Advanced Topics in Nonlinear Control Systems

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^Wfl fll3il:*31l3fli Series Editor: Leon 0. Chua

ADVANCED TOPICS IN NONLINEAR CONTROL SYSTEMS Editors

T. P. Leung & H. S. Qin

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ADVANCED TOPICS IN NONLINEAR CONTROL SYSTEMS

WORLD SCIENTIFIC SERIES ON NONLINEAR SCIENCE Editor: Leon O. Chua University of California, Berkeley Series A.

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Preface This work results from the joint collabration of several researchers who are interested in the recent development of nonlinear control theory. For the last fifty years, a number of textbooks and monographs on nonlinear control theory and design methods have been published. For example, phase plane analysis and describing function are well known in classical control theory. In modern control there are those related with optimal control, using differential geometry and differential algebra methods, variable structure control, Hinfinity control and so on. These have been useful in promoting the advancement of automatic control science and technology. Since 1990 there have been a number of new results and contributions in the area. In view of such, the authors of this book have an aspiration to introduce these to interested readers for arousing broader attention and discussion. Since most systems are nonlinear by nature, such study would have its own significance. This book is mainly targeted at researchers and it can also be of interest to automation engineers. It would be of paramount importance that advanced control theories could eventually be adapted for real engineering applications. This book consists of six chapters. Chapter 1 is devoted to presenting the fundamental theory, the main properties and control aspects for generalized forced Hamiltonian systems. Chapter 2 presents recently developed theories of static and dynamic output feedback stabilization methodology, and design of dynamic compensator for nonlinear control systems. Chapter 3 contains new contributions of developing basic principles for the design and construction of controller for continuous finite-time control systems. In Chapter 4 there are descriptions of modelling, control and simulation of a class of hybrid systems. Chapter 5 deals with the robust and adaptive control of nonlinear systems with nonholonomic constraints, with a case study of constrained robot control. Chapter 6 is concerned with chaos control of nonlinear systems. It concludes theory and modelling of resulting chaos, controlling chaos and synchronizing chaos for nonlinear systems with two or even higher than two dimensions. It is hoped that such coverage will provide readers a rather comprehensive review of some important developments of advanced nonlinear control theory in recent years.

T.P.Leung and H.S.Qin

Contents PREFACE CHAPTER 1 GENERALIZED HAMILTONIAN SYSTEMS D. Cheng 1.1 Introduction 1.2 From Newton's second law to Hamiltonian equation 1.3 Symplectic manifold and Poisson manifold 1.4 Pseudo-Hamiltonian systems and controlled pseudo-Hamiltonian systems 1.5 Pseudo-Poisson manifold 1.6 Integrability 1.7 <»-manifold 1.8 Structure group and its algebra 1.9 Spectrum 1.10 Structure invariance 1.11 Stabilization of excitation control 1.12 Stabilization and Hw control for dissipative Hamiltonian systems 1.13 Summary

41 48

CHAPTER 2 CONTINUOUS FINITE-TIME CONTROL T. P. Leung and Y. Hong 2.1 Introduction 2.2 Classes of finite-time feedback 2.3 Preliminary results 2.4 Finite-time state feedback 2.5 Finite-time observer 2.6 Output feedback 2.7 Convergent rate 2.8 Applications in robot control

52 55 58 62 70 73 76 78

1 4 8 12 16 19 23 26 30 31 37

viii

Contents

2.9 Robust issues 2.10 Conclusions

82 85

CHAPTER 3 LOCAL STABILIZATION OF NONLINEAR SYSTEMS BY DYNAMIC OUTPUT FEEDBACK P. Chen and H. Qin 3.1 Introduction 3.2 Preliminaries 3.3 Stabilization of observable systems 3.4 Stabilization of partially linear composite system 3.5 Stabilization of a special class of systems

88 90 98 104 117

CHAPTER 4 HYBRID CONTROL FOR GLOBAL STABILIZATION OF A CLASS OF SYSTEMS J. Zhao 4.1 Introduction 4.2 Hybrid systems and hybrid control 4.2.1 Hybrid systems 4.2.2 Hybrid control 4.2.3 Switched systems 4.3 Quadratic stability of homogeneous switched systems 4.3.1 Completeness 4.3.2 Quadratic stability 4.4 A hybrid controller for the cart-pendulum system 4.4.1 Background and problem statement 4.4.2 Hybrid controller 4.4.3 Stability analysis 4.4.4 A modified algorithm 4.5 Concluding remarks

129 129 130 130 130 133 133 134 140 140 142 145 152 157

Contents

CHAPTER 5 ROBUST AND ADAPTIVE CONTROL OF NONHOLONOMIC MECHANICAL SYSTEMS WITH APPLICATIONS TO MOBILE ROBOTS Y. M. Hu and W. Huo 5.1 Introduction 5.2 Dynamic model of nonholonomic mechanical systems 5.3 Robust control design based on SMC 5.3.1 SMC based on high-order sliding mode 5.3.2 Dynamic SMC based on high-order sliding mode 5.3.3 Robustness design in the presence of uncertainties 5.4 Applications to the nonholonomic mobile robots 5.4.1 Design of dynamic sliding mode controller for the output tracking of nonholonomic mobile robot 5.4.2 Design of adaptive sliding mode controller for the output tracking of nonholonomic mobile robot 5.4.3 On-line implementations 5.5 Conclusions CHAPTER 6 INTRODUCTION TO CHAOS CONTROL AND ANTI-CONTROL G. Chen, J.-Q. Fang, Y. Hong and H.-S. Qin 6.1 Overview of chaos, chaos control and anticontrol 6.1.1 A brief description of chaos, chaos control and anticontrol 6.1.2 Some basic rationale for utilizing chaos and complex nonlinear dynamics 6.1.3 A brief summary 6.2 Challenges in chaos control 6.2.1 Chaos control: what does it mean? 6.2.2 Challenges in chaos control 6.2.3 Some distinct features of chaos control

ix

161 162 166 166 172 174 182 182 185 187 188

193 194 194 196 197 197 197 199

Contents

3

4

5

Representative chaos control methods 6.3.1 The OGY parametric-variation control method 6.3.2 Feedback control of nonlinear systems 6.3.3 On time-delayed feedback control Anticontrol of chaos: chaotification 6.4.1 Introduction 6.4.2 Problem description and the anticontrol algorithm 6.4.3 Verification of the anticontrol algorithm 6.4.4 External noise control and anticontrol of chaos Some concluding remarks

201 201 206 217 223 223 223 226 229 237

Chapter 1

Generalized Hamiltonian Systems Daizhan Cheng Lab. of Systems Science, Institute of Systems Science Chinese Academy of Sciences, Beijing 100080, P. R. China

1.1 Introduction This chapter investigates the generalized (controlled) Hamiltonian systems. Hamiltonian System is originally from mechanical systems. For a single particle, the Newton's second law says that F F = Ma, equivalently x =— M

(1.11)

where F is the force acting on the particle; M is the mass of the particle; x is the displacement and a is the acceleration. When a conservative mechanical system with n degree of freedom is considered, the Euler-Lagrange equations can be deduced from Newton's second law. They are the following well-known set of second-order differential equations

where

d dL -r

dL r —=Fj,

dt

d(

8qt

. , i = !,•••,n

H

L{q,q) =

T\q,q)-V(q)

1

(1.1.2)

2

D. Cheng

is the Lagrange function of the system, which is the difference of the kinetic energy T(q,q)and the potential energy V{q);q\ and qi i = \,--,n are the generalized coordinates and the generalized velocities respectively. Fj,i = \,---,n are the generalized external forces. The kinetic energy can be expressed as T(q,q) =

±qTM(q)q

for some positive-definite matrix M(q) called a generalized mass. In (1.1.2) we may consider Fj as the controls or inputs of the system (Crouch et. al, 1987), say Fj=uiJ = l,--,n,m and Fj=0,i = m + l,---,n. Then (1.1.2) is called a controlled Lagrange system. By introducing the generalized momenta dL . dqt and the Legendre transform, which produces the Hamiltonian function as n H(q,p)= I.Piqj-L(q,q) i=l the Lagrange equations can be transformed into the set of first-order differential equations .

dH (1.1.3)

pi=—r— + Fii = l,--,n 3qt '' where H(q,p) = ^pTM(xrlp

+ V(q)

is the total energy of the system. These equations are used to describe the dynamics of a very large class of conservative physical systems.

Generalized Hamiltonian

Systems

As for controlled Lagrange system, we assume Fj=Uj,i = l,---,m and i-m + l,---,n . Denote x = (p,q)as

3

Fj=0,

the state variable, the controlled Hamitonian

systems can be expressed alternatively as

x=

0

/

-/

0

nxm V// + 0

SA

(1.1.4)

M

V wy

As a formal generalization, a generalized controlled Hamiltonian system may be assumed as x = M(x)S/H+ yi=hi{x),i

m

Ig/M/ z'=l = \,---,m

(1.1.5)

In recent research articles (Crouch et. al, 1987; van der Schaft 1996; Cheng et. al, 2000b, etc.), for different purpose the structure matrix M(x), input channels g,and the outputs yj are chosen in different way. The choice of M(x)

will be dis-

cussed in detail later. Two of the commonly used ways for setting g,- and yj are 1. gj are general vector fields and y = g V//,equivalently yi=Lg.HJ 2. gj=M(x)VHj

= \,---,m

for some input Hamiltonian f u n c t i o n s / / / a n d yj=Hj,i

=

\,---,m. In fact, such a kind of generalized Hamiltonian systems cover a very general dynamic systems, which are used in Physics, Mechanics, chemical reaction, power systems, biological evolution equation etc. In this chapter we first provide a systematic geometric frame for the systems. The pseudo-Poisson manifold and the co -manifold are proposed as the state-space of the generalized controlled Hamiltonian systems (or controlled pseudo-Hamiltonian systems). A Lie group, called N -group, and its Lie algebra, called N -algebra, are introduced for the structure analysis of the systems. Some properties, including spectrum, structure-preservation etc. are investigated. As an example the theoretical results are applied to power systems. The stabilization of excitation systems is discussed. Then the general stabilization and H^ control of the pseudo-Hamiltonian

4 D. Cheng

systems are studied. Finally, the problem of Hamiltonian realization of dynamic systems is discussed. It intends to answer the question: What kind of systems can be converted into a (pseudo-) Hamiltonian system? Then the Hamiltonian approach can be used to study and manipulate the system.

1.2 From Newton 's second law to Hamiltonian equation It is well known that the Hamiltonian systems are initially from Mechanics (Abraham et. al, 1978; Marsden et. al, 1994). This section outlines the deduction from Newton's second law to Lagrange equations, and then to Hamiltonian equations. Consider a mechanical system with N Particles. Using xj,i = l,---,n = 3N as the Ca.tesian coordinates of N particles. Assume the system has 5 freedom and qi,i = l,--;s are the generalized coordinates. Then (1.2.1)

Xi=Xi(qi,---,qs),i = l, The velocity of x\ is

a=\d(ia Then the acceleration is s a=\

8XJ

Ma)

qa +

dxj

s

la

=I

d2Xj

a=\ fi=\3qaSqj3

. . dx; 1aQ/3+^—(ia dq,a

(1.2.2)

Next, we consider the forces. Let U be the potential function. Then the potential force can be expressed as

du_

Fi = ~ dxj

(1.2.3)

The constrained force Q is consistent with the motion. So ZC/&,-=0

(1.2.4)

Generalized Hamiltonian

Systems

5

Particularly, let the motion be along qr . Then (1.2.4) becomes n

fa.

j=l

dqr

(1.2.5)

Consider the kinetic energy T

1 "

* & / . * & /

2/=i

«=l^«

.

1 * 2

^=1^)5

* . .

a=l/?=l

»

dxj 8xj 5

i=l

(1.2.6)

f c ^

It follows that 8T

1 £

3

2


.

^=i

"

&,- & ,

/=i

1 ' . 2

dqrdqp

a=\

" i=\

&,- dxt °1r

d

1a

(1.2.7) a=l

,=1

ag> 3 ? a

and its derivative with respect to time is d dT dt 8qr

* •• £

dxj fy ,

qa I * i r - - — +

a

• =1 dqr dqa 3 JC; dxj . i, " 3x; 3 Xj . — qp~+q I lOTz-r-*--—^—9/9 TS < ?— a «9^ ^ fyr ^=l,=l dqadqpdqr

s n qa ' ^ I. = l i =I l

(1.2.8)

Similarly, we can calculate dT * l . -^-^YLqalB dc lr ap

.

d2xj dxj Umi———-— dc i=l lrdqpdqa »

(1.2.9)

Subtracting (1.2.9) from (1.2.8) yields d dT dt 8qr

dT dqr

s

=1 a

..

»

dxj dxj

.

I

.

»

d^xt

dxj

(1.2.10)

6

D. Cheng

dxUsing (1.2.5), we can multiply (1.2.2) by — - a n d sum up with respect to /' to get 8qr the same result as the left hand side of (1.2.10). Hence we have d 8T dt 8qr

8T _ 8U 8qr 8qr

(1.2.11)

Define a Lagrange function as L = T-U Then the Lagrange equation is obtained as d 8L 8L = 0,r = l,2,---,s dt 8qr 8qr

(1.2.12)

Next, we convert the Lagrange equation to the Hamiltonian formalism: Define the generalized momenta as 8L Pi=-T—,i = l2,---,s 8qr make a coordinate change: (qi,qi)-*(qj,Pj)and

(1.2.13)

introduce the Hamiltonian function

H(q,p)= ZPjqj-L(q,p) 7=1

(1.2.14)

Note that q^ and pj are independent variables and 9/=9/(9/>/>/)> the following computations: 8H

.

I ' _ fyj 8L Sqj Pj dPi ~ 3qj 8pi

=m

m e nw e n a v e

(1.2.15)

and dH = 8

1i

y

j=\

.^J__9L__ J d

1i

da

i

y

SL dqj 5

y = l %
SL _ . da

(1.2.16)

i

The last equality is from (1.2.11) and (1.2.12). Thus the Lagrange equation (1.2.12) is equivalent to the Hamiltonian equation

Generalized Hamiltonian

dqi _ dH dt dpj dPi dt

Systems

7

(1.2.17)

3H . dqi

For a smooth function H(x) e C°° (/?"), we denote dH ox\

dH(x)

8H oxn

= (—

,•••,—-)

and VH(x) = (dH(x))T. Denote x = (q, p), then (1.2.17) can be expressed as x = JVH,

where J

' 0, 1^

(1.2.18)

Write the kinetic energy (1.2.6) as \

s

s

\

T

(1.2.19)

where "

dxj

dxj

ga,p=^mi^—-z— /=1 8<Ja d1p and M(q) = (gap)

is called the generalized mass. It is clear from (1.2.19) and

(1.2.13) that

3q Then (1.2-14) yields that H = T + U is the total energy of the system. From (1.2.18) we have H = dHJVH = 0 which is the conservation law of the energy.

(1.2.20)

8

D. Cheng

1.3 Symplectic manifold and Poisson manifold This section provides a brief review for some related topics in Symplectic Geometry. It is well known that the symplectic manifold is a suitable geometric structure for investigating a Hamiltonian system (Abraham et. al 1978, Linermann etal., 1987). But when the structure matrix is singular, Poisson manifold, as a generalization of the symplectic manifold, provides a better frame for the generalized controlled Hamiltonian systems (Olver 1986, van der Schaft, 1996). Definition 1.3.1 Given an n-dimensional manifold N. A symplectic structure, co, is a non-degenerate closed skew-symmetric covariant tensor field. The manifold N equipped with a symplectic structure is called a symplectic manifold, denoted by (N,a>). In a local coordinate chart, (U, x), co may be expressed through the local coordinates as co{X, Y) = XTM(0Y,

\/X, Y e Cx (N)

where Ma is a non-singular skew-symmetric matrix, called the structure matrix of co. It is obvious that a symplectic manifold has even dimension, because a nonsingular skew-symmetric matrix can be non-singular only if its dimension is even. Definition 1.3.2 (Darboux' Theory) A two-form co is a symplectic structure on a 2n dimensional manifold N, iff for each psN there exists a local coordinate chart, {Up, (x, y)), such that co=YJdxiAdyu /=1

(x,y)sUp

Note that (1.3.1) means that the structure matrix, Ma in(Up,(x,y))

as

(1.3.1) ,is expressed

Generalized Hamiltonian

Systems

9

( ° !n) V 1n

u

y

Definition 1.3.3 A Poisson bracket on a smooth manifold N is a smooth mapping Cm (N) x C°° (TV) —> C°° (N), with the basic properties: PI. Bilinearity; {cF + c' P, H] = c{F,H}+ C' {P,H\ {F, CH + C'P}= C{F, H] + c' {F, P} for constants c,c'eR P2. Leibniz'Rule: {F,GH}={F,G}H +G{F,H}; P3. Skew-Symmetry: {F,H}=-{H,F}; P4. Jacobi Identity: {F, {G, H}}+ {G, {H, F}} + {H, {F, a}} = 0 A manifold N equipped with a Poisson bracket is called a Poisson manifold, the bracket defining a Poisson structure on N . Definition 1.3.4 Let N be a Poisson manifold. A smooth, real-valued function C:N^>R is called a Casimir function if the Poisson bracket of C with any other real-valuedfunction vanishes identically, i. e. \C, H} = Ofor all H :N —> R. Definition 1.3.5 Let N be a Poisson manifold and H :N —» R a smooth function. The Hamiltonian vector field associated with H is the unique smooth vector field VH on N satisfying VH(F) = {F,H}=-{H,F},

VF-.N-+R

(1.3.2)

The equations governing the flow of VH are referred to as Hamiltonian equations for the "Hamiltonian" function H . Proposition 1.3.6 Let N be a Poisson manifold and F, H: N —> R be smooth functions with corresponding Hamiltonian vector fields VF,VH. The Hamiltonian vector field associated with the Poisson bracket of F and H is, up to sign, the Lie bracket of the two Hamiltonian vector fields: V{F,H}=-[VF,VH]=[VH,VF]

(1.3.3)

10 D. Cheng

Let x = (x , • • •, xm) be local coordinates on M . These basic brackets jV[x) = \c\xj\

i,j =

l,-,m.

are called the structure functions of the Poisson manifold N relative to the given local coordinates, and serve to uniquely determine the Poisson structure itself. For convenience, we assemble the structure functions into a skew-symmetric m x m matrix J(x), called the structure matrix of N. So the Poisson bracket in local coordinate takes the following form {F,H} = dFJ(x)VH(x) Let J(x) = (J,J (x)) be an mxm

Proposition 1.3.7

matrix of functions of

x = {x ,---,xm) defined over an open subset M
i,j = !,-••,m.

(b) Jacobi Identity: Ilf i l dijJ k 1=1

+ JkldiJ'J

+ jJldiJki\=

0, i,j,k =

\,---,m,\/xeN.

Proposition 1.3.8 Let N be a Poisson manifold and x e N. Then there exists a unique linear mapping 71 = 71 t:T*(N^x^T(N%

from the cotangent space of N at x to the corresponding tangent space, such that for any smooth real-valuedfunction H :N —> R, 7T{dH{x)) = VH\x

Generalized Hamiltonian Systems 11 Definition 1.3.9 Let Nbe a Poisson manifold and xeN . The rank of N at x is the rank of the linear mapping n C:T*(N)\X^T(N)\: Proposition 1.3.10 The rank of a Poisson manifold at any point is always an even integer. Definition 1.3.11 Let M and N be Poisson manifolds, a Poisson map is a smooth map :M —» N preserving the Poisson brackets:

\F^'HAM

= F //

( ' W'

\tF,H:N-*R.

Proposition 1.3.12 Let N be a Poisson manifold and VH a Hamiltonian vector field. For each time t, the flow exp(tVfj ):N^>H determines a (local) Poisson map from to itself. Corollary 1.3.13 IfVff is a Hamiltonian vector field on a Poisson manifold N, then the rank of N at exp(tVH)x is the same as the rank of N at x for any tsR. The following is a generalized version of the Darboux' Theorem. Theorem 1.3.14 (Darboux' Theorem) Let N be an m-dimensional Poisson manifold of constant rank2n < m everywhere. At each XQ e N there exist canonical local coordinates (p,q,z) = \p , — ,p ,q ,—,q ,z , - , z j, in terms of which the Poisson bracket takes the form f

{F,H}= Z

8F_8H__dF_dH_

i=\ dj dp1

dp1 dq\

The leaves of the symplectic foliation intersect the coordinate chart in the slices {z = c\,---,z =c[} determined by the Casimir functions z.

12

D. Cheng

The following Theorem shows the relationship between Symplectic structure and the Poisson structure: Theorem 1.3.15 Let (N,a>) be an n-dimensional symplectic manifold. Define a structure matrix W(x) = M~ (x), which determines a bracket as {F(x), GO)} = dF(x)W(x)VG(x),

VF(JC), G(x)

eCx(N)

Then it defines a Poisson bracket on N. Conversely, Let N be a Poisson manifold with the structure matrix of the Poisson bracket W{x). If W(x) is non-singular, we define M^x)

= W~ (x)as a struc-

ture matrix of a tensor field a . Then (N,a>) is a symplectic manifold.

1.4 Pseudo-Hamiltonian systems and controlled pseudo Hamiltonian systems In very recent research works (Maschke et. al, (1998); Sarlashkar et. al, (1996); Ortega et. al (1998)), the stabilization of the forced Hamiltonian systems with dissipation has been investigated. A new approach was proposed, which suggested an energy-based Hamiltonian function as a candidate of the Lyapunov function. The new model suggested by them is x = (J(x)-R(xy)—~ + gu ox jdH y=g dx

(1.4.1)

where J(x) is skew-symmetric and R(x) > 0 is symmetric. Based on their work and some practical examples (Cheng et. al, 1998) the following generalized controlled Hamiltonian system model is proposed x = M(x) y=g

dx BH dx

\- gu (1.4.2)

Generalized Hamiltonian

Systems

13

where M{x) is an «x n matrix with the entries of functions of x . To begin with, we call the following system the pseudo-Hamiltonian system: x = M(x)VH, xeN

(1.4.3)

where Ar is a manifold, M(x) is called the structure matrix. It is known from Linear Algebra that any matrix with constant rank can be uniqueiy decomposed as M(x) = J(x)-R(x)

+ S(x)

(1.4.4)

where J(x) is skew-symmetric and both R(x) > 0 and S(x) > 0 are symmetric and positive semi-definite with the rank of R and the rank of S as the numbers of negaT

tive and positive eigenvalues of M + M respectively. Plugging this decomposition into (1.4.3) yields the energy flow as dH ,dH.TD,.dH ,dH.Tc,.m at ox ox ox ox The first term of (1.4.5) is the dissipation, that is, the energy consumed by the system. The second term is the energy produced by an internal energy source. Now assume the matrix M(x) in (1.4.3) has the decomposition (1.4.4). If S(x) = 0.(1.4.3) becomes x = (J(x)-R(x))VH,

xeN

(1.4.6)

which is called a dissipative Hamiltonian system. If both R(x) = 0 and S(x) = 0, (1.4.3) becomes x = J(x)VH, xeN

(1.4.7)

which is called a conservative Hamiltonian system. Comparing (1.4.7) with (1.2.18), a natural question is: when (1.4.7) can have a constant structure matrix expression? If around any point xeNthere exists a coordinate chart (Ux,x)such that (1.4.7) can be expressed locally as ' 0 / (A x = -7 0 0 0 00

VH,X<EUX

(1.4.8)

14 D. Cheng then (1.4.7) is said to be a Poisson-Hamiltonian system. Following theorem (Olver 1986) shows when the system (1.4.7) is a Poisson-Hamiltonian system. Theorem 1.4.1 (Darboux theorem) System (1.4.7) has a Poisson-Hamiltonian realization iff the elements of the structure matrix M(x) = J(x) satisfy the following condition

z

1=1

Smjk

mU -£-

oxf

dmiri

dm

dxl

dx[

ij

+ mji —f + mki —f-= 0 J

(1.4.9)

If in addition, J(x)is nonsingular, then (1.4.7) becomes a standard Hamiltonian system (1.2.18). Correspondingly, (1.4.2) is called a controlled pseudo-Hamiltonian system; (1.4.1) is called a controlled dissipative-Hamiltonian system; If in (1.4.1) R(x) = 0, it is called a controlled conservative Hamiltonian system. If in addition, J has a local constant realization, it is called a controlled Poisson-Hamiltonian system. Finally, as the constant J is non-singular, the system is called a standard controlled Hamiltonian system. For the controlled pseudo-Hamiltonian system (1.4.2), the energy flow becomes dH .8H.TD..dH m.TQ..dH T — = - ( — ) * ( * ) — + ( — ) S(kx)— + uty (1.4.10) dt ox ox ax ox Comparing it with (1.4.5), the last term is the external energy supply (van der Schaft 1996). The following is an example of the controlled pseudo-Hamiltonian system. It is basical'y from Escobar et. al (1999) with a mild modification by adding an internal source. Example 1.4.2. (Cheng et. al 1999) Consider a boost converter with internal source. The system is described in Fig. 1.1 Let denote the total magnetic flux of the inductor, q the electrical energy stored in the capacitor, Hm (^) the magnetic energy stored in the inductor and Hm {) the electric energy stored in the capacitor. The system Hamiltonian function is 2 ±2 H = H„ + Hm = 1 m 2C 2L

Generalized Hamiltonian

Systems

15

Ms— 1

Fig. 1.1: Boost converter with internal source The system then becomes fi\

4>

UQ

W

-1^

1 0

0 us +

0 0

0

0 MR

V// +

(1.4.11)

which has the general form of (1.4.2). For system (1.4.11) we consider the stabilization problem for a constant control u . Ths technique developed in Maschke et. al (1998) is still applicable provided we can find matrix P(x) (or P(y)) such that gP(x)gT =S(x), (or gP(y)gT: •

S(x))

If the original problem is solvable, then we can choose a state (output respectively) feedback control u = u - Py to stabilize the system at x . The classical Lyapunov function approach can solve the stability problem of the system only for one equilibrium point. Because the Lyapunov function is independent of the system. From the latter discussion, you can see that the new approach can deal with the stability of more than one working points simultaneously. Because the new Lyapunov function is the Hamiltonian function of the system, which is obviously system-depending. In fact, the stable working point is not unique for power systems, which can run at several different working points. So the new approach is

16 D. Cheng more suitable for the systems with multi-stable working points, such as power systems.

1.5 Pseudo-Poisson Manifold Recall the controlled pseudo-Hamiltonian system (1.4.2). We intend to use M(x)as a structure matrix, which may neither be skew-symmetric nor integrable (i.e., locally constant). Definition 1.5.1 Let N be an n dimensional manifold. 1. A pseudo-Poisson bracket is a mapping Cco(Ar)xCco(Ar)i-> C c o ( # ) , satisfying PL bilinearity andP2. Leibniz" rule.} (Refer to Section 1.3 for standard Poisson bracket.) 2. A pseudo-Poisson bracket is said to be symmetric if \F,G) = [G,Fj ;skewsymmetric if{F,G)=-{G,F},for

any F,G e

CX(N).

Now from Leibniz' rule we may define for each F e Cm(N) a vector field, denoted by Xp and called the generalized Hamiltonian vector field determined by Hamiltcnian function F, as L

XFG = {G,F\

\/GGCX(N)

(1.5.1)

It is easy to show that XF is a well defined vector field by verifying Lx

is a

well defined Lie derivative. Since {•,-} is a bilinear form, it can be expressed uniquely as {•,•} = {,-}j + {-,-}s where {-,-}j and {,]s are skew-symmetric and symmetric pseudo-Poisson brackets respectively. ii

n

Now consider the symmetric {,-}s. Let Xp = X 4 i=l L

xF(.Xj) =

It follows that

4j=^j,F\s

d &

• Then i

Generalized Hamiltonian

{G,F}s=l^=l{Xi,F}s^,•=1

&

/

,=i

Systems

17

(1.5.2) &i

Using (1.5.2) for G = F and F = x, yields

{Xj,F}s={F,xj}s= Z {r,-,xy} —

;=i

fy

Plugging it into (1.5.2) yields f

>

{G,F}S=Z

n

n i

) dG dF

{*,,,,} | i | L (1.5.3) /=ly=l ° Mi dxj Similarly, we can show that (1.5.3) is also true for skew-symmetric pseudoPoisson bracket. By bilinearity, (1.5.3) holds for any generalized Poisson bracket. Now we define the structure matrix M(x) of the pseudo-Poisson bracket by its entries as my = \xj,Xj\. Then (1.5.3) can be expressed as S

{G,F}=dGM(x)VF

(154)

One sees that as in the standard Poisson bracket case, the pseudo-Poisson bracket is also determined uniquely by its structure matrix. Recall the controlled pseudo-Hamiltonian system (1.4.2). Now we can connect it with a pseudo-Poisson bracket by taking M{x) as the structure matrix of the pseudo-Poisson bracket. An alternative way to define a pseudo-Hamiltonian system is in a ' reverse way". That is, assume a pseudo-Poisson manifold is given, mimic to the coordinate free definition of the standard Hamiltonian systems (Abraham et. al 1978) we give the following: Definition 1.5.2 1. A pseudo-Poisson manifold is a manifold equipped with a pseudo-Poisson bracket, as defined in Definition 1.5.1. 2. A generalized autonomous Hamiltonian system is a triple (N,{,),H). Its dynamic expression is x = XH . Under this notation the controlled pseudo-Hamiltonian system (1.4.2) can be expressed as

18

D. Cheng

\x =

XH+gu (1-5-5)

T

y

b=g v// Now the question is when will the input channels gt in (1.4.2) be also the generalized Hamiltonian vector fields? The following is an immediate consequence of the definition: Proposition 1.5.3 In (1.4.2) gj are also the generalized Hamiltonian vector fields iff gi(x) = M(x)^i(x)T

(156)

where £,(x) are closed. Moreover, (1.5.6) is coordinate independent. Proof. If (1.5.6) holds, by Poincare's lemma there exists a smooth function Hi such that £j =dHj. By definition it is easy to see that gt(x)=XH

. (If the state

manifold is simply connected the expression is global. Otherwise, it is local.) To see that it is coordinate independent, let y = t/>(x) and its Jacobi matrix be J. Then as a structure matrix we have it in the new coordinate frame y as M = JMJT

(1.5.7)

At the same time we have y = Jx = JMVxH+ rp

m Y.JMWxHt i=\ m

„

= JMJ1 V H + I KJMJTV

_

m

_

Hi = MVH + £ M V # •

i=l

i=l

Assume the conditions of Proposition 1.5.3 hold, then equation (1.5.2) can be expressed as m x = xHr, + £ Xu.u. 0 "'U' t~x yi = {H,Hi)„i=\,-, m

(1.5.8)

Generalized Hamiltonian

Systems

19

which is a coordinate free expression of a controlled pseudo-Hamiltonian system. Remark 1.5.4 When the structure matrix M(x) is skew symmetric and satisfies the equation (4,9), the pseudo-Poisson bracket becomes standard Poisson bracket In this case we have Jacobi identity in both Poisson bracket form or Lie bracket form. Precisely, let F, G, H be three smooth functions. Then {F,{G,H}}+{G,{H,F}}+{H,{F,G}}=0

(1.5.9)

Equivalently, [XF,[XG,XH\+[xG,[xH,XF^+[XH,\xF,XG]rQ

(1.5.10)

1.6 Integrability We consider the integrability of the distribution generated by the generalized Hamiltonian vector fields. First, let AM=span{col(M(x))}

(1.6.1)

Using (1.5.7), it is obvious that AM is a well defined distribution. Similarly, the unique decomposition of Massures that AJ,AR,AS,AJ_R,AS_R:=AP, spanned by the columns of the corresponding matrices, are all well defined distributions. Note that each column of them can not be considered as a vector field because it is coordinate-depending. Proposition 1.6.1 For a given pseudo-Poisson bracket with its structure matrix M(x) set A = span{XH HeCx(N)}

(1.6.2)

A(x) = AM(x)

(1.6.3)

Then

Proof. Since each xeA{X) can be expressed locally as X = M{x)VH e span{col(M(x))}

20

D. Cheng

AM(X)Z}A{X)

. Choosing Hi{x) = xi, it follows that the z'-th column of

which is M(x)VHj(x)eA(x).

M{x), •

Definition 1.6.2 A pseudo-Poisson bracket is integrable if A is involutive and with constant dimension. It is strongly integrable if at each xe.N there is a coordinate neighborhood Ux such that on Ux the structure matrix is a constant matrix, i.e., M{x) = M. Remark 1.6.3 1. The condition "constant dimension" can be removed by using generalized Frobenius theorem and then consider the foliation and leaves produced by A. For statement ease, we assume this regularity through the paper. 2. Darboux theorem assures that the Poisson bracket is strongly integrable. Precisely, under canonical coordinate charts its matrix has the form as 0

10)

M(x) = J = - 7 0 0 0 00

(1.6.4)

Particularly, if M(x) is nonsingular, then W{x) = M(x) determines a symplectic form on the underlying manifold N. Then N with this symplectic form becomes a symplectic manifold (Refer, e.g., Olver, 1986). Definition 1.6.3 A function {x) is called a Casimir function for a pseudoPoisson bracket if {
(1.6.5)

From the definition it is obvious that <j> is a Casimir function, iff, LXtt,=0,VXeA

(1.6.6)

Observe that for the standard Hamiltonian systems or the generalized Hamiltonian systems over the Poisson manifolds the strong integrability is assured. Then the assumption of strong integrability for pseudo-Poisson bracket is natural. In fact, strongly integrable pseudo-Poisson bracket satisfies the Jacobi identity trivially. But it may not be skew-symmetric. The following proposition shows this character clearly:

Generalized Hamiltonian

Systems

21

Proposition 1.6.4 Assume a pseudo-Poisson bracket is strongly integrable and under a canonical coordinate chart the structure matrix is M = J + P, where J is skew-symmetric and P = -R + S is symmetric. Then X

{F,G}

+ X{G,F)

= 2M{Hess{G)PVF + Hess(F)PVG)

(1.6.7)

Proof. X

{F,G} = MV{F, G) = MV(dFMVG)=M(Hess(F)MVG

+ Hess(G)MT VF)

=M(Hess(F)JVG + Hess(F)PVG) - Hess(G)JVF + Hess{G)PVF) While the Lie bracket of Xp and XG is [XG, Xp ]= [ANG, MVF]=M(Hess(F)MVG + Hess{G)MTVF) = M(Hess(F)JVG - Hess(G)JVF + Hess(F)PVG - Hess(G)PS>F) It follows that X

{F,G] + lxF >XG ] = 2M(Hess(G)PVF

(1.6.8)

Similarly X

{G,F} + [XG,XF]

= 2M{Hess(F)PVG)

(1.6.9)

(1.6.8) and (1.6.9) yield (1.6.7). D Next, we consider the decomposition of a pseudo-Poisson bracket. Denote by {v}j the pseudo-Poisson bracket with J as its structure matrix. Thus

{AM ={v}y +{v,}/. ={vb -{v}* + {vk

d-6-10)

Proposition 1.6.5 If'{•,-} is strongly integrable, then {;-}j is a Poisson bracket. Proof. Note that under the canonical coordinate chart of the strongly integrable pseudo-Poisson bracket {,-}j has skew-symmetric and constant structure matrix. The conclusion follows immediately. D The advantage of this Proposition is: In general we don't know whether a pseudo-Poisson bracket is strongly integrable or not. Then we can first verify whether the Mj satisfies the Jacobi identify. If the answer is "no", the pseudo- Pois-

22

D. Cheng

son bracket is not strongly integrable. Otherwise, we may search a transformation y = y(x) with the Jacobian Jy satisfying T JyJJy

Y =J,JyP(x(y))Jy

= COYlSt.

For the further discussion, we make the following assumption A\:ApcA and AjcA. Remark 1.6.7 7. It is easy to see thatAl is coordinate free. 2. Al is true for Poisson bracket. Al holds when M(x)is non-singular. (The systems discussed in Maschke et. al 1998, etc. meet this requirement.) Proposition 1.6.6 If A is integrable and Al holds, then there exists for each pe M a local coordinate chart such that (1.5.8) can be expressed as x = M(x,z0)VxH

+ M(x,z0)

m IV^//^/=1 (1.6.11)

i =0

Proof. Since A is integrable, we can find at each p a local coordinate chart, (x,z),

such that A-spam—,i=l,---,dim(A)\.

[dxt

Note that under such a canonical

j

coordinate chart the structure matrix M can be expressed as r

M{x) =

Mx(x,z) 0

M2(x,z)) 0

It is easy to prove that the assumption Al is coordinate free. It implies that M 2 (x, z) = 0 . (1.6.11) follows immediately. • The coordinate frame, (x, z), in the above proof is said to be friend to A. In fact, (1.6.11) says that each leave of A is XH invariant and the restriction of a generalized Hamiltonian control system on each leave is still a generalized Hamiltonian system.

Generalized Hamiltonian

Systems

23

A particular important case is M = J . It happens at some controlled Hamiltonian systems (van der Schaft, 1996). Then like in standard Poisson manifold case, each leave is a minimum pseudo-Poisson submanifold.

1.7 co -manifold In this section we give a tensor field structure to the state manifold of the generalized Hamiltonian systems and then connect it with the pseudo-Poisson bracket. First, assume the structure matrix of the pseudo-Poisson bracket related to the generalized Hamiltonian system is non-singular. Definition 1.7.1. 1. Let co be a non-singular covariant tensor field of order 2 on a manifold N. Then we call (N, co) an co -manifold. 2. Given an co -manifold{N, co) and let X e V(N) andb: V(N) -» K* (N) be related by ix(co) = co(X,-) = ju We define two isomorphisms: #:V (N)^>V(N)and:V(N)-+V X = /i#,and

n = Xb

(N) as (1.7.1)

Moreover, if X = (dff) , X is denoted as X = XH. 3. ro is integrable if for each xeN there exists a local coordinate chart of x such that the matrix expression of co, denoted by W^, is constant. In other words n n

CO = 2Z Y^MydXi ® dXj My e ^ Now we define a pseudo-Hamiltonian system via an co -manifold as follows. Definition 1.7.2 A pseudo-Hamiltonian system is a triple (N,co,H), where (N,a>) is an co -manifold and H is a smooth function called the Hamiltonian of the system. Its dynamical form is x = XH

(1.7.2)

24

D. Cheng

Assume an co -manifold is given. We define a pseudo-Poisson bracket by setting the structure matrix, M(x), on each coordinate chart as M(x) = W~T(x)

(1.7.3)

where W(x) is the matrix expression of co . We have only to show that the pseudo-Poisson bracket is well defined. Proposition 1.7.3 The matrix M(x) defined by (1.7.3) is a structure matrix. Proof. In fact we have only to show that it is coordinate free. Let y = y(x) be a coordinate transformation. Note that W(y) = J-TW{x)J-yl\x=x{y)

(1.7.4)

Taking inverse and transpose on both sides, (1.5.7) follows.

•

Proposition 1.7.4 Let the pseudo-Poisson bracket be determined by (1.7.3). Then XH defined by (1.5.1) (or by the pseudo-Poisson bracket) coincides with that defined by (1.7.1) (or by the tensor field co . Proof. According to Proposition 1.7.3, we have only to prove it locally. From (1.5.1) we have LX[1 G = {G, H} = dGM{x)VH, VG e Cx (N) Hence XH=M(x)VH

(1.7.5)

On the other hand, from (1.7.1) we get dH = co(XH,-) = XTHW(x) and then XH=(dHW~\x))T

=W'T(x)WH

(1.7.6)

(1.7.5) and (1.7.6) yield the result.

•

Proposition 1.7.4 Let the pseudo-Poisson bracket be defined as above. Then a>(XH,XG)

= {H,G}

(1.7.7)

Generalized Hamiltonian

Systems

25

Proof. On a local coordinate chart we have CO{XHXQ

) = (MVH)T WMSIG = DHMTWMS/G

= DHMVG = {H,G} 0 Next, we consider the singular case. Let co be singular and assume that at each p e N there exists a local coordinate chart, (x, z), such that the matrix expression, W(x,z), of co can be expressed as W=

Wx(x) 0 0

0

(1.7.8)

Then we define a structure matrix, M{x), by M(x) =

/ W^1 (x) 0 0 0

(1.7.9)

Remark 1.7.6 /. In this way we can still define a generalized Hamiltonian system by a triple (N, co, H). 2. When the pseudo-Poisson bracket is determined by (1.7.9), the relationship of (1.7.7) remains true. Similar to Proposition 1.6.8 etc., we may have the following properties, which correspond to their counterparts of pseudo-Poisson bracket case. Proposition 1.7.7 1. H = span{row(W)} is a well defined codistribution. 2. Decomposing Q = Qj + Op , which are skew-symmetric and symmetric respectively, then (1.7.8) holds iff(i) O is integrable; and (ii) fl z> Qj and Q ZD Op. Proof. 1. Let y = y(x) be a coordinate transformation. Then

W(y) = jfw(x)J-y1\x=x(y) The conclusion follows immediately. 2. Note that we can decompose W = Wj + WP , where Wj and WP are skewsymmetric and symmetric respectively. Then Qj and £2p are determined by Wjand WP respectively as f2j =span{row(Wj)} etc. Since this decomposition is

26

D. Cheng

coordinate free, we have only to find one coordinate frame such that (1.7.8) holds. Say, dim(/2) = £ , Q is integrable if there exists a complement, Qc, of i 2 , Qc=span{co]i+\,---,(On} which satisfies the following(Boothby 1986, p 224) da>i= fJ0JAcoi,i

= k + \,---,n;dj eV*(N)

(1.7.10) rs

*s

Then we can find a dual basis Xi,---,Xn

e K(JV)such that A = span{ ,•••, } dx\ dxfc is involutive. It follows that Whas the form of (1.7.8) in a coordinate frame friend to A. • Finally, we consider a local version of pseudo-Hamiltonian vector fields. The following result for standard Hamiltonian vector fields remains true. Proposition 1.7.8 X is a local pseudo-Hamiltonian vector field iff ix is closed. When N is simply connected the conclusion is globally true. Prccf. Using Poincare's Lemma there exists a function H such that ixQ = XTW = dH it follows that X = W ~TVH = XH .

U

1.8 Structure group and its algebra The symplectic group and its symplectic algebra play an important role in the theory of Hamiltonian systems. In this section we propose a generalized symplectic group and its algebra, which will play a similar role for pseudo-Hamiltonian systems. Definition 1.8.1 /. Let N be an nxn matrix. We define N -group, GN, in the following way: GN ={EsGL(n,R)ETNE

= N}

(1.8.1)

Generalized Hamiltonian

Systems

27

LetN € Mn. Define a set NX + XTN = 0}

gN:={Xegl(n,R)

(1.8.2)

Proposition 1.8.2 7. g^ is a Lie sub-algebra of gl(n, R). 2. GN is a Lie subgroup of GL(n, R). 3. The Lie algebra of GN is gN . Proof. 1. gN is obviously a sub-space of gl(n, R). Let X,Y e gn . Then N[X, Y] + [X, Y]T N = N(YX - XY) + (XTYT - YTXT )N = -YTNX

+ XTNY - XTNY + YTNX = 0

which means gN is a Lie-algebra. 2. A straightforward computation shows that GN is a subgroup of GL{n,R) . It is obvi'jus that GN is a topologically closed. So it is a closed subgroup of GL(n, R). Hence it is a Lie subgroup (Varadarajan 1984). 3. We already know that gN is a sub-algebra of gl(n,R). So we have only to show that X e gN iff E = e x

Since e

etx sGNyteR.

e GN .

e GN , X e g{GN) , which is the Lie algebra of GN . Therefore That means (e'x)TNetX

= etxT NetX

=N,teR

Differentiating both sides with respect to t yields XTe'xT

NetX + e'xT Ne'X X = 0

Setting t = 0, we have XTN + NX = 0, which means X e gN. Assume X s gN i.e., X N + NX = 0. It is sufficient to show that E i.e.,

etxT Ne'X =N

eGK,

28

D. Cheng

Using Taylor expansion and X T N + NX = 0, the conclusion follows easily.

D

Examples 1.8.3 1. Let N = Im , we have U-group SU(n) with its Lie algebra su(n) (or special orthogonal group SO(n) with its Lie algebra so(n) for real case). 2. Let N =J =

-I 0

We have symplectic group denoted by Sp(2n,R) with its Lie algebra, called symplectic algebra denoted by sp(2n, R). 3. Let N be a diagonal matrix as N = diag(-1,1,1,1). We have Lorentz group, which are useful in relativity etc. (Clarke 1979). D Through the dimension of g the dimension of G^ can be determined easily. Particularly, we have the following: Proposition 1.8.4 Suppose an nxn matrix N is non-singular. 1. Let N = H be symmetric. Thendim(Gfj) = «(w -1) / 2;Particularly, dim(SO(n,R)) = n(n -1)/2. 2. Let N = K be skew-symmetric. Then dim{Gj^) = n{n + \)l2 ;Particularly, dim(Sp(2n, R)) = n(2n +1) . The following theorem shows that for any N e Mn, GN is nontrivial. Theorem 1.8.5 For any nxn matrix N(n>l), words:

GN is not discrete. In other

dim(Gjy)>0.

Proof. Decompose N as N =H +K where H is symmetric, HT = H , and K is skew-symmetric, KT =-K . We first assume H is singular. Then there exists a E, ^ 0 such that HE, = 0 . T

Consider ££ : Case 1. &TK = 0 : Set X = %f.

