 -- 4 > 0, xi - x2 + 1 > 0, xi - 4x2 + 13 > 0, - 4 * i - x2 + 23 >

0.

Applying the simplex method (for the details see, e.g., [37]) we obtain that the function p(x) = —3xi + 6x2 attains the minimum equal to —33 on D and the maximum equal to 15 on D. Moreover the function p(x) takes the value —33 at the point x = (5, —3) and the value 15 at x = (3,4). Thus, according to Theorem 1.4, the system (1.9) is controllable on D by PC^st iff

det [

0

det I «

(

(

i)

) -

and

c: :o: :

)

| •. A* -■(

(1.10)

1=0.

(

:

)

)

-

Therefore the set of linear systems being of the form (1.9) and controllable on D is the two-dimensional plane defined by (1.10) in R 2x2 Example 1.4 Let us investigate controllability of the cart by PC^^ on the convex domain D defined by the inequality x2 + ( x 2 - e ) 2

< 1,

where e is a real number. In order to find a maximum and a minimum of the function p(x) = x\ on 3D we construct the Lagrange function I ( x , A) = X! - A(x2 + (x2 - e)2 - 1). It is well known (see, e.g., [ 11 ]) that p can have the maximum or the minimum at the point x" only when dL(x*. dL(x\X*) **) = 0, fir

25 Global controllability

and

stabilization

dL(x',X") =0 d\ dX for some A* / 0. Therefore 1 - 2X*x*1 = 0 -2A*(x* - e) = 0 2

W) " (*J - ef - 1 = 0 and we obtain that the function p(x) = xx attains the maximum at the point x*+ = ( + l , e ) and the minimum at a* = ( - 1 , e). If e ^ 0, then x*+,x*_ are not in {a;2 = 0} and, by Theorem 1.4, the cart is not controllable on D. Thus the cart is controllable on D iff e = 0.

1.4

How do control constraints change life ?

So far we have assumed that controls can take arbitrary large values. Prom a practical point of view this is quite a restrictive assumption. More natural considerations imply that controls have to be bounded by some known func­ tions. In other words, the problem is to study cart controllability on a domain D by t h e following class of admissible controls. A ( u _ , u + ) = {«(vMM )*

t--

JO

1.10 Prove that the transient time T7, i.e., the time for the cart to move along the curve 7 £ A(A, D), is expressed by an integral of the second kind, T -

t dx\ i-i x2

1.11 Let Q C D, and let the cart be controllable by PCco^tik—, k+) both on the domain Q and on the domain D. Prove that the cart is controllable by PCcns^k-, k+) on D \ Q as well.

35

Chapter 2 Nonlinear affine system on a plane as a cart garland. We have seen in the previous chapter that almost every linear two dimensional system is equivalent to the cart. This remains true, to some extent, also for nonlinear affine systems on a plane. In this chapter it will be shown, that there is a rich class of nonlinear affine two-dimensional systems being locally equivalent to the cart. A nonlinear system of the class can be considered as a cart garland ,i.e., the whole plane can be covered by the domains where the system is equivalent to the cart. Since we are already able to analyze controllability of the cart, we can analyze controllability of the cart garland.

2.1

About one nonlinear system which is equiv­ alent to a cart.

Consider the affine nonlinear system having the form £(/,&):

x = f(x) + b(x)u,

where f(x), b(x) : R2 —» R2 are two times continuously differentiable mappings, u is a control variable. In systems analysis an equilibrium set of a system plays a very important role. Here we define the equilibrium set of S ( / , b) For this purpose we introduce the function ¥>(x) = det(/(*), &(*)). Definition 2.1 Equilibrium set of a system £ ( / , b) is defined to be v _ 1 (0), where

«J-'(o) = { i a ! ; v(*) = o}. At the points of equilibrium set rank{f(x),b{x)}

< 2.

36 Nonlinear

a&ne system

on a plane

Figure 2.1: One can drive the system along the curve 71 (72) at t h e points x € R 2 such that 0 {f(x) < 0). ~ib{y) denotes the integral curve of the vector field b(x) passing through y. Let us call the right hand side of £ ( / , b) the velocity vector of t h e system. Then at the point x e R 2 , such t h a t ip(x) > 0 ((x) < 0) t h e velocity vector x(t) of the system form an acute (obtuse) angle with the vector obtained by rotating the vector b(x) by | clockwise (Fig.2.1). If, however, y ( x ) = 0, then at the point x the velocity vector of t h e system will collinear to the vector b{x) and using t h e control one can keep t h e system staying at the points of equilibrium set | v - i ( 0 ) ^ 0,

and ,for each x € R 2 , there are T 6 R and e T 6 (x) = x,

where etb : R 2 —► R 2 is the flow generated by the vector field b(x) (i.e., etb(x) = x ( t , x ) , where jfx{t,x) = 6(x(t,x)) and x(0,x) = x), then one can only move along the curves like those in Fig.2.1. Taking u(t) = k for all t e R, with k 6 R, and the new t i m e scale

r=|fc|i,

37 Global controllability

and

stabilization

where | k | = max{fc, —k} we obtain dx

1

~dr~ = ]T\f^

+

s n k

9 ( )K^)-

T h u s for large enough \ k \, one can move along the integral curve ~tb{y) = Wb{y); * € R } for all y 6 R 2 in both directions, i.e., up and down in accor­ dance with the Fig.2.1. It is easy to see that the integral curves {ib(y)\ y £ R 2 } correspond to vertical lines, and y _ 1 ( 0 ) plays the same role for the system E ( / , 6) as "rails' 7 {x2 = 0} for the cart. This gives us a hint about the equiva­ lence between t h e cart and the system S ( / , b) with >p~l(0) and {75(1/); y € R 2 } shown in Fig.2.1. T h e equivalence relation is defined as follows. D e f i n i t i o n 2.2 Two

systems £(/,&):

x = f(x)+

£(/,&):

z = f(z)

b(x)u; + b(z)v

are said to be equivalent (on a domain D C R 2 ) if there exist continuously differentiable functions a(x), /3(x), p(x) : f l - > R and a continuously differ­ entiate coordinate transformation x = $ ( z ) such that: p(x) jt 0, a(x) ^ 0 for all x € D; $ : $ _ 1 ( D ) —► D is one-to-one

and

d*(a ■ (f + /3b))(z) = / ( * ) Vz e $ - J ( D ) , d * $ ( a ■ p ■ l){z) = b(z) Vz G $ _ 1 ( D ) , where d*$(£)(z) = ( ^ i ) " " 1 ^ ( * ( z ) ) for any vector field £ : D ^ R is the jacobian matrix of the mapping $ : $ - 1 (Z?) —> D.

2

; ^

R e m a r k . T h e inverse transformation of 0 } such that D = {e8\x({r)); - d _ ( r ) < 6 < d+{r), 0 < r < 1}. If D is a controllable foliation generated by £ ( / , b), then, under the coordi­ nate transformation x = e s6 (x''(r)), the system £ ( / , 6) has the equilibrium set defined by 6 = 0 and the controllable foliation generated by £ ( / , 6) becomes vertical foliation. Moreover, the following theorem is true. Theorem 2.1 The system £ ( / , 6) and the cart (1.1) are equivalent on a do­ main D c R 2 being a controllable foliation generated by £ ( / , b). Proof. Let D be a controllable foliation generated by £ ( / , b). Then 5n \onip-ii0) ^ 0- Therefore one can choose a parametrization X'-(T) of D n y _ 1 (0) so that 0 ,X2 > 0; ai > 0 is a constant characterizing the herbivore growth rate, the numbers a 2 > 0, c 2 > 0 describe the interaction between herbivores and carnivores; C\ > 0 is the death rate of carnivores when there are no herbivores left; 6i,o 2 are real numbers characterizing a control action on t h e populations dynamic. Clearly only the first quadrant (for xx > 0, x 2 > 0 ) is of interest here. There­ fore we analyze controllability of the system LV on the domain D = {(xi,x2)GR2;xi > 0 , x 2 > 0 } .

41 Global controllability

and

stabilization

T h e first step of the controllability analysis of a nonlinear affine system £ ( / , 6) is t h e calculation of the function ip(x) = det(f(x),b(x)). For the system LV the function _1(0) n D. The system is controllable on D and equivalent to the cart if, and only if, the region darkened by the integral curves of the vector field b(x) emitted from D n 0, 6162 < 0, &!&2 = 0 and b\ + b\ ^ 0 we obtain the following proposition. P r o p o s i t i o n 2 . 1 . 1 The system LV is controllable on D = {x e R 2 ; x a > 0,x 2 > 0}

iffbl + b22^0. Moreover, if bxb2 = 0 and b\ + b\ =f 0, then on D = {x E R 2 ; i j > 0, x2 > 0} the system LV is equivalent to the cart , and D is a controllable foliation generated by LV. T h e proof of this proposition we leave as an exercise for the reader. Here we note only t h a t , if b\b2 = 0 and b\ + b\ / 0, then there are only the possibilities shown in Fig.2.3.

2.2

Controllability of a cart garland.

In t h e previous section a domain D was supposed to be a controllable foliation generated by an affine nonlinear system S ( / , 6). This assumption implies t h a t on the domain D S ( / , b) is equivalent to the cart. In general, a domain D is not a controllable foliation. For instance, in the following example the domain D is t h e whole plane R 2 and is covered by two controllable foliations. E x a m p l e 2.2 T h e system in question is of the form Xi = X\X2 + U,

X2=l+U.

42 Nonlinear affine system on a plane

Figure 2.3: (a) b2 = 0, b\ ^ 0, _1(0) = {x2 = 01/02}; vertical straight lines are the integral curves of the vector field 6. In both cases (a),(b) it is clear that the integral curves of the vector field b passing through D n V _I (0) darken the domain D. The equilibrium set of the system is defined by —1} and on {xi + x 2 < 1}. Hence it is controllable on R2. Thus, in order to investigate the controllability of an affine nonlinear system on a domain D such that < gradtp(x),b(x) > \Dnv-i{0)