Then

Generalized Hamiltonian

HX + XTH = HtfT+tfTH

Systems

= (H%)ZT +Z(H£)T=0

29

(1-8-3)

and

KX+XIK = Ktf1

1

+tf

K

= {tf1KTYHtf1YK

(lg4)

= -(tfTK)T+tfTK = 0 (1.8.3) and (1.8.4) imply that NX + XTN = 0 Case 2. ^TK

* 0: Set X = ^TK

(1.8.5)

. Then

HX + XTH = H^TK+(^TK)TH

= (H^TK-K^(H4)T

=0

(1.8.6)

and KX + XTK = KtfTK

+ (&TK)TH

= KtfTK-K^TK

=0

(1.8.7)

(1.8.6) and (1.8.7) imply (1.8.5) too. We conclude that if H is singular, there is at least one non-zero solution of (1.8.5). That is, d i m ^ ) > 0. Next, we assume H is non-singular. Then there is a non-singular P, which T

performs a canonical congruent transformation as P HP = Ik@{-In-k)'-= Y be a skew symmetric matrix. Set

D • Let

X^PDYP"1

(1.8.8)

It is easy to see that (1.8.8) satisfies HX + XTH=0

(1.8.9)

For any skew-symmetric Y. Now we choose Y such that X is also a solution of the following KX + XTK = 0

(1.8.10)

If K = 0, any Y works. Assume K * 0, plugging (1.8.8) into (1.8.9) we have P-T(PTKPDY-YDPTKP)P~1

=0

(1.8.11)

30

D. Cheng

Choosing Y = P KP yields a common solution of (1.8.9) and (1.8.10), which is a non-zero element in gN . • T

Remark 1.8.6 1. In fact, if X is a solution of NX + X N = 0, then it is also a solution of (1.8.9) and (1.8.10) simultaneously. Because the symmetric - skewsymmetric decomposition is unique. 2. Consider the dimension of the solution (1.8.9), one sees that (1.8.10) provides all the solutions of (1.8.9).

1.9 Spectrum Symplectic eigenvalue theorem is extremely important in both theoretical study and applications (Abraham et al. 1978, Feng et al. 1994, Libermann et al. 1987 etc.). The following proposition shows that they remain true for generalized symplectic group. Mimic to the proof in (Abraham et al. 1978) we have the following: Proposition 1.9.1 (Symplectic Eigenvalue Theorem) Assume N is non-singular, S(,GN.IfAsa(S),then

-e
Proof. By definition, we have N'lSTN

= S~l

Then the characteristic polynomial of S is P(X) = det{XI - S) = det[N~l (Al - ST )N] = det{XI - s~l) = det(S~l (AS -1))=det(S~l

)det(A(- -S))

(1.9.1)

A

= det(S~l )Andet(A(- - S))=det(S~l

)AnP(-)

A

A

Using (1.9.1), if A0 is the eigenvalue of S with multiplicity k, then P(A)=det(S-1 )A"P(\)=0l-/0 That is

A

)* Q(A)=(A-Ao)k

(-^-\)k AQ

Q(X) A

Generalized Hamiltonian

P{\) = A{~-\t^\) A

where q(—) = det{S)——r A

AQ

A

Systems

31

(1-9-2)

A

is a polynomial of —. Now replacing — in (1.9.2) by

jn-k

A

A

A yields P(X) =

(-^—X)kq(x) AQ

which means — is the eigenvalue of 5 with multiplicity t > k. Same argument XQ

shows k > t. Hence t = k.

•

Theorem 1.9.2 (Infinitesimally Symplectic Eigenvalue Theorem) Let SE. gN, and X e o\s) with multiplicity k. Then -Xe. cr(s\X* e a{s) and -X* e cr(s) with same multiplicity. If n is even and 0 e er(s), the multiplicity is even. Proof. Note that since s = -N~ s N, the characteristic polynomial of 5,p(i), satisfies p{X) = det(A/ - s) = (- l) n p{- X) The conclusion follows. The following is two immediate consequences of the previous two theorems.

•

Corollary 1.9.3 /. det(S) = ±1, VS e GN. 2. Trace(s) = 0, Vs e gN.

1.10 Structure invariance An outstanding property of the standard Hamiltonian system is that it is structurepreserving: Let (M, w) be a symplectic manifold, and XH(x) a Hamiltonian vector field with its integral curve e'x

(x). Then (Abraham et. al 1978)

32 D. Cheng (e'y

~x

{x))*Q) = CO

(1.10.1)

Moreover, Denote the Jacobian matrix of XH (x) and e'x

(x) by Jx

and Je re-

spectively, then under the canonical coordinate frame, where Wa=J =

I 0

we have Jx e sp(2n, R). That is JxJ

+ JJx=0

(1.10.2)

JTeJJe=J

(1.10.3)

and Je e sp(2n, R). That is

This property has many applications, e.g., symplectic algorithm (Feng et. al 1994, Sanz-Serna et al. 1994, Wang 1994). This section will be devoted to extend it to the pseudo-Hamiltonian systems. First we have the following formula: Proposition 1.10.1 Let X e V(N) and £ e

T2(N),

which is the set of the sec-

ond order covariant tensor fields, and W(x) = W? (x) is the matrix representation of £ in a local coordinate chart. Then we have Lx(^ = (LX(WuM)) J

riX riX T + W(x)^- + (^-y W(x) ox ox

where the first term is an nxn matrix with Lx(WiJ{x')),i,j

(1.10.4)

= \,---,n as its entries.

Proof. Express £, in local coordinates as n n

£ = £!>//(*)<&, ®afr/ Then L

x(Z) = E H\Lx(«>ij)dx, ® dxj +G>ijLx(dxi)®dxj

+o)idxi ®Lx[dXj))

(1.10.5)

Generalized Hamiltonian

Systems

33

Using the equality Lx{axj)= d =dXt to (1.10.5), (1.10.4) follows immediately. • Now we consider an co -manifold, assume co is integrable and let W be its local constant matrix expression. We consider the problem of structure-preserving. For the remaining of this section we consider only the generalized Hamiltonian systems on co -manifold. Proposition 1.10.2 Let XH{x) curve e'x

be a Hamiltonian vector field with its integral

(x) and denote the Jacobian matrix of XH (x) and e'x (x) by J x and

Je respectively. Assume co is integrable and W is its constant matrix expression. Then co is structure preserving with respect to the trajectory of Xfj, iff, one of the following three equivalent conditions holds: CI. Jx^Sw

(1-10.6)

JeeGw

(1.10.7)

C2.

C3. (WW_T+I)Hess(H)

=0

(1.10.8)

Proof. Using (1.10.4), Lx{co) = Q iff (1.10.6) holds. The conclusion for CI follows. Now locally X can be expressed as X = MVH . Then Jx = MHess(H) It is easy to see that (1.10.6) implies that WJx +JXW = 0 . Substituting Jx by the above expression yields (1.10.8). To see the equivalence of (1.10.6) and (1.10.7) recall the key identity (Abraham et.al 1978)

^4)V)4^M4

34

D. Cheng

Now (e* )*(a>) is independent of t iff Lx{o}) = 0 . Note that the matrix exXH )*(o)) is J~TWJ~l and Lx{co)=WJx

pression of (e'

+ Jyw

• T n e conclusion

X

H

follows. The following corollary is an immediate consequence of (1.10.8). Corollary 1.10.3 a> is invariant with respect to X^for

D

any H e C°°(M), iff,

T

W = -W . From Corollary 1.10.3 we know that we can not expect anything new but the standard Poisson bracket for the "universal structure preserving". Fortunately, for a given system we are only interested in the structure-preserving for the particular vector field X = XH. In other words, we are only interested in such a problem: When the integral of X,et (x), is an N -structure preserving mapping. Mimic to symplectic mapping, we may call such mapping N -mapping. Then (1.10.8) provides a necessary and sufficient condition for such invariance. We may express the structure-preserving property in either tensor field version or in pseudo-Poisson bracket version. The following proposition shows that these two forms are equivalent. Corollary 1.10.4 Let a be a non-singular quadratic tensor field and {•,•} be the pseudo-Poisson bracket induced by co as in above. Then (e'x )*co = eo

(1.10.9)

iff (e'x )*{F,G} = {F,G},

VF,GGC°°(/?")

Proof. We have only to prove it under a local coordinate frame. Then (e*x )*{F,G} = dFJeMJ*VG Hence (1.10.10) holds iff JeMJTe = M . But we have

(1.10.10)

Generalized Hamiltonian

JeMJe = M <-» JeW~ljJ

Systems

35

= W~x <-> WJX +JTXW = 0{eTx )*a = co

Now for a structure-preserving system we can define a generalized symplectic algorithm as Deflnition 1.10.5 A discrete algorithm for a structure-preserving system z

k+\ =(zk)

(1.10.11)

is called a generalized symplectic algorithm, if the Jacobian d dzk

(1.10.12)

7

M$

We consider the following example. Example 1.10.6 Consider following system x=x+3y+x^

+4xy + 4y^

1 3 12 0 o 2 y=--x--y--x^ -2xy-2yz-

(1.10.13)

The Jacobian of the vector field J/-/ is J

/ =

' l+2x+4y -0.5-x-2y

3 + 4x + 8j> -1.5-2^-4^

We try to find an integrable tensor field, which has its matrix expression as TV by solving NJf+J'fN

=0

It is easy to see that the solution is of one dimension as N =r reR V2 4 , So the structure invariant tensor field, E,, is expressed as Z = dx®dx + 2dx®dy + 2dy®dx + 4dy®dy

(1.10.14)

36

D. Cheng

According to Proposition 1.10.2, it is invariant with respect to system (1.10.13). • Before end, we leave searching structure-preserving algorithm for further study. But we give some examples to show that some algorithms preserving symplectic structure (Feng et. al 1994, Wang 1994) also preserves other structures. Example 1.10.7 Assume the system x = XH has an invariant structure determined by N. Equivalently, it satisfies (1.10.14) for any N. 1. Consider mid-point Euler scheme zk+l=zk+sf(Czk+l+(I-C)zk),C

+ ±In

(1.10.15)

where s is the step length. Then it preserves any structure. In fact, from (1.10.15) it is easy to figure out that Jf =(I-0.5sJf)~l(I

+ 0.5sJf)

Then we have only to show that (I+0.5sJTf)(I-0.5sJTfrl N(I -0.5sJfTlV Note that (I-0.5sJj-)~

+ 0.5sJf) = N

and (I + 0.5sJf),

etc., are commutative, so we have

only to show that (I + 0.5jT)N(I

+ 0.5sJf) = (I -0.5JTf)N(I

-0.5Jf)

(1.10.16)

It follows easily from (1.10.14) that (JTf)kN

= (-\)kNJkf

Using (1.10.17), (1.10.16) follows immediately. 2. Consider fourth order Runge-Kutta method

(1.10.17)

Generalized Hamiltonian Systems

37

v

2=/(x«+0-5'$vl) v3=f{x„+0.5sv2) (1.10.18) x

+5

V

V

/ V

V

n+l = *« C"1 1 +/"2 2 +/ 3 3 +-"4 4> 4

It is easy to figure out the Jacobian as J,

I+

aJf+bJf+cj}+dJf

(1.10.19)

where a=s

c=

s3([i3+2{i4)/4

(1.10.20)

Using (1.10.17), the following equation is necessary and sufficient for (1.10.18) to be structure-preserving:

(I + bjj + dJJ)2 -{aJf + cjj)2 = I

(1.10.21)

1.11 Stabilization of excitation control As an example of the generalized controlled Hamiltonian systems we consider the excitation system. It provides a description for the application of the pseudoHamiltonian approach to an engineering problem.

38

D. Cheng

S =

6)-G)Q

• a0 D D( . °>0EqVs co=——P M mm M (co-con) " Mr. —sino x dY.

/ i n n (1.11.1)

_L4+_LJapLFjCOf*+_L; T

d

T

dO

*dT,

T

dO

where 8 : the rotor angle; co: rotor speed; t

Eq : internal transient voltage; PM: mechanical power; M : inertia coefficient of a generator; D: damping constant; EaVs Pe = —T—sin
and denote

a

= ^-Pm,b M

= — , c = - ^ - , r f = _L and e = — ^~^-Vs M MxdZ Td Td0 Xdl

system (1.11.1) can be rewritten as

. Then

Generalized Hamiltonian

V x

2

(0 0 0

0 10

39

0

-a + cx^smx-^

0 b 0 2 0 0 e/cJ J -ccosx^+cdxy/e

-10 0

Systems

0

x

(1.11.2)

0 0 0 i/^o W Choosing a suitable output, (1.11.2) can be written in a standard form as x

\ 2 = (J(x)~R(x))-- ox + g(x)u v*3y 7 dH y=g dx x

(1.11.3)

where '0 0 0^ R(x) = 0 b 0 -10 0 0 0 e/c ooo

0 0

o Io

j{Xy-

l/T

d0.

c

cd x 3 cosxj +• d0 ?d0

T

and „. ,

2

cd

/ / ( X ) = -CJC3COSA:I - O X J H

1 2

X3 + —

2e Now we set an extended Hamiltonian function as

*

2

2

He (xe ) = H(x) + Hs (77) = H{x) -urj where xe =(x\,x2,x2,rj) as

and 77 satisfies the source system (Maschke et. al, 1998)

T] = y = us

40 D. Cheng The extended system becomes ^ 0 x3 \T1 J

10^

f

\A

0

0 0 0

-10 0

0

0 6 0

0 0 0

-i/rdO

0 0 e/c

0

\o 0 0)

(0 0 VTdQ)

W

r0\\ o

dHn

(1.11.4)

dxa 0

Set f

l

K = -(J(x)-R(x))- g

0

^

(1.11.5)

0

=

c/eTd0 Then C(x) =

-xj and eZ rfO 8C(x)

K(x).

dx

Next, we modify the source system such that C(x) can be a Casimir function. Follow Maschke et. al (1998): ' j ^

f f

J T

W

KJ

It is ready to verify that T]-C(X)

JK

\

T

K JK

(

R T

KR

= TJ

RK

X\dHc

T

K RK J) dxa

(1.11.6)

XT, is a Casimir function. Then on eTdO

the submanifold t] - C(x) = CQ = const, we have the restricted Hamiltonian function Hr as H = H(x) - u ( — — * 3 + c0) e7 J0 1 j cd j uc -X3 - M C Q = —cxicosx] —axi + T—^.T J^-, —XZ T + XX 3 ! ' 2 2 2e 3 eT,dO with x as its critical point.

(1.11.7)

Generalized Hamiltonian

Systems

41

Now x, as both the critical point of the Hamiltonian function Hr and the equilibrium of system (1.11.2), is the solution of the following equation: *2=0 (1.11.8)

a-c*3sinxi=0 -dxy +ecos*j+

w=0 l

d0

Next we show that under certain condition x is really a minimum point of Hr Consider the Hessian of Hr, a straightforward calculation shows that

Hess{Hr) =

CX3COSXJ 0 csinxj 0 1 0 csinxj 0 cd/e

New Hess(Hr) is positive definite, iff dx~2 cos3cj > esin X\ Therefore, x is a local minimum provided (1.11.9) is satisfied. We refer to (Cheng et. al, 1999; 2000a) for some details.

1.12 Stabilization and Hx control for dissipative Hamiltonian systems The following two propositions have been proved for standard Hamiltonian systems. (van der Schaft 1996, etc.). Using parallel arguments, one can see that they remain true for the controlled dissipative Hamiltonian systems. Proposition 1.12.1 If the Hamiltonian H(x) for Hamiltonian control system (1.4.1) is boundedfrom below, then (1.4.1) is passive.

42

D. Cheng

Proposition 1.12.2 If the equilibrium point x = 0 of Hamiltonian control system (1.4.1) is a strict local minimum, and system (1.4.1) is zero state detectable, then system (1.4.1) is stabilizable. Lemma 1.12.3 LetR bean nxn symmetric matrix, and g is an nxm matrix. Then there exists an mxm symmetric matrix P such that R + gPg1 >0 if and only if Span{R} + Span{g} = R" Proof. (Sufficiency) Choose a coordinate frame such that R=

0 0

where k = rank(R). Let g = 81 . Then rank(gl) = n-k . Hence there exists a matrix Qmx(„-k) 82 such that }

n-k

gQHence J

gQQTg=

It is easy to see that R + gPg

n-k

82

h

8282

(1.12.1)

> 0, where P = QQ .

(Necessary) Suppose rank(gi)
. Then for any symmetric matrix P we

Generalized Hamiltonian

g\Pg( R + gPgT _ g2pg\ is not positive definite because rank(giPgi Theorem 1.12.4

Systems

43

g\Pgl h+S2pS2. )
a

If the Hamiltonian H(x) has a strict local minimum at

x = 0 and Span{R(x*)} + Span{g(x*)} = Rn

(1.12.2)

then the system is stabilizable. Proof. According to Lemma 1.12.3, there exists a symmetric matrix P such that R(x*) +

g(x*)PgT(x*)>0

(1.12.3)

Then in a neighborhood of x* we have R(x) +

g(x)PgT(x)>0

(1.12.4)

If we select the controller u - -Py, then

dt

dx

dx

dx

r

=#) W))+^/W)?
1

ghx)^-,ueRm,wBRP ax

z = g2(x)—

+ D(x)u

XGR", (1.12.5)

44

D. Cheng

where J{x) is a skew-symmetric matrix, R(x) is a positive semi-definite matrix, u are controls, w are disturbances and H(x) is the system's Hamiltonian function. Using a similar approach as van der Schaft 1993, a sufficient condition can be produced for the controlled dissipative Hamiltonian system (1.12.5) to have the Hx property, i.e. there exists a positive number y and a suitable state feedback control such that the closed-loop system satisfies \l\\z{tfdt
(1.12.6)

and the closed-loop system is asymptotically stable while w = 0 . In general, we asciH

T

sume that ( — ) 7 g2(x)D(x) = 0,\/x&R". dx The representation of z has its practical meaning. The first term represents the 'natural output' through the disturbances channel, and the second term represents the II

n2

dH T

'input'. According to \\z(t)\\ = (

T

dH

) gi{x)g2 (x)

T T

+ M D' (x)D(x)u , the en-

dx dx ergy consumed by the disturbance channel can be represented by the first term, and the external supplying energy can be represented by the second term. We want them as small as possible. Lemma 1.12.5 Assume that the equilibrium point x* = 0 of the controlled dissipative Hamiltonian system (1.12.5) is a strict local minimum, and the system is zero state detectable with respect to outputs y and inputs u,w. If there exists a positive number a satisfying 2 2 ^ + ^gl(x)(DT(x)D(x))-lg((x)-(±-+~)g2(x)gT(x)>0 2 2 2yZ

(1.12.7)

then the H^ control problem of system (1.10.5) can be solved by u = -a(DT(x)D(x))~l

y

Proof. Without loss of generality, assume H(x) be positive semi-definite. According to Proposition 1.12.2 the closed-loop system is obviously asymptotically stable when w = 0 . It is easy to verify that Hamilton-Jacobi-Issacs inequality for system (1.12.5) is satisfied while V(x) = aH(x). D

Generalized Hamiltonian

Systems

45

Proposition 1.12.6 Assume that the Hamiltonian function of system (1.12.5) satisfies the conditions of Lemma 1.12.5 , D D = I, then there exist a and y such that (1.12.7) is satisfied if and only if Span{G2} c Span{Gl} + Span{R} where Gj = g^g\ ,G2 = g2g2 • Moreover, if the condition is satisfied, then the corT

—1

responding Hx control problem can be solved by u = -2a(D D) y . We need the following lemma to prove Proposition 1.12.6. Lemma 1.12.7 Suppose that R and G are two nxn positive semi-definite matrices. Then there exists a constant a > 0 such that aG-R>0 if and only if Span{R} c Span{G}. Proof. The necessity is obvious. We prove only the sufficiency. |~ [" A B~ B~\ , . , "o0 Ol o" and G G= = TT Choose coordinates such that R = 0 / and B D . We first claim that T

T

D is nonsingular. Otherwise, there exists £ * 0 such that £, D = 0 . So £ DE, = 0 and(0 ; 4Ty}(^\ = 0 . Since G > 0 we have (0, £ r ) G = 0 . That is (0, ^ r ) e G J = {x:xTG = 0}. Since Span {R} c Span {G} is equivalent to RL 3 G 1 . It follows that (0, cT) = (0, %T)R = 0, which is a contradiction. Now D is positive definite, so there exists E > 0 such that G - sR > 0 . Choosing CC = \IE it is easy to see that aG — R>0. • Proof of Proposition 1.12.6: According to Lemma 1.12.7, there exists a positive constant c, such that c(R + GO-G2>0

(1.12.8)

46 D. Cheng It is easy to see that there exists a constant a such that c = min{a,a2/2} . a2 SoaR + — Gj - G 2 >c(R + Gi)-G2

^ 0 . Therefore there exist positive constants

a,y -a such that (1.12.7) is satisfied. As an application, we consider the excitation control system (1.11.1) again. It is easy to see that the equilibrium of the system is _ 1 . xl=-sin x =0,2 2

-\,2ad '( ), ec

X3 =—cosQc}) 1

So we should assume

<1, i.e. —— ^ (xd-xd)V}Td

— <1. and it is easy to know that

0<5q
i.e. X3 = — COSX\

X2sinx\ =0

(1.12.9)

{a-cx^sira^ -bx2)sinx\ +xicosx\ = 0 It follows

from

(1.12.9) and (1.11.1) that XT, =—cosx\,x2{t) = 0 and d r. • 1 . —] ,2ad. _ e . . T . , .,., . a-cx?,sinx\ =\J . i.e. x\ =—sin 1{ ),X2 = 0,x-$ =—cos(x\). It is the equilibrium 2 ec d point. So system (1.11.1) is zero state observable. At the same time H(xi,x2,x^) is

Generalized Hamiltonian

Systems

47

bounded from below in the domain D={(x\,x2,X2):0<x\
,X2,XT))-

cx-^cosx^ 0 csinx\ 0 1 0 >0. csirix^ 0 cd/e

So (xi,X2,x^) is a strict local minimum point of H(x\,X2,x-$) . According to Proposition 1.12.2, the 3rd excitation control system (1.11.1) can be stabilized by the controller -y=

c 'rfO

cd

cos*i

X3.

T

dOe

Next we consider the Hx control problem of the excitation system. Example 1.12.8 (Lu et.al, 1993 ) Consider Hx control of the 3rd order excitation system, where J(x),R(x) and g\(x) are as in 11.1, w is the disturbance. r)H = (J(x) - R(x))— + gx {x)u + g2 (x)w ox V*3y

y = g\

(1.12.10) dx T 8H ax T

z2 = J3u u ' 0 0^ where g2(x) = k 0 ,w= 0 /

w

k,l,P arbitrary positive constants. System (1.12.10) 2,

has the form of (1.12.5). Assume y = fa , where / is an arbitrary positive constant. If we select parameter a satisfying the following conditions

48

D. Cheng

a>—* ( / + !) a>

(1.12.11)

i—

f

-e

le

c +\c2+

2

(/

+

1>2

/32TJ0f

2

/^ o

then it is easy to check the conditions of Proposition 1.12.6 satisfied. So the control law can be chosen as a

_ a 2y

2

c

J3 ~ J3 \JdQ

cd COSXj

X3 T

dOe

J

which solves the HK control problem. See (Xi et. al, 2000) for details.

1.13

Summary

This chapter intended to provide a framework for generalized (controlled) Hamiltonian systems, which generalize the existing theory on standard (controlled) Hamiltonian systems. A model for the mostly generalized controlled Hamiltonian systems was proposed. The pseudo-Poisson manifold, which is a manifold equipped with a pseudoPoisson bracket, and the co -manifold, which is a manifold equipped with a quadratic cuvariant tensor field co , are presented as the state-space description of the generalized controlled Hamiltonian systems. They are the generalizations of Poisson manifold and symplectic manifold respectively. Certain properties are investigated. As a generalization of symplectic group and symplectic algebra, N-group as a Lie subgroup of GL(n,R) and its Lie algebra gN were introduced and investigated. They are used for the structure analysis of the generalized Hamiltonian systems. Some properties were investigated. Using them the generalized structure-preserving properties were investigated. As an application, the stabilization of the excitation control systems was discussed. The systems was transferred into a model of the generalized Hamiltonian systems. The energy-based Lyapunov function approach (Maschke et. al 1998) has been used to stabilize the system at a working point.

Generalized Hamiltonian

Finally, the stabilization and the Hx pseudo-Hamiltonian systems were studied.

Systems

49

control problems of the controlled

Acknowledgment Most of the results obtained by the author and presented in this chapter were done under the National Natural Science Foundation G59837270 and National Key Project of China G1998020308, G1998020309.

References [I] Abraham, R. A., J. E. Marsden, (1978), Foundations of Mechanics, 2nd Ed. Benjamin / Cummings Pub. Com. Inc. [2] Arnold, V.I., (1978)Mathematical Methods of Classical Mechanics, Springer, New York [3] - , S.P. Novikov, Symplectic geometry, (1980) in Dynamical Systems IV, Encyclopedia if Mathematical Sciences, Vol. 4, Springer Verlag [4] Boothby W.M., (1986)Introduction to Differentiable Manifolds and Riemannian Geometry, 2nd ed. Academic Press, INC. [5] Brockett R. W., (1977) Control theory and analytic mechanics, in Geometric Control Theory, C. Martin and R. Herman eds., Math.Sci. Press, Brookline,l-46 [6] Cheng, D., et.al, (1998), On generalized Hamiltonian systems, Proc. of ICARCV'98, Singapore, 185-189 [7] Cheng, D., Z. Xi, Y. Hong, and H. Qin, (1999), Energy-Based Stabilization of Fo/ced Hamiltonian Systems with Its Application to Power Systems, Proc. IFAC'99, Beijing, Vol. O, 297-302 [8] Cheng, D.,Z. Xi, Q. Lu, and S. Mei, (2000a) Geometric Structure of Generalized Controlled Hamiltonian Systems and Its Application, Chinese Science, to appear [9] Cheng, D.,Z. Xi, (2000b) Hamiltonian Realization of Control Systems, ACTA Automation, to appear [10] Cheng, D., S. Spurgeon, J. Xiang, (2000c) On the Development of Generalized Hamiltonian Realizations, submitted [II] Cheng, D., and C.F. Martin, (2000d) Normal form representation of control systems, submitted. [12] Clarke,C, (1979) Elementary General Relativity, John Wiley & Sons, Inc., New York

50

D. Cheng

[13] Crouch, P.E., A.J. van der Schaft, (1987) Variational and Hamiltonian Control Systems, Lecture Notes in Control and Information Sciences 101, SpringerVerlag, 1987 [14] Eaton, R., T. Hesketh, D.J. Clements, (1999) High gain tracking control for the nonlinear benchmark systems, Proc. 1999 Hong Kong IEEE Symposium on Robotics and Control, 122-127 [15] Escobar, G., A.J. van der Schaft, and R. Ortega, (1999) A Hamiltonian viewpoint in the modeling of switching power converters, Automatica 35, pp 445-452 [16] Feng,k., Wang, D., (1994) Dynamical systems and geometric construction of algorithms, in Computational Mathematics in China, Contemporary Mathematics, 163, Zhong-Ci Shi, Chung-Chun Yang, Eds. 1-31 [17] Guckenheimer, J. and P. Holmes, (1983) Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields, Springer [18] Isidori, A., (1989)Nonlinear Control Systems, 2nd ed., Springer-Verlag [19] J. Koszul, et. al,(1997), Symplectic Geometry, Science Pub., (in Chinese) [20] Kunder. P. (1994), Power System Stability and Control, McGraw-Hill, Inc., N.Y. [21] Linermann, P. and CM. Marie (1987) Symplectic geometry and Analytic Mechanics, Reidel, Dordrecht [22] Li, Jibin, X. Zhao and Z. Liu, (1994) Theory and Application of Generalized Hamiltonian Systems, House of Science Pub. (in Chinese) [23] Lu, Q., and Y.Z. Sun, (1989), Nonlinear stabilizing control of multimachine systems, IEEE Trans, on Power Systems, Vol.4, No.l, 236-241 [24] Maschke, B.M.J., R. Ortega, A.J.van der Schaft, (1998) Energy-based Lyapunov functions for forced Hamiltonian systems with dissipation, Proc. of CDC98, 3599-3604 [25] Maschke, B.M.J., A.J.van der Schaft, (1992) Port controlled Hamiltonian systems:modeling origins and system theoretic properties, Proc. 2nd IF AC Symp. on Nonlinear Control Systems Design, NOLCOS'92, Bordeaux, 282-288 [26] Marsden, J.E., T.S. Ratiu, Introduction to Mechanics and Symmetry, SpringerVerlag, 1994 [27] Nijmeijer, H., A.J. Van der Schaft, (1990), Nonlinear Dynamical Control Systems, Springer-Verlag [28] Olver, P.J.(1986), Applications of Lie Groups to Differential Equations, Springer-Verlag, New York Inc. [29] Ortega, R., A.Stankovic and P.Stefanov, (1998), A passivation approach to power systems stabilization, IFAC Symp. Nonlinear Control Systems Design, Enschede, NL, July 1-3

Generalized Hamiltonian

Systems

51

[30] Sanz-Serna, J. M., M. P. Calvo, (1994)Numerical Hamiltonian Problems, Chapman & Hall [31] Sarlashkar, J.V., (1996), Hamilton/Lagrange formalisms in stability analysis of detailed power system models, Ph D Thesis, U. of Wisconsin [32] van der Schaft, A., (1989), System theory and mechanics" in Three Decades of Mathematical System Theory, H. Nijmeier and J.M., Schumacher eds., Lect. Notes Contr. Inf. Sci., Vol. 135,426-452 [33] van der Schaft, A., (1996) L2 Gain and Passivity Techniques in Nonlinear Control, Springer-Verlag [34] van der Schaft, A. and B.M. Maschke (1995) The Hamiltonian formulation of energy conserving physical systems with ports, Archiv fur Elektronil und Ubertragungstechnik, Vol. 49, No. 5/6, 362-371 [35]Varadarajan, V.S., (1984) Lie Group, Lie Algebra, and Their Representations, Springer-Verlag, [36] Wan,C.J., D.S. Bernstein, V. T. Coppola, (1996) Global Stabilization of the oscillating eccentric rotor, Nonlinear Dynamics, 10, 49-62 [37] Wang, D., (1994) Some aspects of Hamiltonian systems and symplectic algorithms, Physics D, Vol. 73, 1-16 [38] Xi,Z, D. Cheng, (2000), Passivity-Based Stabilization and Hx Control of the Hamiltonian Control Systems with Dissipation and Its Applications to Power Systems, Int. J. Control, to appear.

Chapter 2

Continuous Finite-Time Control T.P.Leung* and YiguangHong** *:The Hong Kong Polytechnic University, Hong Kong, China **: Institute of Systems Science, Chinese Academy of Science Beijing 100080, China

Notations c'

continuously differ-entiable (functions or vector fields). Euclidean norm of vectors; or induced norm for matrices.

sgn(-)

absolute value of real numbers. sign function.

\fsgn(-). a positive definite homogeneous function of degree 1.

2.1 Introduction This chapter studies the problem of finite-time feedback stabilization in a systematic way. Continuous feedback controllers, especially consisting of terms of fractional powers (or called fractional power control), are constructed to achieve finite-time stability for a class of control systems. Before we introduce why and how we propose finite-time design, the conception of the finite-time stability and stabilizability is described in mathematical terms

52

Continuous

Finite-Time

Control

53

with the motivation introduced then. The detail can also be found in Haimo (1986) and Bhat and Berstein (2000). Definition 2.1.1 Consider the system X=XJC,0,

A0) = 0, x^R"

(2.1.1)

where f:DxR+ —>R" is continuous on an open neighbourhood D of the origin x=0. The equilibrium of (2.1.1) is finite-time convergent if there is an open neighbourhood U of the origin and a function T:U/{0} —> (0,°o), such that, Vxe U T(x)s,(0,x) =0. Here T is called the settling time. The equilibrium of (2.1.1) is finite-time stable if it is Lyapunov stable and finite-time convergent. When U=D=R", then the origin is a globally finitetime stable equilibrium. Definition 2.1.2 Consider the system x=f(x)+G(x)u,

X0) = 0, x e R", u e R"".

(2.1.2)

where J, G:D—>R" are C1 smooth on an open neighbourhood D of the origin x=0. The equilibrium of (2.1.2) is finite-time stabilizable via state feedback if there is a control law u=p(x,t) such that the equilibrium x=0 of the closed loop system x = f(x) + G(x)M(x,t)

(2.1.3)

is finite-time stable. If so system (2.1.2) is called finite-time stabilizability (via state feedback). Definition 2.1.3 Consider the system x = f(x) + G(x)u, f(0) = 0,xzRn,ueRm y = h(x), 6(0) = 0.

(2 1 4)

where f G, h: D—> R" are C1 smooth on an open neighbourhood D of the origin x=0. The equilibrium of (2.1.4) is finite-time stabilizable via output feedback if there is an output feedback control law of the form

u=n(£,t\

£=Mo(£,y),£zRn°

(2-1.5)

where £ = JUQ(£,y) is a system to be constructed according to concrete situations, such that the origin (x,<^) = (0,0) of the closed loop systems consisting of (2.1.4) and

54

T. P. Leung & Y. Hong

(2.1.5) is finite-time stable. If so system (2.1.4) is called finite-time stabilizability (via output feedback). In fact, the purpose of finite-time control is to design control devices in order that we can achieve the control aim in a (given) finite time. It is certainly useful in control engineering because convergence in finite time is quite faster than exponential decay in suitable regions. Moreover, the significance of finite-time control synthesis is also shown in the aspects related to control accuracy since after a finite time, we can exactly, not approximately to some degree, reach the real target. In addition, it has more advantages in improving transient behaviors and robustness because of its nonsmooth structures. In fact, in the control of many practical controlled systems like robots, many realistic factors should be taken into consideration. For example, the cost of control must be minimized. The control hardware and software prefer to be simple and reliable. The operating environment is full of uncertainties, requiring high robustness in design. In some cases, high speed is needed in real-time operations, reducing the influence of large time-delay (in telecommunication), and improving transient processes. In summary, the design of finite-time control becomes more and more important on account of many actual demands such as high-speed, control accuracy and robustness. However, the design is not very easy to be studied, largely due to the lack of corresponding effective analysis tools. In most cases, time-invariant systems and time-invariant (continuous) feedback laws are preferred in the analysis and synthesis, for both theoretical and practical concerns. Thus in the sequel, we usually use i=fix),

A0)=0, xeR".

(2.1.6)

instead of system (2.1.1), and feedback u is usually taken in the form of u=ju(x) or In this chapter, we will pay more attention on state feedback finite-time control than on output feedback finite-time control. In fact, the results of output feedback finite-time stabilization are largely dependent on the results of state feedback finitetime stabilization. In conjunction with finite-time observers, state feedback finitetime control may lead to an output feedback finite-time stabilization control (see Section 2.6). The problem of state feedback finite-time stabilization of a dynamic system has been studied first in terms of optimal control (referred to the book of Athans and Falb), where the feedback laws are usually discontinuous and even time-varying. Then, time-invariant, both discontinuous and continuous, finite-time feedback was proposed by quite a few people from different perspectives (cf. the reference list),

Continuous

Finite-Time

Control

55

especially for double integrator systems. There are some results about continuous finite-time feedback laws, but not many. In many aspects, continuous feedback has some advantages over the discontinuous one. At first, continuity can guarantee existence of solutions for differential equations, which will reduce many theoretical difficulties. Then continuous feedback may remove or minimize the weakness of discontinuous one like chatter phenomena. Moreover, continuous feedback provides more possibility to make a tradeoff in complex practical design. The rest of this chapter is organized as follows. Some simple forms of finitetime controllers are then reviewed in Section 2.2. Then some essential results are introduced for the analyses at a later stage in Section 2.3. Based on these, finite-time feedback stabilizability and controller forms are proposed in Section 2.4 where a relationship with small-time local controllability is presented. Furthermore, finite-time convergent observers are obtained and therefore output feedback finite-time stabilizing control laws for a class of second-order systems are given in Sections 2.5 and 2.6. In Section 2.7, some results about the finite-time convergent rate are illustrated. An application to robot control using the above theory is exhibited in Section 2.8; while in Section 2.9, some advantages of nonsmooth techniques such as robustness are introduced. Finally, concluding remarks are given in Section 2.10.

2.2 Classes of finite-time feedback Consider system (2.1.2) with x&R. Haimo (1986) showed that its finite-time feedback can be presented as u =-j{x)-K\x\asgn(x), K>0, 0 < a< 1 where sgn(-) denotes the sign function. Because of the simplicity, the finite-time stability of first-order systems can be studied easily. Note that it is a fractional power controller. So far the problem of state feedback finite-time stabilization has largely been restricted to the so-called double integrator system: y = u, which is described in the following state space form *1=*2 x2=u,

*i(°)=*io

„.

,. *2(°)=*20

(2.2.1)

n

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T. P. Leung & Y. Hong

The problem was initially explored in the context of optimal control and controllability. Although the results about finite-time convergence and stabilization have been provided in different ways, they cannot form a complete theory. In fact, even for second-order nonlinear systems, asymptotic stabilization and stabilizability have drawn much research attention. As for general high-order nonlinear systems, stabilizability and finite-time stabilizability still require further studies. In fact, there are many different classes of finite-time design for control systems. Some of them are listed as follows. Class 1 Time varying control. /./ As well known, the controllability (of linear systems) is defined based on the existence of finite-time convergent (or stabilizing) controllers. Thus, for linear systems x = Ax + Bu and 7>0, there is u(t,x0,x) = -bTe~A

T 'G

- 1

T->

(7> A *

0

j ,

At

where X{0) = XQ, the initial condition, and G = l^e bb e dt, the controllability grammian (cf. Wonham 1979). For double integrator x =u, the finite time control can be proposed in the form of u = 12[Tx} 0 + (T2 + l)x 2 o VT4 1.2 For system (2.2.1) and a given time T, the control scheme minimizes the T

0

control energy cost function J(u) = U u(t) dt, given by (see the book of Athans and Falb for how the following feedback is derived) "(*o, 0= -^ (3x10+2x20r)+ — (2c10+^2o7)/

(2.2.2)

1.3 Coron (1995) proposed some periodic time varying controllers for finitetime stabilization design. Class 2 Time-invariant nonsmooth feedback. 2.1 Discontinuous feedback. The time-optimal control with \u\
(2.2.3)

The control law can be derived with Pontryagin Principle and a standard optimal procedure (cf. the book of Athans and Falb).

Continuous

Finite-Time

Control

57

2.2. Continuous feedback (unbounded). For instance, a class of continuous timeinvariant controllers was constructed for double integrators as follows (Haimo 1986) u = -|x,| ai sgn(xj-\x2\ a2 sgn(x2)

(2.2 A)

where 0a2/(2-a2). Another classes of continuous state feedback stabilization control laws are proposed by Bhat and Berstein (1998). u = -\x2\asgn(x2)-\a\a,i2~a)sg<
(2.2.5)

2a

where a = Xi+[\/(2-a)]\x2\ ' sgn(x2),0
(2-2.6)

where (/>a is defined as above and sat(-) denotes saturation function. Class 3. Other techniques. For example, James (1991) gave a stochastic infinitedimensional method for finite time observer design of finite-dimensional systems. In this chapter, we will focus on the time-invariant continuous finite-time feedback design. Definition 2.2.1 Consider system (2.1.6). If its vector field f(x) is (local) Lipschitz condition, there is a positive number Lo such that, in a neighborhood U of the origin, W)-Ax)\\
4x^c,xeRn.

Remark 2.2.2 It is easy to see that finite-time stability implies asymptotic stability. In addition, iff(x) in (2.1.6) andf(x)+G(x)fi(x) in (2.1.2) are smooth, then the two systems cannot be finite-time stable. In other words, they must be nonsmooth. Usually, in system (2.1.2), f(x) and G(x) are smooth or even analytic, then the designedfeedback has to be taken in a nonsmooth form. If so, the closed loop system is lack of (local) Lipschitz conditions.

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T. P. Leung & Y. Hong

Remark 2.2.3 The uniqueness in forward time of the solutions to the systems with non-Lipschitz feedback can be studied as in the work ofKawski(1989). For convenience, we give some definitions for the sequel. A function V(x) is (local) positive definite (or negative definite) if there is a neighborhood U e Ft" of the origin such that V(x)> 0, (or V(x) < 0) ifx^OeU;

V(0) = 0

A function V(x) is (local) positive semidefmite (or negative semidefinite) if there is a neighborhood UeF? of the origin such that V(x)>0, (orV(x)<0),xeU;

Kf0J=0

A function V(x) is radically unbounded if Urn//K//-wV(x)=oo. Set sig(y)a = \y\asgn(y), a>0, for simplicity.

2.3 Preliminary results Our approach will be based on the properties of homogeneous systems. Therefore, we will introduce the concepts related to homogeneous systems, which can be found in the papers in the reference list. Definition 2.3.1 Consider n-dimensional system (2.1.6). Dilation Ae is a mapping, depending on dilation coefficients r = (r],...,r,), which assigns to every real e > 0 a global diffeomorphism j£l'->r"\xh...,xn)

=

(£rlxh...y»xn)

where xL...,x„ are suitable coordinates on /?" andrh...,rn are positive real numbers. In the sequel, where no confusion arises, we will simply use "with respect to the dilation (r b ...,r„)" instead of "with respect to the dilation AE ". A scalar function V(x) is homogeneous of degree a> 0 with respect to the dilation (rh...,r J if V(s^x\,...,sr"xn)

=£ P F(JC)

Continuous

Finite-Time

Control

59

A vector field fix) = (fi(x),...,f„(x))T is homogeneous of degree k e R with respect to the dilation r = (rlr...,r^, if fi(£rlxh...,£rnxn)

= ffix),

i = 1,...,«; s > 0

We call (2.1.6) a homogeneous system if fix) is a homogeneous vector field. Set I\x) = (M"* 1 +---+\xn\m/'"

)Vm,m>max{rx,...,rn}

Clearly, we have 1). I\x) > 0, and IJx) = 0 if and only if x = 0; and 2). r{srix\,...,Ernxn) = sljx), s> 0. Ths following lemma is adapted from the work of Rosier (1992). Lemma 2.3.2 Suppose that system (2.1.6) is homogeneous of degree k with respect to dilation (rh...,r„),fis continuous andx = 0 is its asymptotically stable equilibrium. Then, for any positive integer q0 and any real number q > q0max{rh...,r„}, there is a CJ homogeneous function V of degree q with the same dilation as that off such that V is positive definite, radially unbounded (i.e., V(x)—>+<x> as //x//—> +<x), and V (x)/(2.i.6) <0for allx^O. In the following, we usually take q0 = 1. Lemma 2.3.3 Consider the nonlinear system described in (2.1.6). Suppose there is a C2 (continuously differentiable) function V(x) defined in a neighborhood U e R" of the origin, real numbers c>0 and 0
v(r 0 JV_ar V x c(l -a)

(2.3.1)

for all x0 in some open neighborhood of the origin. IfU=R" and V(x) is also radically unbounded, the origin of system (2.1.6) is globally finite-time stable.

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T. P. Leung & Y. Hong

The following lemmas were proposed by Bhat and Berstein (1997), Hong et al. (1999) and Hong (1999). Lemma 2.3.4 Assume system (2.1.6) is homogeneous of degree k with dilation ArBfis continuous andx = 0 is its asymptotically stable equilibrium. Let V (x) be the C homogeneous Lyapunov function of degree q of system (2.1.6) as described in Lemma 2.3.2. Then there are q > 0 such that V {x\2X6)<-c0V{x)(a°+kVa°

(2.3.2)

-min{ru...,rn}
(2.3.3)

Moreover, if then the zero solution of system (2.1.6) is finite-time stable. Proof. From Lemma 2.3.2, there is a C1 homogeneous function V(x) of degree q with the same dilation as that off, which is positive definite and radically bounded, and V(x)\ is negative definite. We have 1(2.1.6) c\r{x)q


(2.3.4)

where c\ = min V(z) and c 2 = max V{z). r(z)=i r(z)=i It is not difficult to see that V (x) = [dV(x)/dx]fix) is homogeneous of degree q+k with the same dilation as that of fix). Thus in analogy with (2.3.4), there exists c i> 0 such that c x r(x)q+ that

< -V(x) and therefore there is a constant c0 > 0 such V(x)<-c0V(x)

(q+kVq

where 0
li+r -> 0, i = !,...,«, as f-> 0. Then the zero solution of

Continuous Finite-Time Control 61

i=X*H/(*), /(0) = 0

(2.3.5)

is (locally) finite-time stable. Remark 2.3.5 Here system (2.1.6) can be regarded as a homogeneous approximation system of system (2.3.5). Proof of Lemma 2.3.4. By Lemma 2.3.2, there is a C1 homogeneous function V(x) of degree q with the same dilation as that of/ which is positive definite and satisfies [dVldx\fix)< -c0V(x) iq+k)lq . The derivative of V(x) along the trajectory of system (2.3.5) is (q+k),q

V (x) < -c0V(x)

+6Y{x)

(2.3.6)

where 5V(x) = Z",= t[dV(x)/dxi] ft (x). Then, for s> 0, we have S:'(B1xl,...,er"x„)

= lsq'r

~ ^

i=\

since fi (sn xi,...,er" xn) = o(s

ft {s^xh...,ernXn)

=o(^*)

( 2 . 3 .7)

dx

i

+r

'). Thus there is a neighborhood of the origin,

r

U = {x e R": x, = s zh i=l,...,n, 0 <s <£Q, Z = (z,,...,z„)r, »

(z) = 1} with £Q > 0

small enough, such that, for any x e U , \SV(x)\ = \5V{e^xh...,sr"Xn)

| <£0_ £i^ 2

min

y(zf«+W

(2.3.8)

zeS 0

Therefore, F (X)<-[CQ/2\ V(x)(q+k)lq holds when xe U . Thus, the assertion follows immediately by Lemma 2.3.3. Lemma 2.3.6 If the equilibrium of a nonlinear system is globally asymptotically stable and locally finite-time convergent, then it is globally finite-time stable. Proof. Asymptotic stability of a nonlinear system makes the system convergent to any given neighbourhood of its equilibrium in finite time, and certainly convergent to the region of finite-time convergence. Thus, after the state of the system is already in the region of finite-time convergence, then it reaches the equilibrium also in finite time. So the conclusion can be obtained.