± 0,

and

5 n f 1 ( 0 ) = u,{u,}, where {u;,} is the collection of the connected component of Z>n - 1 } , where {xt + x2 < 1}, {xi + x2 > — 1} are controllable foliations generated by the system. Definition 2.4 Let £ ( / , 6) be an affine nonlinear system, and let D C R2 be such that < gradip(x),b(x) > |fi n ^(o) + °A cart garland representing the system £ ( / , b) on D is said to be the collection {Di}, where Di is the largest controllable foliation such that Di C D and Di corresponds to u>, which is a connected component of D n ,} cart garland. Making use of the results of Chapter I, one can relatively easy control the system £ ( / , b) on every Di from {Di}. If cart garland covers D, i.e. D C U,Z),, then S ( / , 6) is controllable on D. T h e o r e m 2.2 If cart garland representing the system E(/, 6) on D covers D, then S ( / , b) is controllable on D. Proof. Let x, x be any two points in the domain D. Since D is connected, there is a regular curve -y C D, such that x 7 (r) : [0,1] —> D, x 7 (0) = x, x 7 (l) = x. 7 is a compact subset in D and D C U,D,. Hence there exists a natural number N such that 7

cuf=i A „

where A., C Z) is a controllable foliation generated by £ ( / , 6). The covering {ZVJjLi °f 7 induces the covering of the interval [0,1] C U ^ J r ^ j T , ] , where

44 Nonlinear

affine system

on a plane

T0 = 0,TN = l,ry_, < TJ for ; = 1,2,. ..N, and X 7 ( T ) : [TJ-UTJ] -> Dir Thus a^fo) C A , n Dil+i {j = l , . . . , t f - 1) and X''(TJ-1),X-; to Dj if there exist x £ D , , x e Dj and an admissible control u(t) € A, such that u(t) steers the system from x to i, i.e. xu(T,x) = x for some positive real number T (recall that xu(t,x) is the solution of the initial-value problem x = f(x) + u(t)b(x), x(0) = x), x u ( t , x ) e D for all t £ [0,T] and xu(t, x)eDi andt

U Dj U{D\

(U fc Di)}

G [0,T\.

T h e digraph of S ( / , 6) on D will be denoted by TWM(A,D) if A is either C or PC, PCcon,t).

(or by

TWib)(D)

Recall t h a t a digraph is called strongly connected if from any vertex one can reach a preassigned one by moving along the edges of the digraph in accordance with their orientation. Clearly, if £ ( / , b) is controllable by a class of admissible controls A on D C R 2 , then r S (/, 6 )(A,D) is strongly connected. T h e inverse proposition becomes true when the boundary dD satisfies certain trap free conditions. D e f i n i t i o n 2.6 ( s t r o n g trapfree c o n d i t i o n ) A boundary dD of a domain D is said to satisfy strong trap free condition (for a system, S ( / , b)) provided that: 3D is a union of non intersecting dD is transversal

regular loops without

crunodes;

to -1(0);

any connected component of the set 3D \ y - 1 ( 0 ) is transversal to the vector field b(x),i.e., for any connected curve 7 C dD\ |^-i(0)nS / ° and b(x) ^ 0 V x 6 R2. Let D C R2 be a bounded connected domain with the boundary dD being a finite set of regular loops satisfying strong trapfree condition. Then the system E ( / , 6) is controllable on D if, and only if, i and belonging to D. The sets {£*,} are controllable foliations generated by S ( / , b). Hence, on D, E(/, 6) is equivalent to the cart and is controllable. Moreover, since T^y^A, D) is strongly connected, there exists a set of curves {Cij}-> where £y joins the sets D, and Dj, such that £ ( / , 6) is controllable on U«{& U £ , } • Then, it remains to prove that if the conditions of Theorem 2.3 are fulfilled, then one can reach Uy{£y U Dj} from any point x 6 D \ [Oij{dj U Dj}] and vice versa. Let us take any point x 6 D \ [U,,.,{Cij U Dj}]. Then the segment of the integral curve 7f,(z) emitted from x and belonging to D has a. non empty intersection with some connected component 7 C dD \ 0 and x, 5 e Ut,j{(>} u ^ i } Q.E.D. It is an easy exercise to design the controls u(t),u(t) mentioned in the proof of Theorem 2.3 (see formula (2.3)). Strong trapfree condition does not al­ low to apply Theorem 2.3 to controllability analysis of a system £ ( / , 6), with b(x) ^ 0 V x G R2, on a domain D with "holes" which do not break off the equilibrium set y - 1 (0) of £ ( / , 6). Trapfree condition can be weaken when the vector field b(x) is integrable on D. The latter implies that there exists a neighborhood 0(D) of D such that there is a two-times differentiable real function hb : O(D) —* R \ 0 such that dhh(x) = n(x) < b±(x),dx >, fi(x) ^ 0 for all x G 0(D) ,where dhb(x)

=

£ ^

X 1

+^ d x

2

,

bx(x) = (-6 2 (x),6 1 (x)f

47 Global controllability

and

stabilization

dhb(x) denotes t h e differential one-form corresponding to the first derivative of hb(x), i.e., dhb(x) =< gradhb(x),dx > , where dx = (dx1,dx2)T.