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2.4 Finite-time state feedback In the section, we consider the finite-time stabilizability of second order analytic affine nonlinear systems that are small time locally controllable (STLC) (Hermes, Sussmann and others have studied the concept in detail). Consider systems in the form of

j*i=/i(4

x={Xhx2f

[x2=f2(x)+g(x)v,

(241)

v s R

where g(0) * 0,f{x), i = 1,2 are analytic v/ithf(0) = 0, /' = 1,2. Lemma 2.4.1 System (2.4.1) is STLC, then it can be transformed into the form: \xl=Ax^+h0{x2)+xlhl(xl

h0{x2)=^j

(242)

x2=u Without loss of generality, we take X = 1. In the following we will depart from system (2.4.2). We continue to study system (2.4.2), and here we take the odd integer^' > 0. At first, we consider a simple case, where a global result can be derived, \h=X*

(2.4.3)

Similarly, the feedback can be constructed in light of the homogeneous state feedback as does in above section, say u = -sig(xifl

-sig(x2)a2,0
(2.4.4)

where (rur2) = (l+/'-a2, 1), ax = a2l{j+\-a^), and k = a2-\ < 0. Theorem 2.4.2 The feedback law (2.4.4) is a globally finite-time stabilizing controlfor system (2.4.3). Proof: At first, we take a Lyapunov function in the form of

^='iM

il+or

+[(i+«iV2]x|

Continuous

whose derivative isV = -(\+a\)l2\x2\a2

Finite-Time

Control

63

^ 0 . To keep V=0, it must be xx = 0 orx 2 =

0. In the case of xx = 0, we have x2 = 0, while in the case of x2 = 0, we have x 2 = 0 and then xi = 0, according to the closed loop system with u in the form of (2.4.4). By LaSalle's invariant set theorem, we know the closed loop system is asymptotically stable. Then, notice that system (2.4.3) under the feedback law (2.4.4) is homogeneous of degree k < 0. Again from Lemma 2.3.4, the finite-time stability of the closed loop system can be obtained. Then return to system (2.4.2) with a rewritten form

L = ^ + ^ ( 4 Ao(*) = *l*l(*)+*o(*2)

(2.4.5)

\x2=u Still take u in the form of (2.4.4), and from the above discussion, we have J*l=*2 {x2=-sig{xlYl-sig{x2f2

(2.4.6)

where a2 =a.\rxlj < 1 defined as above. Forx * 0, we have h0 {sr\Xx,sr2x2) = o{sk+r\). Invoking Lemma 2.3.6, we obtain that system (2.4.5) is (locally) finite-time stabilizable. In summary, we have an important result: Theorem 2.4.3 Analytic second order system, (2.1.1), is (locally) finite-time stabilizable if it is STLC. Moreover, the system admits a homogeneous approximation system which is finite-time stabilizable by homogeneous feedback. Since when j = 1, it can be finite-time stabilized because it can be linearized to become the double integrators in the form of system (2.2.1), which is one of the most important classes of control systems. Here we would like to present other forms offinite-timecontrollers: Class 1: Take the feedback law u = -llsig(xi)a^ -l2\xipsig(x2)a2 with 0 < ai,a2,a3 < 1, l\, l2 > 0, and 2a.\ = a2(l+ai)+2a3.

(2.4.7)

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T. P. Leung & Y. Hong

Proposition 2.4.4 (2.4.7) is a globally finite-time stabilizing control for system (2.2.1). Proof. The proof can be divided into standard two steps, which exemplifies a procedure to deal with finite-time analysis. At first, we take a Lyapunov function in the form of ^ = > l N 1 + a i + [ ( l + ai)/2]*2 whose derivative along the trajectory of the closed loop system is F = - ( l + ai)/ 2 |xi| Q r 3| X 2 |«2
-l2sig(x2f2

(2.4.8)

with 0<« f , a2 <\,ai=[a2/(2-a2)], is a finite-time stabilizing feedback law as (Bhat and Berstein, 1997) derived. However, it is a little different from the controller reported in (Haimo, 1986) where ax > a2/(2-a2). Class 2: Take a feedback law as u = -hsigil^sigix.y+sigix^'

a a 1Vv

f -

(2.4.9)

where V2 < a < 1, v> 0, and /, > 0, / = 1,2. Then we have Proposition 2.4.5 (2.4.9) is a globally continuous finite-time stabilizing controller for system (2.2.1) when l,> 0 selected large enough. Proof: Take a Lyapunov candidate function V{x) H+a+

^

f U K ('2(1/aV-iK*)1/a)*

Continuous

Finite-Time

Control

65

which is homogeneous of degree (l+a)/a with dilation (l/a,l), and positive definite because the second term in V(x) is nonnegative on account of the following inequality {hsig{xxf+x2\l2^a\+sig{x2fa)

>0

when l2sigix{f+x2 * 0. To make clear, Kcan be expressed as

W 1 + a + — (N ( 1 + a ) / a -/2 l ( 1 + a ) / a l N 1 + a )

n&=— 1+a

\+a +h(Xla)x,{x2+l2sig{xxf)

Its derivative along the trajectory of the closed loop system under the feedback law (2.4.9) is V = -l2\xxfa+sig{x{)a{x2+l2sig{x{f) +l2il'a)X2(x2+l2sigixlT)+u(sigiX2fla)+l2il'a)Xl)

which is also homogeneous of degree 2 with dilation (1/ct, 1). It is easy to see that, when x ^ O and x2+l2sig(xif = 0 (or equivalently, sig(x2flla)+l2(V a)xi = 0), we have V (x) < 0. Define the unit sphere B0 = {x e R2:JJx) = 1} and B+ = {x 6 R2: -h\x1\2a+(sig(xl)a+h(Va)X2)(x2+l2sig{xir) > 0} and M = max [-/2M2a+(*/g(x1)a+/2(1/a)x2) (*2+Mg(*i)a)] > 0 XBBQ

M2=

max xeB0r>B+

(sigixJ^hlVi^sigilf'^sigixtf

+sig(x2r*y2^<\ Because (l2(v,a)sig(xiy+sigix2ra)) Va)

(l^^+sigix.f^)

>0

(1/a>

when U x\ *—s/'g(jc2) and B0r\B+ is a compact set, we have M2 > 0. Take l{> MXIM2, then V (e)<0 when eeB0. For any (tee/? 2 , there are a constant s(x)>0 and e=(el5 e 2 )e5 0 such that x ={s{x)Va

e,, e (x)e2)T. Therefore, V (x) = s 2 V (e)<0

66

T. P. Leung & Y. Hong

when x*0 due to homogeneity. Thus the conclusion follows by Lemma 2.3.4 because the equilibrium of the closed loop system is asymptotically stable and homogeneous of degree l-(l/a)<0. From the conclusion, we know that the following forms or K = -/>(*. 3 V) [1/15] , are available for system (2.2.1) when we take A sufficiently large. Moreover, if we take v•• = 2a-l, the feedback form of (2.4.9) is the same as that of (2.4.8). In some sense, this class may be viewed as an extended form of discontinuous time-optimal control. In fact, if we take v = 1 and or=l/2, feedback law (2.4.9) becomes u = -l\sgn{l2x\+sig{x2)2). Class 3: This class is in a generalized form of the controller proposed by Bhat and Berstein (1998). However, the finite-time analysis here is much simpler. Consider a class of controllers in the form of: u = - hsig(x2)a -Itsigi <j> (*))[a/( 2"a)],

/, > 0, i = 3,4;

(2.4.10)

where (/>(x) =x,+[l// 3 (2— a)]sig(x2f'a, 0 < a< 1. Proposition 2.4.6 (2.4.10) is a globally continuous finite-time stabilizing controller for system (2.2.1). Proof: The proof can still be proceeded in two steps. At first, we make a diffeomorphism transformation: zf= <j> (x), z2 =x2 to express the closed loop system as

h=z2+hra[-^(z2)°

- f «sW*))"'(2-o)] '3

Z

• =4l 2r^

/M

'

'3

z2=-lsig{Z2f

-l^ig^Y1^

and then select a Lyapunov candidate function for system (2.4.11) V t =

U ^ ) 2

I ,2/(2-^1 2 ' 2

(2-4.11)

Continuous

Finite-Time

Control

67

which is C1 and positive definite. With simple calculation, its derivative along the trajectory of the closed loop system is ^l(2All)=-T-| Z >| 2a/(2 " a) ^| 1 " a +^(z 1 ) a,(2 - a) Z2 -/3|z 2 | 1+a <3

= - f (W(zy(2-«Wfe)(1-a)/2 4 \4x+a)l2f - ^ N1+* ^3

2

4

< _ ^ I |Z2|'+«
0 < a < 1,

(2.4.12)

where y/: and y/2 satisfy the following two conditions: 1). They are continuous with y/,(0) = 0 andyy/fy) > 0, Vy *0 e Rfor i = 1,2; 2) They are C1 with [dyfy)/dy] *0,i = 1,2 in a neighborhood of the origin. Proof. The first step shows that the equilibrium of the closed loop system is asymptotically stable. Take a C1 Lyapunov candidate function, in an explicit form V(x) = ^in(sig(xlr/(2-a))dxl+^

(2.4.13)

It is positive definite and its derivative along the trajectory of the closed loop system is V=~x2y/2{sig{x2f)<0

68

T. P. Leung & Y. Hong

With LaSalle's invariant set theorem, the equilibrium of the closed loop system is asymptotically stable. Moreover, since near the origin dy/, (y)/dy ^ 0, i = 1,2 (in fact, d\fft (y)/dy > 0, / = 1,2 can be derived), then y/, (y), i= 1,2 can be extended in the form of ly+o(y), I, > 0, i = 1,2. In this way it is easy to see that the equilibrium of the system is locally finite-time stable by (2.4.8) and Lemma 2.3.6. Thus it is globally finite-time stable according to Lemma 2.3.6. The functions i//, (y), i— 1,2 provide more freedom in finite-time design, which can further modify and refine dynamic responses of the controlled systems and overcome some limitations in practice. For example, they can be chosen as any bounded sigmoid functions such as th(y) = [(ey-e'y)/(ey+e'y)], arctan(y), and sat(y), the saturation function, to build (global) bounded finite-time stabilizing controllers. In other words,

u = -arctanisigixif^-satisigixjf),

0< a< 1

is a bounded finite-time state feedback law. Remark 2.4.8 As a matter of fact, in light of the above classes, more finite-time controllers can be analyzed and constructed in a simple way by combining the above 'basic' types. For example, u = -l3sig(x2T -Itsatisigifaf^),

h >0,i= 3,4;

with <j>a = Xi+[1/ l3(2-a)]sig(x2)2'a, 0 < a < 1, is a finite-time controller. Moreover, based on the above design ideas, it is not difficult to build other classes of (bounded) finite-time stabilizing feedback laws. Up to now, for high order systems, few results have been obtained. In fact, with the above idea we can further study high order systems. Here for simplicity we only consider a third order system which is described as x{=x2 •x2=X2

(2.4.14)

XT)=U

In what follows, we only give finite-time feedback in an explicit expression for third order systems in the form of (2.4.14), and for high order systems, the feedback forms can be obtained similarly. Theorem 2.4.9 The following feedback law

Continuous

Finite-Time

Control

69

u(x)= -liisigixs) "2 +sig(l2sig(llsig(xly +sigix2y' af2aAW)

"2) <3«-2«2<*i -D-2

(2.4.15)

is a globally continuous finite-time stabilizing controller for system (2.4.14) if v > 1, v2 > 0, ?/3 < a< 1, h> 0 to make p(xhx2) finite-time stabilizing feedback, and l3 > 0 is selected suitably large enough. Proof: Take v> 1, then w{xxj2) = sig(p(x^2)Yl(2a-X) is C1, where p is taken of form (2.4.9) as a finite-time feedback for the double integrator system x i= x2, x 2 = p. Take 1. Then denote

W{x)={)


which is nonnegative on account of ^x){x3-p{xix2)) > 0 when x3 *p(xijc2). Since p(x\jc2) is a stabilizing feedback law of JC i = x2, x 2 = p, from Lemma 2.3.2, there is a Lyapunov function Va{x\jc2) such that V0(x\jc2) is C1 homogeneous of degree v+2a-l > 1 with dilation (I,a) and x

p0(xlyx2)--

2

d(xx,x2)

is negative definite with respect to (x\jc2)Take Lyapunov candidate function V{x) = W(x)+V0(xi^c2). It is easy to verify that V(x) is positive definite and moreover, homogeneous of degree v+2a-l with dilation (1,0,2 a-1). Its derivative along the closed loop system under the feedback law (2.4.15), which is also homogeneous of degree v+3a-2, is •/ N dW V{x) = —— axi

dW dV x2+-—x3mx)u+p0(x1^2)+-—(x3-p(xijc2)) 0x2 0x2

which can be rewritten in a compact form: V (x) = V (x)+$x)u, where

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T. P. Leung & Y. Hong

-/ v dW dW . . dV f . V[x) = —— x2+ -— x3+po(xu x2)+ -— {x3-p(xl^2)) OX\

OX2

OX2

Clearly, when 0}; and

Be ;ause BonB+ is a compact set and #x)(sig(x3)v* sighted)

v

i ) > 0, v2 > 0

when x3-p(xlrx2) * 0, we have M2 > 0. Take h >MXIM2 > 0, then V (e)<0 for all e eB0. Thus, for any 0* xeR2, there are a constant £>0 and eefl 0 such that x = £ e. Therefore, V (x)= £v+2a'1 V (e)<0 when x * 0 due to homogeneity. Then the conclusion follows from Lemma 2.3.4. From the conclusion, we know that the following forms u = -*x(x3+k2(xx5+x245' 7 ) ,/9 ) 3 ' 5 , or

u= -*xxx3,5+k2(xx+x29nf9,

are available for system (2.4.14) when we take kx and k2 are suitably sufficiently large.

2.5. Finite-time observer In the above section, finite-time controller design is proposed. In this section, the effort to construct finite-time observer will be made. Definition 2.5.1 Consider system (2.1.6). A finite-time observer of (2.1.6) is a system in the form

such that, for any initial condition x(0) and £(0) in a given domain, there is a positive number T dependent on x(0) and £(0) so as to make £(t)= x(t) when t >T.

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71

In fact, it is possible to obtain the other classes of finite-time observers in a simple form like

Ci=C2-h\C\-y\ai^\-y\
<7,>o, ^>o, Ci(o)=C10 -y)

- 2 >0, *2 >0, C 2 ( 0 ) < 2 0

Clearly, when ax=a2 = 1, it is nothing but the classic Luenberger observer [22]. Thus, it can be viewed as a generalization of the linear observer. Let e\= £\-x\, e2 = Qi-xi, ind e = (ei,e2)r, then the error dynamics generated by systems (2.2.1) and (2.5.1) is given by kl=e2-kl\el\aUig{el\

ei(o)<10-x10

Theorem 2.5.2 The equilibrium of the error dynamics (2.5.2) is globally finitetime stable for any kt > 0, i = 1,2 andfor all O<0-2<1,

2cr 1 -o- 2 =l

(2.5.3)

Proof: As noted before, the homogeneous degree and dilation coefficients of system (2.5.2) are {a2l<7\) -1 and (l/cr b 1), respectively. Also note that condition (2.5.3) implies V2
l

-lp.

ei

(2.5.4)

which 13 positive definite, and radically unbounded since k2 > 0 and a2 > 0. The derivative of (2.5.4) along the trajectories of (2.5.2) is V*(e)= - ^ 2 ( l + o - 2 ) h r i + ( J 2

(2.5.5)

which is negative semidefinite since kx > 0. Now note that V* (e) s 0 implies ex = 0 which, in turn, implies e2 = 0 using (2.5.2). By LaSalle's invariant set theorem, the equilibrium of the homogeneous system (2.5.2) is asymptotically stable.

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Similarly, other forms of observers can be found such as

£ < 2 -hnk\

-yfx.

CT

I >°> *i >°>
' i2 = « - *2 ^ 2 ,

(2.5.6)

CT

2 > °' *2 > 0» ^2 (°)= ^20

where y/h i= 1,2 are functions: y/j(z)z > 0, z * 0, dy/t{z)ldz* 0, with ^(0) = 0, /' = 1,2, and 0 <
+

(2.5.7)

^2M

>> = *! where at and 6„ / = 1,2 are given constants. In fact, consider the following compensator

Ci = C2 ~ k\ \e\ T1 sSn\e\

)-a2el+b[u

£2 =a\i\+a2^2+b2u-k2\e^2

(2.5.8)

s n

g {e\)-hel "^^lhr 1 sSn{e\\

where e = £-x with e = (ej, e2)r, and <JX and cr2 are defined as before. The error dynamics is given by

h = ei - h h |CTl ssn{e\)a e

k

e

2s

a e

2i

(2.5.9)

n e

a

a

e

h = 2 2 " 2\ \T % ( \)~(*3 - l K - 2*l | l r

1

s ne

S i l)

Letting e{ = e2-a2ex gives ex =et2-k^el\

l

sgn(el) (2.5.10)

e

2 s n e

a

a

h. = -*2 l i r s ( i)- (h - \ - l n Thus letting k3 = ax+a2 makes (2.5.10) exactly in the form of (2.5.2). Therefore, the equilibrium of (2.5.9) is finite-time stable.

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2.6 Output feedback Based on finite-time observers, we consider finite-time stabilization via output feedback for nonlinear system (2.1.2). We first note that if a proposed finite time observer can obtain the real value before the settling time Te. \fu = /^x) is a finite-time state feedback, we take an output feedback law in the form of K = //(<£) with ^determined by (2.5.1). Then we have fKQ = MLx), *>Te

(2.6.1)

Thus, if u = /4x) is a state feedback finite-time stabilizing control law for (2.2.1) with settling time Tx(x0), then the equilibrium of the closed loop system via the above output feedback is also finite-time stable with its settling time no greater than Te + Tx(x(Te (e0)). However, to prove that (2.6.1) is finite-time feedback, we have to prove that during t e (0, Te), the closed loop system x

=J[x)+G(x)^0

cannot flow up. In fact, in most cases, before the finite-time convergence of the observer, the system under the control cannot escape in a finite time (Hong et al, 2000). A lemma is given at first: Lemma 2.6.1 Consider continuous functions V,(x), V2(e), and V (x,e) (xelf and e e K1), which are with the same homogeneity degree with respect to (rlt...,r^, (rh...,r^, and (rh...,rn r,,...,r^, respectively. Suppose that Vj and V2 are positive definite with respect to vectors x and e, respectively, and V (x,0) = 0. Then for any given bj > 0, there is a positive constant b2 such that v(x,e)
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T. P. Leung & Y. Hong

Therefore, for linear systems with b\ = 0, clearly, with (2.5.8), the following control law bxu = -al^ra2^2+/j(Cu Cz) is an output feedback finite time stabilization law provided u(*lrx2) is a state feedback finite-time stabilization law for the double integrator system. Furthermore, consider the following nonlinear control system

• x2 =/oV*l, x 2/ +M

(2.6.2)

is locally finite-time stable where f0 (x) is any C1 function defined in the neighborhood of the origin of/?2 and satisfies ^o(O) = 0. Using homogeneous techniques, we also show that our control law for double integrator system results in a closed loop system with certain robustness property with respect to a class of nonlinear perturbation. Theorem 2.6.3 Consider the control system (2.6.2) and the control law

' £l =^2 -^lhr 1 s?Ae\) tZ2=u-k2\e^2

(2.6.3)

sgn^

where aua2, ku and k2 are defined as in Theorem 2.5.2 and \i(xt, x2) is a finite-time state feedback stabilizing control law for the double integrator system satisfying fj(e ^ x,jc2) = ea2CT1/Ax) Then the equilibrium at the origin of the closed-loop system composed of (2.6.2) and (2.6.3) is locally finite-time stable. Proof: In the coordinates of (x, e), the closed loop system is given by

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Finite-Time

Control

75

X\=X2

x2 = M{*\ +^X2+e2)+fo h=e2~kl\el\

(*)

ls n e

(2.6.4)

g \ l)

e

7 = ~k2 | e l |°"2 sSn{e\)+ /o M

Consider the case when^> = 0

*2 = /J\xi + e], ^2 + ^2) e1=e2-A:1|e1|CTl5'gn(e1) e

(2.6.5)

b , U ^'•Jg«(«l) 2 ==_-AJ: 2R

By the assumption on ju(x\, x2), and the discussion in Section 2.3, the equilibrium at the origin of (2.4.6) is asymptotically stable. Moreover, the right hand side of (2.6.5) is a homogeneous continuous vector field of degree (a2/ai)-l with respect to the dilation (1/oi, 1, l/oi, 1). Thus, by Lemma 2.3.4 and

f{x,e)=

/oto 0

f0{£ °l XU£X2) = O(E)

The assertion follows. Since f0 does not have to be known exactly, Theorem 2.6.3 indicates that the output feedback homogeneous finite-time stabilizing control law has certain robustness property with respect to a class of nonlinear perturbation. Some extended results can be obtained, too, with the help of the following lemma. Lemma 2.6.4 Suppose that the given observer is finite-time stable and the state feedback u = ju(x) can render (2.2.1) finite-time stable. Consider the system x = F(x,e) =fix)+G(x)ju(x+e), x e It, F(0,0) = 0,

(2.6.6)

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T. P. Leung & Y. Hong

with initial condition x(0)=Xo and e=x-xe R". If there is a C1 proper and positive definite function V:Rn—>R+ with V(0)= 0, such that V 1(2.2.6) <-G1(\\x\\)+G2(\\e\\) where GJ:[0,°O)—>[0,<X>), i = 1,2, are continuous functions that are strictly increasing with G/Oj=0, i=l,2, then the output feedback law p.(Q is finite-time stabilizing. Then, we have Theorem 2.6.5 If a given observer is finite-time stable and the controller n(x) can make (2.2.1) finite-time stable and homogeneous, then the output feedback law of form (2.6.1) is also finite-time stable, namely, the closed loop system (2.6.6) is finite-time stable. As a matter of fact, homogeneity makes the analysis much easy because nice results about it (referring to Hermes 1991; Rosier 1992), where the asymptotic stability implies the existence of C1 smooth Lyapunov function of the considered system. For general finite-time stable systems, we cannot assure that they admit suitable smooth Lyapunov functions.

2.7 Convergent rate In finite-time analysis, the convergent rate can be measured by settling time T. The smaller T is, the faster the convergence rate is. The following are the results for double integrater systems to illustrate the main ideas from different viewpoints. Theorem 2.7.1 Consider system (2.2.1). If a continuous control u=w(x) makes the closed loop system homogeneous of degree k < 0 and finite-time stable, then «= f?w(xhx2/P) does, too. Proof: Consider the closed-loop system •^1 = * 2 9

*2

/ x ft \ l>x2/fl)

= w x

Define x{ =xx, x{ = x2l/i, and t' =fi t, then the system becomes

(2.7.1)

Continuous

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Control

77

dx{

~dJ~

2

(2.7.2) dxo / \ v df ' which is exactly in the form of the closed loop system with u = w(x), which is finitetime stable, and therefore system (2.7.1) is, too. Remark 2.7.2 The result suggests that, if the settling time of the system x ;= x2, x 2=w(x) with initial condition Xi(0)=Xi0; x2(0)=0 is Th then the settling time of the system x \=X2, x 2 -pw(xhx2/p) with the same initial condition is T2=T]/J3. Namely, the convergent speed is increased. Another result is as follows. Theorem 2.7.3 Consider system (2.2.1). If a continuous control u=w(x) makes the closed loop system homogeneous of degree k < 0 and finite-time stable, then u= J3w(xi/b,x2/b) does, too. Proof: Consider the closed-loop system

\x2=pMxxl

P,x2l

p)

Define x{ = xxip and x{ = x2ip, then the system becomes 1*1'=*2' \x2'=w{x') which leads to the conclusion. Remark 2.7.4 Thus, if the settling time of the system x i=x2, x 2=w(x) with initial condition Xi(0)-Xio; x2(0)=x2o is T\, then the settling time of the system x ;= x2, x 2=Pw(x1/p,x2/p) with initial condition Xi(0)=Px!0, x2(0)=Px2o is still Th Namely, the convergent speed is increased when P> 1.

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2.8 Applications in robot control The section is try to propose simple finite-time controller for robots (cf. Hong et al. 1999 for detail). Consider the well-known model of rigid robot manipulators: M(q)q + C(q,q)q+G(q)=T,

q e FT

(2.8.1)

where j is the vector of generalized coordinates and z is the vector of external torques (here as the control input variables); M(q) denotes the inertia matrix, C(q,q)q is the vector of Coriolis and centrifugal torques, and G(q) represents gravitational torques. The following is some of its well-known properties: 1. M -2C is skew-symmetric; 2. C(xy)z = C(x,z)y and \\C(q, q )|| < L\\ q ||. Notice that the terms M(q), C(q, q) and G{q) are C1 smooth with respect to q, q in the robotic dynamics (2.8.1), so according to Remark 2.2.3, the control torque r have to be taken in a nonsmooth form in order to realize finite-time design. The purpose of following sections is to give concrete feedback laws for robot control in different cases. For convenience, denote sigixf = (^1\asgn(x2),...,\xn\asgn(x„)f e FT, x e FT with a> 0, and p(x)a = I , = ," |x,|a, a> 0. Lemma 2.8.1 A nonlinear system in the form of

h=x2 '

{Z.H.Z)

x2=-N /!«*(*! )Pi +I2sig{x2f2

with 0 < ah a2 < 1, ai= a2/(2-a2), h,h > 0 and N a positive definite matrix, is globally finite-time stable. Proof: Take a Lyapunov function candidate in the form of V*= hp(xi)l+al

+[{\+al)l2\x2TN~l

x2

whose derivative is V* =-(l+ori)/2p(jC2) 1+Qr2 < 0. To keep V* = 0, it must be x2 = 0, which leads to x 2 = 0 and then x{ = 0, from the closed loop system. By LaSalle's invariant set theorem, the system, of homogeneity degree -\<(ai-l)/(ai+l)<0 with

Continuous

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Control

79

(2/(1+a{), 1) as its dilation vector, is asymptotically stable. Thus, the conclusion follows by Lemma 2.3.4. When xteR, i=l,2, the control law is consistent with the feedback u=-llsig(x1)al

-42sig{x2)a2, «i = a2l{2-a2)

proposed in Section 2.4. Theorem 2.8.2 Given a desired position qd. Then state control law can be given in the form of T= G{q)-llsig{q^ld)al

-l2sig{qp

(2.8.3)

where 0 0, l2> 0, which can make (2.8.1) globally arrive at q infinite time. Proof: Denote xx=q-qd, x2=x x=q, and x=(xiT, x2T)T- Then the closed loop system is M(q)+C(q^2)x2+lisig(xl) a\ +l2sig(x2) "2 = 0

(2.8.4)

Notice that (2.8.4) is not homogeneous so the results of lemmas cannot be used directly. Take a Lyapunov function

V = i x2TM(q)x2+ - 4 — p(Xiy+ a\ 2

1 + cci

Then V =-x2T[hsig(x{)a\ +l2sig{x2)a2

}+llx2Tsig{xl)a\

=-l2x2Tsig(x2) a2 =-l2pix2)l+a2

<0

To keep V =0, x2 =0, which implies X] =0. Thus, according to LaSalle's Invariance Principle, (2.8.3) is a stabilizing feedback, namely, the closed loop robotic system is globally asymptotically stable. The following is to show that (2.8.3) is a finite-time stabilizing feedback. Consider the closed loop system,

80 T. P. Leung & Y. Hong

Xy =X2 x =-M{q)-l\c{q,x2)>C2

X2

+ l\sig(xi)*l +l2sis(x2f2

= -M{qdY\lxsig{xxY\

-l2sig{x2yx2]

+

(2-8-5> f0(gd,x},x2)

where f0(qd, *,, x2) =

-M(qd+x1)'lC(q^2)x2-M(q,xl)xl[hsig(xiri+l2sig(x2T2]

and M (q, xi)xx = M{q)A-M{qd) , according to the dynamics of robots. Note that (2.8.5) is time-invariant because q* is a fixed vector. Consider system *1=*2 x2=-M(q)~l

liSig(xxY\

+l2sig(x2}*2

(2.8.6)

which is finite-time stable by Lemma 2.3.4 and also homogeneous of degree -1< a 2 - K 0 with dilation (2/(l+oc,), 1,..., 2/(l+a0, l)TeR2". Notice that/ 0 is 'higher degree', namely, for any x * 0, f0[erixl,er2x2 lim — ^ a £->0 £ 2

^ =0

,(xi,x2)*0

Thus, near the origin, by Lemma 2.3.4, (2.8.5) is locally finite-time convergent on account of the finite-time stabilization of (2.8.6). Moreover, since the origin of system (2.8.4) is also globally asymptotically stable, so from Lemma 2.3.4, feedback (2.8.3) is a globally finite-time stabilizing controller. In fact, the gains, /;, / = 1,2, can be substituted by diagonal matrices: L'= diag(l,l,...l"), ;'=1,2, respectively, because sig(x,)aL'xi = L' = f(x,)i+a, i = 1,2. Sometimes, we cannot measure the velocity directly or accurately. In this case, we should consider effective ways to design measurement feedback. Theorem 2.8.3 Consider a dynamic outputfeedback law

Continuous Finite-Time Control 81 T=G{q)-lxsig{Qa\

-l2sig(C2)a2

(2.8.7)

(2.8.8) ^2 =M(q)-l[-G(q)-C{q,^2

+T]-k2sig{exY2

where 0
2 = A(x)~MW

j [llsis(xl

+e

\Yx+

l si

2 s{x2 + e2Y2

(2.8.9)

T

el=e2-klsig(eif l e2=f2(x'e)-k2sis(e\Y2 where fi = -M(q)'lC(q^c2)x2-M(q,xl)x1[l1sig(xi+e1)a1+l2sig(x2+e2T2] and f2 =-M(q)~i[C(q, X2)x2-C{qjC2+e2){x2+e2)~\ with M (q, xipci defined as above. Considering ai=a2/(2-a2), a2 =crl, a2 = 2ax-\, system x

\=x2

x2 =M\qdj

[ - / ^ ( x i +e1)Qrl -l2sig{x2

+e2Y2 (2.8.10)

=e

k s

e

l

h 2~ l 'S\ ir

e2=-k2sig{exYl is homogeneous with respect to dilation (l/or2, ..., \lcc2, 1,...,1, \lai,—,\la.i, 1>--->1) and of degree a2-\. It is easy to see that system (2.8.6) is finite-time stable and

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T. P. Leung & Y. Hong

ex=e2-hsis{e\Yx

eteR",i

= 1,2

e2=-k2sig{e\Y2 is, too, with a procedure similar to Theorem 2.5.2. Therefore, (2.8.10) is finite-time stable because after a certain finite time, e = 0 and x+e will be x exactly. Alsc, we can easily find that in system (2.8.9), fu f2 are 'higher degree' terms near the origin, {xxT, x2T, e/, e2T) = (0,0,...,0)e/?4". Thus, Lemma 2.3.4 can guarantee that system (2.8.9), or equivalently, system (2.8.1) with (2.8.7) and (2.8.8) is finitetime stable. The control in (2.8.1) can be taken as T= G-lisig(xx)a\ -l2sig{C,2)a1. In fact, other forms of homogeneous finite-time feedback laws can be used in robot control.

2.9 Robust issues The chapter has discussed finite-time stabilization and stabilizability for a class of controlled systems, based on nonsmooth techniques. In addition to the aspect of finite-time convergence, in this section, the robustness of the nonsmooth controllers is emphasized. In fact, as mentioned before, robustness is also a concern in our approach, considering the working environment of robots full of uncertainties. For simplicity, we only analyze the robustness in the case study of robot control. In practice, there are internal and external uncertainties or errors in robot systems, and deviations caused by modeling and measuring. Consider a practical model, instead of the nominal system (2.8.1) without uncertainty, M(q)q+C(q,q)q

+ G{q)+F(q,q)=T^w(t),

w = (wh...,w„)T

(2.9.1)

There are several classes of uncertainties in the dynamics. At first, there are parametri; uncertainty, which make M, C, G become M ,C ,G , unknown friction, which is usually modeled as F = F0+Fi( q ) where F0 is the dry friction and Fx is defined as viscosity friction, and bounded external disturbance. Thus, the robust concerning is quite important in practical design. Fortunately, the above approach can be easily applied to the situation when there are some uncertainties in robot systems. /. Robust against smooth terms in the dynamics (locally). Consider M(q) q + C{q, q)q +G(q)+F1( q)=T

(2.9.2)

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Control

83

where F ^ q), which is smooth and Fi(0) = 0, is viscosity friction and may be unknown. If we design a linear controller and a linear observer in the forms of

r=G{q)-ll{q^d)-l2q

|C 2 =M(q)-l[-C{q,^2

-G(g)]-*2«1

we have to make /1; l2, ku and k2 large enough to suppress the friction and other inertial forces in order to guarantee the convergence to some range. However, if we employ the nonsmooth technique, that is, r = G(q) -Iviglq-cf)

a

\ -l2sig( q ) «2

Cl=C2-kls>g{elFl c2=M(qyl[-c{q,c2)z2-^)]-hsigM72 where a, and o„ / =1,2 are defined as above. In this case, any smooth terms, including F\{q), become 'higher-degree' terms, so lu l2, ku and k2 need not be selected as very large numbers. 2. Robust against constant bounded disturbance (globally). In fact, the control also keeps some robust properties of bang-bang control. Consider a robot model with external disturbance (with bounded amplitude): M(q) q +C(q, q ) q +G(q) = i+w(0,

w = {wu...,wn)T

(2.9.3)

where ||w,{f)|| ^ W,j = 1,..,« denote the uncertainties. Now we still consider that T is taken in the form of (2.8.3). Roughly speaking, when li\qj-qdj\ai >W, (2.8.3) is still valid. In other words, the possible steady-state error range must be \qj-qdj\< (W/li)hal ,j = \,...,n. Therefore, when fixing h> W, the smaller ax means smaller steady-state error. Namely, we can adjust au instead of the gain lu to minimize the error. 3. About gravity compensation. The more independent the control law is of the robot base parameters, the more robust it is. feedback (2.8.3) only contains G which is related to the knowledge of the considered robot. We can study the term related to G with the above two aspects. Consider

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T. P. Leung & Y. Hong

M{q)q + C{q, q)q + 6{q)= r

{2.9 A)

where M ,C ,G contain some uncertainties, and they will be the nominal terms, M,C,G, when there is no uncertainty. Still use rin the form of (2.8.3), then we have M(q)q + C(q, q)q = G(q) -4'fii&q-q*)"l -42sig{ q ) «2 The only troublesome term is G=G-G

in regulation. However, its influence

can be minimized according to the analysis above because G is bounded. In fact, because G(q) is bounded, we can use feedback in the form of T =-/,«gfo V ) "1 -lisigi q) a2

(2.9.5)

whose structure is simpler and completely independent of the robot parameters. Here G{q) is regarded as a bounded disturbance, whose influence can be minimized according to the above analysis. In fact, form (2.9.5) is preferred in practice on account of its simplicity and robustness, especially in the environment of micro-gravity in space. 4. Tradeoff. Unlike the way adding boundary-layer in the design of practical sliding-mode control, the tradeoff between the continuity and robustness and highspeed of the form of bang-bang control, can be made with those adjustable numbers a, and 05, i = 1,2 in the approach. The continuity is also important in reality on account of the minimization of the chatter phenomena and some limitations of equipments. Return to a practical model of robot systems. Take r= G(p)+r0 with r0 as the nonsmooth part in feedback. Set G = G-G , w =w+ G +F0, then (2.9.1) can be rewritten as M{q)q + C{q, q)q + Fx (q)=r0+ Mf)

(2.9.6)

Notice that on the left hand side of (2.9.6) the terms are all smooth, and w (t) can be viewed as a new version of bounded disturbance. Thus, combining the above aspects, we can find that the above approach can be used to deal with a practical model with both uncertain parameters and external disturbances. As we know, the behavior of a smooth system with smooth control can be dominated by its linear part, if any, in local sense. Nonsmooth control, continuous or discontinuous, can provide the properties that linear control cannot, which exhibits the 'real' nonlinearity and its features positively. For example,

Continuous

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Control

85

1). Nonsmooth PD control is better than the conventional one because it is robust with respect to even linear terms, and also inherits the advantages of conventional PD control, for instance, simple and reliable structures. 2). Finite-time observers can be designed with nonsmooth techniques. That means that we can get the values of the real states in finite time. 3). It is also proved that some smooth systems, such as the ones with nonholonomic constraints, or underactuated links, or some nonlinear interconnections, cannot be stabilized by smooth feedback, but maybe by nonsmooth one. 4). Optimal control has a close relation with nonsmooth control. 5). Bounded control, sometimes, is needed in practical cases like actuator saturation and chaos control. 6). There are many applications for the design with good transient behaviors and real time. Sliding mode control and other discontinuous control have been studied intensively for this purpose (Su and Leung 1993; Utkin 1992), which can reach the sliding surface in finite time. To render a controlled system to arrive at the aim in finite time, terminal sliding mode control is discussed, too (Venkataraman et al. 1991; Yu et al. 1996). However, nonsmooth feedback may cause serious chatter phenomena and sometimes boundary layer techniques are needed.

2.10 Conclusions In the recent decade, the subject of nonsmooth analysis has grown rapidly, due to the recognition that nondifferentiable phenomena are more widespread and more fundamental to applications than had been thought. In fact, it has come to play a role in optimization, differential equations, mechanics and control theory, and is also in keeping up with the coming to the fore of several other types of nonlinear and irregular behaviours: finite-time convergence, robustness, catastrophes, and chaos. Here, we have focused on a simple class of nonsmooth control: continuous feedback control with fractional powers. In future, the potential research tasks are to find effective finite-time feedback laws for high order general nonlinear systems and the condition which is easily checked whether or not the considered systems are finite-time stabilizability. Furthermore, special finite-time control is needed for special practical plants. The control scheme should combine the general knowledge of finite-time stability and some characters in concrete situations. In addition, if possible, a complete theory for finite-time convergent differential equations should be made, including the unique-

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T. P. Leung & Y. Hong

ness of solution in forward time, finite-time stable region analysis, and finite-time control design for time-varying or time-delay nonlinear systems. Acknowledgement This work is supported in part by Natural Science Foundation and Project 973 of China.

References [I] Athans, M , and Falb, F., Optimal Control: An Introduction to Theory and Its Applications. New York: McGraw-Hill, 1966. [2] Beckenbach, E., and Bellman, R., Inequalities, third printing, Berlin, Heidelberg, New York: Springer-Verlag, 1971. [3] Bhat, S., and Bernstein, D., Continuous finite-time stabilization of the translational and rotational double integrators, IEEE Trans. Automatic Control, vol. 43, pp. 678-682, 1998. [4] Bhat, S., and Bernstein, D., Finite-time stability of homogeneous systems, Proc. ofACC,pp 2513-2514, 1997. [5] Bhat, S., and Bernstein, D., Finite-time stability of continuous automomous systems, SIAM J. Control and Optimization, vol. 38, pp. 751-766, 2000. [6] Canuda de Wit, C , Siciliano, B., and Bastin, G., Theorey of Robot Control, London: Springer-Verlag, 1996. [7] Celikovsky, S., and Nijmeijer, H. On the relation between local controllability and stabilizability for a class of nonlinear systems, IEEE Trans. Automatic Control, vol. 42, pp. 90-94, 1997. [8] Coron, J.M., On the stabilization in finite tiem of locally controllabel systems by means of continuous time-varying feedback law, SIAM J. Control and Optimization, vol. 33, pp. 804-833, 1995. [9] Haimo, V., Finite time controllers, SIAM J. Control and Optimization vol. 24, pp. 760-770, 1986. [10] Hermes, H., Homogeneous coordinates and continuous asymptotically stabilizing feedback controls, in Differential Equations, Stability and Control, (S.filaydi ed.), New York: Marcel Dekkere, pp. 249-260, 1991. [II] Hong, Y., Finite-time stabilization and stabilizability of controllable systems, preprint, 1999. [12] Hong, Y., Huang, J., and Xu, Y., On an output feedback finite-time stabilization problem, IEEE Trans. Automat. Contr., vol. 46, pp. 305-309, 2001.

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Finite-Time

Control

87

13] Hong, Y., Xu, Y., and Huang, J., Finite-time control for robots manipulators, preprint, 1999. 14] Hong, Y., Yang, G., Bushnell, L., and Wang, H., Global finite-time control: from state feedback to output feedback, Proc. of IEEE CDC, 2000. 15] James, M. R. Finite time observer design by probabilistic variational methods, SIAM. J. Control and Optimization, vol. 29, pp. 955-967, 1991. 16] Kawski, M., Stabilization of nonlinear systems in the plane, Syst. Contr. Lett., vol. 12, pp. 169-175, 1989. 17] Khalil, H., Nonlinear Systems, Prentice Hall, Upper Saddle River, NJ, 1996. 18] Rang, E., Isochrone families for second order systems, IEEE Trans. Automatic Control, vol. 8, pp. 64-65, 1963. 19] Rosier, L., Homogeneous Lyapunov function for homogeneous continuous vector field, Syst. Contr. Lett., vol. 19, pp. 467-473, 1992. 20] Ryan, E., Singular optimal controls for second-order saturating system, Int. J. Control, vol. 36, pp. 549-564, 1979. 21] Su, C. and Leung, T., A sliding mode controller with bound estimation for robot manipulators, IEEE Trans. Robotics and Automation, vol. 9, pp. 208-214, 1993. [22] Sussmann, H., A general theorem on local controllability, SIAM J. Control and Optimization, vol. 25, pp. 158-194, 1987. [23] Utkin, V., Sliding Modes in Control Optimization, Berlin: Springer-Verlag, 1992. [24] Venkataraman, S., and Gulati, S, Termial sliding modes: a new approach to nonlinear control systems, Proc. of IEEE Advanced Robotics, pp. 443-448, 1991. [25] Wonham, W., Linear Multivariable Control, New York: Springer-Verlag, 1974. [26] Yu, X., and Z. Man, Model reference adaptive control systems with terminal sliding modes, Int. J. of Control, vol. 64, pp. 1165-1176, 1996.

Chapter 3

Local Stabilization of Nonlinear Systems by Dynamic Output Feedback Pengnian Chen* and Huashu Qin *: Institute of Systems Science, The Chinese Academy of Sciences, Beijing 100080, China : Division of Mathematics, China Institute of Metrology, Hangzhou 310034, China

3.1 Introduction Stabilization of nonlinear systems by dynamic output feedback is one of the most important problems in nonlinear control theory. It has been received considerable attention since the middle of 1980's, but its advance is rather limited (see, for instance, Levis etal, 1987; Kokototvic, 1999). Roughly, dynamic output feedback stabilization may be divided into three classes: local stabilization, semi-global stabilization and global stabilization. In this chapter, we shall only study the local stabilization of nonlinear systems by dynamic output feedback. It is very known that the main difficulty in local stabilization of nonlinear systems by dynamic output feedback is that the nonlinear systems may have linearly uncontrollable and (or) unobservable critical modes. Based on the concepts of the local uniform complete observability and the stability in approximation, we shall present some methods of stabilization by dynamic output feedback for several classes of nonlinear systems which have linearly uncontrollable and (or) unobservable critical modes. Although the method of design, based on the local uniform complete observability, had already been presented by Tornambe (1994), the sys-

88

Local Stabilization

of Nonlinear Systems

by Dynamic

Output Feedback

89

terns considered in the chapter are more general than those in Tornambe (1994). The stability in approximation is an important concept and has many applications in theory of stability (Hahn, 1963). As the authors' best knowledge, the first application of the concept to stabilization of nonlinear systems by dynamic output feedback was presented by Chen, Qin and Zhu (1999) and Chen, Qin and Huang (1999) etc. The classes of nonlinear systems considered in the chapter contain locally uniformly completely observable system, minimum phase nonlinear systems, partially linear composite systems and so on. Some of these systems may be nonminimum phase. Due to space limitation, the problems of the semi-global and global stabilization by dynamic output feedback are not considered in the chapter. For the study of these problems, the reader, who is interested in these problems, is referred to the papers of Marino and Tomei (1991), Esfandiari and Khalil (1992), Khalil and Esfandiari (1993), Battilotti (1994), Mazenc and Praly (1994), Teel and Praly (1994,1995). In addition, stabilization of nonlinear systems by static output feedback is also not discussed here. For this topic, the reader, who is interested in this topic, is referred to the papers of Tsinias and Kalouptsidis (1990) and Byrnes, Isidori and Willems (1991). Throughout the chapter, only the local stability and the local stabilization at zero are considered, so the statement "at zero" will be omitted later. For convenience, some notations used in the chapter are introduced as follows: Rk[x] denotes the linear vector space consisting of all homogeneous polynomials of degree k with real coefficients in xijc2,- • -TX„, where x = (xi^c2,- • -^nf, and Ck[x] is the complexification of Rk[x]. In denotes the unit matrix of order n, C the open left half complex plane, and C* the closed right half complex plane. If P is a matrix or a linear operator, then u(P) denotes the spectrum of P, that is, the set of the eignvalues of P, listed according to multiplicity (see, for instance, Wonham 1979). cr(C^i,B) denotes the set of zero points of the linear control system 2(Cy*,.B). 0(||x||w) denotes any cf function defined on a neighborhood of the origin of R" that vanishes at the origin together with its partial derivatives up to order A?— 1. Z*= {0,1,2,...}. The rest of this chapter is organized as follows. Section 3.2 introduces some mathematical preliminaries, which contain spectrums of induced operators and stabilizability and detectability of dynamically extended systems. Section 3.3 deals with the dynamic output feedback stabilization of locally uniformly completely observable nonlinear systems. In Section 3.4, based on the concept of the stability in approximation, the dynamic output feedback stabilization of partially linear composite systems is considered. In Section 3.5, the dynamic output feedback stabilization of a special class of nonlinear systems is considered.