< dhb(x),£ > denotes < gradhb(x),£ — hb{etb(x))

> for f e R 2 . It is easy to see that =< dhb(x),b(x)

>=fi(x)

< b±(x),b{x)

>= 0

for all x € D. Hence, kb(x) is a, constant on an integral curve of the vector field b(x). hb{x) is called an integral of the vector field b(x) on D. D e f i n i t i o n 2.7 ( w e a k trapfree c o n d i t i o n ) Let D C R 2 be a bounded do­ main, and let the vector field b(x) be integrable on D. Let arg

extrD(hb)

denote the set of points where the function hb(x) has local extremums, i.e., a connected segment 8 C D of an integral curve of the vector field b(x) is in arg extrD(hb) iff there is an open connected neighborhood 0(8) of 8 such that either hb(x) < hb(8) for all x € 0(8)11 D or hb(8) < hb(x) for all x E 0(8)C\D, where hb(9) is a value taken by the function hb on 8. The boundary dD is said to satisfy weak trapfree condition iff, for every con­ nected component 8 C arg extrjj(hb),

ffrnf

■J(O)#0

holds. T h e following theorem gives necessary conditions for a system E ( / , 6) to be controllable on D C R 2 T h e o r e m 2.4 Suppose D is a domain in R 2 and £ ( / , b) is an affine nonlinear system, such that < grad |sn»>-i(o) 5* ° an^ tfle vector field b(x) is integrable on D. If the system S ( / , b) is controllable on D, then sup hb = sup hb, 3D Drv-^o) inf hb = dD

inf

Dnip-^o)

and dD satisfies weak trapfree condition, vector field b(x) on D.

hb

where hb(x)

is an integral of the

48 Nonlinear affine system

on a plane

T h e proof of Theorem 2.4 is analogous t o t h e proof of T h e o r e m 1.2, and is left for the reader as an exercise. It is clear from t h e geometrical point of view, that if the conditions of Theorem 2.4 are violated, then there are mysteries and prisons in D. Theorem 2.4 and Theorem 1.2 give only necessary conditions for controllability on D. In order to formulate more general sufficient conditions of controllability on D, we need to use weak trapfree condition and t h e notion of digraph of £ ( / , b) on D. T h e o r e m 2.5 Let a domain D C R 2 be bounded, let £ ( / , b) be an affine non­ linear system, such that < gradip(x), b(x) > \r>nv-l{o) ¥" 0 and ^le vector field b(x) is integrable on D. Let the digraph o / £ ( / , b) on D be strongly connected. Suppose further the boundary dD is a finite set of disjointed regular loops being transverse to -1(0). Assume sup hi, = sup hi, 3D Drv-Ho) inf hi, = inf hi, 9D Dn^-'fo) and dD satisfies weak trapfree condition, where hi,(x) is an integral vector field b(x) on D. Then the system £ ( / , b) is controllable on D.

of the

As an exercise it is recommended for the reader to prove Theorem 2.5 using the idea of the proof of Theorem 1.3. Consider examples illustrating the application of the results obtained. E x a m p l e 2 . 3 . The system ii =

x2-tp(x1), x2 = «,

where ip(xi) is a differentiate function defined for all x^ G R, is equivalent to the cart and hence controllable on R 2 . Indeed, under the coordinate transfor­ mation z\ = s i , z2 = x2V =

dip(xi) Jk (X2

ij>(xi), _

^ ( * l ) ) + «:

the system becomes *i = H

i2 = v. From the geometrical point of view, R 2 is a controllable foliation generated by t h e system (Fig.2.5).

49 Global controllability

and

stabilization

Figure 2.5: T h e integral curves of the vector field b(x) are vertical lines. It is easy to see, t h a t R 2 is shadowed by the vertical lines emitted from ¥>_1(0) =

{x2 = i>{xr)}. E x a m p l e 2 . 4 . T h e system X1 = —X2U, X2 = 1 + X\U

is controllable on D = {Rj < x\ + x\ < R\} (0 < Ri < R2), since all conditions of Theorem 2.4 are met. On the other hand, D is covered by a cart garland representing t h e system on D. E x a m p l e 2.5. Given the system i\ = x2 + QU, x2 = t a n x i + ui it is required to determine the values of the constant a for which the system is controllable on R 2 . In our case _1(0) *s disconnected and since the domain D = R 2 is simply connected, it is necessary to check t h e condition of Corollary 1 of Theorem 2.2. We obtain < grad-

cos-" x\

+ 1,

and if | or | > 1, then the function < gradip(x),b(x) > will always be negative. T h u s t h e conditions of Corollary 1 of Theorem 2.2 are valid. Since for | a | > 1 any integral curve of the vector field b(x) intersects y> _1 (0).

50 Nonlinear afEne system on a plane

Figure 2.6: D = {x2 > 0,xi > 0} is a controllable foliation generated by the system under consideration. Example 2 .6 We investigate the controllability of the system i i = x\

ln(xix 2 ) 2 \

1" x i

ln(xiX2)

+ Xi«,

ln(xix 2 ) 2 ln(xiX2) x2 1- x2u 4 2 on the domain D = {xi > 0, x2 > 0}. For this system we have

|/(») I | 6(x) | Sgn

ff } „ < I(X} b[X)

'

,

>

for all x £ D n )[grad |6(x) then < gradip(x),b+(x) >< gradtp(x),b-(x) and mysteries in D (Fig.2.9).

u+(i);

> > 0. Hence there are prisons

If, for some point x G D either 7i,+ (x)nDny> -1 (0) = 0 or 'yb_(x)nDr\tp~1(0) = 0, then, in accordance with the Jordan curve theorem [31], 7&±(x) splits the

55 Global controllability and stabilization

)[gradip(x)]j_.

1b-{x) I

Figure 2.9: The region Q is either a prison or a mystery for the system on D.