90

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3.2 Preliminaries In this section we will present some lemmas on spectrums of the induced operators and on stabilizability and detectability of dynamically extended systems.

3.2.1 Spectrums of induced operators Let x = (JCI^C2,. . .jc„f and He R"x". Define QH: Rk[x]-+ Rk[x], by QkH(p(x)) = (4-pW)Hx,Vp(x)sRk[xl ox QH is called the induced operator on Rk[x] by H.

(3.2.1)

Lemma 3.2.1 Let H e R"xn and QHk the induced operator on Rk[x] by H. Let A.],Z2,..., and Aj, are the eigenvalues ofH. Then a{QkH) = {XeC\X= I kfakieZ*,i /=1

= \,2,...,n, 1*/=*} i=\

(3.2.2)

Proof: Since we only study the spectrum of QHk, we can assume that QH is a linear opeartor defined on Ck[x]. First suppose H is diagonalizable, that is, there exists T e C"*" such that T~ HT = diag{X\,X2,...,Xn) where Xh i = 1,2,...,«, are the eigenvalues of//. Write A = diag{kuX2,...,^S. Let QAk be the induced operator on Ck[x] by A. Then we claim that cr{QkA) = {XeC\X= I kiAj,kieZ+,i /=1

= ia,...,n, ! * / = * } /=1

(3.2.3)

and °(QH) =
(3-2.4)

Local Stabilization of Nonlinear Systems by Dynamic Output Feedback 91

* 1

n

2 "

n , £ £ ; = £ , £ / e Z ,i = l,2,...,n}

is a basis of Ck[x], and QkK(x^x^ ...*$•) = (•£-(*?*% •••**„" ))M*l + ^ + ... +

x

"'

x

2

••-"£ ))h*2

{^-(x'?xkK..xkn"))Xnxn dx„

= (A:1/l1 + £ 2 ^2 + — + ^ n ^„)^i 1 ^2 2 --- : c n"

This implies that every polynomial in the basis is an eigenvector of Qk and that k

k

k

kiX\+k2Aq+. • •+k„A„ is an eigenvalue with respect to the eigenvector Xj' x22 —x„" , so the (3.2.3) holds. Let pi(x),p2(x),..., and pix) be a basis of Ck[x\. Let M = (mjj)M be the matrix of Qk with respect to the basis, that is,

= ZmyPtW,]

QA(PJW)

= 1,2,...,/

;=l

This means that dp; (x) I — f — H x = I mjjpj(x),j = \,2,...,l Let ^/(x) = p^T~1x), / = 1,2,...,/. Then q\(x), q2(x),...,qfcc) is also a basis of Ck[x]. By a simple calculation and (3.2.1), we obtain, fory = 1,2,...,/, 0 H fa I (*)) = — d

#* =—"

Pj(y) k i = —^TKy\y=T-\

^

H*\

T-l

I , (T-\ = ZmyPiV lx)

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P. Chen & H. Qin

I i=\ This follows that M is also the matrix of QHk with respect to the basis qi(x),.. .,q£x). Therefore we have a(QkH) = a(M)

q\(x\

= a(QkA)

that is, (3.2.4) holds. That (3.2.3) and (3.2.4) hold mean that Lemma 3.2.1 holds for diagonalizable H. For the general case, choose matrix Am = (%)„x„, such that H+Am is diagonalizable, m = 1,2,..., and Am —>0,/n—>oo Such matrix Am exists, m= 1,2,.... Clearly, we have H= Let X\m, A^m,...,^

lim (H + Am). /w—>oo

be the eigenvalues of H+Am. Choose a subsequence of

{/w}^=1-, written still as {w}^ = 1 , such that \Xim } ^ = 1 converges, /' = 1,2,...,«. Let Ajm —> Xx:,m —> oo,/ = 1,2,...,n Then by the continuity of the spectrums of linear operators, we know that the set {Ai,A2,...,A„} is just the spectrum of//. We have just proved that CT

ZkiAim,kieZ+,i=\,2,...,n,Zkj=k}

(2w+ A ) = i^C\l= m

i=\

i=\

where Q/'n+Am is the induced operator on Ck[x] of H+Am. Again by the continuity of the spectrums of linear operators, we have *(QkH>

lim -(Qk m—>oo

m

)

Local Stabilization

of Nonlinear Systems

= lim {XeC\X=Y.kiXim,ki&Z+,i m—»oo i=\ n = {/leC|/l= YjkiXim,kj&Z ;=1

by Dynamic

=

Output Feedback

93

\,2,.-,n,Y.ki=k} i=\

n ,i-\,2,...,n,'Z,ki=k} /=1

This concludes the proof. The expression (3.2.2) of the spectrum of Qfj is referred to Huang and Rugh (1992) and Kang and Huang (1997). It takes an important role in checking the observability of the dynamically extended systems and the stabilizability of partially linear composite systems by dynamic output feedback in the following. Remark 3.2.2 The operator QH can be extended, in a natural way, to a linear operator QH

m

on (Rk[x])m as follows: QkH{p\(x))

^ l * " - ! ^ where p(x) = (p1(x),p2(x),...,pm(x))T. mal polynomial.

QkH{p2ix))

Clearly, QH and Qfj m have the same

mini-

3.2.2 Stabilizability and detectability of dynamically extended systems Given the linear system U(C^i,B) i = A£ + Bu,

where | e / ? r , u&BT, and yzBf; AzRr*\ cally extended system of 2(C^4,5)

B&Rrxm and CeRpxr

S'AS+Bn,

(3.2.5)

. Consider the dynami-

94

P. Chen & H. Qin

T]2=D2HDI^

+ T13,

(3.2.6)

1-2,

iJit=D2H'^D^

+

rjh

/

,/-l,

where ^ e / T and «,e/?, / = 1,2,...,/; u is the input, and y is the output; HeR"*n, D^lf and D2e IT"". In order to simplify the expression of (3.2.6), we write (3.2.6) as the following compact form 77 = Aer) + Beu,

(3.2.7)

y = cerj, where 77 = {£, 77/,..., rjif, and A D2Dl ^e =

B 0

0 Im

...

0 0 (3.2.8)

1 2

D2H - DX l l

D2H ~ Dl

0 a 7

l /«

Be=[0 ... 0

0

a

- Im 2 w • .. a}Irm J X M M 7

iji/uxm

ce=[c 0 ... 0] IfXfJ where /x = /-+//«. Lemma 3.2.3 Consider the system (3.2.7). Let a(C,A,B) be the set of zeros of ',A,1). Let Letp(s) £(C,A,3). p(s) = s'-(a s -(a/s''+... s+ aj minimal polynomial ofQd, which ls'~'+... + a2s+a 1) be the n is the induced operator ofH on RjfxJ. Assume that

Local Stabilization

of Nonlinear Systems

by Dynamic

(i) Z(C,A,B) is stabilizable and detectable; (ii) a(C,A,B)na(QHj) nC+ = 0; (Hi) IfD, *0andD2 *0, thenj = 1 and Vco e H-coln Rank

Output Feedback

95

Cna(H)cna(C,A,B),

L\

BD2 0

A-coIr = n + r, C

where o(H)c is the complement ofo(H). Then system (3.2.7) is stabilizable and detectable. Proof: The stabilizability of (Ae, Be) is obvious. For the detectability, it is sufficient to show that \/co e C^, Rank

Ae-coIM

-r + lm

C„

(3.2.9)

Let co&C If co <£ a\Cy4,B), then r + lm>Rank

'A e-°>lH C

e J

YIK

'A-alr C

B~ + (l-l)m 0

(3.2.10)

-r + lm, which implies that (3.2.9) holds. Let cos a (C^4,B). Then by the condition (ii), mi. c(Qii). If £>, = 0 or D2 = 0, then by the condition (i) r+lm>Rank

Ae-ooIM L

= r + Im,

C

e

Ao-C0lv

= rank e .

C

r +lm . (3.2.11)

which also means that (3.2.9) holds. Now let D\ * 0 and D2 * 0. Then by the condition (iii),y'=l. This implies that, by (3.2.2), a(Qd) = o(H) and ffleo-'(C^,B)no(fl)c nC1". In this case, we have

96

P. Chen & H. Qin

Aa -col,, rank s n = rank C e where D\ Let

H-coIr 0 0

- rank(H - coln),

Ac-coIM C„

=[Dl0...0]„Xff 0

2 D2H

0 0

0 .. . 0 ... . 0 .. .

D-yHl-\

0

0

J

n 0

h

D

P=

' « • •

0 0 0 0

m

Then 0

l

n 0 -D2

P~l =

h

0 .. . 0 0 .. . 0

0 1m~ . 0 0 0 .. . 0

D2H

D-yH/ - l 0

0

By a direct computation, we obtain P-1

0

o /,

\H -a>In

D

0

l Ac-coIM

0

C

e

(3.2.12)

Local Stabilization

H-coh BDo 0

of Nonlinear Systems

0 A-coIy B 0 -col £>,

0

0

0

0

0

C

0 a l

\m 0

m

by Dynamic

0

0

0

0

97

0

*m -col m "27/H

Output Feedback

0

.(ai-co)Im

0

0

Then it follows from (3.2.12) that

rank

col,

H-colh -rank 0 0 H-coIn 0

• rank

0

= rank

I,

.-o M Ac-coI C„

L\ Ac-aaM

P-n

0

H - coIn

A

BD2 0

A-coIr

+ lm-n

C

= r + ml, coeC+ n < r ( / / ) c

na*(C,A,B).

(3.2.13)

In deriving the last equality of (3.2.13), we have used the condition (iii) and the fact that p{cd) * 0. This completes the proof of Lemma 3.2.3. Remark 3.2.4 In Lemma 3.2.3, the condition (iii) is implied by conditions (i) and (ii) whenever Di = 0 or D2 = 0. Therefore, system (3.2.7) is stabilizable and detectable if condition (i) and (ii) hold, and £>; = 0 or D2 = 0.

Lemma 3.2.5 Let m = p. If E(C,A,B) has relative degree {rhr2,...,rm} with m

£ r j = / . then system (3.2.7) is controllable and observable.

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P. Chen & H. Qin

Proof: This is an immediate consequence of Lemma 3.2.3 and the fact that m

2(C^4,B)having relative degree {rur2,...,rm} with £ / , =rimplies that a(C^4,B)= i=l

0. The proof of the following Lemma 3.2.6 is only routine checks and omitted. Lemma 3.2.6 a{CJ.,B) = a(Ce^ie,Be).

3.3 Stabilization of observable systems In this section, we shall consider the local stabilization by dynamic output feedback for nonlinear systems which may have linearly uncontrollable critical modes, but do not have linearly unobservable critical modes. Consider the nonlinear systems x = yf(x) w + 5e(x)u W y = h{x),

(3.3.1)

where xeR", uslT and yeEF;f.U-+R", g:U^>B"xm, and h:U-^l€ are smooth functions, fiO) = 0, h(0) = 0, and [/is a neighborhood of x = 0. We first introduce the concept of the local uniform complete observability of system (3.3.1), which is a generalization of the concept of observability of linear systems. Define the following differential operators: y = h(x)

y=y{x,u)~(j(x)+g(xyu) ox

/'+1W''+1>or,.,»a>,.,„w>) dx

i= Write

,=o duU) l,2,3,---,n-2.

(3.3.2)

Local Stabilization

of Nonlinear Systems by Dynamic

Output Feedback

99

yeHyT,yT,---x/n-W)T,ue=(uT,u^...^"-2V)T. Clearly, yeeJR"p and ueefl(""1)m. System (3.3.2) can be written as ye=y/(x,ue), where y/e C(mitnA)m,

(3.3.3)

FTP), y<0,0) = 0.

Definition 3.3.1 System (3.3.1) is called locally uniformly completely observable ifx can be locally solvedfrom (3.3.3), that is, there exists a C° function q, q(0,0) = 0, such that on some neighborhood ofx = 0,ye = 0,ue = 0 x=

q(ye,ue).

Remark 3.3.2 The concept of local uniformly completely observability is referred to Tornambe (1992). Let A-(ddx,)f(x))jx = 0, C=(d/3c)h(x)/x=0. It can be proved that system (3.3.1) is locally uniformly completely observable if and only if (C,A) is an observable pair of matrices (Chen, Han and Zhang, 1997), so the condition of the concept is weaker than that condition of global uniform complete observability (see, for instance, Teel and Praly, 1994). In order to study the stabilization of system (3.3.1) by dynamic output feedback, we first introduce the following Lemma 3.3.3. Lemma 3.3.3 Let A, Eh Lp and b be defined by the following (3.3.4) and D e Rpxnp with 0 1p 0 0 0 Ip

- 0 - 0 >Ek =

A= 0 0 0 0

0 •• 0 ••

h 0

'hip o o ... o hip o o ... o ,b = Jnlp 0 0 ... o

kl, k"-lI,

(3.3.4)

100

P. Chen & H. Qin

and D elf "p , where Ip is the unit matrix of order p, and k > 0, and /, > 0 (i = 1,2, ...,n), such thatp(s) = s"+lls"~'+...+l„ is a Hurwitzpolynomial. Then there exists k* > 0 such that when k > k , A-kEkLp+bD is a Hurwitz matrix. Proof: Let AL = A-Lp. By a straightforward computation, the eigenpolynomial of AL is (p(s)y. Thus AL is Hurwitz matrix. Let k > 0. By a simple computation, we have Ej;1 (A - kEkLp + bD)Ek = kAL

+^L-bDEk, k

This implies that Lemma 3.3.3 is true since Urn —bDEi. =0 . Now we consider the problem of dynamic output feedback of (3.3.1). Theorem 3.3.4 If system (3.3.1) is stabilizable by C°° state feedback and locally uniformly completely observable, then it is stabilizable by dynamic output feedback. Proof: Consider the dynamically extended system of (3.3.1) x = f(x) + g(x)Al, Aj = A . 2 ,

7 _,

(33 5)

-

^ n - 2 - ^n-1> A„_l = U,

y = h(x), where A, e BT, i= 1,2,..., w-1, and u e/T xm Let /l=(V, A/,---, IV-IV- Since system (3.3.1) is stabilizable by C°° state feedback, in the light of the principle of adding integrator stabilization, system (3.3.5) is also stabilizable by C° state feedback, that is, there exists a C°° function a, a(0,0) = 0 defined on some neighborhood of x=0, X= 0, such that the closed loop system consisting of system (3.3.5) and the feedback law u = a(x,X) is asymptotically stable.

(3.3.6)

Local Stabilization

of Nonlinear Systems by Dynamic

Output Feedback 101

Because system (3.3.1) is locally uniformly completely observable, there exists a C° function q, q(0,0) = 0, such that, on some neighborhood of x = 0, ye = 0, and ue = 0, x=

q(ye,ue).

For system (3.3.5), it is clear that ue =/L Thus we have x = q(ye,A).

(3.3.7)

It is obtained by replacing the expression (3.3.7) of x into (3.3.6) that u = a{q{ye,X),X).

(3.3.8)

Let fXye,X)r a(q(ye,A),Jl). Since the closed loop system consisting of (3.3.5) and (3.3.6) is asymptotically stable, the closed loop system consisting of (3.3.5) and the feedback u =p{ye,X) is also asymptotically stable. Let;/"' =y(-"\x,u, u,.. .,w("_1)) be the w-th derivative ofy. Define

ky..W

= /HHx,u,l.-.,u^^^^n^,

(3.3.9)

where q(ye, X) is defined by (3.3.7). Let w, = (wn,yva,--;WjP)T(=Rp, i = 1,2,...,«, w = (w1T,w2T,...,w„T)Tand

Construct an observer ofye as follows: M>\

= w2 + Mi (y ~ wi),

\\>2 = w3 + k Ijty — wi), (3.3.10) w«-l=wn+*'l"1'iil(y-wi).

w„ = * " / „ ( y - w 1 ) + ^(w,A), u = ^(w,A), where £ > 0, /, > 0, / = 1,2,...,n, such that p(s) = s"+lif'l+...+l„ is a Hurwitz polynomial. Then the closed loop system consisting of (3.3.5) and (3.3.10) is

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P. Chen & H. Qin

x = f(x) + g(x)Ah (3.3.11)

W] =M>2 + A / l ( j - W j ) , M>2

= W3 + k2 / 2 ( ^ - wj),

w„=*%0'-wi) + ^ w > ; 1 ) ' where A: > 0 is to be determined later and ;y = /z(x). Now we show that system (3.3.11) is asymptotically stable. For convenience, we write the closed loop system of (3.3.5) with u = £Kye,X) as a compact form: i = F{z,ye),

(3.3.12)

where z = (xT, XT)T. According to our construction, system (3.3.12) is asymptotically stable. By using the notations in (3.3.4), the system (3.3.12) can be expressed as z = F(z,w\ w = Aw + kEkL(y - w{) + b<j>(w, X),

(3.3.13)

where L = {ljp l2Ip ... IJP)T and y = h(x). It is easy to see from (3.3.9) and (3.3.11) that for (3.3.13) yW=fcye,A,fi(w,A)).

(3.3.14)

Let e = ye-w. Then, according to (3.3.14), system (3.3.13) is transformed into z= F(z,ye-e), (3.3.15) e = (A- kEkL)e + b{jj>{ye, X, 0(ye - e, X)) - {ye - e, X)).

Local Stabilization of Nonlinear Systems by Dynamic Output Feedback 103

By the definition of $w,A), we have ikve, ^ P(ye - e, X)) - {ye - e, Z))\e=0 = 0. Then we can assume that kye.-*>Piye

~ e>*•))~¥
where D =—(jj> (ye,A,J3(ye-e,X))—fiye—e,/t))|c=0 and oe R(ye,Z,e)

=

-D-fQj-(kye,yi,/3(ye-cTe,A))-^ye-ae,A))\x=(xda

with #(0,0,0) = 0. Thus system (3.3.13) can be rewritten as z=

F(z,ye-e), e

e = (A-kEkL

(3.3.16) + bD)e + bR(ye,A,e)e.

By Lemma 3.3.3, there exists k >0, such that when k> k , A —kEkLp+bD is a Hurwitz matrix. According to Lemma 4.3 of Byrnes and Isidori (1991), system (3.3.13) is asymptotically stable. This implies that system (3.3.1) is stabilizable by D dynamic output feedback. Example 3.3.5 Consider the system *\ ~ x2 + x3

+M

>

3 z

(3.3.17)

*3=-*3+*l*2> y=x\. It is easy to see that (3.3.17) can not be exactly linearized. We now show that system (3.3.17) is locally uniformly completely observable. By a direct computation, we obtain the derivatives of y

104

P. Chen & H. Qin

y = X2+X2+u,

(3.3.18)

(2) 3 3 yK ' =XT, -x~ +M-x, +x\x2+u. It follows from (3.3.18) that x\ = y and x-> + xi + u - y = 0, 3

3

.

v

m

(3-3-19)

X3 - x j -x^ + yx2 +u + u-y ' =0, It is clear that system (3.3.19) is locally solvable with respect to x2 and x3 at x2 = 0, x3 = () Let x2 = ^i(y, 7 , y(2), M, M ) and x3 = q2(y, y , yi2\ u,ii)be the local solution. Then we have x\ =y, x

2 = <7i (J, J, J ( 2 ) »",«)>

^3 = <]2(y>y>yi2)

>«.«)•

This means that system (3.3.17) is locally uniformly completely observable. It can be verified that system (3.3.17) with the feedback U X\ X3 IS asymptotically stable, so it is stabilizable by state feedback. According to Theorem 3.3.4, the system (3.3.17) is stabilizable by dynamic output feedback.

3.4 Stabilization of partially linear composite system Systems considered in this section may have linearly uncontrollable and unobservable critical modes. The local stabilization by dynamic output feedback for these systems is more difficult to be handled than that for those systems discussed in last section. In this section, we shall only consider the stabilization of partially linear composite systems, which are an important class of systems that may have linearly uncontrollable and unobservable critical modes (Saberi et al, 1990). The partially linear composite system can be described as the form i=/(z,£), 4 = AZ + B(F(z,€) + G(z,4yu),

y=c{.

(3.4.1)

Local Stabilization

of Nonlinear Systems

by Dynamic

Output Feedback

105

where zeFT, £eRr, uelT andyeRf; AeRr"n, B e Rr*m and Celf";f: U-+ R", G : U^gnxm a n d F. ^ ^ a r e s m o o t h functions,/(0,0) = 0, det(G(0,0))**>, F\0,0) = 0, F(0,^) = 0(| | ^ |2) and £/ is a neighborhood of (z£) = (0,0). Throughout this section, we will assume that p=rankC>m=rankB. System (3.4.1) can describe a large class of nonlinear systems. In general, it is not exactly linearizable. It may not only have linearly uncontrollable and unobservable critical modes and but also be nonminimum phase. Our study is closely related to the concept of asymptotic stability in approximation and Malkin's stability theorem (see, for instance, Hahn, 1963), so we introduce them first. Definition 3.4.1 Consider the system x = F(x),F(0) = 0,

(3.4.2)

where x e R", F:U—> R" is a C1 function and U a neighborhood ofx = 0. System (3.4.2) is said to be asymptotically stable according to the N,h order approximation if there exists an integer N > I such that the following system

x = F(x) + 0(lx\\N+1) is asymptotically stable. It is said to asymptotically stable in approximation, if there is an integer N > 1 such that it is asymptotically stable according to the N' order approximation. For a practical point of view, the concept of asymptotic stability in approximation is very important, since a system, which is asymptotically stable but is not asymptotically stable in approximation, may become unstable even under perturbation of very high order. The asymptotic stability in approximation means some robustness with respect to higher order perturbed terms. The following Lemma 3.4.2 is a special case of Malkin's stability theorem and its proof is omitted here. For general Malkin's stability theorem, may see the book of Hahn (1963). Consider the system z = h(z,y), K ,yh y = Ay + p(z,y),

(3.4.3)

where zeR", yeR™, and AGR""""; h andp are smooth functions with h(0,0) = 0 and P(0,0) = 0.

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Lemma 3.4.2 Suppose the following conditions holdfor system (3.4.3) (i) Rea(A)<0; (ii) The system z = h(z,0)

(3.4.4)

1

is asymptotically stable according to the N* order approximation for some integer N>1; (Hi) p(z, 0) = O(lfet'), andp(O.y) = 0(//y/f). Then system (3.4.3) is asymptotically stable. For the proof of Lemma 3.4.2, see, for instance, the book of Hahn (1963). Now we study the problem of dynamic output feedback of the system (3.4.1). Lemma 3.4.3 Suppose the following conditions holdfor system (3.4.1): (i) The system i = /(z,0)

(3.4.5)

is asymptotically stable according to the N' order approximation for some integer N>1; (ii) £(C,A,B) is stabilizable and detectable; (Hi) F(Z,O) = O(IMDThen system (3.4.1) is stabilizable by a linear dynamic output feedback of the followingform w = Sw + Ry, (3.4.6) u = Jw + Ky. Proof: Since G (0,0) is invertible by assumption, condition (ii) implies that Z{C, A, B G(0,0)) is stabilizable and detectable. Thus there exist matrices S, R, J and K of appropriate dimensions such that all eigenvalues of the matrix A

e=

A + BG(0,0)K BG(0,0)J RC S

(3.4.7)

have a negative part (Wonham, 1979). The closed loop system consisting of (3.4.1) and (3.4.6) can be written as i = /(*,£), , TJ = A lj + p

s (Z,TJ),

(3-4-8)

Local Stabilization

of Nonlinear Systems

by Dynamic

Output Feedback

107

where n=(f, wT)\ andpe(z,rj)=(v<^i)T, O7/with itiz,n)=B(G(z,®-G(0,0))(Jw+Ky)+BF(z,®. Then by the condition (iii), we have W(z,0)

= BF(z,0) = O(\\z\\N+l), |2)-

HO, 77) = B(G(0,£) - G(0,0))( Jw + Ky) + BF(0,£) = 0(U Thus system (3.4.8) satisfies all conditions of Lemma 3.2.1 and is asymptotically Q stable. This completes the proof of Lemma 3.4.3. Remark 3.4.4 In the dynamic compensator (3.4.6), we may let K = 0 (Wonham, 1979). Theorem 3.4.5 Supose the following conditions holdfor system (3.4.1): (i) System (3.4.5) asymptotically stable according to the N"1 order approximation for some integer N > 1; (ii) E(C, A, B) is stabilizable and detectable, and a (C,A,B)nAfiN has no points on the imaginary axis, where Affl

={AeC\A= YJ^-i^i i=\

with X\,A2,---,Xn the eigenvalues of—(z,0)| _n,k;sZ dz '*-u +

(Hi) Either D2 = 0 or Va> e a (C, A, B) nC H-coIn rank

BD-) 0

,z' = l,2,---,w,l< £<7V}; /=1 c

na(H) ,

Dl A-col,. = n + r C

(3.4.9)

where D, = —f(z,E)lz = OE-O, and D2 = — F(z,£)/z = 0£ = o, and a(H)c is the comdg dz plement ofo(H).

108

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Proof: Let F ( Z ^ ) = F W ( Z ) + ^,(Z,#),

(3.4.10)

where /^(z) e (Rj[z])m, and $>(z,# has the property that

If/ > TV, then by Lemma 3.4.2, system (3.4.1) is stabihzable by dynamic output feedback and Theorem 3.4.2 holds. Now let j
i = /(*,£), £ = A5 + B(F{z,4) + G(z,&Ai), :

(3.4.12)

I i=\ y = C?; In order to prove Theorem 3.4.5, it is sufficient to prove that system (3.4.12) is stabihzable by dynamic output feedback. We introduce the following coordinate transformations for system (3.4.12): Hi =rii(z,4,T}l, — ,Tji_uAi) = Fi_l(z,4,T}x, — ,r}i_1) + G(z,4)Ai, where >' = 1,2,...,I, and F0(z,Z) = F(z,Z), Fi(z,4,m,-,rJi)

=

-D2Hi-lD14

+ [~(Fi_1(z,4,li,??h-,rji_1)

+

G(z,4)Ai)]f(z,4)

(3.4.13)

Local Stabilization

of Nonlinear Systems

by Dynamic

Output Feedback

109

(3.4.14) l

+ I (1^-Fi_l(z,Z,7}h-,T}i_l))(D2H>- Dl{ r=\ dTlr

+ T?r+il

with Xt = G\Z, dim-FUz, #, m, • • -. tb-ii), i = h2,.. .,/. Now we claim that, for / = 0,1,...,/, = {Q}Hm)i{F^\z))

Fi{z,£,Xuill,---,rli)

+

(l>i(z,E,rlw--,rli)

(3.4.15)

where <j>, has the property that


(3.4.16) l|2

^•(0,1,7!,—,7/) = 0( (#,7/i,•••.J?/) )• For the definition of QJH

,see Remark 3.2.2.

We prove the claim (3.4.15) and (3.4.16) by induction. According to (3.4.10) and (3.4.11), the claim (3.4.15) holds for / = 0. Assume that (3.4.15) holds for / = k, 0 < k < I. By (3.4.14), we have Fk+1(z,E,r?l,...,r?k+l)

=

-D2HkDli

+ [^((e^;W)A:(^[y'](z)) + 4 ( ^ ^ ' 7 l . - ^ A ) + Gfe#)^+1)]/(z,#) + t ^ ( ( e ^ m ) A ( ^ 1 ^ ) ) + ^ ( ^ # , ' 7 l , - , ^ ) + G(z,#)4 + 1 )](/l# + 5 7 i ) k p> + I [-^((e^OT)A(FW(z)) r=\ 0rlr

+

=l£(ei,w)*(FW(z))]//z

^+1(z,#,71,-,^+l)

= (QJHm)k(F[jkz))

+

^(z,#,// 1 ,-,7 / t ))](£» 2 //'- 1 Z)i#

+ ^k+l(^m,-^k+l)

where \+1= G"\z,#)(;/k+,-Ffa£TJ,,...,

%)) and

+

7r+i)

(3-4.17)

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P. Chen & H. Qin

**+l(*>&7l.->>7Jt+l) -D2HkDlZ+[j-(QJHm)k(F[Jkm(f(z,t)-Hz)

=

+[— (.k(z>^Tii>-~>7ik)+G(z>£)h+i)]f(z^)

H-^((QJHytn)k(Fti\z))+
k+l(z,0,---,0)=[—(Q{{Jk(F[jXm(f(z,0)-Hz) + [— (^(z,0,-,0) + G(z,0)4 +1 )]/(z,0)

where A*+1 = - G "1(z,0)Fi(z,0,0,...,0). If/ = 1, then F^(z) = D2z and, by (3.4.18), h+i(°>t>Tih--->rik+i)

=-D2HkDl{HJ-(QJH,m)hD2z)]Az,Z))\z=0

dz d_ + [-(^(z,^/7l,-,^) + G(z^)A^ +1 )]/(z^))| z=0 d

+ [— Wjfc(0)^»7i,-,i/jfc) + G ( 0 > ^ + 1 ) ] U ^ + B7i) + I h^*(0,£i7i,-" 7*)](02#r_1^ + W

(3.4.19)

Local Stabilization

of Nonlinear Systems

by Dynamic

Output Feedback 111

< 3 - 4 - 20 )

=o(\({,m,-,nk+if),

where \k+l = G'^O^iq^rF^O^rji,.. .,rjk)). The last equality in (3.4.20) has used the fact that [[d/dz](£HjF(D#m,Q)\, =o = fyltD&CXWtf). If/ > 1, then D2 = 0 and, by (3.4.18), ^+1(0,^,^,-,^+1)=[^(ei)m)*(/W(z))]/(z^)|z=0 dz HfWk(z,t,TJh---,T?k) ^dz

+

G(z,t)Ak+1)]f(z,{))\z=0

+ [ ^ ( ^ ( 0 ^ , / 7 i ) - , ^ ) + G(0,^)Ayfc+1)](^^ + JB^1) + S [^k(0^,rjh-,rjk)](D2Hr-lDl4 r=\ Crlr =

+

Tjr+l)

(3.4.21)

0(\({,nh~;nk+lfl

where Ak+\ is the same one as in (3.4.20). From (3.4.17),(3.4.19) (3.4.20), and (3.4.21), it follows that (3.4.15) and (3.4.16) hold. Let TJ = (rji7,^,...,^7)7 and A — (A,/, A2T,...,AiT). Obviously, rf is invertible with respect to A, on some neighborhood of z = 0, £ = 0, and A = 0. From the definition (3.4.13) of /ft, /'= 1,2, ...,/, we obtain n, = D2Hi-lDl{ = D2Hi-lDl{

+ Fi(z,{,m,-,Th) + rji+l,i =

+ G(z,{)Ai+l

l,-l-l, l

l-\

i=\ l l-\ = D2H ^Z+^aiTn+Feiz^Ti^-^ l

i=l

where

+ Giz&v

(3.4.22)

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I I Fe(z,£,77i,---,?7/) = F/(z,£,/7i,---,77/)+ Ia/G(z,£)A/- Hami i=l i=\ I

= Fl(z,Z,m,-,rU)-

ZO/FMCZ.^ITI.-.I/M)

(3-4-23)

/=1 The last equality above follows from (3.4.13). Therefore, under coordinate transformation (3.4.13), the system (3.4.12) is transformed into the form i = /(z,£), r) = Ae7] + Be(G(z,t)v + Fe(z,Ti)),

(3.4.24)

y=cen, T

7 T

where rj = (f,Tjl ,...,Tj, ) , Fe(z,rj) = Fe(z,£„r]x,...,r]i) and (Ce, Ae, Be) is given by (3.2.8) In terms of (3.4.15) and (3.4.23), we obtain

i=l

'

z'=0 l-\

= (/>l(z,£,Til, — ,Tji)-YJai+li{z,Z,rix,-,rii))

(3.4.25)

;=0

Obviously, by (3.4.16) and (3.4.25), Fe (z,0) = 0(|z| y + 1 ), Fe (0,77) =

0{\rf).

Since the assumption (ii) holds, according to Lemma 3.2.1 and Lemma 3.2.3, E (Ce^4e,Be) is stabilizable and detectable. In addition, by Lemma 3.2.3, a\CeAe,Be) =

a\CJ,B).

Moreover, in system (3.4.24), D2 = [d/dz~]Fe(z,rf)\z = 0^ = 0 = 0. Thus it can be seen that system (3.4.24) satisfies the condition (i) and (ii) of Theorem 3.4.5 if£(Ce^ie,Be) replaces £(CJ,B).

Local Stabilization

of Nonlinear Systems

by Dynamic

Output Feedback

113

By repeating the process above no more than N+l-j times, we can get from (3.4.1) a system which has the form of system (3.4.24) with Fe(z,0) = 0(\\z\\N+i), Fe(0,n) = 0(||T7||2).

(3.4.26)

We have shown that system (3.4.24) with the condition (3.4.26) is stabilizable by dynamic output feedback at the begining of the proof. This completes the proof of Q Theorem 3.4.5. Corollary 3.4.6 Suppose system (3.4.1) satisfies conditions (i) (3.4.5) is asymptotically stable according to the N ' order approximation for some integer N> 1; (ii) Z(C,A,B) is stabilizable and detectable; (Hi) <j(C,A,B)nC+ = 0. Then system (3.4.1) is locally asymptotically stabilizable by dynamic output feedback. Proof: This is an immediate consquence of Theorem 3.4.5 and the fact that the condition a(C^.,E)r\C? =0 implies that cr*(Cy4,B)ri(C/jliO'(g^)) has no points on Q the imaginary axis, and that the condition (iii) of Theorem 3.4.5 holds. Now we consider the dynamic output feedback stabilization of nonlinear minimum phase systems of the form

x =fyc)+g{x)u

(3.4.27)

y = h(x) where x e R1, u, y e R",fe C°(U, Rl),t{0) = 0,ge U is an open neighborhood of x = 0.

C°(U, ^m), h e C°(U, IT), and

Theorem 3.4.7 Let the system (3.4.27) have relative degree {ri,r2,...,rm} atx = 0 and its zero dynamics is asymptotically stable according to the N1 order approximation for some integer N > 1 Then system (3.4.27) is asymptotically stabilizable by dynamic output feedback. Proof: We can assume without loss of generality that system (3.4.27) has uniform relative degree rt = r2= ... = rm = r since this can always be achieved by performing dynamic extension on system (3.4.27). Consider the dynamically extended system

114

P. Chen & H. Qin

x = f{x) + g{x)A, X = u,

(3.4.28)

y = h(x). In order to prove Theorem 3.4.7, it is sufficient to show that system (3.4.28) is stabilizable by dynamic output feedback. Clearly, system (3.4.28) has uniform relative degree {r+1, r+1,...,r+1} and has the same zero dynamics as (3.4.27). By using an appropriate coordinate transformation, (3.4.28) can transformed into the form (Byrnes and Isidori, 1991) * = /() (*>£)>

<3A29)

£r=Zr+l i r + 1 = r Z £ I ^ + F ( z > 0 + G(z,|)«,

wherez e FT, fcFT, E,eBrxm, i = 1,2,...,r+l, £= {&,&,...,&??, I = n+mr, and/ 0 , F and G are some C° functions defined on some neighborhood of (£A) = (0,0) with the property that det(G(0,0)) * 0 and F(0,£) = Ofl|£||2). The zero dynamics of (3.4.29) is i = /o(z,0) which is asymptotically stable according to the N lh order approximation, since by the assumption the zero dynamics of (3.4.29) is asymptotically stable according to the N"1 order approximation. System (3.4.29) can be written as the form of system (3.4.1) i = / 0 (2,£), £ = AZ + B(G{z,ftu + F(z,&), y^O;. with

(3.4.30)

Local Stabilization

of Nonlinear Systems

0 Im 0 •• • 0 0 Im - •

by Dynamic

0 0

115

' 0"

A=

,B = 0

Output Feedback

0

0 •• • / » m E E E \ 2 3 V+l fjtfi

0

_V

//X7M

C = \fm 0 - Ol'/WX// where//=(rt-l)/«. It can be easily verified that E{C^i,E) of (3.4.30) is controllable and observable, and a(C, A, B) = 0 . Thus system (3.4.30) satisfies all conditions of Corollary 3.4.6 and is stabilizable by dynamic output feedback. This completes the proof of Theoa rem 3.4.7. Example 3.4.8

Consider the system Z = -Z^ +

g2+U2,

(3.4.31)

y=h~2hAlthough system (3.4.31) does not have the form of system (3.4.1), we can transform it into the form of (system (3.4.1) by introducing the dynamic system u = X\,

(3.4.32)

Consider the closed loop system of (3.4.31) and (3.4.32). Let rjx = z2+Xx. Then the closed loop system of (3.4.31) and (3.4.32) is transformed into

i=-z3+
£> =24l +
116

P. Chen & H. Qin

which has the form of system (3.4.1) withflz,® = -z3+^22+(Tjrz2)2, G(z,£) = 1, F(z,# = 2z(-z3+62+(71-2^1-4:2-z2)2) and "0" "0 1 0" A = 2 1 1 ,B = 0 , C - [ l 0 0 0 1

-2

0}

where £ = ( £ , ^2,JJI) • Now we verify that (3.4.33) satisfies all conditions of Theorem 3.4.5. It is easy to see the system z=Xz,0) = -z3+z4

is asymptotically stable according to the fourth order approximation, so the condition (i) of Theorem 4.1 satisfied. By a direct computation, we obtain a(C^i,B)={l/2} and H =— Az,0)|z = 0 = 0 for system (3.4.33). Then a(C, A, B) f] (UAj=x (QJH)) = 0 . dz Thus the condition (ii) of Theorem 3.4.5 is satisfied. We can also see that the condition (iii) of Theorem 3.4.5 is satisfied since D2 = (— F(z,£))|(z ^ = 0 = 0. Thus system dz (3.4.33) is stabilizable by dynamic output feedback. Since F(z,0) = -2z4 = 0(|z|4), system (3.4.33) satisfies all conditions of Lemma 3.4.2. Then using Lemma 3.4.1, we design a linear dynamic compensator for (3.4.33) as follows: u = Jw + Ky, (3.4.34)

w = Sw + Ry, with J = [-5

-13

- 4 ^ = 0,5': -6

-9 -8 -11

0" ~-5~ 1 ,R = - 5 -1 -1 1

Combining (3.3.32) and (3.3.34) gives the control law for system (3.3.31) as follows u = X\, X\ = Jw, w = Sw+Ry.

Local Stabilization

of Nonlinear Systems

by Dynamic

Output Feedback

117

Example 3.5.2 Consider the system f

l = z 2 - z l ( z i 2 + z 2 ) + £2'

z

2=-zl-z2(zl2+z2)

+

£'

2

(3.4.35)


l

2-zl(zi2+z?)> 1

1

2 (3 4 3 6 )

z

2=- z l- z 2( z jf + z 2 ) '

Now we show asymptotic stability in approximation. Let K(z 1 ; z 2 ) = ^ ( z 2 + z 2 ) . Then V is positive definite on R2. It is easy see that K(z 1 ,z 2 ) = - ( z 1 2 + z | ) 2 . which implies that zero dynamics (3.4.36) is asymptotically stable according to the fourth order approximation. Thus by Corollary 3.4.6, system (3.4.35) is stabilizable by dynamic output feedback. Since F(zuz2,%) = ^ W and F(z,,z2,0) = z24 = 0(\\(zu z2)\\) in system (3.4.35), by Lemma 3.4.2, we only need to employ the static output feedback u= ~2y.

3.5 Stabilization of a special class of systems In Section 3.4, one of the main assumptions is that the system i = /(z,0)

118

P. Chen & H. Qin

is asymptotically stable according to the N,h order approximation for some integer N > 1. In this section, we will continue to study the problem of dynamic output feedback stabilization of system (3.4.1) without this assumption. But we only consider a special case of system (3.4.1), i.e., the dimension of z in (3.4.1) is one. Thus the system considered in this section is i = /(z,£), i = A£ + B(G(z,Zyu + F(z,4y) y = C£.

(3.5.1)

where zeR, £eRr, ueFT zndy&KF; AeRr*r, BeRr*m and CeR"*r;f. U-+R", G:C/-> FT""", and F: U^FT are smooth functions, 7(0,0)= 0,det(G(0fi))* 0,F(0,0) = 0,F(0,# = 0(||£||2) and U is a neighborhood of (z,£ = (0,0). We first need to study the stability of a class of systems before we study the stabilization of system (3.5.1). Consider the system z = h(z, yv), y = By + Gl(z, y)y + G2 (z, y)h{z, y).

(3.5.2)

where z e R,y e FT, B e #"*'"; h{z,y), Gx(zy) and G2(z,y) are soomth functions on some neighborhood of z = 0 and £= 0; /»(0,0) = 0, 0^0,0) = 0 and G2(0,0) = 0. Lemma 3.5.1 Consider the system (3.5.2). Assume that the following conditions hold: (i) i== h(z, 0) is asymptotically stable; (ii) a(B) aC~ Then system (3.5.2) is asymptotically stable. Proof: Let a =—h(x,y)\ A A. dz '^^^=0,^=0 If a * 0, then by the condition (i),a < 0. It is clear that Lemma 3.5.1 holds for this case. Now let a = 0. Then system (3.5.2) has a center manifold y = A?), which satisfies the equation

Local Stabilization

of Nonlinear Systems

^ ^ / ? ( z , / r ( z ) ) = Bx(z) + G1(z,x(z)Mz)

by Dynamic

Output Feedback

+ G2(z,x(z))h(z,x(z)).

119

(3.5.3)

dz Clearly, we can express h(z,y) as h(z,y) = h(z,0)+G0(z,y)y, 1

(3.5.4)

8

where G0(z,y) = \—h(z,x\ 0dz

da .

According to (3.5.4), the equation (3.5.3) can be rewritten as ^-(h(z,0)

+ G 0 (z,;r(z)Mz))

OZ

(j.D.D)

- Bn(z) + Gx(z,n{z))n{z) + G2(z,x(z))(h(z,0)

+

G0(z,x(z)Mz)).

From (3.5.5), we obtain y = TAX) = y^z)h(z,0), where W(z)

= [ 5 + Gx (z, n(z)) + G2 (z, 7c{z))G0 (*,*(*)) - ^

dz

G0 (z, *(z))] "* ( ^ ^ dz

- G2 (z, x(z))).

By the reduction principle of stabilty, the stability of system (3.4.3) is equivalent to the stability of the reduced system z = h(z,0) + G 0 (z,;r(z)Mz,*-(z))/2(z,0) = (1 + G 0 (z,;r(z)Mz,*(z)))/*(z,0). which is asymptotically stable at z = 0 since \jAz,iiz)) = 0(\z\). This completes the Q proof of Lemma 3.5.1. In what follows, we write

where j[z,Q is defined in (3.5.1). Lemma 3.5.2 Suppose the following conditions holdfor system (3.5.1): (i) The equilibrium z = 0 of the system

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P. Chen & H. Qin

z = f(z,0)

(3.5.6)

is asymptotically stable; (ii) Z(C,A,B) is stabilizable and detectable; (Hi) 0e{/3}na(C,A,B); (iv) F (z,0) = O(llzlf). Then system (3.5.1) is stabilizable by dynamic output feedback. Proof: Since system (3.5.6) is asymptotically stable, /? < 0 It is clear that system (3.5.1) is stabilizable by dynamic output feedback if j3 < 0. Now let /? = 0. For system (3 A l ) , we construct a dynamic system u = A,

(3.5.7)

X = u. Let TJ = G(z,^)A+F(z,^). Then the system consisting of (3.5.1) and (3.5.7) has the form i = /(z,£), % = AZ + Br], i = £(G{z,4)A. dz

(3.5.8) +

F(z,4)))f{Z,4)

+ (— {G{z, 4)1 + F(z, ZMA where•k= Let

Z + BTJ) + G(Z,

&u,

G\z,§{rrF(z,§). ' 0" 'A B~ ,ce=[co] Aa = ,Ba = e 0 0 ' e }m_

Then system (3.5.8) can be rewritten as i=/(*,£), W = Aey/ + Be(G{z,4)u + Fl(z,¥){A4 y=cey/, where y/ =

(^T,TJT)T,

+ Br1) + F2{z,¥)f(z,4)),

(3.5.9)

Local Stabilization

of Nonlinear Systems by Dynamic

Fx{z,¥)=—(G(z,£)A

Output Feedback

121

+ F(z,£)),

F2(z,W) = ^-(G(z,Z)A + F(z,Z)). oz 1

with X = G (Z,®(TJ-F(Z,®). Since the conditions (ii), (iii) and (iv) hold, by Lemma 3.2.3, we can prove that I (Ce^ie,BeG(0,0)) is stabilizable and detectable. Then there exists a linear dynamic compensator of the form w = Sw+Ry, u=Jw.

(3.5.10)

such that the matrix A--

Ae

BeG(0,0)J

RC„

S

(3.5.11)

is asymptotically stable. The closed loop system consisting of (3.5.9) and (3.5.10) can be written as (3.5.12)

i = /(*,£),

where g=(yf, wT)Tand B„ ((G(z,^)-G(0,0))7w+F 1 (z,^)(^^ + JB/7)), Gx{z,C)C--

G2{z,of{z,$y-

B„ 0

F2{z,¥)f{z4).