56 Nonlinear

a£&ne system

on a plane

simply connected domain D into two disjointed parts. Hence D has prisons and mysteries, and the system is not controllable by A[u~, u+] on D. Q.E.D. The result of Theorem 2.7 can be generalized in various directions. In our study, the generalization is effected by passing to domains which are not ad­ missible and _1(0) n D is not connected. This generalization is based on the concept of cart garland modified as follows. Definition 2.9 LetT,(f,b) be an affine nonlinear system, letA[u~,u+] of admissible controls. Let D C R 2 be such that: u+(x)

> u~(x) for all x € D;

< gradip(x),b{x) u±

be class

> |s n ^-'(o) + °i

± " | l ( f ) | ^ n < /(*)>*(*) > f°r allxeDD

v,-i(0).

Then a cart garland representing £ ( / , 6 ) under A[u~,u+] on D is said to be the collection {Di}, where Di is the largest subset of D such that £ ( / , 6) is controllable by A(u~,u+) on Di and Di corresponds to w; which is a connected component of D n -1(0), such that Di n ;. T h e controllability of the system S ( / , b) by A[u~, u+] can be analyzed on each of {Di} with the help of Theorem 2.7. Thus from D C U,D, follows the controllability of £ ( / , 6) on D by A[u~,u+}. T h e o r e m 2.8 If cart garland representing the system £ ( / , b) under on D covers D, i.e. D C U;D;, then £ ( / , b ) is controllable on D by

A[u~,u+] A[u~,u+].

The proof of Theorem 2.8 is completely analogous to the proof of Theorem 2.2 and is left as an easy exercise for the reader. For study the controllability by A [ u ~ , u + ] on a damain D with "holes" the concept of digraph rs(/,6)(A[u~,u + ],D) will be used. T h e digraph

Tnm{A[u-,u+},D) is defined in the same way as Ts^ib^(A,D), A[u~,u+] in Definition 2.5.

but A should be replaced by

If the system S ( / , b) is controllable by A[u~,u+] on D, then T^fb)(A[u~, u+], D) is strongly connected. As we have already seen in previous sections, un­ der some trapfree conditions the controllability follows from t h e fact t h a t T S (^ ( A [ u ~ , « + ] , D) is strongly connected. For t h e class A[u~,u+] of admissi­ ble controls a strong trapfree condition is defined as follows.

57 Global controllability

and

stabilization

D e f i n i t i o n 2 . 1 0 ( s t r o n g trapfree c o n d i t i o n ) A boundary dD of a domain D is said to satisfy strong trap free condition (for a system S ( / , b) ) provided that dD is a union of non intersecting and non-self-crossing regular loops and for any connected curve 7 6 dD \ V _ 1 (0) there is a parameterization X 7 ( T ) : (0,1) -> R 2 , such that det[-5J-i,6+(x^T))]det[^p,6_(^(T))]

< 0

for all T e (0,1). T h e next theorem gives a criterion for t h e system S ( / , 6) t o be controllable by A [ u ~ , u + ] on D, with dD satisfying strong trapfree condition. T h e o r e m 2 . 9 Let S ( / , b) be an affine nonlinear system on a domain D C R 2 . LetA[u~,u+] be class of admissible controls. Let the following conditions hold: u+(x)

> u~(x)

for all x £ D; < grad bn^-i(o) ¥= °;

< f(x),b(x) >

for all x £ D fl -1(0). Suppose further dD satisfies strong trap free condition. Then the system S ( / , 6) is controllable by A[u~,u+] on D if and only if the digraph T•£,(jj,)(A\u~,u+],D) is strongly connected. T h e idea of t h e proof of Theorem 2.9. is completely the same as that of the proof of Theorem 2.3 We recommend the reader to prove Theorem 2.9 independently. Let t h e vector fields b+(x),b_(x) be integrable on D, i.e., there exist a neighbor­ hood 0(D) and continuously differentiable functions h+(x), h-(x) : O(D) —» R 2 , such t h a t dh±(x) = fi±(x) < [b±(x)]L,dx > for all x e 0(D), where n±(x) ± 0 for all x 6

0(D).

Then necessary conditions for t h e system S ( / , 6) to be controllable by A[u~, u+] on D C R 2 can be formulated in terms of extremums of the functions h+(x), h^(x). T h e o r e m 2 . 1 0 Let £ ( / , b) be controllable on a domain D C R 2 by Let the vector fields b_(x),b+(x) be integrable on D. Then

A[u~,u+].

sup ft+ = sup h+, sup h- = sup h_, 9D Snvj-^O) 9D Dnip-l(0) inih+ = inf h+, inf/i_ = _ inf h_ dD Dn^-^o) 3D Dnv-'fO) and, for every connected component 8 C arg extr[,(h+)

v.-l(o) + 0.

or 8 C arg e x i r s ( / i _ ) , 0n

58 Nonlinear afSne system on a plane We recommend the reader to prove Theorem 2.10 independently using the idea of the proof of Theorem 1.2.

2.4

Dynamic of populations u n d e r e x t e r n a l influence.