It can be readily verified that Gi(z,M= O(\\(z,0\\2), G2(z,Q = 0(11(2,011). Therefore system (3.5.12) satisfies the conditions of Lemma 3.2.3, and is Q asymptotically stable. Thus Lemma 3.5.3 holds. Theorem 3.5.3 Assume that the following conditions hold: (i) z=f(z,0) is asymptotically stable;

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(ii) Z(C,A,B) is stabilizable and detectable; (Hi) 0€{P}na(C,A,B); 'p-a Dx = 1+r, a e a(C,A,B)nC\ (iv) rank BD2 A - a>lr

ca*P

Then system (3.5.1) is stabilizable by dynamic output feedback. Proof: Define an w-dimensional dynamic compensator for system (1. l ) a s lows: u = X,

(3.5

i = pPi + v. The closed-loop system composed of (3.5.1) and (3.5.13) is z = /(z,
(3.5

X = px+v, Let n = G(z,$A+F(z,$, then n = [ - ( F ( z , 0 + G(z^)A ) ]/(z,|)| ; i = G _ 1 ( z ^ ) ( 7 _ F ( '4)) + [ — ( F ( * , 0 + G(z, nzmAt

+

Brj)\A=G-l(z4){r}_F{z4))

+ G(z,£)(aA + v) [|(F(z,£) +

+

[-(F(z, 0

(3.5

G(z,^)]f(^)\x._G,izM!1_F{z4))

+

(Kz&XMAt

+

+ ^ - n z ^ ) ) + G(z^)v = D2DlZ +firi+ Fl(z,4,Ti) + G(z,tyv

^

.

^

^

Local Stabilization

of Nonlinear Systems

by Dynamic

Output Feedback

123

where D, = —/(z,4)|z = 0,£=o, A> = — F(z,$\z = 0,t=o, and oq oz

+

[A (/ r (z ^ ) + G ( Z j | H ) ] ( ^ + ^ ) | /i=G _ 1(z ^ )(/7 _ i7(z ^ ))

-/3F(z,£). By a direct computation, we have F^zflfi) = E ^ W . O ) + G (*.°)W( z >°)|;^->(z,0)F(z,0) - /^( z >°) = J D 2 /(z,0)-aF(z,0) + [^(F(r,0) - D2z + G(z,0)A)]/(z,0)|;l=_G.(z>0)F(z Q) = £> 2 (A + 0(|z| 2 )) -/?(Z) 2 z + 0(|z| 2 )) + 0(|z| 2 ) = 0(|z| 2 )

and ^(0,^,17) = -D2D^ + D2f(0,^)

-oF(0,^) + O(||(|,f;)|2)

= o(||(^)|2) Let y= {^,rf)T. Then according to (3.5.15), system (3.5.14) is transformed into

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P. Chen & H. Qin

y, = Aen + Be(G(z,Z)v + Fl(z,W)),

(3.5.16)

y=ce¥. with (Ce, Ae, Be) defind by A

e=

A D2L\

B pim ,B»

0

,c=[c o]

m. In addition, it can be prove by Lemma 3.2.3 and Lemma 3.2.6 that, in (3.5.16), E(Ce, Ae, Be) is stabilizable and detectable and a\Ce, Ae,Be) = a\CA,B). Therefore (3.5.16) satisfies the conditions of Lemma 3.5.2 if Z(C, A, E) is reD placed by Z(Ce, Ae, Be). This implies that Theorem 3.5.3 holds. Now we apply Theorem 3.5.3 to a class of minimum phase nonlinear systems described by x = f(x) + g(x)u y = h(x), where xerf, u, yeJr,feC(U, open neighborhood of x = 0.

(3.5.17)

/?'),/0) = 0, geC°(U, FJxm), heC°(U, R"), U is an

Corollary 3.5.4 Let (3.5.17) be a minimum phase nonlinear system having a relative degree {ri,r2,...,rm}. Suppose the manifold of its zero dynamics is one dimension. Then the equilibrium point of (3.5.17) at the origin is is stabilizable by dynamic output feedback. Proof: As shown in the proof of Theorem 3.4.7, by adding an integrator, (3.5.17) can be transformed into the form of (3.5.1) with cr(C^4,B) = 0 . The rest of the proof D is as same as the proof of Theorem 3.5.3 and omitted. Example 3.5.5 Consider the system

Local Stabilization

of Nonlinear Systems

by Dynamic

Output Feedback

125

z = -z->exp{-y-r} + sin£,\ + g~ ,

r £=&>

(3.5.18)

£2=2£l+£2+z5+M> ? = £• where the expression exp{ -y-r} = 0 if z = 0. r It ;s easy to see that system (3.5.18) has relative r = 2 and its zero dynamics is i = -z3«ap{~} r

(3.5.19)

which is asymptotically stable. Note that zero dynamic (3.5.19) is not asymptotically stable according to the Y'1 order approximation for any integer N>\. But according to Corollary 3.5.4, it is still stabilizable by dynamic output feedback. A dynamic compensator for (3.5.18) can be designed as follows: We first introduce the dynamic system u = A,,

(3.5.20)

i = v, and the coordinate transformation TJ = z5+A

and transform the closed loop system of (3.5.18) and (3.5.20) into 3 z = -z

1

2

exp{—p-r} + sin^i + S,~ ,

PI £2=24I+§2+T,,

TJ = 5Z (-z exp{-j-r} + sin%\+%~) + v, r\

(3.5.21)

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P. Chen & H. Qin

Then we construct a linear dynamic compensator of the form v = Jw, w = Sw + Ry, for I{Ce^ie,Be)

with 0 "0 1 0" A.= 2 1 1 ,Be = 0 ,C = [l 0 Oj 1 0 0 0

By a simple design, J, S and R are given as follows: -4 J = [-9

-9

-4\S

= -5 -10

1 1 -9

~4 0" 1 ,R = 7 -4 1

Finally, a dynamic compensator for system (3.5.18) is given by u = X, X = Jw, w = Sw + Ry.

References [1] Battilotti, S., Stabilization via dynamic output feedback for systems with output nonlinearities, Systems Control Lett., vol. 23, pp. 411-419, 1994. [2] Byrnes, C. I. and Isidori, A., Local stabilization of minimum phase nonlinear systems, Systems Control Lett., vol. 11, pp. 9-17, 1988. [3] Byrnes, C. I. and Isidori, A., New results and examples in nonlinear feedback stabilization, Systems Control Lett., vol. 12, pp. 437-442, 1989. [4] Byrnes, C. I. and Isidori, A., Asymptotic stabilization of minimum phase nonlinear systems. IEEE Trans. Automat. Control, vol. 36, pp. 1122-1137, 1991. [5] Byrnes, C. I., Isidori, A. and Willems, J. C , Passivity, feedback equivalence, and the global stabilization of minimum phase nonlinear systems. IEEE Trans. Automat. Control, vol. 36, pp. 1228-1240, 1991.

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of Nonlinear Systems

by Dynamic

Output Feedback

127

[6] Carr, J., Applications of Centre Manifold Theory, New York: Springer-Verlag, 1981. [7] Chen, P., Han, Z. and Zhang, Z., Dynamic output feedback stabilization of affine nonlinear systems, Acta Automatica Sinica, vol. 23, pp. 338-344. [8] Chen, P., Qin, H.,and Zhu, Q. M., Dynamic output feedback stabilization of partially linear composite systems, Preprints of the 14th World Congress of IF AC, vol.E, pp. 345-350, 1999. [9] Chen, P., Qin, H., and Huang, J., Stabilization of a class of systems by dynamic output feedback, Proceedings of the 38th Conference on Decision and Control, pp. 4891-4896, 1999. [10] Esfandiari, F. and Khalil, H., K., Output feedback stabilization of full linearizable systems, Int. J. Control, vol. 56, pp. 1007-1037, 1992. [11] Hahn, W., Theory and Application of Lyapunov's Direct Method, Englewood Cliffs, NJ: Prentice-Hall, 1963. [12] Huang, J. and Rugh, W. J., Stabilization on zero-order manifolds and the nonlinear servomechanism problem, IEEE Trans. Automat. Control, vol. 37, pp. 1009-1013, 1992. [13] Isidori, A., Nonlinear Control Systems: An Introduction, 3rd ed., New York: Springer-Verlag, 1995. [14] Kang, W. and Huang, J., Calculation of the minimal dimension k-th order robust servoregulator, IEEE Trans. Automat. Control, vol. 42, pp. 382-385, 1997. [15] Khalil, H.K. and Esfandiari, F., Semi-global stabilization of a class of systems using output feedback, IEEE Trans. Automat. Control, vol. 38, pp. 1412-1415, 1993. [16] Kokotovic, P. and Arcak, M., Constructive nonlinear control: progress in the 90s, Preprints of 14th World Congress of IF AC, plenary and index volume, pp. 49-77, 1999. [17] Levis A. H. et al, Report on the workshopxhallenges to control-a collective view, IEEE Trans. Automatic Control, vol.32, pp.275-285, 1987. [18] Marino, R. and Tomei, P., Dynamic output feedback linearization and global stabilization, Systems Control Lett, vol. 17, pp. 115-125, 1991. [19] Mazenc, F. and L. Praly, L., Global stabilization by output feedback: examples ana counterexamples, Systems Control Lett, vol. 23, pp. 111-125, 1994. [20] Saberi, A., Kokotovic, P.K. and H. J. Sussmann, H. J., Global stabilization of paitially linear composite systems, SIAMJ. Control Optimiz., vol. 28, pp. 14911503, 1990. [21] Teel, A. and Praly, L., Global stabilizability and observability imply semiglobal stabilizability by output feedback, Systems Control Lett., vol. 22, pp. 313325, 1994.

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[22] Teel, A. and Praly, L., Tools for semi-global stabilization by partial state and output feedback, SIAMJ. Control Optimiz., vol. 33, pp. 1443-1488, 1995. [23] Tornambe, A., Output feedback stabilization of a class of non-minimum phase nonlinear systems, Systems Control Lett., vol. 19, pp. 193-204, 1992. [24] Tsinias, J. and Kalouptsidis, N., Output feedback stabilization, IEEE Trans. Automat. Control, vol. 35, pp. 951-954, 1990. [25] Wonham, W. M , Linear Multivariable Control: A Geomtric Approach, 2nd ed., New York: Springer-Verlag, 1979.

Chapter 4

Hybrid Control for Global Stabilization of a Class of Systems Jun Zhao School of Information Science and Engineering Northeastern University, Shenyang, Liaoning, 110006, P.RChina

4.1 Introduction Hybrid systems are dynamical systems involving both continuous-valued and discrete-valued variables. In recent years, great attention has been paid to the study of hybrid systems. This is because hybrid systems have extensive background. A vast number of practical systems can be modeled as hybrid systems. Hybrid systems have been extensively studied also because the fast development in computer science and technology makes it possible to implement switching and hybrid control in practice. This chapter studies the global stabilization of a class of systems by means of hybrid control. The organization of this chapter is as follows. In Section 4.2, we briefly review some recent developments of hybrid systems, focusing on stability of switched systems. A necessary and sufficient condition of quadratic stability for homogeneous switched systems is given in Section 4.3. As an application example of hybrid control technique, we design a hybrid controller for the cart-pendulum system in Section 4.4. Section 4.5 contains concluding remarks.

4.2 Hybrid systems and hybrid control In this section, we give a general description of hybrid systems and hybrid control, and review some recent developments, focusing on stability of switched systems.

129

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J. Zhao

4.2.1 Hybrid systems Hybrid systems are dynamical systems involving both continuous-valued and discrete-valued variables. Typically, the evolution of a hybrid system is given by equations of motion that generally depend on both continuous and discrete variables. These equations contain mixtures of logic and discrete-valued or digital dynamics, and continuous-valued dynamics. The continuous dynamics of such systems is generally given by differential or difference equations (or equivalently vector fields). The discrete dynamics is often governed by a digital automation or input-output transition system with a countable (usually finite) number of states, or higher-level supervisory commands. Hybrid systems containing control variables are referred to hybrid control systems. There are many kinds of definitions for hybrid systems. A fairly general one was given by Branicky in [4].

4.2.2 Hybrid control For a control system, if the control is not of the conventional form, such as, open loop, static or dynamic state feedback, etc, and instead, the control depends on switching or supervisory commands, etc, the control is often considered hybrid control. There is no clear bound between conventional control and hybrid control. Hybrid control is not only applicable to hybrid systems, but also applicable to ordinary systems. It is commonly understood that hybrid control often provides better robustness and safety than conventional control, and is suitable for dealing with problems with state and control constraints. Besides, hybrid control can be used to solve many problems that cannot be solved by single continuous control or discrete control. For example, hybrid control can stabilize some systems that cannot be stabilized in ordinary sense. On the other hand, in many cases, we have to use switching or supervisory control. For instance, if we are limited to the use of certain pre-designed controllers, we have to switch between them to achieve some goal.

4.2.3 Switched systems Switched systems constitute a special class of hybrid systems. A switched system consists of several subsystems, which are ordinary continuous or discrete systems, and a switching law. Since most of hybrid systems can be modeled as switched systems after some proper technical treatment, switched systems are often taken as simplified models of hybrid systems. Therefore, there has been considerable interest in studying switched systems. A continuous switched system can be described by a differential equation of the form

Hybrid Control for Global Stabilization

x=fa

(x)

of a Class of Systems

131

(4.2.1)

where xeR" is the state, {fp: p&P} is a family smooth vector fields defined on /?", which is parameterized by some index set P, usually a subset of Z+, a: [0, oo)-»P is a switching law (or called switching signal), usually piecewise constant. The value of a at any time t may depend on t or x(?) or both, or generated by supervisory commands. Each system x = fj(x), ieP, is called a subsystem of switched system (4.2.1). Let t0 < ti <.. .< tk <... be a strictly increasing sequence of time. Suppose that a = ijeP is constant on [tj tJ+i) and ij^ij+i for ally = 0,1,2,.... System (4.2.1) evolves on [tj tj+i) following the vector field / . , that is, the trajectory of System (4.2.1) satisfies •> • / + i

';

x(tj) = Hmt^t.x(t) x(t0) = x0. In this case, the subsystem

x

=/

(4 2 2)

^-L-L>

(x) is said to be active. It is easy to see that j

the trajectory x(t) of System (4.2.1) is continuous and continuously differentiable everywhere except tj,j = 1,2,.... Unlike ordinary systems, switched systems possess some unusual and interesting behavior. A switched system might be unstable under certain switching laws even if all of its subsystems are stable. On the other hand, a switched system might be stable under some switching laws even though none of its subsystems is stable. Examples can be found in [3]. Therefore, conventional analysis and design approaches must be extended to be applicable to switched systems. As in the study of ordinary systems the issue of stability of switched systems is considered to be of great importance. There are three basic problems concerning stability of switched systems, which have been extensively studied in recent years. Pi oblem 4.2.1 Stability under arbitrary switching laws. It is obvious that a necessary condition for (asymptotic) stability under arbitrary switching laws is that all subsystems are (asymptotic) stable. However, this condition is not sufficient as mentioned earlier. The existence of a common Lyapunov function for all subsystems is a necessary and sufficient condition for switched linear systems to be asymptotically stable. This condition is sufficient for switched nonlinear systems to be asymptotically stable, and is also necessary if some other mild assumptions are imposed. There have been a lot of results based on common Lyapunov function method (refer to the references).

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Problem 4.2.2 Stability under certain classes of switching laws. Since the class of switched systems that are stable under arbitrary switching laws is, in general, very limited, much attention has been turned to the study of stability under certain restricted (but useful) classes of switching laws. The goal is to identify these switching laws. There has been no general method to solve this problem. The multiple Lyapunov function method introduced by Branicky [3] seems to be an effective method. Roughly speaking, if there exists a "Lyapunov-Mke" function V, ( not real Lyapunov function) for each Subsystem x = fix), satisfying that V\ (x(t)) < 0 on each time interval where the subsystem is active and V, is monotonically nonincreasing on the switching time sequence for which x = f(x) is "switched on", stability of switched system (4.2.1) can be guaranteed. Asymptotic stability can be obtained if some other conditions are added. As a consequence, if all subsystems are asymptotically stable, then the switched system (4.2.1) is asymptotically stable as long as the switching is slow enough. By introducing the concept of dwell time which is the lower bound on all active interval lengths for all subsystems, this result can be expressed as: There exists a positive constant such that if the dwell time is no less than the constant, then the switched system is asymptotically stable. There are a few ways to explicitly calculate the constant. Problem 4.2.3 Design of stabilizing switching laws. This is a design problem rather than a stability analysis problem. If at least one of subsystems is asymptotically stable, then the problem is trivial. There are two major approaches to deal with the problem. (a) Single Lyapunov function technique This technique includes designing a switching law and finding a Lyapunov function such that the derivative of the function is negative along the trajectory of the switched system. There is no general way to find such a Lyapunov function and to design a switching law. Some results concerning quadratic and homogeneous Lyapunov functions have been reported. For switched linear systems a quadratic Lyapunov function can be easily found and a switching law can be determined if a convex combination of subsystem matrices is stable. This condition is also necessary if the switched system consists of only two subsystems. (b) Multiple Lyapunov function technique The multiple Lyapunov function technique mentioned earlier is also applicable to the design of a stabilizing switching law. The common strategy is to partition R" into several sub-regions /2, corresponding to each subsystem. In each Q try to find a positive definite function V, which decreases along the trajectory of corresponding

Hybrid Control for Global Stabilization

of a Class of Systems

133

subsystem. By imposing some additional conditions on Vt, the switched system can be shown to be asymptotically stable.

4.3 Quadratic stability of homogeneous switched systems In this section, we study quadratic stability of homogeneous switched systems.

4.3.1 Completeness In the study of stability of switched systems, the concept of completeness is useful. Definition 4.3.1 A set of continuous functions { v1,v2,...,vk }, where v,-: R"->R, is called complete if for any x e R", there exists an i e {1,2,..., k } such that V,(JC) < 0 . In addition, the set { vhv2, ...,vk } is called strictly complete iffor any x e R", x * 0 there exists an i e {1,2, ...,k} such that v,(x) < 0 . The concept of (strict) completeness of functions is a generalization of the concept of ^ negative) nonpositive definite functions. Let us consider the switched nonlinear system * = /,•(*)>

/ = l,2-.*

(4.3.1)

where xeR". As a straightforward consequence of the direct Lyapunov method, asymptotic stability of system (4.3.1) can follow from the completeness of a properly chosen set of functions. Theorem 4.3.2 System (4.3.1) is asymptotic stable if there exists a positive definite radially unbounded function V(x), such that the set of functions { L f v , J = 1,2,..., k) is strictly complete, where i , y is the Lie derivative of V along f , ' j

This result means that strict completeness of certain set of functions implies asymptotic stability. However, it is usually very difficult to check strict completeness. Our goal here is to find an alternative condition. To do this, we need the Fritz John condition for nonlinear programming problems. Lemma 4.3.3 Given a nonlinear programming problem

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J. Zhao

Max s.t.

f(x) sj(x)>0J

= l,2,---m,

(4.3.2)

hj(x) = 0J = \,2,-,l, where f Sj, hj : R"—>R are smooth functions. Ifx is a local optimal solution to the problem, then there exist constants /Uo,fa,..., /J„, not all zero, and constants Xh X2, ..., X\such that

fiisi(x*) = 0,i = l,2,---,m, fij>0,i

=

(4.3.3)

\,2,---,m,

where V stands for the gradient operator. 4.3.2 Quadratic stability We are only interested in finding conditions for quadratic stability. First of all, we give the precise definition of quadratic stability. Definition 4.3.4 System (4.3.1) is said to be quadratically stable if there exist a positive definite matrix P and a switching law with the state feedback form i = ifx) such that the derivative of the function V — xTPx is negative along the trajectory of System (4.3.1) for any x 0. For quadratic stability we have the following result, which can be easily proved by Theorem 4.3.2. Lemma 4.3.5 System (4.3.1) is quadratically stable if and only if a positive definite matrix P exists such that the set of functions {xTPfj(x), j = 1,2,..., k} is strictly complete. In the sequence, we assume that System (4.3.1) is s homogeneous system, that is, there exist r,- >0,j= l,2,...,k, such that fj(ax) = arJfj(x),j

= \,2,-,k

(4.3.4)

holds for any a> 0. Suppose P is a positive definite matrix. For any fixed i e {1,2,..., k} and x e R", x 0, define the set

Hybrid Control for Global Stabilization

Mi(x)={j\xTPfJ(x)

of a Class of Systems

= 0,j*i}

135

(4-3-5)

and vector fields H , = Pf i + -^—Px -XT ^—Pxx J

J

J

dx

J7

= 1 ,2,-,k.

(4.i.0)

dx

If Hj,j = 1,2,...,A; are not all zero at x, choose one of the largest linearly independent subsets of vectors in {Hj,j e Mt (x)} (empty set if Hj = 0,j e Mt (JC)), denoted by H ,• ,-,H , LetH = (H,- ,-,H ,• ) , S,= {ji,...Js},Qi = Mi\Si. In order to avoid checking strict completeness, we transform strict completeness into a nonlinear programming problem. Theorem 4.3.6 System (4.3.1) is quadratically stable if and only if a positive definite matrix P and an i e {1,2,..., k} exist which make the nonlinear programming problem Max s.t.

xT T

x

Pfj(x) Pf

xTx-1

(x)>0

(4.3.7) j*i,

=0

has no feasible solution or the optimal value is negative if a feasible solution exists. Here, a feasible solution means a vector in /?" satisfying all the constraints in the problem. Proof: Sufficiency. If (4.3.7) has no feasible solution, then the set |x| xT Pf j (x) > 0,xT x -1 = 0,j = 1,2,-- ,k, j * i\ is empty. Thus, for anyx0e./?", XQ^O, an integer qs{ 1,2,..., k}, q^i must exist such that [jc0r/||JC0 IWPf, (xo/\\x0\\) < 0 . Since fq{x) is homogeneous, we know x0TPfq(x0) < 0, which means that {xTPf,j = 1,2,..., k } is strictly complete. Therefore quadratic stability follows from Lemma 4.3.5. If (4.3.7) has a feasible solution and the optimal value is negative, then for any xe/J", x ^ 0, [xT/\\x\\]Pfi(xT/\\x\\y<0 must hold as long as [*%||]J#<*/||x||)>0, j= 1,2,...,A, j^i . Again from the homogeneity of f(x) we have xrPf£x)<0, which means that {xTPfj,j = 1,2,.. .,k } is strictly complete. Necessity. Suppose System (4.3.1) is quadratically stable. Lemma 4.3.5 says that {xTPfj,j = 1,2,.. ,,k} is strictly complete. If (4.3.7) has a feasible solution x0, that is,

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J. Zhao

xlPfj{x0)>0,j*i, ft

ft

—

x0TPfi(xo) < 0 follows immediately from the strict completeness of { xTPf,j = 1,2, ..., A;}, and thus the optimal value is negative. Now, we give the main result of quadratic stability. Theorem 4.3.7 System (4.3.1) is quadratically stable if and only if a positive definite matrix P and an integer i exist, which make the following two conditions hold. Condition 1. Hj-xTPfjx-H(HTH)-1(HTHj-HTxTPfix) T

x Pfj = 0J=jlJ2,-.js,

= 0, (4-3.8)

xTPfj>0,j*i, xTx-i=0

has no solution or any solution x satisfies x Pf< 0. Condition 2. Yj±iXj(x)H

j = 0, (4

xTVj*0.j*,.

-3-9)

xT x-\ = 0

has no solution or any solution x satisfies x Pfi < 0, where Zj(x)>

0, Z

Xj{x)

= \,

J*' X j{x)x1

.T Pf j = 0,j*

i,

Proof: Necessity. Assume that System (4.3.1) is quadratically stable. From Lemma 4.3.5 we know { xTPfj,j = 1,2,...,* } is strictly complete. If (4.3.8) or (4.3.9) has solution x, then, xTPf >0,j^i. Therefore xTPfi<0 follows from the strict completeness of {xTPf j=\,2,...,k}. Sufficiency. Suppose that (4.3.8) and (4.3.9) have no solutions or have solution x satisfying xTPf < 0. If (4.3.7) has no feasible solution, then quadratic stability follows from Theorem 4.3.6. Now, suppose that (4.3.7) has a feasible solution, then a global optimal solution must exist because all feasible solutions are limited to the sphere xTx = 1. According to Theorem 4.3.6 we only need to show that the optimal value is negative.

Hybrid Control for Global Stabilization of a Class of Systems 137

Let x be an optimal solution to (4.3.7). From Fritz John condition there exist constants Ay, / = 1,2,..., k, not all zero, and a constant /7 satisfying

3/f

d

ff

-

h(Pfi +-±rPx)+

Z Aj(Pfj +-fj-Px) + / « = 0, (4-3-10)

IjXTpfj=0J*i, Ij>0,j

= l,2,--,k.

Multiplying (4.3.10) by xT from left and taking xTx = 1 into account, we have Ai(xTPfi+xT

dfT

d

-

fJ

—'— Px)+ I A:xT—+-Px dx J 8x jH

+ jI=0

(4.3.11)

Now, we split the proof into two cases. Case 1 At ^ 0. Denote Aj = A .• / Aj, j=£i and // = /7 / Aj .(4.3.10) becomes d

d/T

ff

(Pf +-1—Px)+ dx

,

I XJ/(Pfj+-/-Px) -^j •> ox

+ px = 0,

T

(4.3.12)

XjxT Pfj =0,j*i, Aj > 0,j = 1 , 2 , • • • , * ,

V

'

(4.3.11) gives V = -(xTPfj+xT

—'— Px)- I AJXT-^—PX dx -^j J dx

(4.3.13)

Substituting this into (4.3.12) resulting in Hj-xTPfjx+

X AjHj=0

(4.3.14)

Note that Aj = 0 if/' £ M{x), and for anyy e Q(x), 3a • (x), m &Sj{x) such that

H •(*)=

I meS,.(x)

«/

m

(x)Hm(x)

138

J. Zhao

Ht-x

PfiX + XJes.(x)*-jHJ +

= Hi-^Pfix

+ ZjeS.(x)XjHj+^j&Qiix)Xj^meSiix)ajJx)HmJ

= Hi-x1 Pfix + T.jeSi(x);ijHj = Hi-xTPfix

XjeQiWljHj

+

^

^

Zmesi(x)Hm{Zj£Qi(x)Zjajm(x)

+

ZmeSi(x)SmHm=0

where sm = 2 m + I.jeQ.(x)

x a

jj

(*)•

Writin

S (4.3.15) in the matrix form

gives (8, 1

Ht -x

^

(4.3.16)

PfjX + H V

•> s J

Multiplying (4.3.16) by H from left yields

H1 Ht-H1rTvxT1 PfjX + H1 H

(4.3.17)

=0 \

•> s J

Therefore

= -{HTH)-X{HTHi

-HTxTPf{x)

(4.3.18)

Substituting (4.3.18) into (4.3.16) we arrive at Hi-xTPfix-H(HTH)-l(HTHi-HTxTPfix)

=0

(4.3.19)

which is exactly the first equation of (4.3.8). Thus, xrPft < 0 follows from Condition 1. Case 2 Xj = 0. Note that F • are not all zero, we can denote

Hybrid Control for Global Stabilization

A' j, =

= — ,M>/"

of a Class of Systems

139

^m^i vw

^m^i^m Rewriting the first equation of (4.3.10) as

8fj

(4.3.20)

I A,.(Pfj+-J-Px) + Mx = 0 and multiplying(4.3.20) by xT from left, we have T

/" = - Z ^

1 J +x

VI

VJ i J —ox — Px

(4.3.21)

Substituting (4.3.21) into (4.3.20) and noticing ^xTPf = 0, we have (

T

T

df

— PX -X1

Pf : +

J

J

dx

J

T

\

8f

J — PXX

=0

(4.3.22)

dx

which is exactly the first equation of Condition 2. Thus, xTPf < 0 follows from Condition 2. Corollary 4.3.8 If System (4.3.1) is a switched linear system, that is, f/x) = Ape, then System (4.3.1) is quadratically stable if and only if a positive definite matrix P intpvpr i prist and an integer exist which make the following two conditions hold. Condition 1. BiX-H(HT

H)~l HT BjX-x7

xTBjX = 0,j =

jlJ2,—,js,

BiXx = 0, (4.3.23)

T

x BjX>0,j*i, xT x-\ = Q has no solution or any solution x satisfies xTBx < 0, where Bj = A/P+PAj, H = ( B ; ,B ,• J\ J2

• ' ; > •

140

J. Zhao

Condition 2. xTBjx>0J*i,

(4.3.24)

xT x-\ = 0 has no solution or any solution x satisfies xTB;x < 0, where fyx) > 0, Ej*i fyjc)=\, XJ{X)XTBJX = Q,j^

i.

As another corollary, we can easily show the well-known convex combination condition [23]. Corollary 4.3.9 If Systems (4.3.1) is a switched linear system withf/x) = Afc, and in addition, there exists a convex combination Zi = I Pi Ai: Zi = i Pi = L Pi^ 0, which is stable, then System (4.3.1) is quadratically stable.

4.4 A hybrid controller for the cart-pendulum system In this section, we design a hybrid controller for the cart-pendulum system.

4.4.1 Background and problem statement The cart-pendulum system is one of the classic mechanical systems that have been studied extensively in control area during the past few decades. In the early stage, this system was used to understand control theory for educational purpose [10,13]. Later on, it was studied to demonstrate the validity or usefulness of design techniques and algorithms. Recently, particular attention has been paid to the study of underactuated mechanical systems which are systems with fewer actuators than degrees-of-freedom. As many underactuated robot systems have quite similar dynamics to that of the cart-pendulum system, the cart-pendulum system has been studied as a simplified model of underactuated robot systems by some authors [31,32,33]. Though the dynamics of the cart-pendulum system appears quite simple, many standard techniques of nonlinear control are ineffective. For example, the system is not feedback linearizable or even has no constant relative degree corresponding to the swinging energy of the pendulum as an output. Also, the controllability distribution of the system does not have constant rank. Many geometric properties of the system are lost when the pendulum moves through horizontal positions. Therefore, the study of the system is much more difficult than it seems to be. The most common approach in the study of the cart-pendulum system is the socalled energy-based method. Since there is no continuous feedback of 2^-period in angle which can render the upward vertical position of the pendulum globally as-

Hybrid Control for Global Stabilization of a Class of Systems

141

ymptotically stable, discontinuous feedback has been considered [32,35]. It has been shown that the system can be stabilized for almost all initial states except the stable manifold of the downward vertical position of the pendulum. Here, we focus on the global stabilization of cart-pendulum system. The main difficulty is to swing up the pendulum from the downward vertical position and keep the cart stable as well. We consider the well-known cart-pendulum system shown in Fig.4.1.

cart

W Fig.4.1

The cart-pendulum system

The dynamics are given by ^ 2L , sin 9 + F (M + m)x + ml cos 09 = ml(9)

(4.4.1)

i'cos 9 +19 = gsin 9,

where Mand x are mass and position of the cart which moves along a horizontal axis, m, I, 9 are mass, length and angular deviation from the upward vertical position of the bar, which can rotate around a point fixed on the cart, and F is a horizontal force applied to the cart. Applying the partial feedback linearization technique we can easily rewrite the cart-pendulum system as x=v, v = u, 0 = w, w-sind-ucosO, The total energy of the pendulum is given by

(4.4.2)

142 J. Zhao E(t) = -w2(t)

+ cos0(t)

The control goal is to make the origin globally asymptotically stabilizable.

4.4.2 Hybrid controller The design of the controller consists of three stages. First of all, find a neighborhood of the origin

u=\x,v,e,w)T

\x\
(4.4.3)

with positive constants a, b,c,d, c
(4.4.4)

V = M,

such that the states x and v are steered to the origin in finite time. This can be easily realized because the subsystem (4.4.4) is controllable. The third stage is to steer any state of the whole system (4.4.2) to the neighborhood U so that M/ can be applied. This can be realized by a hybrid controller given by the following algorithm. Before we give the algorithm, we introduce some notations. Let h, r be positive constants, h
\ Aw=o>h\

[0(t\),if

w(t\) = 0,for some ?j >tQ,and

[ r,

h<0(t\)
otherwise

and max \h,8\ 0{t\), h,

n


if ve(fj) = 0, for some t^>tQ,and otherwise

h<0(t\)
We will see in the algorithm that if such t\ exists, it must be unique.

Hybrid Control for Global Stabilization

of a Class of Systems

143

Algorithm. Step 0 Apply us to steer x and v to zero. Step 1 Given initial states x0=0, v0 = 0,6b = flfo) and w0 = w(/o)> choose Tx> 0, T2 > 0 and T3 > 0, where T\ is an upper bound of the time in which the pendulum moves from 0 =;rto 0* = min {3^/2, 0\w = o> rf with w(t) > 0 under all possible nonnegative constant control u satisfying w>0 if w(t0) = 0, T2 is an upper bound of the time in which the pendulum moves from 6 =n\o 0* =3^/2 with E0 =(w2o/2)+ COS&Q >( 0 under all possible nonpositive constant controls u satisfying (l/3)(casc-£(0)) < M < 0, r 3 is an upper bound of the time in which the pendulum moves from 0=0 to 0*=n/2 with £„=( wV2) +COS0Q>( Sl2) +1 and w(t)> 0 under all possible nonnegative constant controls u satisfying 0 (^/2)+l, go to step 4. Step 3 If 00^/ror 0Q = n, x0w0 > 0, set u = 0 until 0= nandxw <0, and then set current states as initial states and restart Step 3. Otherwise calculate 1 + Irf2_£ £ ^2

2_

(4.4.5)

Case 1. 0O = n, x0 < 0 and w0 > 0. Apply controller II = q

(4.4.6)

until 6* =6**= min{3n/2, 0\w = o >n), and then switch the control to w = -Cx. When v = 0, set current states as initial states and then return to Step 2. Case 2. 0O = it, x0 = 0, w0 < 0 or <9„ =;? x0 > 0, w0 ^ 0. Apply controller

«=- q

(4.4.7)

until 0 =0*= max{n/2, 9\w = o <#}, and then switch the control to u = Q . When v = 0, set current states as initial states and then return to Step 2. Step 4. Case 1 w>ox0 > 0. If 9^n, set u = 0 until 0 = ^ and set current states as initial states and restart Step 4. Otherwise calculate

144 J. Zhao \ a Eft - cose -mini——, >, if wn > 0, J 1T2

3

a

EQ-COSC[

mm {——,

\

°

(4.4.8)

>, if wn
and apply controller (4.4.9)

u = Cn until -n

if

WQ

>0,

9=\ —n if

WQ

<0,

(4.4.10)

then switch the control to u = -C2. When v = 0, set current states as initial states and then return to Step 2. Case 2. wox0 < 0. If #7^0, set u = 0 until 0 =0, and set current states as initial states and restart Step 4. Otherwise calculate

min \^-AA,E°-C0SC\,if C

w0>0, (4.4.11)

3=1

2

. \ a d d -mini.—— — \T2 2 2

E - cose 3

} if wn <0, ^ 0

and apply controller (4.4.12)

u=d until —n if w 0 > 0, 0=

3 —K (or equivalently

(4.4.13)

1 n~), if WQ <0,

then switch the control to u = -C3. When v = 0, set current states as initial states and then return to Step 2.

Hybrid Control for Global Stabilization

of a Class of Systems

145

Remark 4.4.1 In Step 0 we use Us to steer x and v to zero, which will be neededfor the stability analysis in next section. According to different initial energy the algorithm works in the following way. (1). cosc< E0 < d2 12 +1. Apply zero control until the trajectory enters the domain of attraction U and then apply local controller ul. (2).

EQ<

cos c.

Step 3 is taken. We first use u=0 until 6 = n andxw< 0. This is always possible because E0< cos c. Then, we split the condition 9 = n and xw< 0 into two cases: Case 1. 6>0 = n , XQ < 0, w0 > 0 , and Case 2. ff0 = n , XQ = 0 , w0 < 0 or GQ=TC , XQ>Q

, 2

(3). E0>d /2

WQ

< 0 and apply different controllers to the system.

+ l.

Step 4 is taken. w0 is obviously not zero. We design the controller in two different cases. Case 1: x Q w 0 >0 and Case 2:

XQWQ

<0. We first set control u to zero

until 9 = n (in Case 1) or 9 =0 (in Case 2). This is also always possible because EQ> d2/2 + 1. Then, apply different controllers to the system. It is worth while to point out that in the algorithm, once the trajectory enters the domain of attraction U, Ul will be used forever, no matter the trajectory exits U or not. In each iteration of the algorithm, the control is first set to some constant value C. While 9 reaches a specific value, the control is then switched to -C. When v reaches 0, the iteration ends. Therefore, the control is of the bang-bang form and at the end of each iteration, v must be zero.

4.4.3 Stability analysis In this section we show that the hybrid controller given above indeed globally stabilizes System (4.4.2). We begin with the analysis of the validity of the controller. Lemma 4.4.2 (a). Suppose the pendulum starts from 9(0) =nwith w(0)>0. Then the time T=T(w2(0),u) in which the pendulum rotates from 9=nto 6=9 =min {3n/2, 9 \w = o >^j with w(t)> 0 is identically bounded from above for all nonnegative constant controls. One such a bound is

146 J. Zhao (b). Suppose E(0)>(di/2)+l. If the pendulum starts from 6(0)=nwith w(0)>0, then the time T=T(w2 (0),u) in which the pendulum rotates from 6=n to 9*=3n/2 with w(t)> 0 is identically bounded from above for all nonpositive constant controls u satisfying (l/3)(cosc-E(0))
+ (2/3)COSC

).

(c). Suppose E(0)>(d2/2)+l. If the pendulum starts from 6(0)=0 with w(0)>0, then the time T=T(w2(0),u) in which the pendulum rotates from 6=0 to 9=6=7i/2 with w(t)> 0 is identically bounded from above for all nonnegative constant controls u satisfying 0
-(w2 (/) - w 2 (t0)) = u(sin 0(to) - sin 0(t)) + cos 0(to) - cos 0(t)

(4.4.14)

Therefore d0 lF=Wit)

(4.4.15) 2

= ±y w (to) + 2u(sin 0(to) - sin 0(t)) + 2(cos0(tQ) - cos 0(t)) Thus,

ffi-SH '<>=#*

whereiA.6,u)= ^w2(t0)

if w(t)>0j0
1

(4.4.16)

W s

-fitt W O V\-rlr^ (0,K)

* W °»'0 * ^ <\

+ 2u(sin0(tQ)-sin0)

+ 2(cos0(tQ)-cos0)

.

2

(a). By letting t0 = 0, 6{t0) =K, tx=J=T(w (0), u) be the time at which 9{tx)=6(T) =6* = min{3n/2, 0\w=o>n} and w(t) > 0, we have r-r(w2(0X.,)-lf

. Tjw2(0)-2usin

de

(4A17) 0 - 2(1 + cos 0)

Hybrid Control for Global Stabilization

of a Class of Systems

147

A simpie calculation shows that E = -wucos0

(4.4.18)

which indicates that the energy E of the pendulum does not decrease if w>0, w(t)>0 and n<0<6*. Therefore, if w2(0)>2, 0* =3n/2 must hold for all nonnegative constant control u. In this case, since T(w2(0),u) decreases as either w2(0) or u increases, T\w2(0),u) < T(2,u) < 71(2,0) for all w2(0) > 2 and u > 0. It is obvious that w(7T2,0)) = 0. Therefore, we only have to estimate T = T(w2(0\u)

_d0

= j% -<]w2(0)-2usin

(4.4.19)

0 - 2 ( 1 + cos 0 )

with w2(0) < 2, w(T) = w\0, a =0, n 0. Now, fix u and w(0) and let I(0)=w2(O)-2usin

0-2cos0-2

(4.4.20)

(4.4.15) guarantees 1(d) = 0. Since ^-4r=2usin0 + 2cos0 y(0) given by w 2 ( 0 ) + 2 \ \ + u2 - 1 y(0) =

—-—,

n <0< arctan {u) + x,

^ J «"*»(») w 2 (0) + 2( Vl + " 2 -lXa-0) , a -arctan

(4.4.22) arctan (u) + n < 9 < a,

(H)-K

Hence,

. tarctan (u)+x

"&

L2 ( 0 ) + 2 rv^"-il-£^\

) arctan (a) d

+ f«

'arctan

\

(u)+ir

A w (0) + 2[ Vl + " 2 2

a - arctan (u) - rt

-lXa-0)

148

J. Zhao

2arctan(u)

a- arctan(u) - n

Vw2(0) + 2Vl + w2 - 2 + w(0)

TJW2(0) + 2-JI + U2

-2

<2

L 2 (0) + 2[ Vl + « 2 To make further estimate, we express a in terms of w(0) and u. It follows from (4.4.18) that = \t Edt = u(sin0(tQ)-sin0(t))

E(t)-E(t0)

(4.4.24)

'0

Recalling w{T) = 0 and noticing (4.4.15) we have w2(0)-2usina-2

= 2cosa

(4.4.25)

Substituting p = sina into (4.4.25) and then squaring it give (4w 2 +4)p2 +4(2-w2)up+w4

-Aw2 =0

(4.4.26)

_ (w 2 (0)-2)«- A /4« 2 -w 4 (0) + 4w2(0)

(4.4.27)

Since p < 0, the solution of equation (4.4.26) is

2(M2+1)

and thus a = n-arcsinp. One can easily show that arcsinx < 2x for all xe[0, 1]. From (4.4.23) we have 7>2(0),^-

2arcs

JnE Vw (0) + 2(Vl + « 2 -1) 2

_

2arcsin(-p) 2

Vw (0) + 2(Vl + « 2 -1) Vw2(0) + 2(Vl + t/2 -1) (2-w2(0))u + iJ4u2 -w 4 (0) + 4w2(0) " / , / j= (u1 +l)Vw2(0) + 2(Vl + « 2 -1) 2

(4.4.28)

Hybrid Control for Global Stabilization

of a Class of Systems

149

Applying the inequalities Vi + « 2 - 1 > M 2 / ( 2 + « ) and w2+5> 4 + 2 K results in

T< 2,™ w - ( 2 - w 2 ( 0 ) ) « + r(wi(0),«)<2 2

A/4« !

2

-w4(0)+4w2(0)

( U +i)J w 2 (0)+^i V

2

(2-w (0))i< |2 + «

(,2+D

2

2

iu

2

/I—^~ 2 <— V4 + 2« + " : (« +D

4M

„2+li|

+

-w4(0)+4w2(0) 2 w2(o)+2« 2+u

4 M 2 - w 4 ( 0 ) + 4w 2 (0)

2

2

2+ u 2

lJ

w

2 ( 0 ) + 4«2 H2 + 5

<

_2

^ ^ 7 ,

2 (w + l )

(w +l) 4

(«

2 +

l)

l(a2+5)(4a2-w4(0)+4tt>2(0))

tt2+iy

V4 + 2w+—

2

<

2

2

M 2 W 2 ( 0 ) + 5 W 2 ( 0 ) + 4M2

V« 2 + '

M +1

-V« 2 +;

(4-4-29)

<4A/5"

(b). From £(0)>(c^/2)+l and (4.4.24) we know that the pendulum can indeed reach 6 =3^/2 under all nonpositive constant controls u satisfying (1/3)(cosc£(0))<w<0. Therefore w 2 (0) - lusin 9 - 2cos 0-2 = 2E(0) - 2usin 6 - 2cos 9 •*4ivn^2 i a > — E(0) + — cose - 2cos 6 3

(4.4.30) v

3

Note that cosd<0, the conclusion then follows by applying (4.4.17) with 6^=3^2. (c). Since 0< u< min{d 12, (fl2), we know from (4.4.24) that the pendulum can indeed reach 6 =n/2. Now, for fixed u, the minimum of the function w 2 (0)2usin&r2{\-cosO) restricted to de[0, n/2] occurs at W =arctan(u), and the minimum satisfies

'

150 J. Zhao wl (0) - lusin d + 2(1 - cos 0 ) = w2 (0) + 2 ( 1 - sec 0)

= w 2 (0) + 2(l-Vl + « 2 ) >rf2 -2w 2 1 ,

(4.4.31)

2 The conclusion then follows from (4.4.16). Remark 4.4.3 The same bounds can be given, respectively, for the following cases. (a). The pendulum rotates from 6=nto 8=8 =max{n/2, 8/w = 0<^} with w(t)<0 and under all possible nonpositive controls. (b). The pendulum rotates from 6=nto 8=6*=n/2 with E(0)>(d2/2)+l, w(t)<0 and under all nonnegative constant controls u satisfying 0(a^/2)+l, w(t)<0 and under all nonpositive constant controls u satisfying -min {d/2, cP/2} cose if £(0) >(^/2)+l. Lemma 4.4.5 After each iteration of Step 3 or Step 4, the energy E increases at least by \ usind \for the case E(0) ((<sP/2)+l is similar. Without loss of generality, we suppose x(0)<0. Also suppose the pendulum moves from 9=n at t=0 with w>0 and w>0, and at t=T reaches 6=8*= min{3^/2,^w=0>^'}• By the algorithm, control is switched to -Ci at t = Tand lasts until t = 27" when v reaches zero. We now show that during the time interval [0,27] the energy increases at least by u\sin8*\. To do this, it is sufficient to show that £(27) is no less than £(7) because £(7)-£(0) = u\sind*\ according to (4.4.24). If JiT)=8* =3a/2, (4.4.24) gives that

Hybrid Control for Global Stabilization

of a Class of Systems

£(27;) - E(T) = -u(sin0(T) - sin0(2T)) > 0

151

(4.4.32)

If 6(7) =0' <3TZ/2, W(7) = 0 must hold. Let 0=(3n/2)-0', then 0* or (3ri2)+/}< 6(27) < 2;r(equivalently P - 7i 12 < 6(27) < 0), which still guarantees (4.4.32). Now suppose the claim is false, that is, 6(27) g[0, 0*]U [(3^/2)+/?, 2 ^ . For any f e[T, 27], if 6>(7) =6(7) = 0*, (4.4.24) gives £ ( H = £(7), which implies w(f ) = w(7) = 0. Since w (i) = sin0(t)-(-C{)cos0(t) < 0, whenever 0 e [;r, 0*], we know the pendulum can not reach 6(27) upward from 6(7). Therefore, the pendulum must arrive at 6(27) through 0=0. In particular, there exists h&{T, 27) such that 6(?,)= (37d2)+p. Applying (4.4.24) once again we get E(tx)-E{T) = -u(sin0CT)-sin0(ti)) = 0. However, E(h) = 1/2 w\h)+cos((3ri2)+j3)>cos((3td2)+/J)> 0. This is a contradiction. Lemma 4.4.6 At the end of each iteration of Step 3 or Step 4, \ x \
(4.4.33)

= i(0) + c 1 r 2

where

r cx if o
\-Cl

if

T
Noticing -a<x(0)<0 and (4.4.5) we know \x(2T)\ < a. Lemma 4.4.7 IfE(0) 0, such that | sin0 \ > q for all 0 in the iteration. Proof. It is obvious that \sin0*\ increases with the iteration because the energy increases according to Lemma 4.4.5. At the beginning of the iteration, we know from (4.4.2) that | w (0)| = \u\ > 0. The existence of q is then evident.

152

J. Zhao

Now we give the main theorem. Theorem 4.4.8 System (4.4.2) is globally asymptotically stabilized by the hybrid controller given in Section 4.4.2. Proof: Global attractiveness follows from Lemma 4.4.2-4.4.7. More precisely, the trajectory of system (4.4.2) starting from any initial point enters, in finite switches and finite time, the neighborhood U, and then is steered to the origin by the local controller ut. Local asymptotic stabilizability is guaranteed by the local controller Uj.

4.4.4 A modified algorithm In the algorithm given in Section 4.4.2, the control is set to zero between adjacent bang-bang controllers. Energy remains the same during these periods. This is a kind of "waste" of time and as a result the algorithm may converge slowly. To overcome this drawback we modify the algorithm as follows. Step 0. Apply us to steer x and v to zero. Step 1. Given initial states x0=x(t0), v0 = v(to),fy>=0(to) and w0 = w(t0) and choose s> 0, T4 > 0, T5 > 0 and T6 > 0 (see Remark 4.4.9 below). Step 2. Calculate the energy £ 0 =( W V2)+ cos90. If cose < E0 <(cP/2)+l, set u =0 until (x,v,Q,w)r enters U, and then switch to u\. Otherwise if E0 < cose, go to step 3. If E0 >(d2/2)+l, go to step 4 Step 3. Case 1. One of the following conditions holds. \w o > 0,

wn =0, (4.4.35) \?-
n<x,

W Q< 0 ,

(4A36>

3* —<e

and

Q<2X-C,

Hybrid Control for Global Stabilization of a Class of Systems 153 WQ <0,

(4.4.37) c<0n<--

Calculate

mm

l+ d E

(«-0)

2~0

// (4.4.34)or (4.4.35) holds , (4.4.38)

C 4 =1

. l(«-*o) ! - £ 0 , if(4A.36)or (4.4.37) holds Tl

'

4

Apply the controller )C 4 // C 4 > £ U=

C A =

(4.4.39)

lo if CA <e until mini 3^/2,6»| w=0 >^k!/(4.4.34)or(4.4.35)toWi , 0=0* = 3TT/2 ,

i/(4.4.36)/ioto ,

max{:,0\w=o<7r/2}, if (4.4.37) holds . If C 4 < e, set current states as initial states and restart Step 3. If C 4 ^ £, switch the control to u = -C4. When v = 0, set current states as initial states and then return to Step 2. Case 2. One of the following conditions holds. w0 < 0 ,

Till < e 0 < 3 n 12,

(4.4.40)

w0 = 0 ,

7C <0Q<

3tf 12

(4.4.41)

154 J. Zhao

\w0

>0, (4.4.42)

| 3 ; r / 2 < 6>0 , and

Jw0 >0,

(4.4.43)

[0Q <7tl2,

Calculate a + XQ

[+

mm

-d2-Eun

2 4

, if •,if

(4.4.40) or (4AA\)holds (4.4.44)

C<

min \

a + xn 1 - E, *° , °- \ , if (4.4.42) or (4.4.43) holds.

Apply controller u=C5 =

\-C5ifC5>e 10

(4.4.45)

ifC5<e

until max {jtll,0

\W=Q<

n \

0 = 0* = min {2n-c,0\w=Q>3?i

nil,

if (4.4.40)or (4.4.41) holds I2\, if {AAA.2)holds , if (4.4.43) holds .

If C 5 < s, set current states as initial states and restart Step 3. If c 5 5: s, switch the control to u =-C5. When v = 0, set current states as initial states and then return to Step 2. Step 4. Case 1.

J w0 < 0 , [x/2<0o Calculate

(4.4.46) <3nl2,

Hybrid Control for Global Stabilization

C,=min ^ 0 6

^EQ-COS^I

of a Class of Systems

155

(4A4?)

r,5

Apply controller u=C(.= \

C 6 if C6>e _

[o // C6<s,

(4.4.48)

until 0=ri2. If C 6 < s, set current states as initial states and restart Step 4. If C~6 > e, switch the control to u =-C6. When v = 0, set current states as initial states and then return to Step 2. Case 2. fw0 > 0,

(4.4.49)

U/2<80<37r/2 Calculate \a + xQ

EQ

C-j = min •

- COS C

{T2

4

(4.4.50)

Apply controller u=C1 = \ C [Oif

l l

l

Cl £

~ C7<£

(4-4-51)

until 0=3nf2. If C 7 < e, set current states as initial states and restart Step 4. If C 7 > e, switch the control to u = -C7. When v = 0, set current states as initial states and then return to Step 2. Case 3. Either fw0 > 0, \?>7tl2<e 0<2x or 0 or

(4.4.52) <9Q
156

J. Zhao

Jw 0 < 0 ,

(4.4.53)

\2>KI2<0 Q<2TT or

O<0Q<X/2,

holds. If 0O = 0, calculate Co = min •

a-x0 2

T

d d2 '2' 2

EQ - COSC

(4.4.54)

3

and CQ

.

.a

= mm i

+ x

0

d2

d

—, —, '2' 2 T2

EQ-COSC

(4.4.55)

where T3 is the constant given in Section 4.4.2. Apply controller Cg

if

WQ

> 0 and Cg > e,

" = 8 = 1-Cy ifwQ<0and 0 otherwise c

Cg>e,

(4.4.56)

until 0=f;r/2 I 3 K11

ifw0>0, (or equivalently - nil ) if W Q < 0 .

(4.4.57)

Then, if u = 0, set current states as initial states and restart Step 4. Otherwise, switch the control to u = -C8. When v = 0, set current states as initial states and then return to Step 2. If 0O^O, set M=0 until 0= 0 (or equivalently, 2n) or 6=7d2 or 0=3nt2, whichever holds first and then set current states as initial states and restart Step 4. Remark 4.4.9 In the modified algorithm T4 is an upper bound of the time in which the pendulum moves from 0oto 6 = min{3n/2, 9\w = 0>7t) with w(t)>0for all 7t/2<90 <3n/2 and under all possible nonnegative constant controls, T5 is an upper bound of the time in which the pendulum moves from 90 to 9 =3n/2 with initial energy E0> (d2/2)+1, w(t)2 0for all n/2<0o <3n/2 under all possible nonpositive constant controls u >(cosc-E0)/4, T6 is an upper bound of the time in which the pendu-

Hybrid Control for Global Stabilization of a Class of Systems 157 lum moves from 60 to 6 = max{c,8\w = 0} with initial energy E0 < cose , w(t)<0 , c< 9o<7z /2 and under all possible nonnegative constant control u satisfying 0< u< (l-E0)/4. s is chosen as

< • { « Moll >

e < mm {—— ,

A
^ j

Simulation results are shown in Fig.4.2 and Fig.4.3.

4.5 Concluding remarks Quadratic stability of switched nonlinear systems has been studied in this chapter. The key point is to avoid employing strict completeness which is usually very hard to be verified. Instead, we first transform the quadratic stability of systems into an equivalent restricted nonlinear programming problem and then give a necessary and sufficient condition for quadratic stability with Fritz John condition. It is easy to verify this condition because only algebraic equations and inequalities are involved. As an application example of hybrid control technique, a globally stabilizing controller for the cart-pendulum system has been designed in this chapter. Comparing with the existing results, this result has two features. Firstly, we give a globally asymptotic stabilization result. To the best of our knowledge, no global asymptotic stabilization results for the cart-pendulum system have previously been reported. Secondly, before switching to a local stabilizing controller, we switch between bang-bang controllers. Therefore it is practically easy to realize the control. Although the method is based on energy, we do not limit ourselves to keeping the energy increasing (in the case of E0(c?2/2)+l) all the time during an iteration. However, the controller is designed in such a way that at the end of each iteration the energy increases (in the case of E0 < cose) or decreases (in the case of£ 0 >(^/2)+l) comparing with the initial energy of the iteration. This gives us a big "freedom" of choices of controls, which allows us to steer the position and velocity of the cart to the pre-described neighborhood U at the end of each iteration. The constants Tu T2 and T3 (in Section 4.4.2) and T4, T5, T6 (in Section 4.4.4) play an important role in the design of the controller. It is clear that the smaller these constants, the faster the trajectory converges, especially in the first few switches. Some choices of 7\, T2 and T3 were given in the chapter though they might be more or less conservative. It may be possible to extend the method to other mechanical systems, such as underactuated robot systems.

158

J. Zhao Swinging up control of the cart-pendulum system

f 5

10 Fig.4.2

15 20 25 30 t(sec), sampling step=0.1sec

35

Control switching 0.4 0.2

—

0

—~~

0.2 U.4

f\

I \ -

" -

"

0.6 0.8

-

-

-1

-

•

1.2 1 a.

10 Fig.4.3

15 20 25 30 t(sec), sampling step=0.1sec

35

40

References [1] A.A.Agrachev and D.Liberzon, Lie-algebraic conditions for exponential stability of switched systems, in Proc. IEEE Conf.Decision and control, pp.2679-2684, 1999.

Hybrid Control for Global Stabilization

of a Class of Systems

159

'2] A.Back, J.Guckenheimer and M.Myers, A dynamical simulation facility for hybrid systems, in Hybrid Systems, R.Grossman et al. Eds., 255-267, 1993. ^3] M.S.Branicky, Multiple Lyapunov functions and other analysis tools for switched and hybrid systems, IEEE Trans.Automat.Contr., vol.43, pp.475-482, 1998. [4] M.S.Branicky, V.S.Borkar and S.K.Mitter, A unified framework for hybrid control: model and optimal control theory, IEEE Trans.Automat.Contr., vol.43, pp.31-45, 1998. [5] M.S.Branicky, Stability of switched and hybrid systems, in Proc. 33rd IEEE Conf.on Decision and Control, 3498-3503, 1994. [6] R.W.Brockett, Hybrid models for motion control systems, in Essays on Control Prospectives in the Theory and its Applications, H.L.Trentelman and J.C.Willems, Eds. Boston, MA: Birkhauser, 1993, pp.29-53. [7] C.C.Cheng and J.Hauser, Nonlinear control of a switching pendulum. Automatica, Vol.31, No.6, pp.851-862, 1995. [8] W.P.Dayawansa and C.F.Martin, A converse Lyapunov theorem for a class of dynamical systems with undergo switching, IEEE Trans.Automat.Contr., vol.44, pp.751-760, 1999. [9] J.A.De Dona, S.O.Reza Moheimani, G.C.Goodwin and A.Feuer, Allowing for over-saturation in robust switching control of a class of uncertain systems, in Proc. IEEE Conf.Decision and control, pp.3053-3058, 1999. [10] Z.Feng, Z.Yin and H.Chen, Microprocessor-Based Controller for double inverted pendulum, In Proc. IFAC 10th World Congress, pp.237-240, 1987. [11] J.F.Frommer, S.R.Kulkarni and P.J.Ramadge, Controller switching based on output prediction errors, IEEE Trans.Automat.Contr., vol.43, pp.596-607, 1998. [12] K.Furuta, M.Yamakita and S.Kobayashi, Swing-up control of inverted pendulum using pseudo-state feedback. Proc. Instn. Mech. Engrs., Vol.206, pp.263-269, 1992. [13] K.Furuta, T.Okutani and H.Stone, Computer control of a double inverted pendulum. Comput. Elect. Engng., Vol.5, pp.67-84, 1978. [14] S.Geva and J.Sitte, A Cart-Pole experiment benchmark for trainable controllers. IEEE Control Systems, Vol.13, No.5, pp.40-51, 1993. [15] A.Gollu and P.P.Varaiya, Hybrid Dynamical Systems, in Proc. 28th IEEE Conf. Decision and Control, pp.2708-2712, 1989. [16] J.P.Haspanha and A.S.Morse, Stability of switched systems with average dwelltime, in Proc. 38th Conf.on Decision and Control, 2655-2660, 1999. [17] A.Hassibi and S.Boyd, Quadratic stabilization and control of piecewise-linear systems, in Proc. American Contr. Conf, pp.3659-3664, 1998. [18] A.Iftar and U.Ozguner, Overlapping decompositions, Expansions, Contractions, and Stability of Hybrid Systems. IEEE Trans.Automat. Control, Vol.43, No.8, pp.1040-1055, 1998.

160

J. Zhao

[19] P.A.Iglesias, On the stability of sampled-data linear time-varying feedback systems, in Proc.33rd Conf.Decision and Control, pp.219-224, 1994. [20] A.Isidori, Nonlinear Control Systems, 3rd Edi., Springer-Verlag, Berlin, 1995. [21] I.Kolmanovsky and N.H.McClamroch, Hybrid feedback laws for a class of cascade nonlinear control systems, IEEE Trans. Automat. Contr., vol.41, pp. 1271-1282, 1996. [22] M.Krstic, I.Kanelakopoulos and Petar Kokotovic, Nonlinear and Adaptive Control Design, John Wiley & Sons, New York, 1995. [23] D.Liberzon and A.S.Morse, Basic problems in stability and design of switched systems, IEEE Contr. Syst. Magazine, vol.19, 1999. [24] R Marino and P.Tomei, Nonlinear Control Design, Prentice Hall, 1995. [25] A.S.Morse, supervisory control of families of linear set-point controller-part 1: exact matching, IEEE Trans.Automat.Contr., vol.41, pp.1413-1431, 1996. [26] T.Ooba and Y.Funahashi, On a common quadratic Lyapunov function for widely distant systems, IEEE Trans. Automat. Contr., Vol. 42, 1697-1699, 1997. [27] K.M.Passino, A.N.Michel and P.J.Antsaklis, Lyapunov stability of a class of discrete event systems, IEEE Trans.Automat.Contr., Vol.39, pp.269-279, 1994. [28] P.Peleties and R. DeCarlo, Asymptotic stability of m-switched systems using Lyapunov-like functions, in Proc. American Contr. Conf., pp.1679-1684, 1991. [29] R.Sepulchre, M.Jankovic and P.V.Kokotovic, Constructive Nonlinear Control, Springer-Verlag, Berlin, 1997. [30] E.Skafidas, RJ.Evans, A.V.Savkin and I.R.Peterson, Stability results for switched controller systems, Automatica, vol.35, pp.553-564, 1999. [31] M.W.Spong, The swing up control problem for the acrobot, IEEE Contr. Syst. Magazine, vol.15, pp.49-55, 1995. [32] M.W.Spong, Underactuated Mechanical Systems. In Control Problems in Robotics and Automation, pp.135-150, Springer, 1998. [33] M.W.Spong and L.Praly, Control of underactuated mechanical systems using switching and saturation, In Control Using Logic-Based Switching, pp. 162-172, 1995 [34] P.P.Varaiya, Smart cars on smart roads: problems of control, IEEE Trans.Automat, Vol.3 8.pp. 195-207, 1993. [35] Q.Wei, W.P.Dayawansa and W.S.Levine, Nonlinear controller for an inverted pendulum having restricted travel. Automatica, Vol.31, No.6, pp.841-850, 1995. [36] H.Ye, N.Michel and L.Hou, Stability theory for hybrid dynamical systems, IEEE Trans.Automat.Contr., vol.43, pp.461-474, 1998. [37] J.Zhao and M.W.Spong, Hybrid Control for Global Stabilization of the CartPendulum System, Automatica (to appear).

Chapter

5

Robust and Adaptive Control of Nonholonomic Mechanical Systems with Applications to Mobile Robots Y.M.Hu*

and

W.Huo*

^Department of Automatic Control Engineering, South China University of Technology, Guangzhou, 510641, E-mail:auymhu@scut. edu. en

P.RChina.

*The Seventh Research Division, Beijing University of Aeronautics Astronautics, Beijing, 100083, P.RChina

and

5.1 Introduction Nonholonomic mechanical systems have profound engineering background such as mobile robots, vehicles and spacecraft. Nonholonomic mechanical systems, although controllable, cannot be stabilized by smooth static or dynamic state feedback control laws. This fact makes the control of general nonholonomic systems very difficult, and stimulates researchers to construct time-varying and discontinuous feedback controllers for some special nonholonomic systems. Many sophisticated control approaches, such as smooth time-varying strategy, time optima" control law, discontinuous control law and iterative learning control, etc, have been proposed for various nonholonomic control plants. However, most of current control approaches for nonholonomic systems are dependent on complete knowledge and special structures of the controlled plants, which may practically produce poor dynamic responses in the presence of system uncertainties or parameter variations. The design of robust and adaptive controllers for general nonholonomic mechanical systems with uncertainties is thus emphasized in recent surveys (Hu, Zhou and Pei., Kolmanovsky and McClamroch). 161

162

Y. M. Hu & W. Huo

Sliding mode control (SMC) is a special discontinuous control technique applicable to a broad variety of practical systems (referred to the book of Utkin, and the intioductory papers of Young and Ozguner). This approach is mainly based on the design of switching functions which are used to create a sliding manifold. When this manifold is attained, the switching functions keep the trajectory on the manifold, thus yielding desired system dynamics. Since nonholonomic mechanical systems can not be stabilized by smooth time invariant static or dynamic feedback laws, the SMC technique becomes appealing in the control of nonholonomic systems. A few of researchers have investigated the design of sliding mode controllers for a class of nonholonomic systems, which heavily depend on the existence of suitable Lyapunov functions and special structures of the systems (see, for instance, Bloch and Drakunov). Even if the necessary Lyapunov function is constructed, however, the resulted sliding mode motion is only locally ensured. Although considerable progress has been made for the control of nonholonomic systems during the last decade, the literature on uncertain nonholonomic systems is sparse. The aim of this chapter is to study the robust and adaptive control problems of uncertain nonholonomic mechanical systems. We first present the reduced model of uncertain mechanical control systems with classical nonholonomic constraints, and review some properties of uncertain nonholonomic control systems. Based on the backstepping design approach, high-order sliding mode and differential flat systems theory, we then develop some adaptive and robust control strategies for a class of nonholonomic mechanical control systems with mismatched uncertainties. Finally, we apply the proposed approaches to a two-wheel driven nonholonomic mobile robot. Numerical simulations and on-line implementations are performed to show their efficiency. The proposed approaches are not only adaptive and robust to uncertain parameter variations or uncertainties, but also have the potentiality to weaken the chattering phenomenon of traditional sliding mode control systems.

5.2 Dynamic model of nonholonomic mechanical systems Many nonholonomic mechanical systems can be described by the following dynamic equations M(q)q + V(q,q)q + G(q) = B(q)r + A T (q)X

(5.2.1)

and the nonholonomic kinematic constraints A(q)q = 0

(5.2.2)

where q e R" and r e Rr are respectively the generalized configuration and the control input vectors; X & Rm is the constraint force vector; M(q)e R"x" is a

Robust and Adaptive

Control of Nonholonomic

Mechanical Systems

. ..

163

symmetric positive definite matrix; V(q,q)q e R" is the term which may include centripetal and Coriolis forces; G(q) is the gravitational force; B(q) is a n x r full rank input matrix; A(q) i s a m x n full rank matrix associated with the constraints. Let Nt = Nj(q)(i = \,...,n-m) be a set of smooth and linearly independent functions such that A(q)Nl=0,i

= l,...,n-m

(5.2.3)

and A be the distribution spanned by the vectors Nt (i = 1,..., n-m), then from (5.2.2) it follo'vs that q e A, that is, there exists an (« - m) - dimensional pseudo-velocity vector v = (Vj,... vn_m )T such that q = Nv where N = N(q) =

(5.2.4)

(Nl(q),...,Nn_Jq)).

Differentiating (5.2.4), we obtain q = Nv + Nv

(5.2.5)

Substituting (5.2.5) into (5.2.1) and premultiplying by N r , we have NTMNV

+ (NTMN+NTV)V

+ NTG=NTBT

(5.2.6)

That is, the nonholonomic mechanical systems (5.2.1) and (5.2.2) can be reduced to the following form \q = Nv r~ ~

~

~

(5.2.7)

[MV + VV + G = BT

where M =NTMN; V =NTMN +NTV; G = NTG; B =NTB

(5.2.8)

Since the above base matrix N is independent on the uncertainties of the mechai ical system, it is easy to prove the following properties PI). M is an (n-m) x (n - m) symmetric positive definite matrix; P2). M - 2V is a skew symmetric matrix; P3). Let 4" e RM be the parameter vector of the mechanical system (5.2.1), then Mv + Vv + G = 0(q, q, v, v)^

(5.2.9)

164

Y. M. Hu & W. Huo

where 0(q,q,v,v) e /? ( " m)x// is independent on the parameters of the system. As a typical example of nonholonomic mechanical systems, we consider the two-wheeled plat mobile robot as shown in Figure 5.1. Two independent but identical dc motors are used to steer the wheels, the corresponding dynamic equations and kinematic constraint are given as follows mx = —(rj+r 2 )cos^-/lsin^ R -my = —(ri+r, )smd + Xcos6 R

(5.2.10)

xsm6-ycos6

(5.2.11)

Fig.5.1

=Q

Two-wheel steering mobile robot

where (x,y) is the position of the center P corresponding to the rear axis, and 9 is the orientation with respect to the X -axis. TX and r 2 represent, respectively, the torques produced by the two driven motors. 2R and 2D are respectively the diameter and distance between the two rear wheels. Other nonlinear dynamic model can also be developed using the Lagrangian formulation (see, for instance, Canudas de wit, Siciliano and Bastin, 1998). It is to rewrite the above model in the reduced form (2.8) with q =(x, y, 9)T, r =(r,, r2f, and

Robust and Adaptive

'm

0

0N

0

m

0 , B=

,o o / 0 , Select N(q)

Control of Nonholonomic

'cos# 1

sin#

7? v

Mechanical Systems

...

165

cos#N sin# , F = 0, G = 0, ^ = (-sin6> cos0 o)

D

as 'cos#

°) 0 Io u

N(q) = sin#

then, one can develop controller based on the reduced model q = Nv NTMNV

+ NTMNV

=

NTBT

(5.2.12)

with the above coefficients. However, motor can not behave ideally to produce torque or velocity as the control law needed, actuator dynamics should be considered in the complete robot dynamics to improve control performance especially in the case of high-velocity movement and fast varying loads. Suppose that the steering motors used are the same types, then the dynamic equations for brushed DC motors can be written in the form •Tl+kJci=Ul

" T: + JJfv rp

fJn,

(5.2.13)

rp

whereh 0 ,r,k e denote, respectively, the armature inductance, the armature resistance and the back-EMF coefficients, co = ((o],eo2)T represents the angular velocity vector defined as

1

-D

(5.2.14) V"27

:„ P is the ratio of the decelerating gear, kT is the torque constant, and u - {ux, u2)\T is the vector of applied voltages to the motors. Differentiating the second equation of (5.2.12), and substituting (5.2.13) and (5.2.14) into the equation, we obtain the following model

[q = Nv {H(q,v,v)v + V(q,v)v + K(q)v = u where the control input u is selected as the voltage vector, and

(5.2.15)

166

H=-

Y. M. Hu & W. Huo

l^

RK

IDP-k,

KmD

V

Rr 2Dp-kT

=•

0/

mD mD

h^ ,K = kj

-1 0 /

R

1

D

1

-D

(5.2.16)

The output variables are generally selected as Y=

= h(q):

\*u

y + hsind

(5.2.17)

that is, the position of the central point Q.

5.3

Robust control design based on SMC

5.3.1 SMC based on high-order sliding mode As discussed above, nonholonomic mechanical systems can be reduced to the following form x = f(x) + B(x)u

(5.3.1)

where x e R" and usRm are respectively the state and input vectors; / and B(x) = (bx ,...,bm) are sufficiently smooth functions with corresponding dimensions. To design a sliding mode controller with sliding order r = {rx,...,rm )T such that the above system (5.3.1) has proper dynamic behavior, we need to determine a highorder sliding mode control law and a sufficiently smooth w-dimensional function S = (S,,.., Sm)T : R" -+Rm such that (I). Each components, exists at leastr, -th order continuous derivatives; (II). The sliding motion constrained on the set Q defined by the following equations S, = S,

:0H)

: 0 , (/=!,.., m)

(5.3.2)

is asymptotically stable (III). The following regular sliding condition (',-i)i rank{VS,, VS,, • • •,VS« '' |/ = \,...,m) = r

(5.3.3)

is satisfied, where r = rx H— + rm is called the total sliding order. (VI). The desired high-order sliding motion on Q can be achieved in finite time.

Robust and Adaptive

Control of Nonholonomic

Mechanical Systems

...

167

Obviously, the above high-order SMC is a natural generalization of the conventional SMC(i.e., rx = ••• = rm = 1, r =m). However, tools in the conventional SMC designs can not be easily applied to the high-order SMC design, since, in general, the derivatives of the switching function components 5 ; , • • •, SJrr' (i = 1,..., rri) may be dependent on the derivatives of inputs and the fact that the relative degree components are not one may cause instability of the system. To derive the mathematical description of high-order sliding mode motions, we suppose that the designed switching function S satisfies the following conditions (HI). Lb.LkfSt=0 forallxandO<£
Lb(LrflS,)

-

Lb

(L'f'sy (5.3.4)

mxm

z

ls

L

^V /~ »> - V /~

,s

m ).

is nonsingular for all x. The above conditions (HI) and (H2) imply that the system (5.3.1) has strongly relative degree {rx,...,rm} from the control input to the sliding manifold. Such kind of conditions are natural in the input/output linearization of nonlinear control systems (see, e.g., Sastry, 1999). Under the above conditions (HI) and (H2), we have S\k) =LkfSi,Q
(5.3.5) (5.3.6)

which imply that the r, -th derivatives of each switching function components st are dependent on the control. Obviously, once the sliding motion is achieved, we have SJk^ (x) = 0 (z' = l,..., m\ k = 0,1,..., rt -1), that is Si-k\x) = LkfSi=0,

i = l,...,m;k = 0,l,...,ri-l.

(5.3.7)

which implies thatS}r''(x) = 0. The equivalent control ueq can thus be uniquely determined by (5.3.6) as follows ue=-Es-lKs

(5.3.8)

168

Y. M. Hu & W. Huo

where Ks=(L}Su...,LySmf

(5.3.9)

It should be noted that the equivalent control ueg in (5.3.8) is actually determined by the rt -th order derivatives of the switching functions, that is, the equations sfr'\x) = 0(i = l,...,m). It thus generalizes the conventional equivalent control method (see, e.g., Utkin, 1992), since in the case of at least one /-, > 1, the conventional equivalent control can not be directly solved from the conventional equivalent control equation S(x) = 0. Let the control be the following form u = Es-\v + Kzzs) + ueq whereK z = diag(Kzl,...,Kzm),

Kzi=(Kzi,...,Kzfl)(i

(5.3.10) = l,...,m)arerrth

dimensional

s;ain vectors to stabilize the new state componentsz si ; whilev = (vj,...,v m ) r is the nonlinear sliding mode control such that the desired high-order sliding motion can be achieved in finite time. On the other hand, as well-known in nonlinear control theory (see, e.g., Sastry, 1999, and Isidori, 1995), we know that the following row vector functions VS, (x), VLfSx (*),..., V l V ' S , (x) Sm(x),VLfSm(x),...,VLj-1Sm(x) are linearly independent, and the regular sliding condition (5.3.3) is thus satisfied. Consider the nonlinear state transformation z = ( z j ,£T)T = T(x) as follows zs =(zsl,...,zsmf

=Tz(x) = (Sl,LfSl,...,L}'lSl;...;Sm,LfSm,...,Lrf1Sm)T

£ = T( (x) = (Ci ,-,C„.7)T

(5.3.11)

= (r, (x),. ..,Tn_7 (x))T

where £ = T^ (x) is chosen to be the smooth functions such that z = T(x) is a diffeomorphism, then from (5.3.5), (5.3.6) and (5.3.10) we transform the system (5.3.1) into the following canonical form

Robust and Adaptive

Control of Nonholonomic

2 si = (Ai+biKzi)zsi+b,vi,i £ = p(zs,0 + q(zs,C)v

Mechanical Systems

...

169

= l,...,m

where (Aj,bj)(i = l,...,m)sre controllable canonical pairs; and | p = [{LfTk) + {Lbj Tk (x))(Es~lKzzs {q = {LhTk{x))E?

+ ueq)] o T~x (z) e Rn~T

o r ' W e * ^

The switching function S(x) is rewritten in the new state variable z in the form S(zs,0 = S(x)°T-\z)

(5.3.13)

The first subsystem in the above canonical form (5.3.12) describes the dynamics of switching variable 5 , while the second subsystem, once the sliding motion is attained, determines the sliding modes. Since the inputs and switching variables are decoupled, one can easily determine the gain constants Kzi and sliding control law v, such that the system can realize the sliding motion with desired dynamics due to the controllability of the pair (A^ty). Let Kzi = (Kzi ,...,Kr<~x)be chosen such that X' = KrflArrl

+••• + Kl2iX + K°zi

(5.3.14)

are Huwitzian polynomials, then the asymptotic stability of the first subsystem in (5.3.12) is obviously guaranteed. Next, we consider the design of sliding mode control law so that the desired high-order sliding motion can be achieved in finite time. Let 5, =S<"-1) -Mrs?-2)

—-M?St

-n)St

(5.3.15)

where /// (J = l,...,rt -1) are constants such that the following (r, -1)-th order systems Sj*-* -M?-^

--M?S,

-M]S,=0

have desired eigenvalues. Then, from (5.3.12) and (5.3.15) we obtain -/=(->"/.•••,-//,•'

= (rn\ ,...,-$

,l)z«

,\)[{At + bjKzi )zsi + bjVj ]

(5.3.16)

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Y. M. Hu & W. Huo

r -1 = I (KJzi -MJi)S{iJ)+K%Si 7=1

+Vi

(5.3.17)

Thus, under the following control law

v, = - E W -M/)Sy) -K°z,Si -y,E, -S,Sign(E,)

(5.3.18)

7=1

we have Ei=-riEi-SlSign($l)

(5.3.19)

where yt > 0 and£, > 0 (/' = l,...,m), which implies that the desired sliding dynamics (5.3.16) can be achieved. Finally, we need to determine switching function S(x) such that the sliding mode system has proper dynamic behavior. Once the sliding motion is attained, the dynamic equation of sliding motion is obviously given as follows C=P?(ZS,O

(5320) (,i ,)

S,. = S ' j = - = l S, - =0, i = \,...,m where p = [{LfTk) + {LbTk{x))E-\Kzzs

-Ks)]oT~\z)

(5.3.21)

Select the switching functionS(z s ,£) to satisfy the regular sliding condition, e.g., rank{d{Si,S1,---,Syi'X))ldzs\i

= \,...,m) = r

(5.3.22)

then one can solve r state variables zs = G(£) from the 7 constraints 5, = St (r -1)

= Sf' =0 in (5.3.20). The sliding motion is thus described by the following equation

C-p0(zs,O zs=G{Q where the function zs = G(£) can be regarded as the input of the sliding mode system. If the following zero dynamics

Robust and Adaptive

Control of Nonholonomic

Mechanical Systems . ..

171

(5.3.24)

^ = A,(0,O

is asymptotically stable (i.e., the system is of minimum phase), then the sliding mode system is locally asymptotically stable. To present an explicit solution of the above high-order SMC design problems, we consider the following nonlinear SISO canonical control systems

(0

f0~\

h-^

x=

0

0

( 0 ^ (5.3.25)

x + 0 u+

0

1

a(x).

where x = (x,,...,x„) r e # " andMe/?are respectively the state and control vectors; a{x) is a scalar function. Let the switching function be given in the form s{x) = cxxl+ — + ckxk + xk+x =Cx, (0
*0)

(5.3.26)

then, we obtain AJ) _ clxJ+l+{n

k)

\s

+ ckxJ+k+xj+k+lJ

= cxxn_k+x+-

=

0,l,...,(n-k-l)

+ ckx„+u + a(x)

(5.3.27)

which implies that the conditions (HI) and (H2) are satisfied forr, = n-k (i.e., the sliding order is n-k). The nonlinear transformation (5.3.11) and the equivalent control (5.3.8) in this case are thus respectively given as follows z s =Ts(x) =

(z0,...,zn_k_1)7

Z

} ~ C l Xj+\

H

+C

k


eq=~C\Xn_k+x

X

j+k

+

X

j+k+\

(j =

0,...,n-k-\)

(5.3.28)

=(x„_k+l,...,x„) CkX„-a

= -C1Cl

Ck£k-(Z

(5.3.29)

With the above state transformation and the following control u = ueq + K°zz0 +••• + K"-k-'zn_k_x

+v

we obtain the canonical forms as follows 7 'j^ J = 0,1,...,«- * - 2 z

n-k-\

• K°zz0+-

+

Krk-lzn-k-\ + V

(5.3.30)

(5.3.31)

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Y. M. Hu & W. Huo

Gj -Cj+iJ-h-,k 1 Ck = K°zz0 +... + K"z-k-izn_k_l -c^x

(5 3 32) ckCk +v

Let K{ (j = 0,...,n-k-l) be the coefficients of the desired Huwitzian polynomial (5.3.14) and the SMC law be given by (5.3.18), then the system (5.3.25) can realize the desired sliding motion, as indicated above. What we need to do is the selections of proper constants c'j(j = \,...,kt;i = \,...,m) such that the sliding mode has prescribed dynamic behavior. During the sliding motion, from (5.3.26) and (5.3.27) we have zj =c]xJ+l+-

+ ckxJ+k+xJ+k+l

=0, j = 0,1,...,(«-k-1)

(5.3.33)

From (5.3.32) and (5.3.33) we can obtain the sliding mode equation as follows

J f •/' = ^ +1 'J = 1, "•'k ~l [Ck =-C\£i

c

(5.3.34)

kCk

which ran be stabilized with desired dynamic response so long as the corresponding k -th order characteristic polynomial Ak +ckZk~l +--- + c2Z + cl = 0 is assigned to have desired eigenvalues. Consequently, as conventional SMC, high-order SMC can also be explicitly designed for the class of nonlinear SISO or MIMO canonical systems (for the case of MIMO, see, Hu and Chao, 2000).

5.3.2 Dynamic SMC based on high-order sliding mode Since nonholonomic mechanical control systems can not be exactly linearized via smooth state and input transformations, two approaches are generally used to simplify the control design problem. The first one is to construct proper nonlinear state and input transformations to transform the reduced model (5.2.7) into the canonical chained or power forms. Another approach is to apply dynamic extension algorithms to ensure that the controlled plant is differentially flat to a linear controllable system. However, the first approach is difficult to implement in the presence of parameter or system uncertainties since the constructed transformations are heavily dependent on the exact knowledge of the controlled plant. Without loss of generality, we consider the following Fliess SISO canonical system

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Control of Nonholonomic

Mechanical Systems

...

173

a, =<J-,

(5.3.35)

=CT

°>1

P {r)

Q(crl,---,ap,&p,u,u,---,u ,x)

wheredQ/d&p

* 0 . Let u-vx,

=0

VJ =v/ + i(/' = l,...^-l)andv

= w b e the dynamic

compensate variables, then (5.3.35) can be rewritten in the new state variables
p-\

a

=

p

Op = 2 o ( ° V v, =v 2 Vi

= v

Vy

=W

• • ^

P

^ \ , -

•,vy,w,x) (5.3.36)

r

where Q0 is differentiate with respect to all its variables and dQ0 ldw*Q. For the sake of simplicity, we consider the design of SMC with sliding order p. Let the switching function s be given in the form .5 = 0^, then, from (5.3.36), we have \s(i)=ai+1(i (p)

[s

= l,...,p-l)

=Q0(.(?i,---,o-p,vl,---,vr,w,x)

(5.3.37)

Let Z = s(p-»-MP-is(r-V-...-Ss

-Mls

(5.3.38)

and the reaching law be given as

^-rZ-ssigniZ) where /iJ\j system

= l,...,p-l)

(r>o,s>o)

(5.3.39)

are constants such that the following ( p - l ) - t h order

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Y. M. Hu & W. Huo

Ap-l-Mp-]Ap-2

(t2X - / / ' = < )

(5.3.40)

has desired eigenvalue distribution, then, from (5.3.37), (5.3.38) and (5.3.39) we obtain Qo(ai,--,ap,v\,--,vr,w,x)-np~

ap

M^'M

°"2

= -rf-SSign(?)

(5.3.41)

A dynamic SMC law can be obtained from the above equation (5.3.41) since dQ0 ldw*0. The asymptotic stabilization of the output y = ox is thus achieved via the proposed dynamic SMC approach.

5.3.3 Robustness design in the presence of uncertainties 5.3.3.1 Model reduction of nonlinear uncertain systems As discussed above, high-order SMC can be designed for a wide class of nonlinear systems. However, due to the fact that some essential parameters such as inertial, payload, and the motor constants are generally unknown or uncertain, therefore it is necessary to consider the uncertainties in the design of SMC. This subsection will concentrate on the development of robust SMC design based on the backstepping approach, which allows the mismatched uncertainties. In the presence of uncertainties, the nonlinear system (5.3.1) can be rewritten in the fom JC = f(x) + Af(x,p) + [B(x) + AB(x,p)]u

(5.3.42)

where A/ and AB(x) = {Abx,...,Abm) represent, respectively, the uncertain parts off and B(x) = {bx,....,bm);p&Rl denote the uncertain parameter vector. Apart from the assumptions (HI) and (H2), we suppose that the uncertainties A/ and AB(x) satisfy the following conditions (H3). There are integer numbers at (i = \,...,m) such that L^L'y~lSl * 0 for some x and p; (H4). There are integer numbers^ (i = \,...,m) such that L^Ly'^S,*0 somey'e{l,...,w},

for

xandp.

If there is at least an integer/e{l,...,/w}such t h a t c r ^ ^ o r yi
then the

Robust and Adaptive Control of Nonholonomic Mechanical Systems . .. 175

above assumptions imply that the uncertain part Af or AB does not satisfy the socalled generalized matching conditions. Without loss of generality, we suppose 1
= l,...,«)

(5.3.43)

m

£i =Kg +AK# + Y(e$j

+Ae

i!j)uJ,

l = 7 + \,...,n

(5.3.44)

where r

T

Ksj=L'j-Si,

—1

AKSI.=LyLj-

Sj,i = l,...,m;

Kg =LfT-p+j, AKg =LyT7+J, A£j

=

j =

\,...,n-r;

(&esij )mxm = (^Abj Lf Si ) m x m >

E( =(egj)(n-7)xm=(LbjTl)(n-7)xm

>

AE^ = (Ae^j )(„-fym = (LdJ)j 7} ) ( „_? )xm ; At]p+J-l

= L¥Ly+J-lS„

0 < ; < r , - C T ; - l ; i = l,...,m.

Other terms are defined as before. Under the control of (5.3.10), the above subsystems (5.3.43) and (5.3.44) are respectively reduced to the forms if =zf + 1 ,l<*<<7,-l; (SF),: Z'.+J

= Z°,+J* +AT}^-\

*? =KZIZSI+ATJ?-

0<j
-a, - 1 ; i = \,...,m

(5.3.45)

+V,+A£,V

C = X{zs ,D + *Z + (Ec + AEC )E;\ where Es and Es + AES are assumed to be invertible, and Atf~l =AKsi +(Aen,....,Aeim)EJi(K2zs A£, =(Aen,....,Aeim)E;\

-Ks), i = \,...,m; i = \,...,m;

(5.3.46)

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Y. M. Hu & W. Huo

Z(z„0

=

AZ =

Ki+EcE;\Kzzs-Ks); AKc+AEiE;\Kzzs-Ks)

? (UJ » (x) / Once the sliding motion is achieved, then5', = 0(z = l,...,w;7 = 0,l,...), that

is z) = 0 , i / =0, \<j
i = \,...,m

(5.3.47)

From (5.3.45) and (5.3.47), we obtain the equivalent control veq=-(Im

i r l T +AEsE; r\Atf-\...,A?7 m»- )

(5.3.48)

Substituting (5.3.48) into (5.3.46), we obtain the sliding mode equation £ = Z(Zs,0 + *Z

(5-3-49)

where Ax =

Az-(E(+AE()E;l(i+AEsE;ly\An?-\...,ATiZ-l)T

The robust stability of (5.3.49), as discussed in the section 5.3.1, can be guaranteed by selecting proper switching function via conventional design approaches in nonlinear continuous control systems. What we need is to design SMC law v so that the desired sliding motion can be achieved in the presence of uncertainties. 5.3.3.2 High-order SMC design for nonlinear uncertain systems Consider the functions Ei(i = l,...,m) defined in (5.3.15), then from (5.3.45) it follows that

= Kzizsl +A?li +v, +AElv-rxrfzf+1 where

A^A^-TV^^1 m

Let the norm of any vector a be defined as |a| = £ | a , | > anc* suppose i=l

(5.3.50)

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Control of Nonholonomic

Mechanical Systems

...

177

(5.3.51)

|A77,|
(5.3.52) 7=1

we have m

m f

i

il

IS,S,.<-X[ r,.S?+(^-A/7,M)|S,|J 1=1 (=i i-i <=i

;=i

+ //| S /| +
7=1

^-(^„-'"/ma,^M)2:^,2 r,-l (=1

)|S,| (5.3.53)

;=i 7=1

where Xm;„ = winy,-, ? W = waxx,, Smin = min8i, 5max = maxSi, AEM I

I

I

I

AEM = maxAElM , ArjM = maxAr]iM i

i

Therefore, ifAEu is relatively small, then we can select proper parameters St and X, in (5.3.52) such that rmin-mymaxhEM

>0

8mm -m8maxt^M

~ A1M

r,-\ J^J+I '^MZ Ztf 1=1

•Kzlzsl

>0

(5.3.54)

From (5.3.53) it follows that the desired sliding dynamics (5.3.16) can be achieved in the presence of bounded uncertainties. Hence, ifKJzi =///0=l,...,r, - l ; / = l,...,/w) , then from (5.3.17) and (5.3.54) we know that the SMC law (5.3.52) has a low switching gain.

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Y. M. Hu & W. Huo

5.3.3.3 High-order SMC design via backstepping approach Backstepping approach has been extensively applied to the design of robust and adaptive controllers in recent years. However, almost all the controls developed by this approach are based on the triangular assumption of the system model. Unfortunately, the uncertainties in (5.3.45) do not satisfy, in general, such triangular conditions. In the following, we will develop a robust controller based on the backstepping approach. Only the Lipschitzian-like bounds of uncertainties are used in the design, and the triangular structure assumptions are thus eliminated. Let = (0l,...,(\...,0lm,...,!z)T = T(z) be defined as follows tf=z?-a>?(z),...,z*-1)

(l
= l,...,m)

then<j> = T(z) is a diffeomorphism so long ascof (z),...,zf~])(l
(5.3.55) = l,...,m) are

(5.3.56)

for the subsystems {SF)t, then V} =z)zf =-p,{z))2

+ z\{z2 +Piz}) ( A >0)

(5.3.57)

If z2 + ptz) =0, then (5.3.57) implies that the components z) are asymptotically stable. However, in general, this is not true. To ensure that z) andz 2 proper asymptotic behavior, we introduce the auxiliary variables ] =z),
2

(OTO)] =0,o)

=-Plz\ )

+ptz)have (5.3.58)

and Lyapunov functions V?=V>+{tffl2

(5.3.59)

then, we have 2, Vl=-2PiV>+4>}4>f 4>2 =z] -^L-zf dzi where

=-*/ -Pitf

+
(5.3.60) (5.3.61)

Robust and Adaptive

Control of Nonholonomic

tf = z? -a)?,a>? =-) -Pltf

Mechanical Systems ...

+ ^zf

179

(5.3.62)

dz,

From (5.3.61) we know that <j>] and $f are also asymptotically stable if $ = 0. In general, at the k-th (1 < A: < cr, -1) step, we obtain a series of /fc-th order systems lk

_

Pi ~Zi

*=?a»*

M

la

j

(l<*
j+x

(5.3.63)

Z

i

and Lyapunov functions

V" = iV/ Hti? /2 (I = U,I» )

(5.3.64)

y=i

such that K;* = -2PiV* + tftf+l,(\
m)

(5.3.65)

where a!+\zl..,zf)

= -tf-1 -Pltf

+ Z^rz{+\

(l
i = l,...,«) (5.3.66)

At the (cr, + 7) - th (0 < _/' < r,- - cr, -1) step, due to the presence of uncertainties, we select the corresponding auxiliary functions co°l+J and the Lyapunov functions V°'+j as follows

<'+>+1 =-t?'+H -P,trJ + T 1 ^ V ^ , ' + 1 /=1

^+J,'=ffZ

V/Htf'+J)2'2

(5-3.67)

*Z;

(i = l,...,m)

(5.3.68)

/=i

which satisfy i?+J=z?+J+l+Atf.+J-*= - ^ + y - ' - ptfrt

£ - - ^ y - Z,'+1 /=1 <^, +
(5.3.69)

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Y. M. Hu & W. Huo

V?+] = -2PiV°>+1 + 4>?+jtf>+J+1 + £tf+l An?+'-1 /=o

(5.3.70)

At thefinalstep, the auxiliary functions $ and the Lyapunov functions

V? =r£v/+(tf)2/2 = zTz/2

(5.3.71)

7=1

satisfy, respectively, the following relations fi =Kzlzsi + A^'- 1 +v,- + AE.-V-

(5.3.72)

Y^T^

r-2A

r

7=1

^/

/~l

(5.3.73)

'i

_/+!