The Lotka-Volterra model considered in Example 2.1 is not realistic because the prey growth is unbounded in the absence of predation and the growth rate of the population of one type of species depends only on the density of the population of the other species. To be more realistic these growth rates should depend on both the prey and predator densities. For example, a more realistic prey-predator populations equations might take the form [ ^%■ = = xi(c xi(c — — dx dxxt — — xx22 ■ ■ r(xi)) r(xi)) + + 6i«, 6i«,

1

pp ■

1

g-r(x )) + bb2iu, ^^■ = xX2{-h 2{-h + g-r(x11))

where c, d, h and g are positive constants; 6j, b2 axe real numbers characterizing an external influence on the population dynamic; u is a control and/or an external influence and r(A) is one of the following predation terms;

n(A) = ^ , , 2 ( A ) = ^ , , 3 ( A ) = ^

p

,

where A, B and a are positive constants. Other examples of prey-predator population models are discussed in [36]. PPi, PP2, and PP3 will denote the system PP with r(xi) being replaced by ri(x1), r2(xi) and r3(xi), respec­ tively. In this section controllability of the system PP is investigated. As a first step we construct equilibrium set y - 1 (0) of the system PP. The function 0 , x 2 > 0 } .

59 Global controllability and stabilization Let 62 = 0 and bx =£ 0. Then V-1(0) = {x2 = 0}U{h = g-r(x1)}.

(2.6)

The following proposition gives criterion of controllability of PPV (1/ = 1,2,3) with 62 = 0, 61 ^ 0. Proposition 2.4.1 Let b2 = 0 and 6j ^ 0. The system PPi is controllable on R^. if, and only if, h/g < The system PP2 is controllable on R^_ if, and only if, h/g < The system PP3 is controllable on R^_ if, and only if, h/g
0} has a nonempty intersection with y>_1(0) n R+. (2.6) yields

^- 1 (0)nRi = {h = gr(x1)}. Since n(A) < AjB, r2(A) < A/(2B), for v = 1,2,3,

r3(A) < aA VA > 0 and lim A ^ +00 r„(A) = 0

{x2 = const} r\ 0 iff h/g < A/B;

{x2 = const) n ^ ^ O ) / 0 V const > 0

iff h/g
- 0, then the systems PPi,PP2,PP3 lable on R+.

are not control­

60 Nonlinear Proof. Recall that, for real functions

on a plane

afiine system

i>(t),£{t),

jp{i) ~ £(i) as r —* oo means lim;-,,*, | w = 1. Since lim i?„(A) = A - K ■ h

for

u =

1,2,3,

A—*oo

we obtain, for A — nh ^ 0 and (xi, x 2 ) £ V - 1 ^ ) ' x2 ~

r(cxi — dx?) as xi —► oo. A-nh Thus there is a positive real number Q such t h a t {det (Xl

* ) = consi; x s > 0,x 2 > 0} n ^ ( O )

= 0

(2.7)

for all const > Q. Hence PPi,PP2,PP3 Exercise 2.2).

are not controllable on R^. when A — re ■ h ^ 0 ( see

If J4 —re■fe= 0, then either iJ„(A) = 0 or there exist a positive real number C and a natural number q > 1 such that ^(xa)--^

Xj[

V

i/ = 1,2,3

(2.8)

as xj —* oo. Hence there exists a positive real number Q such t h a t (2.7) hold. Therefore PPUPP2,PP3 are not controllable. Q.E.D. If re < 0, then the system PP2 can be controllable on R^_ ( see Exercise 2.8), but the system PP\ not. P r o p o s i t i o n 2.4.3 If K, < 0, the system PPX is not controllable

on R+.

-1

Proof. Equilibrium set
0, t G R, where et(rV denotes the set {et(z; z G V}. (Hi) an equilibrium x' G R2 is called asymptotically stable ( in the large) if x" is stable and there exists a neighborhood Q of x* (Q = R 2 ) such that lim et(z = x'

t-»+oo

Vz G Q.

Sometimes an equilibrium which is asymptotically stable in the large is called globally asymptotically stable. (iv) an equilibrium x" G R2 is said to attract a domain D C R2 if lim e z = x* Vz G D.

t->+oo

Consider the system T,(f,g) : x = f(x) + ub(x), where x G R2 and f(x),b(x) are C°° vector fields on R 2 , i.e., f(x) and b(x) have continuous derivatives of any order; u is a control value. In this section we are interested in the global smooth stabilization problem. Definition 3.2 A system S ( / , b) is said to be globally smoothly stabilizable at x* G ip'^iO), where _1(0) is equilibrium set o / S ( / , 6), iff there exists a C°° feedback u = v(x), such that x* is the asymptotically stable in the large equilibrium of the closed loop system x = f(x) + v(x)b(x).

71 Global controllability

and

stabilization

Sometimes a system £ ( / , b) which is globally smoothly stabilizable can be called smoothly stabilizable in the large also. T h e next theorem which is, in fact, a corollary of Theorem 2.1 gives us a sufficient condition for a system to be smoothly stabilizable in the large. T h e o r e m 3.1 If equilibrium set V _ 1 (0) of a system £ ( / , b) is connected, < grad(p(x),b(x)

> |v-i(0) ^ 0

and. Vz € R 2 3t € R

such that

etbz £ tp'1 (0),

then £ ( / , 6) is smoothly stabilizable in the large at any point x* £ -1 (0). Then , by Theorem 2.1, there is a coordinate trans­ formation $ : R2 - t R2, z = * ( i ) , such t h a t *(x*) = 0, * is a C°° function, and

4,tt(/)(*) = « ^ r ) ^ +/?{«) ( j ) ,

0 functions.