^ Z i z « + v , +AE,-v- X—7-z,7=0

;=i &,.

j

(5.3.74)

Finally, let V = ^lVjr' be the Lyapunov function for the general combined system i=i

(5.3.45), then we have I.\-2plVlr'-1+r':£'tf>+JAT,?+J-1

V= ;=i

7=0

1-ltfto /V z / z s , + v,.+4£,v-I—i' M

I Suppose that the uncertain terms AT)?'+J conditions ^L^z;

l

(5.3.75) ,/+l

&

satisfy the following bounded

A £ j S 4 £ w S i £ •;M

(5.3.76)

7=0

that is, the uncertain terms only need to satisfy Lipschitian-like constraints. Consider the following SMC laws

Robust and Adaptive

v, =-Pifp

Control of Nonholonomic

Mechanical Systems

r <~ldcor' + E — ^ z , ' + 1 -SiSign(tf),i

-Kzlzsi

...

181

(5.3.77)

= l,...,m

then, from (5.3.75), (5.3.76) and (5.3.77) we obtain m { V<-Z\-2PiV /=ll

+

m

r.

ZAEm z'=l

r,

T

r

m

'?

> + Ltzl

ji

r

z-Si

X

da>j

r

I V 7=1 I

m (

< I {- 2{Pi - L, )Vp - (Si - AEiM mSmax W i=\

2 m

'

J=I

'

'

'

I

f

< S { - 2 ( A -I,)K;< - ( * ,

- ^ m * ^ ^

(=1 2

+ TP m ax(^M + 2 ^ M W >=i

^ - 1 2P; - 2 1 , - ^ t f - / ^

^

+

1

T ^

M

W

+

N >

+ L^V (5.3.78)

-U^-AEiMmSmax)\^ where >
_/+i

^ # M ; Pmax = wax A ; Smax = wax J, (5.3.79)

Consequently, if the switching/>,,£, gains are selected such that

182

Y. M. Hu & W. Huo

(

2Pl-2L,-AElMK-p„ 8i-AEum8max

m

AElM + ZAEM |>0 I j=\ J J ,i = \,...,m

(5.3.80)

>0

then, fr jm (5.3.78) we know that the sliding motions can be achieved. The SMC laws (5.3.52) and (5.3.77) have their own advantages. The SMC law (5.3.52) can be obtained directly from the bounds of uncertainties, and may need high gain switching due to the constraint (5.3.54). While the SMC law (5.3.77) designed by backstepping approach may need small gain switching, and thus reduce the chattering of SMC systems. Besides, unlike the conventional backstepping design, we only need the uncertainties to satisfy Lipschitian-like conditions. With the similar backstepping design approach as above, we can also develop adaptive controller for nonholonomic mechanical systems due to the linear property (5.2.9).

5.4

Applications to the nonholonomic mobile robots

To show the efficiency of the above control strategies, we will consider the robust and adaptive control problems of nonholonomic mobile robot systems (5.2.12) and (5.2.15). 5.4.1 Design of dynamic sliding mode controller for the

output

tracking of nonholonomic mobile robot Consider the dynamic model (5.2.15) and the output (5.2.17), we introduce the integral compensator v = w, then we have Y = DX (0)v Y = D2(0)v2v + Dx(0)w

(5.4.1)

Y = (D2 (0)w2 - Dx (0)v22 )v + 1D2 (0)v2w + DX (0)H~l (u-Kvwhere

DJ9) = Let

cos 6 —hsmO^ smO

h cos 6 ,

D2(0) =

f-sin# cos 0

-hcos0^\ -hsin0

Vw)

Robust and Adaptive Control of Nonholonomic Mechanical Systems ...

P = (D2 (d)w2 - D1 (6>)v2 2 )v + 2D2 (9)v2 w

183

(5.4.2)

z4=0

(5.4.3)

u = HDX (6>)_1 (v - P) + Kv + Vw

(5.4.4)

Y = P + DX {6)H-X (u-Kv-

(5.4.5)

ZX=Y,

Z2=Y,

Z3=Y,

then, we have Vw)

It is to show that the transformation Z = T(q,v,w) defined by (5.4.3) is a diffeomorphism, and the original system is transformed into the following form 7=7 7 =7

(5.4.6)

Z 3 =v 39 i 4 = — S(q)v dq

The zero dynamics of (5.4.6) is 80

z4=^%)D,-'WZ2=0 dq

(5.4.7)

and is thus stable. ( V

Letr,=

^

" " be the desired output, and a = Zx - Yd be the tracking error \yid)

vector, then

a = Z,-Yd,

& = Z2-Yd,a

= Z,-Yd

(5.4.8)

We consider the SMC design for the following three cases Case 1: SMC with Sliding order (1,1) In this case, we select the switching functions = a + m2& + mxa , and the SMC law as follows •

v = Yd

-m2a-/MjCT-ksign{Z)-SZ,(k,5>0)

(5.4.9)

where E = S. • Case 2: SMC with Sliding order (2,2) In this case, we select the switching function S = & + mxa, and the SMC law as follows

184

Y. M. Hu & W. Huo

v = Yd-mlcr-{i1(a + ml&)-ksign(Z)-8Z,(k,8>0)

(5.4.10)

where Z = S + /xxS. •

Case 3: SMC with Sliding order (3,3) In this case, we select the switching function S = a, and the SMC law as follows (5.4.11)

v = Yd - n2a - HXG - ksign(Z) - SZ, (k,8 > 0 ) where 27 = S + n2S+pxS . All the above SMC Laws satisfy the following reaching law

(5.4.12)

Z = -ksign(Z) - SZ

In the presence of parameter uncertainties, we can also select k and 8 sufficiently large to guarantee the existence of the desired motions. Such parameter estimations are somewhat complex and omitted here. Numerical simulations are respectively performed for the above controllers, with the desired outputYd(t) = (yld,y2d)T =(3shU,-5cosf) r , and the following nominal parameters m = 50kg, R = 0.lm, D = 03m, I0 =0.6kg-m2, L = 2.03m//, ke =0.02, kT =0A9\Nm/A,

J3 = 7l,

h0 =2.03mH,r = 5Alfi

21-error; z2-errof

z1 error

if.

z2 error

i \

"' " desired /

\ \

actual

/ /

v

^ ^_

S

-6 -4

Fig.5.2

Tracking error curves of SMC with sliding order (3,3)

Robust and Adaptive

Control of Nonholonomic

Mechanical Systems ...

angular and linear velocity

actua control u 80

Av1 / \\ •'

/ . /A v / / /

\ ' V

A •' \\ /

\ / V

A2 A

/ A

I-

A / \i /

A,

V,

Fig.5.3

185

\ /

\ / V

A ;A \

60

/ /,

40

-d V

A\

s^s

/"^

A\

.A

A\

/

\\ //

%J\ \

1

\j

\\ 1! \

A'/ \ !

\j

u2

A

20

vA ' A

/

Control curves with sliding order (3,3) dynamical sliding surface and control

dynamical sliding surface and control 70 10 0

\

-

S1 w1

• •^--•^

-10 -20

•

-30

Fig.5.4

Switching functions and the integrator curves

Due to limited space, we only give the simulation results for the sliding order (3,3) SMC, which show that the proposed SMC has an evident effect on the chattering reduction and ideal dynamic response.

5.4.2 Design of adaptive sliding mode controller for the output tracking of nonholonomic mobile robot Consider the two-wheel driven mobile robot in Fig.5.1. The reduced dynamic model is given by (5.2.15). Unlike the kinematics, the dynamics of the mobile robot have some uncertain parameters such as the mass of the mobile robot, the moment of inertia and diameter of the driven wheels. To guarantee the position tracking, a robust controller should be designed to make the practical velocity output perfectly satisfy the required velocity input vr. We now develop a robust adaptive controller to overcome the influences of the above kinds of uncertainties. Assume that the parameters of driving motor are well known precisely and m,I0,R are unknown but with bounded variations, let£=($!,£2,£3)Tbe the

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parameter vector and 0(q, q, v, v) = {0tj )GR2X3

be the linear parameterized matrix

defined, respectively, as follows Rm j3-kT


_^0 ^2=

P-k,

~,Cs=

K-P

(5.4.13)

R

h, <£..= — v , + — v,, 0„= ——v? + 11

2

1

2

1'

12

2 D

2

v,

2 D

2

(5.4.14)

0n=vx+Dv2,02i=vx-Dv2 21

2

i

22

2

2D

2D

then, we can rewrite the left side of (5.2.15) in the form H(q,v,v)v + V(q,v)v + K(q)v =

0(v,v,v)£

(5.4.15)

Let the output be given by (5.2.17) and the desired out be Yd, select the switching function as follows (5.4.16)

S = v + Xv = v —v. w h e r e v = v - v d , vr=vd-X-v v^=-

, and

hcos0

hsind'

-sin#

cos#,

(Yd-ke),(k>0,e

= Y-Yd)

(5.4.17)

is the control to achieve the asymptotic tracking of the given output. Define the Lyapunov function

v = -(sTHTHS+cTr-lc)

(5.4.18)

where r is positive definite, g = g - £ is the estimation error, then from (5.2.15), (5.4.15) and (5.4.16) we obtain V=STHTHS+£Tr-l£ =STHT (HV -

Hvr )+irr~1c

=STHT (u-Vv-Kv-

Hvr)+iTr~lC

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Control of Nonholonomic

Mechanical Systems ...

=STHT (u -Y(v,v,vr ]C)+iTr~1C

187

(5.4.19)

Let the control and parameter adaptive laws be respectively given by u = Y{v,v,vr )C-KDS-

Kssign(HS)

£ = ~rT0THS

(5.4.20)

then V=

(5.4.21)

-S'H'KpS-S'H'KsSigniHS)

Hence, if we select K D and K s properly (e.g., KD= H,KS

= SI(S > 0) , then

from (5.4.21) and the Lasalle's invariance principle we know that S - » 0 and <^—>£"as t—>x>. v=v-v d —»0is then true by (5.4.16). Fig.5.5 gives the control and compensator time curves. Both of them have no serious chattering as in the conventional SMC systems. The desired trajectory is Y

d(0 = (yw,y2d)T

=(3sin?,5-5cosO r

and the parameters are the same as those in last subsection. The gains in (5.4.17) and (5.4.20) are given as follows A: = 2 0 , r = diag[0.l,03,0.2],KD=lOH,

'2

\ = 2-I

V

J2

10

Fig.5.5

15

Velocity tracking error and voltage control input curves

5.4.3 On-line implementations In order to investigate the availability of dynamic SMC approach to the control of mobile robots, a two-wheel driven mobile robot platform is built to perform

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related on-line implementations. Each wheel is driven by a dc motor, and a P-II 233 computer is used as a high-level processor. The related parameters are identified and given in the above two simulation examples. The desired output trajectory is an ellipse Yd=(yld,y2d)T = (asino}lt,bcoso)2t)T. Fig.5.6-Fig.5.9 give the tracking error and control curves of the on-line experiment for the high-order sliding mode controller with the parameters a)x=0.\rad I s, a) 2= 0.2 rad/ s, a = 250 mm, and b = 300 mm. Both of the numerical simulation and on-line implementation results do show that the high-order SMC has much better performance, and can greatly weaken the chattering of the SMC systems. Besides, we also find that, with the same electrical constants, the greater the sliding order is, the better the dynamic response will be.

x-JM 0

10

Fig.5.6

20

30

40

50

60

Fig.5.7

Tracking error ex | CMMnamrl

Tracking error e2

|

1000

ii . L

-1000

-2000

. ^ r -

. W .J 0

10

Fig.5.8

5.5

71/1 A. A

i\i

.„.-.-

,„

T 20

30

40

50

Control to motor 1

0

10

Fig.5.9

20

30

40

SO

Control to motor 2

Conclusions

To overcome the drawbacks of SMC, some researchers have recently concentrated on the development of the so-called high-order SMC approach, which not only

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Control of Nonholonomic

Mechanical Systems ...

189

generalize the conventional SMC theory, but also has considerable application potentiality especially for nonlinear control plants with dynamic actuators and sensors. By combining the high-order SMC technique with other robust algorithms such as backstepping algorithm, it can deal with mismatched uncertainties, alleviate chattering phenomenon and improve the control precision. This chapter has developed the robust and adaptive control strategies for nonholonomic mechanical systems. The canonical form of the high-order SMC systems is developed by using the input/output linearization technique. The design problem is thus decomposed to the design problems of two low dimensional subsystems: the switching functions and their related derivatives determined linear subsystem, and the high-order sliding modes described nonlinear subsystem. Two kinds of high-order SMC laws are also presented to achieve desired high-order sliding mode motion. Numerical simulations and on-line experiments have been performed to show that the proposed approach has considerable potentiality to weaken the chattering phenomenon of SMC systems. It should be noted that, apart from the physical background of high-order SMC applications, the dimensionality of sliding mode dynamics is lower than that of the traditional SMC. This property is particularly useful for the analysis and design of sliding modes. Further work will be concentrated on the robust and adaptive control design in the presence of both mobile robot and driven motor uncertainties. The high-order SMC approach is also applicable for a wide class of uncertain nonlinear control systems. It is also necessary to develop more general design method of switching functions and find proper conditions under which a high-order SMC with proper sliding order can be applied. Acknowledgement The work was supported by the National 863 Plan (#980519), the NSF (#69974015�), the Guangdong NSF (#990258) and the Education Department of Guangdong Province.

References [1] Astolfi, A., Discontinuous control of nonholonomic systems, Systems & Control Letters, vol.27, no.l, pp37-45, 1997. [2] Bartolini, G., Ferrara, A., and Giacomini, L., A robust control design for a class of uncertain nonlinear systems featuring a second-order sliding mode, Int.J.Control, vol.72, no.4, pp321-331, 1999.

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3] Bartolini, G., Ferrara, A., and Usai, E., Chattering avoidance by second-order sliding mode control, IEEE Trans, on Automatic Control, vol.43, no.2, pp241246, 1998. 4] Bloch, A.M., Stabilizability of nonholonomic control systems, Automatica, vol.28, no.2, pp431-435, 1992. 5] Bloch, A.M., and Drakunov, S., Stabilization and tracking in the nonholonomic integrator via sliding modes, Systems & Control Letters, vol.29, no.2, pp91-99, 1996. 6] Canudas de Wit, C , Siciliano, B., and Bastin, G., Theory of Robot Control, London: Springer-verlag, 1998. 7] Chiacchiarini, H.G., Desages, A.C., Romagnoli, J.A., and Palazoglu, A., Variable structure control with a second order sliding condition: application to a steam generator, Automatica, vol.31, no.8, ppl 157-1168, 1995. 8] Colbaugh, R., Barany, E., and Glass, K., Adaptive control of nonholonomic robotic systems, J.Robot. Syst. vol.15, no.7, pp365-393, 1998. 9] D'Andrea-Novel, B., Campion, G., and Bastin, G., Control of nonholonomic wheeled mobile robots by state feedback linearization, Int. J. of Robotics Research, vol.14, pp543-559, 1995. 10] Dong,W.J.,and Huo, W., Adaptive stabilization of uncertain dynamic nonholonomic systems, Int.J.Control, vol.72, no.18, ppl689-1700, 1999. [11] Dong,W.J.,and Huo, W., Time-varying adaptive control of uncertain dynamic nonholonomic systems, Control Theory and Applications (in Chinese), vol.20, pp325-328, 1999. [12] Dong,W.J.,and Huo, W., Time-varying stabilization of uncertain dynamic nonholonomic systems, Acta Automatica Sinica (in Chinese), vol.25, pp402-405, 1999. [13] Dong,W.J., Xu, W.L., and Huo, W., Trajectory tracking control of dynamic nonholonomic systems with unknown dynamics, Int.J.Robust Nonlinear Control, vol.9, pp905-922, 1999. [14] Dong,W.J., Xu, Y.S., and Huo, W., On stabilization of uncertain dynamic nonholonomic systems, Int.J.Control, vol.73, no.4, pp349-359, 2000. [15] Eideeb, Y., and Eimaraghy, W.H., Robust adaptive control of robotic manipulator including motor dynamics, J.Robot. Syst., vol.15, no.ll, pp661-669, 1998. [16] Filippov, A.F., Differential Equations with Discontinuous Right-hand Side, Kluwer, Dordrecht, the Netherlands, 1988. [17] Fliess, M., Levine, J., and Martin, P., Flatness and defect of nonlinear systems: introductory theory and applications, Int. J. of Control, vol.61, ppl327-1361, 1995. [18] Fridman, L., and Levant, A., High-order sliding modes as a natural phenomenon in control theory, in Robust control via variable structure and Lyapunov techniques, Franco, G., and Luigi, G., (eds), London:Springer-verlag, 107-133, 1996.

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[19] Guldner, J., and Utkin, V.I., Stabilization of nonholonomic mobile robots using Lyapunov functions for navigation and sliding mode control, Proc. of the 33rd IEEE Conf on Decision and Control, pp2967-2972, 1994. [20] Hu, Y.M., Lee, C.K., and Xu, J.M., Robust output tracking of nonlinear constrained systems, Control Theory and Applications, vol.13, Suppl.l, pp27-31, 1996. [21] Hu, Y.M., Zhou, Q.J., and Pei., H.L., Theory and applications of nonholonomic control systems, Control Theory and Applications, vol.13, no.l, ppl-10, 1996. [22] Hu, Y.M., and Chao, H.M., High-order sliding mode control of nonlinear control systems with application to mobile robots, Proc. of the VSS'2000, Australia, ppl25-134, Dec.2000. [23] Hu, Z., Research on Robust Control of Nonholonomic Mobile Robots, Ph.D Thesis, South China University of Technology, P.R.China, 2000. [24] Isidori, A., Nonlinear Control Systems, 3rd edition, London:Springer-Verlag, 1995. [25] Jiuig, Y.A., Hesketh, T., and Clements, D.J., High order sliding mode control of uncertain linear systems, Proc. of the 14th IF AC World Congress, Beijing, PRC, vol.G, pp437-442, 1999. [26] Jiang, Z., and Nijmeijer, H., Tracking control of mobile robots: a case study in backstepping, Automatica, vol.33, ppl393-1399, 1997. [27] Kolmanovsky, I., and McClamroch, N.H., Developments in nonholonomic control problems, IEEE Control Systems, vol.15, pp20-36, 1995. [28] Krishnan, H., and McClamroch, N.H., Tracking in nonlinear differentialalgebraic control systems with applications to constrained robot systems, Automatica, vol.30, no.12, ppl885-1897, 1994. [29] Laumond, J.P., (Ed.), Robot Motion Planning and Control, London: Springerverlag, 1998. [30] Levant, A., Sliding order and sliding accuracy in sliding mode control, Int. J. Control, vol.58, no.6, ppl247-1263, 1993. [31] Li, Q.X., Hu, Y.M., Pei, H.L., and Zhou, Q.J., Robust output tracking of mobile robots, Control Theory and Applications, vol.15, no.4, pp515-524,1998. [32] Li, Z., and Canny, J.F., (Ed.), Nonholonomic Motion Planning, Nowell, MA:Kluwer, 1993. [33] Pomet, J.B., Explicit design of time-varying stabilizing control laws for a class of controllable systems without drift, Systems & Control Letters, vol.18, ppl47158, 1992. [34] Retchkiman, Z., and Khorrami, F., The time optimal control problem for a class of nonholonomic systems: a case study, Preprints of 13th Triennial World Congress, San Francisco, USA, pp449-453, 1996. [35] R'os-Bolivar, M., Zinober, A.S.I., and Sira-ramirez, H., Dynamical sliding mode control via adaptive input-output linearization: a backstepping approach, in Robust control via variable structure and Lyapunov techniques, Franco, G., and Luigi, G., (eds), London:Springer-verlag, ppl5-35, 1996.

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[37] Sastry, S., Nonlinear Systems Analysis, Stability and Control. New York: Springer-verlag, 1999. [38] Sira-Ramirez, H., On the dynamical sliding mode control of nonlinear systems. Int. J. of Control, vol.57, no.5, ppl039-1061, 1993. [39] Steven, C.C., and Lin, C.L., A general class of sliding surface for sliding mode control, IEEE Trans, on Automatic Control, vol. 43, no.l, ppl 15-119, 1998. [40] Su, C.Y., and Stepanenko, Y., Robust motion/force control of mechanical systems with classical nonholonomic constraints, IEEE Trans, on Automatic Control, vol.39, pp609-614, 1994. [41] Su, C.Y., and Stepanenko, Y., On the robust control of robot manipulators including actuator dynamics, J. Robot. Syst., vol.13, no.l, ppl-10, 1996. [42] Tarn, T.J., Bejczy, A.K., Yun, X.P., and Li, Z.F., Effect of motor dynamics on nonlinear feedback robot arm control, IEEE Trans, on Robotics and Automation, vol.7, no.l, ppl 14-121, 1991. [43] Utkin, V.I., Sliding Modes in Control and Optimizations, New York:Springerveilag, 1992. [44] Wang, C.L., The Robust Stabilization of Nonholonomic Dynamical Systems, Ph.D Thesis, Beijing University of Aeronautics and Astronautics, P.R.China, 1999. [45] Wang, W.H, Soh, C.B, and Chai, T.Y., Robust stabilization of MIMO nonlinear time-varying mismatched uncertain systems, Automatica, Vol.33, nol2, pp2197-2202, 1997, [46] Xu, S.J, Rachid, A., Sliding mode controller design for linear systems with mismatched uncertainty, Proc.ofthe 14lh IFAC World Congress, Beijing, PRC, Vol.G,pp431-436, 1999. [47] Yang, J.M., and Kim, J.H., Sliding mode motion control of nonholonomic mobile robots, IEEE Control Systems Magazine, vol.19, ppl5-23, 1999. [48] Yang, J.M., and Kim, J.K., Sliding mode control for trajectory tracking of nonholonomic wheeled mobile robots, IEEE Trans, on Robotics and Automation, vol.15, no.3, pp578-587, 1999. [49] Young, K.D., Utkin, V.I., and Ozguner, U., A control engineer's guide to sliding mode, IEEE Trans, on Control Systems Technology, vol.7, no.3, pp328342, 1999. [50] Young, K.D., and Ozguner, U., Sliding mode: control engineering in practice, Proc.ofthe 1999 ACC, San Diego, CA, USA, ppl50-162, 1999.

Chapter 6

Introduction to Chaos Control and Anti-Control Guanrong Chen'

Jin-Qing Fang®,

Yiguang Hong®

Hua-Shu Qin®

^Department of Electrical amd computer

Engineering

University of Houston, Houston, TX 77204, USA ^Department of Electrical

Engineering

City University of Hong Kong, Hong Kong SAR, P.R. China ®China Institute Of Atomic P.O.BOX275-27,

Beijing 102413,

Energy P.R.China

^Institute of Systems Science, Academia Sinica, Beijing 100080,

P.R.China

6.1 Overview of chaos, chaos control and anticontrol Chaos control and anticontrol technologies promise to have a major impact on many novel, time- and energy-critical applications, such as high-performance circuits and devices (e.g., delta-sigma modulators and power converters), liquid mixing, chemical reactions, biological systems (e.g., in the human brain, heart, and perceptual processes), crisis management (e.g., in power electronics), secure information processing, and critical decision-making in political, economical, as well as military events. This new and challenging research and development area has become a scientific interdiscipline, involving systems and control engineers, theoretical and experimental physicists, applied mathematicians, physiologists, and above all, circuits and devices specialists. Both control and anticontrol of chaos can be analyzed using chaos and bifurcation theories, and can be implemented by suitable design of controllers, device, and circuitry.

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6.1.1 A brief description of chaos, chaos control and anticontrol Chaos refers to one type of complex dynamical behaviors that possess some very special features such as being extremely sensitive to tiny variations of initial conditions, having bounded trajectories in the phase space but with a positive leading Lyapunov exponent, possessing a finite Kolmogorov-Sinai entropy, a continuous power spectrum, and/or a fractional topological dimension, etc. Very often, chaos coexists with some other complex dynamical phenomena like bifurcations, fractals, and strange attractors. A typical chaotic attractor generated by a power system model is shown in Figure 6.1 Due to its intrinsic dynamical complexity, chaos was once believed to be neither controllable nor predictable, therefore useless. However, recent research advances have demonstrated that chaos not only is (long-term) controllable and (short-term) predictable, but also can be beneficial to many real-world applications. In fact, control and anticontrol of chaos have become a rallying point for an important segment overlapping engineering, physics, mathematics, and biomedical science. Chaos control refers to the situation where chaotic dynamics is weakened or eliminated by appropriate controls; while anticontrol of chaos means that chaos is created, maintained, or enhanced when it is healthy and useful. Anticontrol of chaos is also called chaotification, meaning to chaotify an originally non-chaotic system (i.e., to make it chaotic). Both control and anticontrol of chaos can be accomplished via some conventional and nonconventional methods such as microscopic parameter perturbation, bifurcation monitoring, .entropy reduction, state pinning, phase delay, and various feedback and adaptive controls [1].

6.1.2 Some basic rationale for utilizing chaos and complex nonlinear dynamics There are many practical reasons for controlling or ordering chaos. First of all, chaotic (messy, irregular, or disordered) system response with little meaningful information content is unlikely to be useful as chaos can lead systems to harmful or even catastrophic situations. In these troublesome cases, chaos should be reduced as much as possible, or totally suppressed. For instance, stabilizing chaos can avoid fatal voltage collapse in power networks and deadly heart arrhythmias, can guide disordered circuit arrays (e.g., multi-coupled oscillators and cellular neural networks) to reach a certain level of desirable pattern formation, can regulate dynamical responses of mechanical and electronic devices (e.g., diodes, laser machines, and machine tools), can help well-organize an otherwise mismanaged multi-agency corporation to reach a stable equilibrium state whereby achieving optimal agent performance, etc. Ironically, recent research has shown that chaos can actually be useful under certain circumstances, and there is growing interest in utilizing the very nature of

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chaos, particularly in some novel time- and/or energy-critical applications. The most motivative reason is the observation that chaos permits a system to explore its every dynamical possibility: when chaos is under control, it provides the designer with an exciting variety of properties, richness of flexibility, and a cornucopia of opportunities. Figure 6.2 visualizes how by varying a constant feedback control gain within a simple quadratic map, period-doubling bifurcation and chaos can be created and then be stabilized to a variety of equilibria of different periods. Traditional engineering design always tries to reduce irregular dynamical behaviors of a system and, therefore, completely eliminates chaos. However, such overdesign is usually accomplished at the price of losing great flexibilities in achieving high performance near the stability boundaries, or at the expense of radically modifying the original system dynamics. On many occasions, this proves to be unnecessary. It has been shown that the sensitivity of chaotic systems to small perturbations can be used to direct system trajectories to a desired target quickly with very low and ideally minimum control energy. This can be crucial for navigation in the multiplanetary space system. A suitable modification of chaotic dynamics such as stability conversion or bifurcation delay not only can significantly extend the operational range of machine tools and jet engines, but also may enhance the artificial intelligence of neural networks, as well as increase coding/decoding efficiency in signal and image communications. Other application examples of chaos control and anticontrol technologies include designing high-performance circuits and devices (e.g., delta-sigma modulators, automatic gain control loops, and power converters), achieving chaos synchronization for information processing, pattern recognition, and secure communications, forming various wave patterns and self-organized behaviors in oscillator arrays and neural networks, delaying bifurcations in electric power systems and energy convection loops, and performing crisis management and critical decision-making in political, economical, as well as military events. Fluid mixing is another good example in which chaos is not only useful but actually very desirable, where two fluids are to be thoroughly mixed while the required energy is minimized. For this purpose, it turns out to be much easier if the dynamics of the particle motion of the two fluids are strongly chaotic, since it is otherwise difficult to obtain rigorous mixing properties due to the possibility of invariant two-tori in the flow. This has been one of the main subjects in fluid mixing, known as chaotic advection. Chaotic mixing is also momentous in applications involving heating, such as in plasma heating for a nuclear fusion reactor. In this process, heat waves are injected into the reactor, for which the best result is obtained when the heat convection inside the reactor is chaotic. Within the context of biological systems, the controlled biological chaos appears to be important with the way a human brain executes its tasks. There has been some suggestions that human brain can process massive information instantly, in which the ability of human beings in controlling the brain chaos could be a fundamental reason. The idea of anticontrol of chaos has been proposed for solving the problem of driving responses of a human brain model away

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from the saddle-type of equilibrium, so that undesirable periodic behaviors of neuronal population bursting can be prevented. Also, some recent laboratory studies reveal that the complex variability of healthy dynamics in a variety of physiologic systems has features reminiscent of chaos. For example, in the human heart, the amount of intracellular Ca2+ is closely regulated by a coupled process in a way similar to a system of coupled oscillators. Medical evidence reveals that controlling the chaotic arrhythmia in an appropriate way can be a new, safe, and auspicious approach to the design of a smart pacemaker for regulating heartbeats. Motivated by many of such potential real-world applications, current research on control and anticontrol of chaos has become intensive. In the theoretical aspect, chaos control and anticontrol are posing a new challenge to both system analysts and control engineers. This is due to the extreme complexity and sensitivity of chaotic dynamics, which can cause many unusual difficulties in long-term predictability and short-term controllability of chaos. A controlled chaotic system is inherently nonautonomous, which cannot be converted to an autonomous system in most cases since the controller as a time function is yet to be designed. Possible time-delay, noise, and coupling effects often make a controlled chaotic system Lyapunovirregular and topologically extremely complex. As a result, many existing theories and methodologies for autonomous systems are no longer applicable. On the other hand, at me technical level, chaos control and anticontrol have also posed new challenges "o circuit designers and instrument specialists. A successful circuit implementation in a chaotic environment is generally difficult, due to the extreme sensitivity of chaos to parameter variations and noise perturbations, and the nonrobustness of chaos to the structural stability, within the physical devices. Notwithstanding many technical obstacles, both theoretical and technical developments in this area have gained remarkable progress in the last few years. For instance, some unified control methods have been developed under the nonautonomous Lyapunov stabilization theory; some rigorous anticontrol techniques, even for spatiotemporal systems, have been initiated; some novel chaos based encryption approaches have been advanced; and some chips of chaotic circuits have been made toward commercialization [1,2].

6.1.3 A brief summary In conclusion, the emerging field of chaos control and anticontrol is very stimulating and full of promise; it is expected to have far-reaching impacts with enormous opportunities in industrial and commercial applications. New theories for dynamics analysis, new methodologies for control, and new circuitry design for implementation altogether are calling for new efforts and endeavors from the communities of nonlinear dynamics, controls, and circuits and systems [2].

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6.2 Challenges in chaos control

6.2.1 Chaos control: what does it mean? What does "chaos control" mean? Similar to conventional systems control, the concept has first come to mean stabilizing unstable periodic orbits (including unstable equilibria) of a dynamical system. However, it is now commonly agreed that it can also be understood as a process or mechanism that enhances existing chaos or creates new chaos in a dynamical system when it is beneficial, and suppresses it when it is harmful. Engineering and scientific methods are used to manipulate the transition between chaos and order and, sometimes, between chaos and chaos. When the control of chaos involves eliminating or weakening chaos, it is usually a process of stabilizing unstable periodic orbits or reducing the leading positive Lyapunov exponent of the dynamics of the chaotic system. However, the scope of chaos control also encompasses such control tasks as self-similarity, pattern and symmetry forming or breaking, birth or delay of bifurcations, birth or shapes of limit cycles, etc. Moreover, it includes the notion of generating chaos, in a sense creating instability into a perhaps originally stable system, depending on the need of the application at hand. Apparently, the notion of chaos control is neither exclusive of, nor conflictive with the conventional control systems theory and practice. Instead, it aims at better managing the dynamics of a nonlinear system on a wider scale, with the hope that more benefits may thus be derived. In addition, chaos control seeks to develop new theories and methods that are particularly useful for chaotic systems in various disciplines.

6.2.2 Challenges in chaos control To appreciate the challenge of chaos control, consider two potentially chaotic systems—the logistic map and a model of an electric power system. The Logistic Map A simple, yet typical, example of discrete-time chaotic systems is the onedimensional logistic map x

k+l=f(xk>Py-=Pxk(l-x0

(6-2-1)

where p>0 is a real variable parameter. Since each state of the system is a function of/?, i.e., xfc =x]i(p), a change ofp will significantly alter the dynamical behavior of the system. By solving the algebraic equation x = f(x,p), one can find the two equilibria of the system, 0 and x* ={p-\)lp. Further examination of the system

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Jacobian, J = dfldx = p-2px, reveals that the stabilities of these equilibria indeed depend on p. It is well known that the logistic map has period-doubling bifurcation leading to chaos [16]. At this point, it is illuminating to pose the following questions: Is it possible (and, if so, how) to find a simple (say, linear) control sequence, {u^} to be added to the logistic system: xk+i =f(xk,p)

= pxk(l-xk)

+ uk

(6.2.2)

such that, to mention just a few, (i) the limiting chaotic behavior of the period-doubling bifurcation process is suppressed? (ii) the first bifurcation is delayed to take place, or some bifurcations are changed either in form or in stability? (iii) the asymptotic behavior of the system becomes chaotic (the anticontrol problem), when the parameter/? is currently not in the bifurcating range? An Electric Power Model As a real-world engineering example, consider the simple power system shown in Figure 6.3. This model is described by the continuous-time dynamical model 9 = a> d) = \6.6661sin{6L -<9 + 0.0873)F£ -0.1667^ + 1.8807 0 = 496.8718F^ -166.667cos(0 L - 0 - 0 . 0 8 7 3 ) ^ -666.6667cos(0 L - 0 . 2 0 9 4 ) ^ -93.3333F^ + 33.3333/7 + 43.333 VL =-78.7638F^ + 26.2172cos(<9/,

-9-0M24)VL

+ 104.8689cos(6>/, -0.1346)F£ + 14.5229F/, -5.2288/J-7.0327 where 9 is the rotational angle of the power generator, with angular velocity a . In this power system, the load is represented by an induction motor, M, in parallel with a constant PQ (active-reactive) load. The variable reactive power demand, p, at the load bus is used as the primary system parameter. Also in the power system, the load voltage is VLZ.6L with magnitude ^L and angle ZG^, the slack bus has terminal voltage EZQ (a phasor), and the generator has terminal voltage denotedE m Z9. Typical chaotic behavior of this model is shown in Figure 6.1 Similar to the logistic map discussed above, a few interesting control problems are: (i) can the limiting chaotic behavior of the period-doubling bifurcation process

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be suppressed? (ii) can the first bifurcation be delayed in occurrence, or some bifurcations be changed either in form or in stability? (iii) can the voltage collapse be avoided or delayed through bifurcation or chaos control? Many nonconventional control problems such as those prompted above have posed a real challenge to both nonlinear dynamics analysts and control engineers.

6.2.3 Some distinct features of chaos control To further understand the very nature of chaos control and anticontrol, it should be helpful to highlight some distinctive features of chaos control theory and methodology, in contrast to other conventional approaches with regard to such issues as objectives, perspectives, problem formulations, and performance measures. 1. The target trajectories in chaos control can be unstable equilibria or unstable periodic orbits (limit cycles), perhaps of high periods. The controller is designed to stabilize these unstable trajectories or to drive them to switch from one orbit to another. This inter-orbit switching can be either chaotic -» regular, chaotic -> chaotic, regular —> chaotic, or regular —> regular, depending on the application of interest. Conventional control, on the other hand, does not normally investigate such interorbit switching problems of a dynamical system, especially, not those problems that involve guiding the system trajectory to an unstable or chaotic state by any means. 2. A chaotic system typically has a dense set of unstable orbits embedded within it, and is extremely sensitive to tiny perturbations in its initial conditions. Such a special feature, useful for chaos control, is not available in linear or non-chaotic nonlinear systems. 3. In most conventional control schemes, one usually works within a state space framework. In chaos control, one deals with parameter space and phase space as well, where Poincare maps, delay-coordinates embedding, parametric variation, entropy reduction, bifurcation monitoring, etc., are some typical, but nonconventional tools for study. 4. In classical control, a target for tracking is usually a constant vector in the state space, which is generally not a state of the given system, and the terminal time for the control is usually finite (e.g., the elementary concept of "controllability" is typically defined using a finite and fixed terminal time, at least for linear systems and affine-nonlinear systems). However, in chaos control, a target for tracking is not limited to constant vectors in the state space: it often is an unstable periodic orbit of the given system. This generally requires only tiny control to achieve, but technically can be quite difficult due to the instability of the target. Moreover, in chaos control the terminal time is infinite to be meaningful and practical, because most nonlinear dynamical behaviors such as equilibrium states, limit cycles, attractors, and chaos are asymptotic properties.

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5. Depending on different situations or purposes, performance measures in chaos control tasks can be different from those of the conventional control. Chaos control uses criteria such as Lyapunov exponents, Kolmogorov-Sinai entropy, power spectra, ergodicity, and bifurcation changes, in addition to what the conventional control usually emphasizes: robustness of the system stability or control performance, optimality of control energy or time, ability in disturbance rejection, etc. 6. Chaos control includes a unique task - anticontrol, required in some unusual applications such as those in fluid mixing, secure communication, and biomedical engineering activities described in Section 6.1. Anticontrol tries to retain or even enhance chaos to improve performance. Bifurcation control is another example of chaos control, where a bifurcation point is expected to be delayed in case it cannot be avoided or stabilized. This delay may significantly extend the operating time or system parameter range for a time-critical control process such as chemical reaction, voltage collapse of electric power systems, and compressors stall of gas turbine jet engines, etc. These examples are in direct contrast to traditional control tasks such as the textbook problem of stabilizing an equilibrium position of a nonlinear system. Due to the inherent association of nonlinearity and chaos to various related issues, the scope of chaos control and the variety of problems that chaos control deals with is much more diverse than usually imagined. These include creation and management of self-similarity, symmetry, pattern formation, sizes of attractor basins, and birth, amplitudes, and change of bifurcations and limit cycles, etc., in addition to the classical target tracking and system regulation. It is also important to point out an additional distinctive characteristic of a controlled chaotic system that differs from an uncontrolled chaotic system. This characteristic emerges from the situation where a control input is involved in a dynamical system. A controlled system is usually nonautonomous and cannot be reformulated as an autonomous system by defining the control input as a new state variable, since the control is physically not a system state variable and, moreover, it has to be determined via design for the control purpose. Therefore, a controlled chaotic system is intrinsically much more difficult to design than it appears. This observation raises the question of extending many existing theories and techniques from autonomous system dynamics to general controlled dynamical systems, including such complex phenomena as degenerate bifurcations and hyperchaos in the system dynamics when a control is involved. Unless suppressing complex dynamics in a process is the only purpose for an intended control problem, understanding and utilizing the rich dynamics such as bifurcations and chaos in nonlinear control systems are very stimulating in both theory and applications. Chaos control is a new and challenging subject for study for both nonlinear dynamics analysts and systems control engineers.

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6.3 Representative chaos control methods While talking about chaos control, it is worth mentioning at the beginning of introducing some typical methodologies that, strictly speaking, controlling chaos is not any harder than controlling a general nonlinear system—if chaos suppression is the only concern and if a "brute force" strategy is allowed. Many effective control techniques are indeed available today [1]. Several not-so-typical methods from the conventional control point of view are briefly introduced here, showing some special flavor of chaos in control. Recall that the dynamical behaviors of a nonlinear system, particularly bifurcations and chaos, depend on the values of the system parameter(s). The route to chaos via period-doubling bifurcations for the logistic map discussed in Section 6.2 is a typical case where, as the system parameter continuously varies, the dynamical behavior of the system trajectory shows dramatic changes. Many examples of such parameter-dependent dynamical phenomena can be found from the vast literature. Hence, it is intuitively clear that the system dynamical behavior can be altered by changing some of these parameter values, provided that the parameters are accessible for adjustment. Th 2 idea of varying certain system parameter(s) to control chaos may be traced back to John von Neumann, who pointed out in the early 1950's that small, carefully chosen, preplanned atmospheric disturbances could lead to desired, large-scale changes in the weather after some time. This suggestion was quite reasonable, since the weather dynamics is seemingly chaotic and chaos is mainly characterized by extreme sensitivity to small perturbations. Ever since then, this characteristic of chaos, known as the "butterfly effect," has been viewed or used as a possible means for changing some basic properties of chaos. In recent years, various types of parameter-dependent control methods have been applied to chaotic systems. In the effort of controlling chaos by varying some accessible system parameters), one usually looks for a strategy that requires a small control effort to achieve great effect on the ultimate behavior of the controlled chaos. The objective of a parameter-dependent control can be to suppress chaos completely or to utilize the sensitivity of a chaotic system to rapidly direct its trajectory to a desired state. This target state is typically an unusual periodic state such as an unstable equilibrium or limit cycle. In other cases, parameter-dependent controls are used to switch the trajectory between different chaotic behaviors, as circumstances change during the control process, to achieve some specified performance for the system at different times.

6.3.1 The OGYparametric-variation control method This section further develops the parameter-dependent approach in a unified fashion under a somewhat general framework. The approach taken for controlling a nonlinear dyr amical system is to stabilize one of its unstable periodic orbits embedded in

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an existing chaotic attractor via small time-dependent variations of a variable system parameter [16]. The idea comes from the observation that a chaotic attractor typically has embedded within it a dense set of unstable periodic orbits. Heuristic reasoning easily leads to the suggestion of controlling any of these orbits to a (usually unstable) periodic one, by continuously applying small forces that are designed for the purpose. 6.3.1.1 Using special features of chaos for control Before giving a formal presentation, a simple description of the basic idea and outline of the control procedure using a graphical language is desirable. In this approach, while trying to bring the system orbit to a target, one first locally linearizes the system and then uses the controller to slightly push the system orbit toward the target. At this step, the control method is classical (which may be interpreted as a pole placement approach), but the pushing has to be so small that it does not destroy the structural stability of chaos thereby preserving the chaotic property of the system (pole placement generally is a "brute force" approach which would be too strong for this purpose, and so would consume too much a control energy). Then, the controller stops doing anything but wait, letting the orbit free to travel until it moves back to nearby the target (by the ergodicity of chaos, it will). This step is not typical in classical control, and cannot be applied to non-chaotic systems. Note that the system orbit will move closer to the target using the system energy but not the control energy in this phase of the process. Also at this step, the system chaotic parameters have been changed. To this end, another local linearization of the system is carried out but at the new parameter values, and then a new and small control push applies, so the next cycle of the process begins. Each time, the system orbit moves closer to the target using part of the classical feedback control energy and part of the given system energy, along with the ergodicity of chaos, while preserving the chaotic dynamics until the orbit eventually arrives at the target (and chaos is thus eventually suppressed). This control method is based on a classical approach, but it differs from the classical approach in that it utilizes some very nature of chaos (such as the ergodicity and the structural stability of chaos). 6.3.1.2 The OGY control method For explanation, consider a general continuous-time autonomous system, x = f(x(t),p)

(6.3.1)

where p is a scalar system parameter accessible for external adjustment. Suppose that when p=p* the system is chaotic. Suppose that it is desired to control the system trajectory, x(t), to reach an unstable saddle type of equilibrium [8,11,18,20]. Assume that within a small neighborhood of p , say

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203

(6.3.2)

in which ^pmax > 0 is the maximum allowable perturbation such that as long as p is maintained within this range, neither the chaotic attractor nor the target orbit disappears. In other words, it is assumed that within this small neighborhood ofp , there is no bifurcation point of the target periodic orbit. Let r be a periodic orbit of system 6.3.1 and 2" be a surface of cross section of the underlying Poincare map denoted by P [8,18]. For simplicity, assume that the continuous-time system (6.3.1) is three-dimensional, with coordinates (x,y,z), so that the corresponding surface of cross section is two-dimensional and is orthogonal to the third axis, that is, Z = {\cc p yY eR^ : Y = z0 (a constant)}. Now, let £ be the coordinates of the surface of cross section, which is a 3 x 1 vector satisfying

Zk+\=ptfk>Pk)

(6-3-3)

where Pk=P

+Apk

\ApA<Apmax

Since/? is adjustable at every iteration of the map, p = p ^is chosen to be a constant at each iteration. Then it is possible to determine, from the map P, many distinct unstable periodic orbits within the chaotic attractor, and select the target periodic orbit that maximizes certain desired system performance with respect to the dynamical behavior of the system. For example, sometimes it makes sense to pick a higherorder orbit as the desired one since it visits more regions of the attractor. This can be advantageous because different regions correspond to different physical states of the system. At some other times, one may want to pick as many of such regions as possible. For illustration, suppose that ?f=P{?f,p*)

(6.3.4)

has been selected as the desired, unstable, period-one saddle type of equilibrium of the map P, corresponding to the desired target trajectory of system (6.3.1). To understand the local properties of this chosen equilibrium (a degenerate periodic orbit), one needs to fit the iterations of the map to a local linear approximation of the map about the desired orbit. A linear approximation o f f near %*r and p* is given by

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$k+\*sy+Jk^k-?f)+wkiPk-p*)

(6.3.5)

or (6.3.6)

A

%k+i*JkAh+wkAPk

where

A£ = 4k-fy jk=dP(?f,P*)/dtk

>

A

Pk=Pk-P*

, wk=dP(4*f,p*)/dPk

.