Vz <E R 2 , 7 ( z ) ^ 0

Vz e R 2 and a ( z ) , 7 ( z ) , / ? ( z ) are C°°

Therefore S ( / , 6) is smoothly stabilizable at x* in the large if, and only if, the cart zi = z 2 , Z2 =

«

is smoothly stabilizable at the origin in the large ( see Exercise 3.11 ). The cart is smoothly stabilized in the large by the smooth feedback of the form u(z) = —kizi — k2z2, where h, k2 are positive real numbers ( for a proof see, e.g.,[39] ). Q.E.D. T h e proof of Theorem 3.1 leads to a designing procedure for a smooth stabi­ lizer. T h e designing procedure can be evolved numerically as follows.

72 Nonlinear afBne system on a piane Numerical stabilizer designing procedure. Let all conditions of Theorem 3.1 be fulfilled. Let < gradtp(x),b(x) > |^-i(o) > 0 (otherwise we can replace x by —x ). Then the value u(x) of a smooth feedback which globally stabilizes the system £ ( / , 6) at x* is calculated as follows. Step 1. (calculation of z-i) z2 is a solution of the equation

V(e-^(x)) = 0, to find the value of z2, we need to go along the trajectory generated by the solution of the problem

V = ~Kv), 7/(0) = x,

if

tp{x) > 0,

and to stop when (^(77(22)) = 0. If (x)r ^

'

x and k%, k2 are positive real values.

The proposed designing procedure gives us the stabilizer such that the closed loop system is equivalent to the cart closed by the feedback u(z) = —kxzi — k2z2. We now consider examples in which the feedback globally stabilizing a nonlin­ ear system can be constructed explicitly. Example 3.1 Our aim is to construct a smooth stabilizer for the system Xj = x 2 - t/>(x x )

x2 = u, where ip(xi) is a. smooth function. Following the proof of Theorem 3.1 we obtain the feedback u(x)

= ^at,p l ^ f o - 4>{xi)) - M*i - »I) - *»(*» - ^(n))

(3.i)

where k\, k2 are positive real numbers. The feedback 3.1 globally stabilizes the system at the point x* = (xj,*/>(xj)). Example 3.2 Consider LV system defined in Example 2.1. By Proposition 2.1.1 the system is controllable on R^. if, and only if, b\ + b\ ^ 0 and b\b2 = 0. Given the point x* 6 tp~l (0) we need to design a feedback which stabilizes LV at x* on R+. Moreover, R+ has to be a right invariant set of the closed loop system, i.e., if an initial state of the closed loop system is in R+, then the next states of the system will remain in R+ for all t > 0. Since b\ + b\ ^ 0 and bxb2 = 0 hold, we design a smooth stabilizer for the system i\ = a\Xx — a2x\x2 + 6i«, (3.2) X2 = —CiX2 + C 2 XiX 2 ,

where &i, a x , a 2 ,Ci,c 2 are positive real numbers. For the system LV with &i = 0, b2 ^ 0 smooth stabilizer is designed analogously. Equilibrium set of the system 3.2 is defined by R+ H {(-ci + c 2 xi)x 2 = 0},

74 Nonlinear affine system on a plane

^{o) n Ri = {(xux2) e Ri, xi = —}. Given a point ( ^ x ^ ) £ V_1(0) n R+> under the nonlinear transformation zi = l n ( — X i )

z2 = lnx 2 ,

(3.3)

u = —o — — {a\Xi — a2xix2) bi

61

the system 3.2 becomes «i

=

t),

i2

=

J1 - 1). Cl(e

(3.4)

Thus we have to stabilize this system at the point (O^nij) the image of R^. under the coordinate transformation z-i = ln(—xi),

over

R2 which is

Z2 = lni2-

Cl

We leave for the reader to prove that the smooth feedback v(z) = - c i

e21 — 1 (fcizi + k2(z2 - lnsj)),

(3.5)

where fci,fej are positive real numbers, globally stabilizes the system 3.4 at (0, ln(xj)) ( s e e Exercise 3.1). Thus (3.3) and (3.5) yield a feedback which stabilizes the system (3.2) at the point (^-, x|) over R+ and R+ is a right invariant set of the closed loop system.

3.2

Stable covering and semiglobal stabiliza­ tion.

In the previous section the system £ ( / , 6) under consideration was equivalent to the cart. This assumption allows us to design a smooth stabilizer. At the same time the equivalence between the system E ( / , b) and the cart implies that the equilibrium set -1(0) of S ( / , b) has to be a, connected regular curve. If ¥?_1(0) is not connected, then in general, a smooth system can not be globally stabilized by a smooth feedback. Example 3.3 It is proved in Section 3.1 of Part II that the system

75 Global controllability

and stabilization

ii

=

sin(x 2 + x 2 ),

x2

=

u

(3.6)

is not stabilized by any smooth feedback at any point x* e 0 the derivative set of points and —xu{t, at

£xu(t,x0)

x 0 ) = f(xu(t,

of time which is defined for all t G R + exists for allt G [0,T] except of a finite

xo)) + u(xu(t,

x 0 ))6(x„(i, x 0 ))

for all t £ [0, T] except of this finite set of points; x u ( 0 , x 0 ) = x0.

76 Nonlinear

affine system

on a piane

Having fixed a feedback u(x) such that u(x) £ PS and t h e solution for (3.7) exists for all x0 £ V C R 2 we obtain the flow e f ( / + u l > ) : V —> R 2 generated by the closed loop system x = f(x) + u{x)b(x). denotes the point into which the flow etV+"V steers x0 and e' 0. e ((/+ u fc)( So )

Definition 3.4 Let Q C R 2 . A system £ ( / , 6) is said to be piecewise smoothly stabilizable at a point x" £ 0 such that e»itf+-i*)( Xo ) G U i l l K .

T h a t means t h e existence of 1 < v(l) < j ~ 1 such t h a t e ^ ' + ^ x o ) G K(i).

78 Nonlinear affine system on a pJane The feedback w(x) - u„(i)(x)

for

i £ K(i) \ {uj£{

K}

and 1^,(!) is an invariant set of the system £ ( / , 6) closed by the feedback u„(i)(x). Moreover the feedback u„(i)(x) stabilizes the system E ( / , 6) at x*^) over V^(!j. Therefore X

UD 6 U ^ "

yields the existence of i„(i) > 0 such that eKW(f+"H

»i>ce,'(W(»,)£uS'^

IKK* obtain An+.ain Thus, after all, we

* ^ ) > ° , * ^ - i ) > 0, ..., 0,tj > 0 such that i et*(rt(/+<M«>>>

o et"("-1)'/+u'-{x%-i{0) ? 0.

The cart garland representing the system (3.9) covers the whole plane, and thus the system (3.9) is controllable on R2. Moreover, differentiating the function V(x) = - ■ ((*! - x\)2 + x 2 )

(3.10)

79 Global controllability

and

stabilization

along the solutions of the system (3.9) closed by the feedback

with

u(x) - -(xi-xl)-ip(x)-k-z2,

x*, fc £ R

and

k >0

yields jV{x{t))

= -k-xl

(3-11)

Therefore, by La Salle's invariant principle (see, e.g., [3]), it follows from (3.10), (3.11) t h a t t h e feedback u(x) globally stabilizes the system (3.9) at the point (xj, 0). Hence for any point ( x j , 0) and for any compact set K C R there exists a stable neighborhood (V,U(x),(xI,0)) such t h a t K C V. We now prove t h a t a cart garland, which is controllable on R 2 , is piecewise smoothly semiglobally stabilizable at any point of its equilibrium set. T h e o r e m 3.3 Let £ ( / , 6) be such that > \v-i(0) ¥" °-

< gradip(x),b(x)

Assume that the system £ ( / , 6) is controllable on R2- Then £ ( / , b) is piecewise smoothly semiglobally stabilizable at any point x* £ (£>_1(0). P r o o f . Given a compact set K C R 2 and a point x* £ _1(0) we have to design a piecewise smooth feedback which stabilizes the system at the point x* £ _1(0) over the compact set K. By Theorem 3.2, we need to construct a stable covering of K in order to prove the existence of a piecewise smooth feedback stabilizing £ ( / , 6) at x* £ v l _ 1 (0) over K. Since < gradip(x),b(x)

> | p -i ( 0 ) # 0,

we have v>j

+ l}nQt].

Thus there is a stable neighborhood (Vj,Wj(x),x*),

where Sj C ^ and {a:*; /j > v > j + 1} fl Qj C VJ. Thus we have constructed a stable covering {(VJ^UJ^X),!*); j = l,2,-..,/i}

such that

KCU%

iVj.

Q.E.D. Thus, by Theorem 3.3, any controllable on R2 cart garland is piecewise smoothly semiglobally stabilizable at any point of its equilibrium set. It is proved in Chapter III of Part II that any controllable multidimensional nonlinear sys­ tem is piecewise smoothly semiglobally stabilizable at a point of its equilibrium set provided the system is locally stabilizable at the point. As we shall see in Chapter 2 and Chapter 3 of Part II, in general, controllability does not imply either global or local smooth stabilization.

3.3

S o m e e x a m p l e s of a d a p t i v e stabilizers.

So far we have been dealing with a well-defined system. In other words, the vector fields f(x),b(x) were known. In this section, uncertain nonlinear affine systems are considered.

81 Global controllability and stabilization Let the system be of the form x = u>(x)f(x) + b{x){/3(x) + a(x)u),

(3.12)

where f(x), b(x) are known continuously differentiable vector fields; w(x), P(x), a(x) are unknown continuously differentiable functions satisfying the following in­ equalities 0 < u_ < u>(x) < u>+, (3.13)

\P(x)\ < p+(x),

0 < a_ < a(x) < a + , where P+(x) is a known continuously differentiable real function and a;_,u>+, ct-,a+ are known positive real numbers. In this section the stabilizability of the system (1) at the points of equilibrium set c^ -1 (0), where -1(0) of the system (1) is connected, < grad |„-i(0j ^ 0 and VzeR2

3< 6 R

such that

etbz 6 ¥>_1(0)-

Then, by Theorem 2.1, for any x" 6 (z) = u> oQ-^z), p{z) = P o ^(z), a{z) = a a i~l{z), (z), a(z), /?(z) satisfy the inequalities (3.13). Thus, if R2 is a controllable foliation generated by £ ( / , b), then for the system (3.12) the problem of a stabilizer design is reduced to the stabilizer design for (3.14).

82 Nonlinear

edSne system

on a plane

Let us construct a stabilizer based on the phenomenon of sliding motion. Sup­ pose f>+ > ^

> 0,

(3.15)

where 0,

where v = min{A;a;_