The stable and unstable eigenvalues, As £ a n d / ^ ^ satisfying U s J <1
gikeu,k=ikes,k=° then the Jacobian J^ can be expressed as J

k -\keu,kSu,k+As,kes,kslk

(6-3.7)

The dominant theme of the OGY method is first to monitor the system until its trajectory comes near the desired orbit, i.e, until ^ falls close enough to £*f • Then, one changes the nominal value of the parameter p by a small amount, Ap . This will change the location of the orbit and its stable manifold, such that the next iteration (represented by ££+1 in the surface of cross section) is forced toward the local, stable manifold of the target 4*f • Since the nonlinearity is not contained in the linearized system, the control so designed usually is not able to bring the moving orbit

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to the target. Thus, the controlled orbit will leave the small neighborhood of the target again, and continue to travel chaotically as if there was no control effect on it at all. Nevertheless, due to the ergodicity of chaos, or, because the chaotic attractor has a dense set of unstable periodic orbits embedded within it, sooner or later the orbit ^ will return to be close to £/• At this time, the next cycle of parametric variation and iteration can be applied. This procedure is visualized by Figure 6.4 Now, suppose that ^ has moved to being sufficiently close to £,*s , so that (6.3.6) holds. For the next iterate, < ^ + ] , to fall onto the local stable manifold of £,** , the parameter p

=p*+Ap

has to be chosen such that the next iteration is per-

pendicular to the direction of the current local unstable manifold:

To achieve this, taking the inner product of equation (6.3.6) with gu ^ and applying equation (6.3.7), one has g

ukA£k

Ap

(6 3 8)

r-^Tir

--

<,kwk

where it is assumed that g LW^ * 0 . This is the control formula for determining the variation of the adjustable system parameter/? at each step k = l,2,---. Using the control formula (6.3.8) to consequently nudge the parameter p, one can expect that the target E,\ will eventually be reached. When this happens, this unstable target orbit is said to be stabilized [10,14,17]. Note that this calculated Ap is used to perturb the parameter p only if Ap <Apmax . When the calculated Ap

is greater than Ap a s on a

should be set to zero. Also, when 4k+\ ^ "

'

oca

'

starj

, the variation

te manifold of E,*r , the

variation Ap is set to zero since the stable manifold might lead the orbit directly to the target by its stable nature. The orbit for the subsequent time instants (/.e.,^+2,^+3,-")is

tnen

supposed to approach £,*r at a geometrical rate. How-

ever, due to the errors or inaccuracies incurred in the sequence of calculations and linearizations, the subsequent iterations may tend to fall off the local stable mani-

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folds. In this case, to make sure that the subsequent { ^ } is approaching S,*r, a new4f, has to be calculated following each iteration. It should be emphasized that this control method is based on a classical approach, but it differs from the classical approach in that it utilizes some very nature of chaos (such as the ergodicity and the structural stability of chaos), as discussed at the beginning of this section. 6.3.1.3 Some remarks on the OGY control method It should also be noted that the above derivation is based on the assumption that the Poincare map, P always possesses a stable and an unstable direction (saddle-type of equilibria). This may not be true in general, particularly for higher-periodic orbits. Also, it is necessary that the number of accessible parameters for control be at least equal to the number of unstable eigenvalues of the periodic orbit to be stabilized. In addition, modified or alternative techniques become necessary when some key system parameters are not accessible. Moreover, the technique is successful only if the control is applied after the system trajectory moves into a small neighborhood of the target over which the control formula remains valid. In this case, one may have to wait for a long time until the trajectory returns to this region. In the case of multiple attractors within the phase space, the orbit may be attracted to another region so may never return. Nevertheless, some methods such as the so-called targeting technique [1] can often persuade the system dynamics to quickly approach the region of control. Finally, this parametric variation control method is derived based on the assumption of absence of noise. However, even small noise may lead to some rare chaotic bursts where the orbit wanders far from the stabilized orbit. In a noisy environment, the amplitude of the parameter variation must exceed a noisedependent threshold in order to obtain effective control. This issue is especially important in physical implementation of the control algorithm.

6.3.2 Feedback control of nonlinear systems Feedback control is classic in engineering applications, which has many advantages such as guaranteed and robust stability, precise target tracking, strong ability of noise rejection, systematic design, and, most important of all, no requirement for assessing oftentimes unassessable system parameters. A general approach to controlling a nonlinear dynamical system via feedback can be formulated as follows:

\m=xx,u,t), [y(t) = h(x,u,t),

(639)

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where x(t) is the system state vector, y(t) the output vector, and u(t) the control input vector. Here, in a general discussion, once again it is assumed that all the neccesary conditions on the vector-valued functions / and h are satisfied such that the system has a unique solution in a certain bounded region of the state space for each given initial value XQ =XQQ), where the initial time tQ >0 . Given a reference signal, r(t), which can be either a constant (set-point) or a function (target trajectory), the problem is to design a controller in the statefeedback form u(t) = g(x,t)

(6.3.10)

or, sometimes, in the output-feedback form u(t) = g(y,t) where g is a nonlinear (including linear) vector-valued function, such that the controlled system \x(t) =

f(x,g(x,t),tl

y(t) = h(x,g(x,t)\ can be driven by the feedback control g(x,t) to achive the goal of traget tracking:

!™f I*0-K0|| = 0

(6.3.12)

where the terminal time, / ^ < oo, is predesired according to the application, and ||| is the Euclidean norm of a vector. Since the second equation in system 6.3.9 is merely a mapping, which can be easily handled in general, it is ignored in this discussion by simply letting h{-) = I, the identity mapping, so that;y=x. 6.3.2.1 Engineering perspective about controller design It is very important to point out that in a feedback controller's design, particularly in finding a nonlinear controller for a given nonlinear system, one must bear in mind that the controller should be (much) simpler than the given system [1]. For instance, if one would like to determine a nonlinear controller, say u^ in the discrete-time setting, for guiding the state vector x^ of a given nonlinear control system of the form

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to a target trajectory satisfying a predesired constraint matically it is very easy to use u =0

k k(xk)-fk(xk)

JC^ + J

= &]i(xjl), then mathe-

>

which will bring the original system state x^ to the target trajectory in just one step! As another example, to design a nonlinear controller u(t) in the continuous-time setting 10 guide the state vector x(t) of the nonlinear system x = f(x(t)) + u(t) to a target trajectory x*(t), it is mathematically correct to use u(t) = x* (t) - f(x(t),t) + K(x(t) - x* (0) where K has all its eigenvalues with negative real parts. This controller leads to e(t) = Ke(t), with e(t) = x(t) - x* (?) yielding e(t)—>0 or x(t)—>x*(t), as ?-»oo . One more example is the following: for a given nonlinear controlled system in the canonical form x\(t) = x2(t) x2(t) = x3(t) Xn{t) = f{xX(t\--;Xn(t))

+ u{t)

Suppose that one wants to find a nonlinear controller u(t) to guide the state vector x(t) = (xi(t)---xn(t)y

to a target state, x , i.e., x(t)->x

as ?-><»

It is mathematically straightforward to use the controller u(t) = - / ( * ! (0, -,x„ (0) + kc 0 ( 0 - x) with an appropriate constants row vector. Note that the resulting ^-dimensional controlled system is a completely controllable linear system, x = Ax + bu , with 0 1 0---0 0 0 1-0 A = ; : i'-.i

o o o-o

0

and 6 =

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Therefore, a suitable constant control gain exists for the state-feedback controller, such that x(t) - > J a s t -» oo. All such examples seem to reveal a "universal controller design methodology" that works for any given system. However, a fatal problem with such "design" is that the controller is even more complicated than the given system, and hence has no practical value: it virtually remove the given system and then replace it by a desired one. It is hardly imagine that one can accept a controller for a machine described by /(such as a car) that is bigger than the machine itself (e.g., an airplane), described by something more complicated than the same J). Moreover, if this / is a simplified mathematical model for the real machine, and the controller uses the same mathematical model, then the design merely shows that the controller works for the mathematical model but there is no reason to believe that it works for the real machine a^ well. Therefore, in a feedback controller's design, it is expected to come out with a simplest possible and satisfactory controller: if a linear controller can be found to do the job, use a linear controller; otherwise, try a simple nonlinear controller with such as a piecewise linear or a quadartic nonlinearity and so on. Also, oftentimes, full state feedback information is not available in practice, so one should try to design a controller using only output feedback (i.e, partial state feedback), which means the second equation of (6.3.9) is essential. Whether or not can one find a simple, easily implementable, low-cost, and effective controller for a given nonlinear system for tracking control requires both theoretical background and design experience. 6.3.2.2 A general approach to controller design Return to the central theme of feedback control for a general nonlinear dynamical system (6.3.9)-(6.3.12). A basic idea is first outlined for a tracking control task. Let the target trajectory (or set-point) be x(t), and assume that it is differentiate, so that by denoting x=z(t)

(6.3.13)

one can subtract this equation (6.3.13) from the first equation of (6.3.11), so as to obtain e = F(e,t) where e = x-x

(6.3.14)

and F(e,t):=f(x,g(x,t),t)-z(t)

If the target trajectory 3c is a periodic orbit of the given system (6.3.9), that is, if it satisfies

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x=f(x,0,t)

(6.3.15)

then similarly a subtraction of (6.3.15) from the first equation of (6.3.9) gives e = F(e,t) where e = x-x

(6.3.16)

and F(e,t) := f{x, g(x,t),t) - /(5c ,0,0

In either case, e.g., in the second case which is more difficult in general, the goal of design is to determine the controller u(t) = g(x,t) such that Urn ||e(0| = 0 ?-»oo

(6.3.17)

which implies that the goal of tracking control is achieved: Urn \x(f)-x(i)\

=G

(6.3.18)

It is now clear from Eqs. (6.3.17) and (6.3.18) that if zero is a fixed point of the nonlinear system (6.3.16), then the original controllability problem has been converted to an asymptotic stability problem of this fixed point. Thus, the Lyapunov first and second methods [7,10,12,14,17] may be applied or modified to obtain rigorous mathematical techniques for the controller's design. This is further discussed in the following. 6.3.2.3 Feedback controllers design methods This subsection discusses how a linear or nonlinear controller may be designed for the control of some nonlinear dynamical systems based on the rigorous Lyapunov function arguments. Linear feedback controllers for nonlinear systems In light of the Lyapunov first method for nonlinear autonomous systems and the linear stability theory for nonlinear nonautonomous systems with weak nonlinearities (called linear stability theory), it is clear that a linear feedback controller may be able to control a nonlinear dynamical system in a very rigorous way. Take the nonlinear, actually chaotic, Chua's circuit as an example. This circuit is a simple, yet very interesting, electronics system that displays rich and typical bifurcation and chaotic phenomena. The circuit is shown in Figure 6.5, which consists of one inductor L, two capacitors Cj and Cj, one linear resistor R, and one nonlinear resistor / which is a nonlinear function of the voltage across its two termi-

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nals: g = g(Vc (/)). Let iL(t) be the current through the inductor L, and Vc (?) and Vc (t) be the voltages across Cj and C 2 , respectively, in this circuit diagram. It follows from Kirchhoff s laws that C,— V. (0=— ^,(0-^(0 l dt c\ R 1

c c 2*- ^ 2( 0 = 7?L - ^.(0-^(0 i c

dt

L

+ s(rc (0), c

+iL(t)

C

2

In the consideration of controlling the circuit behavior, it turns out to be easier to first apply the nonlinear transformation xx(t) = Vc^t),. x2(t) = VC2(t), x3(t) = RiL(t),

7 = tlC2R

to reformulate the circuit equations in the following dimensionless form: \x = p[-x + y-f(x)] }y = x-y + z

(6.3.19)

[i = -qy where /?>0,<7>0 , and f(x) = Rg(x) is a nonlinear function represented by f(x) = mQx + -(ml -w 0 )(|x + l|-|x-l|) in whichOTQ<0 and m^ < 0 . It is known that with /? = 10.0,g = 14.87,/WQ =-0.68, and /Mj =-1.27, the circuit displays a chaotic attractor (see Figure 6.6) and a limit cycle of large magnitude. This limit cycle is a large unstable (saddle-type) periodic orbit encompasses the nonperiodic attractor, and is generated due to the eventual passivity of the transistors. Now, let (x,y,x) be the unstable periodic orbit of the Chua circuit (6.3.19). Then, the circuit trajectory (x,y,z) of the circuit can be driven from any current state to reach this periodic orbit by a simple linear feedback controller of the form u

l x—x \ ll 0 =2 = -K y-y *22 ° M-5 z — 'z *33 u

with

x-x y-y z — 'z

(6.3.20)

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A|i>-/>7Wi, #22 ^ 0 , and A33>0 where the control can be applied to the trajectory at any time. A mathematical justification is given as follows. First, one observes that the controlled circuit is, x = p(-x + y - /(*)) - kx x (x - x), y = x-y + z-k22(y-y\ z= -qy-k^{z-z). Since the unstable periodic orbit (x,y,z) one has

(6.3.21)

is itself a (periodic) solution of the circuit,

x = p(-x + y y = x -y + 'z,

f(x)), (6.3.22)

z=-qy. so that a subtraction of (6.3.22) from (6.3.21),with the new notation X = x-x,

Y = y-y,

and Z = z-7

yields x = p(-X + Y-f(x,x))-knX, y = X-Y + Z-k22Y, z= -qY-k^Z,

(6.3.23)

where

f(x,x)--

rriQ(x-x)

x>l,3c>l

mQX-m^x + m^-mQ)

x>i,-l<x
TW 0 (X-X) + 2(/MJ -mQ)

X>\,X<-\

myx-mQX-m^+mQ

—1<JC<1,3C>1

m\(x-x)

-1<X<1,-1<X<1

m^x-rriQX + m^ -itiQ

-l<x<\,x

/WQ(x-?)-2(7ni-OTQ)

niQX - m-^x - m^ + ntQ /MQ(X-X)

<-l

X<-\,X>\

x < - l , - l < x <1 x<-\,x<-l

Define a Lyapunov function for system (6.3.23) by

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+-F2-Y2 + ^-Z2 2 2

It is clear that F(0,0,0)=0 and V(X,Y,Z)>0

for all X,Y,Z

not simultaneously

>

zero. On the other hand, since p,q>0 and ^22'%3 ^ ' > ^ f°ll° w s

= -p[q(X - Y)2 + qk22Y2 +k33Z2]-

q(pXf(x,x) +

tnat

kuX2)

<0 for all X,Y ,and z , if ^ ( ^ , x ) + A:11A'2>0

(6.3.24)

for all x and x . To find the conditions under which (6.3.24) is true, by a careful examination of the nine possible cases for the function f(x,x) shown above, the following common condition can be obtained: &jl >max{-pmQ,-pm^} = -pm^

(6.3.25)

in which wj < WQ < 0, as indicated above. This condition guarantees the inequality (6.3.24). Hence, if the conditions stated are satisfied, then the equilibrium point (0,0,0) of the controlled circuit (6.3.23) is globally asymptotically stable, leading to |x|-»0, |y|-»0, |z|-»0, as t -» oo simultaneously. That is, starting the feedback control at any time on the chaotic trajectory, one achieves Urn |x(/)-3c(0| = 0, lim |x(0-x(?)| = 0, Urn \z(t)-z(t)\ = 0. t->°o t—>oo t—>oo Nonlinear feedback controllers for nonlinear systems It is not always possible to use a linear controller to control a nonlinear dynamical system. So nonlinear feedback controllers are often necessary. Consider the Duffing oscillator \ \y = -p2x-x where p[;p2,q,

-5 - pyy + qcos(cot),

(6.3.26)

and co are systems parameters. It is known that with the parame-

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ters set p\ =0.4,/? 2 =-l.l,T t^T

(6.3.27)

for a terminal time T < <x> For this purpose, consider a nonlinear feedback controller of the form u(t) = h(t;x,x). By adding the controller to the second equation of the original system, one has the following controlled Duffing system: x = y, •x _ (6.3.28) y - ~P2X — x— p^y + qcos{cot) + h(t;x,x). Since the periodic orbit (x,y) is itself a solution of the original system, subtracting (6.3.26), with (x,y) being replaced by (x,y) therein, from system (6.3.28), gives x=Y y = -P2% ~(x

,

~x

,

_ ) - P \ Y + h(t'<x->x\

(6.3.29)

where X = x = x and Y = y-y Next, observe that the controlled Duffing system (6.3.29) is a nonlinear, nonautonomous system. Therefore, the Lyapunov first method may not apply. For this particular case, however, the Lyapunov second method can be applied fairly easily. Indeed, a nonlinear controller h(x) can be designed as follows. Let h(x) = kX + 3x2+3xX2 which is only quadratic and so is simpler as compared to the given cubic oscillator. Then, under the control of this controller, system (6.3.9) reduces to

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215

X =y p2)X-PlY-X3

Y = -(k +

(6.3.30)

and one can easily verify that under the condition k + p2 > 0, which gives a criterion for determining the linear control gain k, the Lyapunov function V{XJ)=^P2X2+}_X4+}_Y2

2

4

2

satisfies V < 0, where equality holds if and only if both X = 0 and Y = 0 . More precisely, one has K = - / ? j F 2 . If 7 * 0 then K<0 . If Y = 0 then the orbit is located on the X-axis, so one has the following two cases: (i) X=0, thus V =0 only at (0,0); (ii) X * 0 , thus the second equation of (6.3.30) shows that 7 * 0 , which means the orbit is moving and will not stay on the X-axis (so sooner or later the case returns to (i)). Therefore, V <0 , where equality holds if and only if both X = 0 and 7 = 0 . This means that the zero fixed point of the controlled Duffing system (6.3.30) is asymptotically stable, so that X -» 0 and Y -» 0 as t -> oo, or the goal |x-5c| —>0 and ir —JC —>0 (t —»oo) is achieved. Some general criteria for controllers design Recall the general tracking control problem described earlier by equations (6.3.13)-(6.3.18). For a general nonlinear and nonautonomous systems of the form x = f(x,t)

(6.3.31)

which is assumed to possess a periodic orbit x of period t >0: x(t + t ) = x(t) for all 0 < / < oo , the goal is to design a feedback controller of the form u(t) = K(x-x)

+ g(x-x,

t),

(6.3.32)

where AT is a constant matrix and g is a (simple) nonlinear vector-valued function, which is to be added to the original system, to obtain x = f(x,t) + u = f(x,t) + K(x -x) + g(x -x,t).

(6.3.33)

The controller is required to be able to drive the trajectory of the controlled sys-

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tern (6.3.33) to approach the target periodic orbit x , in the sense that lim \\x(t)-x(t)\\ = 0

f6 3 34)

where, again, |-| is the standard Euclidean norm. Since the target periodic orbit x is itself a solution of the original system, it satisfies x=f{x,t),

(6.3.35)

and since the feedback controlled system with the controller (6.3.32) is given by x = f(x,t) + K(x-x) + g(x-x,t),

(6.3.36)

a subtraction of (6.3.35) from (6.3.36) gives x = F(X,t) + KX + g(X,t)

(6.3.37)

where X = x-x

and F(X,t) =

f(x,t)-f(x,t).

It is clear that F(0,t) = 0 for all ?e[0,oo). Now, Taylor-expand the right-hand side of the controlled system (6.3.37) at X=0 (i.e., at x = x ), and suppose that the nonlinear controller to be designed will satisfy g(0,r) = 0. Then x = A(x,t)X + h(X,K,t), where A(x,t) = df(X,t)/dx\j^=Q

and h(X,K,t)

(6.3.38)

is the Taylor expansion (truncated,

if necessary, in a design), which is a function oft, K, and 0(X). To this end, the design is to determine both the constant control gain matrix K and the nonlinear controller g(X, t) based on the linearized model (6.3.38), such that X->0 (i.e., ;C-»JC ) as ?->oo.lf this can be done, then when the controller is applied to the original system, as shown in (6.3.36), the goal (6.3.34) can be achieved [9,20]. Theorem 6.3.1 Suppose that in system (6.3.38) h(0,K, t)=0 and A(x,t) = A is a constant matrix whose eigenvalues all have negative real parts. If lim

\KX,K,t)\

IMh° \\x\\ uniformly with respect to /e[0,co), where J is the Euclidean norm, then the control

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u(t) defined in (6.3.32) will drive the trajectory x of the controlled system (6.3.36) to the target orbit x as f —» oo . Theorem 6.3.2 h(X,K,t)

In system (6.3.38), suppose that h(0,K,t) = 0 and that

and dh(X,K,t)/dX

are both continuous in a bounded region \x\<<x>.

Also assume that lim

x

\\ \h°

\\h(X,K,t)\\_

\\x\\ =0

uniformly with respect to ?e[0,°o). If all the multipliers of the system (6.3.38) satisfy 1^1 <1, i = \,...,n V?e[0,oo) then the nonlinear controller (6.3.32) so designed will drive the orbit x of the original controlled system (6.3.38) to the target orbit x as t—><x>. It is clear from te above derivation and discussion that the state feedback control approach described here has the essence of conventional control but has incorporated some special features of a chaotic system (e.g., the unstable periodic orbit of a chaotic system is a solution orbit of the same system, which is utilized for the controller design). Basically, this kind of state feedback control methods are "brute force" strategies as compared with the parametric variation approach introduced earlier. Nevertheless, feedback control technology has proved very successful for chaos control as well as anticontrol (which will be further discussed in a later section).

6.3.3 On time-delayedfeedback control Time-dslay feedback control is another effective approach of the conventional type. A time-delayed feedback control (TDFC) system is by nature a rather special version of the familiar autoregressive moving-average (ARMA) control, or the canonical state-space control systems. Despite some of its inherent limitations, TDFC can be quite successful in many chaos control applications. In order to understand to what extent the TDFC can be successful in chaos control applications, some typical (sufficient) conditions for the TDFC approach are derived in the following, based on Lyapunov stabilization arguments. Consider a general continuous-time nonlinear dynamical system: x = f(x,t),

x(t0) = x0<ERn

(6.3.39)

Suppose that this system has an (unstable) periodic solution, x(t), and is currently

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in a chaotic state. The task is to find a TDFC (with a proper delay-time, r > 0 ), u(t) = K(x(t)-x(t-r)), to be added to f(x,t),

(6.3.40)

such that the controlled system orbit can track the target: lim \\x(t)-x(t)\\ = 0

(6 3 4H

The design problem is then to determine the control gain matrix K to achieve the goal (6.3.41). Since the controlled system is x = f(x,t) + K(x(t)-x(t-r)),

(6.3.42)

and the periodic orbit is a solution of the original system, namely, t = f(x,t)

(6.3.43)

subtracting (6.3.43) from Eq.(6.3.42) yields the error dynamical system e = F(e,t) + K(x(t)-x(t-z)),

(6.3.44)

where e:=x-x

and F{e,t):= f{x,t)-

f{x,t) = f(e + x,t)-

f{x,t).

6.3.3.1 Stabilization problem First, the stabilization problem is discussed for the case where the target is an unstable fixed point, i.e., x =constant and is unstable. Without loss of generality, let x = 0, and the error dynamical system (6.3.44) becomes x = F(x,t) + K(x(t)-x(t-T)),

(6.3.45)

where F(x,t):=f(x,t)-f(0,t). This system has a zero fixed point, F(0,t) = 0, for all t > ?o, and the control objective is to force x ( 0 - » 0 as /->oo . For this simple case, the following local asymptotic stabilization result can be obtained, where J(i) := F {G,t): Theorem 6.3.3 For the error dynamical system (6.3.45), if there are two positive definite and symmetric constant matrices, P and Q, and a constant gain matrix, K, such that the Riccati polynomial matrix

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JT\t)P' + PJ{t) + PKQ'lKT+

PK + KTP + Q

219

(6.3.46)

is either zero or (semi)-negative definite (=0,<0,or <0) , then when |x(f)| 'n (6.3.45) is small enough, it will always approach zero: U(f) ~> 0 a * t —> °o. To verify this condition, construct a Lyapunov function of the form V(x,t) = xTPx + \\_TxT Qxds. Then, since 0 is a fixed point

of F(x,t)

, its Taylor expansion

is

F(x, t) = J(t)x(t) + [H.O.T], where [H.O.T] are higher-order terms in x(0-Thus, V(x, t) = xT (t)Px(t) + xT (t)Px(t) + xT (t)Qx(t) -xT(t= -[Ql/2x(t-T) + xT(t)[JT(t)P

+ Ql/2KTPx(t)]T[Ql/2x(t-T) + PJ(t) + PKQ-lKTP

+ [H.O.T.]TPx + xTP[H.O.T.]<0,

r)Qx(t - r) +

Ql/2KTPx(t)]

+ PK + KTP + Q]x(t) V small | x | .

To this snd, a standard verification using class- K functions [10] for this nonautonomous system completes the verification of the theorem. Corollary 6.3.4 Theorem 6.3.3 holds if condition (6.3.46) is replaced by the Riccati equation P{t) = JT{t)P{t) + P(t)KQ-XKTP{t)

+ PK + KTP + Q

(6.3.47)

which has a positive definite and symmetric solution P(t) > 0, t e [/Q , oo). It suffices to replace the constant matrix P by P(t) in the above Lyapunov function V(x,t), and then observe that there is an additional term, x* (t)P(t)x(t), in V(x,t) of the above proof. Note that in the very special case where the system Jacobian J(t) = J is a constant matrix, a sufficient condition for (6.3.46) and (6.3.47) to hold is that (J,K) is stabilizable. This is well known in linear control systems theory. Clearly, whether or not the above stabilization control is successful depends on if the trajectory error, e(t), can first be forced into a small neighborhood of the origin; if so, the theorem guarantees that the linear delayed feedback controller will

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continuously drive the error signal to zero. Thus, the above result provides a theoretical guideline for the controller design. Tha next theorem demands weaker conditions, but provides weaker results in the sense that it does not guarantee to which target the controlled trajectory moves. Theorem 6.3.5

Suppose that the given system (6.3.39) is autonomous. Let

J(x) = f (x) = df(x)/dxbe

the system Jacobian and assume that J(x)is uniformly

bounded by a constant matrix JQ ,namely, J(x)< J§for all x(t) and all t:tQ. Then, there is always a constant control gain matrix, K, such that the controlled orbit of system (6.3.42) is driven to approach a limit set in the phase space; particularly, for planar systems, the controlled orbit converges to either a fixed point or a periodic orbit as /—»a>. According to the fundamental theory of differential equations, the solution of the linearized system of (6.3.42) is

e(t-s)[J(x)+Kh(x(s)-x(s-r))ds

x ^ e C - W W + ^ o +j' 'o which satisfies \\x(t)\\<e^~t0^J0+KM\xJ+

\t

s J +K

e^-

^ 0 ^K(x{s)-x(s-T))ds

'0

where both x(t) and x(t-z) are chaotic and hence are uniformly bounded in the phase space, so that the integral term above converges (so is bounded). Since the real parts of all eigenvalues of [JQ + K] can be made negative by a suitable K, the first term on the right-hand side of the above inequality tends to zero as t—>oo . This implies that the controlled orbit is always bounded, and is always pointing inward in the phase space. It then follows from the (extended) Poincare-Bendixson theorem that the controlled orbit approaches a limit set in the phase space; in particular, for planar systems, it converges to either a fixed point or a periodic orbit by the classical Poincare-Bendixson Theorem. 6.3.3.2 Tracking problem Now, consider the case where the target is an inherent unstable periodic orbit, x(t) with period t >0, of the given system (6.3.39). In this case, the error dynamical system (6.3.44) becomes

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e = F(x,t) + K(x(t)-x(t-T)) where F(x,t):= f(x,t)-f(x,t)

221

(6.3.48)

. The control objective is to force x(/)->x(?) as

f —»oo

It should be pointed out that this tracking problem using TDFC has a particular property. Proposition 6.3.5 Suppose that the matrix K is nonsingular. Then, the controlled orbit x(t) tracks the inherent unstable orbit x(t) , with x(x,t)-^x(t) , as t—»co, if and only if the delay-time r is (an integer multiple of) the period t „ of x(t). To verify this, it suffices to notice that if x(t) tracks X(7Q) as / -» GO , then F{x, t) -» 0, eit) -> 0, and e(0 -» 0, as / -» oo. Eq. (6.3.48) then gives Kix(t)-xit-T))-*0 implying that x(t)-xit-r)->0 . Since x(f) = 5c(f) by periodicity, the constant r must be (an integer multiple of) the period of the unstable orbit. Th? converse can be easily verified by noticing that \\e\\ < \\Fixit))\\ + K\\xit)-x(0|+|3c(0-x(t-r)|

+||*(/ - r ) - x ( r - r ) | ]

< \\Fixit)\\ + K\\eit)\\ + 0 + \\eit - r)|] -> 0 This property deserves a special attention when dealing with a tracking problem with TDFC: it suggests that the period tv = r has to be known beforehand in order to achieve a successful tracking. Note also that if K is singular, then some components) of the state vector xit) may exhibit periodic behaviors. This may also result in other components exhibiting periodic behaviors as well. Now, the trackability of a TDFC is characterized as follows. Theorem 6.3.6 For the error dynamical system (6.3.48), if there are two positive definite and symmetric constant matrices, P and Q, and a constant gain matrix, K, such that the Riccatipolynomial matrix J T it)P + PJii) + PKQ~lKJP

+ PK + KTP + Q

(6.3.49)

is either zero or (semi)-negative definite (= 0, < 0, or < 0) , then when |e(0| 'n

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(6.3.48) is small enough, it will always approach zero: |e(?)||—>0 as ?—»oo. This result follows directly from the proof for Theorem 6.3.3 with x(t) being replaced by e(t) therein. Similarly, the following result follows from Corollary r3A. Corollary 6.3.7 Theorem 6.3.5 holds if condition (6.3.49) is replaced by that the Riccati equation P = JT(t)P(t)

+ P(t)J(t) + P(t)KQ~lKT

P(t) + PK + KT +Q

(6.3.50)

has a positive definite and symmetric solution P(t) > 0, t e [/Q , °o). Next, a method is suggested for estimating an unknown period constant, t > 0. Chaotic systems are generally very complex, so that it is very difficult to obtain, either analytically or experimentally, the periods of the inherent unstable orbits of a chaotic system. An effective approximation method is by means of the gradient descent approach that can search for the period. To apply this technique, the following criterion is used for search: minimize the performance index J=-

I Ik'n +ih)-x(t0 +ih-r)f "/=l" "

(6.3.51)

where h is the time step length and n is the total number of time-series data used for search. The gradient can be easily derived as 1 Y\*.VQTiri)-A\tQTiri--i)i "r — =—

-=— Z |x(?0 + ih) -x(tQ+ih-r)[

~\r xyiQ-tm-L) x(t^+ ih - r)

(6 3.52)

Then, the algorithm for searching the constant r is formulated as follows: 1. Set a tolerance e > 0 and proper n and h such that large enough amount of time-series data are covered for the estimation. Simulation starts at TQ = 0 . Choose an arbitrary initial guess, r = TQ , and let the chaotic system run freely for nh periods of time. 2. Adaptively adjust tj by Ti+1 Ti

~

0dJ P drt

(6.3.53)

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where /? is a properly chosen positive parameter that may improve the convergence rate. Set i = i + l .3. 3. If J > e, go to Step 2; otherwise, stop. It is clear from the above derivation and discussion that the TDFC method also reminisces many basic features of classical feedback control. It utilizes some chaotic properties such as matching the delay time in the controller to the period of the unstable target periodic orbit. It also utilizes some intrinsic nonlinear dynamical properties such as the Poincare-Bendixon Theorem in the controller design amd analysis. For many other classical and non-conventional chaos control methods, the reader is referred to [1,2,7], among others, and the references therein.

6.4 Anticontrol of chaos: chaotification

6.4.1 Introduction Recently, there has been increasing interest in designing a controller that can generate chaos in an original non-chaotic system. Making a non-chaotic dynamical system chaotic or retaining the chaos of a chaotic system (anticontrol of chaos or chaotification), even for discrete-time dynamical systems (maps), can be very important in many real-world applications, as reviewed at the beginning of this chapter.

6.4.2 Problem description and the anticontrol algorithm This section discusses a general discrete-time dynamical system, originally need not be chaotic, of the form '**+l =/*(**)• XQ - given,

K

Rn

(6A1)

where f^ is assumed to be continuously differentiable, at least locally in a region of interest. The objective is to design a control input sequence {M^ }?> „ such that the output (state vectors) of the controlled system XQ-given, behaves chaotically, in the sense of some commonly used criteria for chaos.

(6.4.2)

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6.4.2.1 Problem description Consider linear state-feedback controls of the form uk=Bkxk

(6.4.3)

where \Bk\ are nxn constant matrices to be determined, without tuning any of the system parameters. Using this uk, the controlled system (6.4.2) becomes f*jfc+l=/*(**) + % * *

(644)

XQ- given, To describe the problem more precisely, some new notation is needed. Let JJ(Z) = /J(Z) + BJ be the Jacobian of the controlled system evaluated at z,y' = 0,l,2,..., and let Tj(x0,...,Xjy.= Jj(xj)Jj_l(xj_l)..Jl(xl)J0(x0)

(6.4.5)

(6.4.6)

Moreover, let /uj = jUj(TJ T•) be the /'th eigenvalue of the jth product matrix [TjTA,

where i = \,...,n and y = 0,1,2,....

Recall that the /th Lyapunov exponent of the orbit \xk fa_n of the controlled system (6.4.2), starting from the given x 0 , is defined, for / = 0,1,•••,«, by Aj(x0)

=

Urn —In Mi(T[Tk)

Urn —In Mi (JQ k—>oo 2«

(XQ ) • • • J\

(xk )Jk (xk ) • • • J0 (xQ ) (6.4.7)

The approach suggested here is divided into two steps. First, in system (6.4.4), one designs the constant matrices [Bk }^ =0 such that all the Lyapunov exponents of the system orbit \xk fan are finite and strictly positive: 0
i = \, — ,n,

(6.4.8)

where c is a predesired constant. Then, one wants this controller to bound the diverg-

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ing system orbits back into a bounded region, so as to drive the system chaotic. To be practical in achieving this goal, namely, for implementation purpose, it is required that the sequence [B^ f£_n of control-gain matrices be uniformly bounded: SU

P

0<&
|]| B J I < M < O O

*"

(6.4.9)

for some constant M, where |.|| denotes the spectral norm of a finite-dimensional matrix, that is, the largest singular value of the matrix. It is shown below that this is possible under a natural condition that all the Jacobians {/^(x^)} are uniformly bounded, i.e., there is a constant N such that SUP

\\fk(xk)UN
ru

0<&
positive feedback control gain such that the matrix

[TQTT

6Q =
an

,m

(6.4.10)

XQ

is initially

d let 7 Q = J Q ( X Q ) . Design a

by choosing a positive number

CTQ

>0

] is finite and diagonally dominant.

For k = 0,1,2,-••, start with the controlled system x ^ + | = fk(xk)

+B x

kk '

wnere

x^ was obtained from the previous step. Step 1. Compute the Jacobian Jk (xk) = fk (x^ ) + crkIn and then let Tk = Jk Tk_\. Step 2. Design a positive feedback controller by choosing the positive number a^ such that the matrix \TflT\-e

In is finite and diagonally dominant,

where the constant c > 0 is the one given in (6.4.8). Step 3. Finally, apply the mod-operation to the controlled system: x

k+\ =fk(xk)

+ Bkxk ( m o d 0

(6.4.11)

The control-gain sequence {Bjl} = {crk-In} , along with the mod-operation, will achieve the desired anticontrol. One remark is that the first two steps of the algorithm ensure that all the

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Lyapunov exponents of the controlled system become strictly positive, so that the system trajectory is expanding in all directions. The third step, however, bounds the system trajectory globally. A combination of these two effects is expected to create complex chaotic dynamics within the bounded region of system orbits. Note also that, instead of Steps 1 and 2, a convenient choice is to simply use u

k ~ ^kxk

=

°kxk

w tn

'

°k = ^

+ e

°

(6.4.12)

for all £ = 0,1,2,---, where N is specified in (6.4.10). Then Step 3, namely, the modoperation (6.4.11) can apply. 6.4.3

Verification

of the anticontrol

algorithm

6.4.3.1 For linear time-invariant systems Consider a given w-dimensional linear time-invariant controlled system with the mod-operation: x

k+\ ~ Axk

+u

k

(mod 1)

(6.4.13)

where the mod-operation is defined component wise, and the controller takes the simple form (6.4.12) with the choice of a constant bound N > \A\, namely, uk=(N + ec)xk

(6.4.14)

in which c is specified in (6.4.8). It is shown below that the controller u^, designed by following the above anticontrol algorithm, will make the controlled system (6.4.13)-(6.4.14) chaotic, in the sense of the following three criteria: A map 0:S^>S, where S is a set, is chaotic if (textbook definition [5,6,8]) (i) <j> has sensitive dependence on initial conditions, in that for any xeS and any neighborhood N of x in S, there exists a 8 > 0 such that m(x)-m{y) >S for some yeN

and m>0, where <j>m is the wth order iteration of <j>, i.e.,

(f>'n •= (j> o • •. o ^ (m times). (ii) (/> is topologically transitive, in that for any pair of open subsets U,VcS, m

exists an integer m > 0 such that (U) fl V * 0 . (iii) The periodic points of ^ are dense in S .

there

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227

The simplest case First, consider the simplest case of the anticontrol algorithm for the onedimensional linear time-invariant system: xjc+Y=axji+uji

(modi)

(6.4.15)

where the controller takes the simple form with the natural choice of an JV>|a|, in which the constant c is specified in (6.4.8): uk=(N + ec)xk

(6.4.16)

First, observe that the controlled system (6.4.15)-(6.4.16) is xk+l = axk + (N + ec)xk = (a + N + ec)xk

(modi)

(6.4.17)

where a + N>0 and, since c>0, (a + N + ec)>\. Next, define a map 0:S*^>S* , where S 1 is the unit circle on the twodimensional plane, by {x) = {a + N + ec)Zx

x e Sl

(6.4.18)

in which Zx is the angle of the point x e y (thus, 0 < Z x < I n and <j> is 2K periodic). Note that this map is a one-variable map since the radius of the circle is fixed and only the angle is variable. Then, it can be verified that the two systems, (6.4.17) and (6.4.18) are equivalent. It is easy to see, by multiplying 2n to both sides, that (6.3.15) is equivalent to x

k+\ =(a + N + ec)xk

(mod2;r)

(6.4.19)

or Zx = (a + N + ec)Zx,

xksSl

which is equivalent to (6.4.18) where the initial state XQ can be arbitrary. It can be verified that the map (6.4.19) is equivalent to the well-known chaotic logistic map x^ + j =4x^(1 -x^) map, where %:=(a + N + ec)>\. comes

by defining x^=—[l-cos(27r^)j

in the logistic

With this change of variables, the logistic map be-

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|[l- COS (2^^ +1 )]=4-i[l- COS (2«^ it )].|l-|[l-c QS (2«-^)]J = [l - cos(2/i&k )]• ll + cos(27r&k )J which has the solution precisely given by formula (6.4.19). To provide one more proof, it is referred to the following definition of discrete chaos [19]: A map $: S -> S, where S is a nonempty set, is chaotic if and only if for any two nonempty open subsets in 5 there is a periodic solution of

'i(x) = i;'lZx. Thus, since <j> is 2n -periodic, one can let^Zx

= Zx + 2£^ , where £ is an integer satisfying 0<£<\g'c J - l . Then, one can

see that the roots {x^ g} of the this equation (i.e., periodic solutions of the map
which means that for two arbitrarily given nonempty open subsets in S , there are two large enough integers kq and £Q such that the periodic solution xk £ of the map has at least one point in each of these two subsets. The general case Now, consider the general case of the controlled system (6.4.13)-(6.4.14). To show that this controlled system is chaotic under the control of uk =(N + ec)xjc, where

Af>|/4| , only the first component

X£+j(l)

in the state vector

x

nee k+\ =Vck+\Q}'"xk+\(nn ^ s to be discussed, without loss of generality. Observe that this component of the controlled system has the expression x

k+\ 0) = [fll \xk (^ + • •' + a\nxk (")]+ (N + e°^>xk & --(a^\+N + ec)xk(1) + [fl|2*k(2) + • • • + a\nxk(ri)\

(mod 1)

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where, since N>JAJ>a^ j and c>0, ax \ +1/4| > 0 and ax j + 1 ^ | + ec > 1. Next, in the phase space spanned by the coordinates x(\),•••,x(n),

consider an

w-tuple map (projection) S*, before the mod-operation is applied at each iteration, where S* is the unit circle on the two-dimensional plane. This projection is defined by ^(x(\),x(2),---,x(n)) = (au+N where Zx is the angle of the point x(l)eS',

+ ec)Zx(\)

x(\)eSl

satisfying 0
(6.4.20) (subject to the

2n mod -operation). Clearly, is 2/r -periodic with respect to x(l). Then, it can be similarly verified that the map (6.4.20) satisfies the three criteria as a chaotic map, and so does the controlled system (6.4.13)-(6.4.14), when it is projected on JC(1) (hence, on each principal direction) of the phase space spanned by the coordinates x(\),---,x(n). As a result, the overall controlled system is chaotic within the entire phase space by superposition; otherwise there would be at least one projection that is not chaotic. 6.4.3.2 For time-varying and nonlinear system For giver, time-varying and/or nonlinear systems, the above proof can be carried out in a similar procedure. Intuitively, if the given system is time-varying or even nonlintar, one has a higher chance to obtained a chaotic controlled system. However, due to the time-varying and/or nonlinear nature of the map, a proof is difficult (but has already been established), which is beyond the scope of this chapter. Finally, it is noted that the mod-operation used above can and has been replaced by some other operations such as piecewise-linear functions or even a sine function (if time delay is also used) in the controller [21-23]. Also, some anticontrol methods have been initiated for continuous-time dynamical systems [22,23] which, however, is still far from being as complete as that for the discrete-time case discussed above.

6.4.4 External noise control and anticontrol of chaos External noise, both white and colored, can be used for control and anticontrol of chaos [24-27]. In fact, external noise not only can control chaos but also can reorganize hyperchaos [26,28] and spatiotemporal chaos [29]. Noise-driving method has became one effective method for both chaos control and synchronization. In this sec-

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tion, two typical nonlinear dynamical systems are used as examples for illustration. 6.4.4.1 White noise control and anticontrol of chaos Consider a radio frequency (rf) plasma coupled with a PLC circuit [24], described by

X2=A cos(27VC2) + Bx\ + Cx? - (Dxr - EJxj +Fn

(6421)

*3 =COI2K

where X\ has the physical meaning as the normalized charge density in the plasma of an rf gas discharge, ^ is its derivative, and x^ is a related time function. Also, Fn is an external noise-driving controller, a and A are driving frequency and amplitude of the external rf power source, respectively, B, C, D, and E are the system control parameters. Gaussian white noise is used, Fn=Fw(t), with

l(Fw) = ° V ' (6 4 22) \(Fw(t)Fw(s)) = 2DFS(t-s) = 0 where Q denotes the mathematical expectation (average), DF is the diffusion coefficient, and m is the Dirac delta function. Eq. (6.4.22) completely determines the statistical feature of the controller. Control of chaos In the right regime of the parameter space, white noise can realize chaos control, as shown in Figure 6.7, where ^ = fi = C = l,£> = 0.075,£ = 0.270975,fl> = 1.745. As seen, the chaotic state (where Dp is stabilized to a period-1 limit cycle by strong white noise with Dp =0.5). The main reason is that strong white noise compensates the dissipation process and plays the role of "signal-to-noise enhancement" (or "stochastic resonance"), where the control via noise actually improves the signal processing. Anticontrol of chaos Anticontrol of chaos, and alternation of periodic-chaotic sequences, can also be achieved by changing the intensity of white noise, as shown in Figure 6.8, where /4 = 5 = C = l,Z) = 0.075,£ = 0.270975,fij = 1.745 with Dp =23.98,24.03,24.031, and 24.05, respectively. Only two transition points are shown in the sequence from period-5 to chaos.

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231

6.4.4.2 Colored noise control and anticontrol of chaos A set of chemical reaction equations are considered [27]: i j = -kxXJX3 + a(x\ r - JCJ ) *2 = A*1X3 -2(*2 X 2 - * - 2 * 3 ) - * 3 * 2 i

+ fl

(*2/ -*2>

(6.4.23)

3=^2x2"A-2x3,

where x j , ^ , ^ are the molar fractions of three reactants, &+;- are the rate constants of the fth reaction, a is the inverse residence time or feed rate, and x\ r and x2 /" are the feed concentrations of xj and x2, respectively. The system is impermeable tO X3 .

The basic features of system (6.4.23) are: (a) it consists of binary collisions only; (b) it is the chemical oscillator in which there is no separation of time scales; (c) it involves only three species. It was found that the parameters, which produced sustained time periodic oscillations (limit cycle) in the system without any control, are *j = \,k2=k_2 =5,* 3 = 0.03333,x2/- =0.2667 , and a =0.03333. Figure 6.9 shows a periodic oscillation of xj for this set of parameters. It is especially interesting to see the realization of chaos control and anticontrol for system (6.4.23) by using colored noise input. Here, the so-called OrnsteinUhlenbeck process is discussed. The unified controlled equations for system (6.4.23) is described by \ ' ' l ' ' [si=-Zei+AFw

(6.4.24)

where /,(*;) represents the nonlinear function of the right-hand side of the system (6.4.23), and A and gj are constants. Consider the particular case where g^ = 1 and s\ = £j = £3 • Let Fw be the Gaussian white noise satisfying the statistics of Eq. (6.4.22). Then, the driven noise Ex is an exponentially correlated colored noise, with properties

f(«l('))=o \(sx (f)*i (s)) = DAexp(-A\(t -s)\),

(6-4-25)

in which (•••) denotes the mathematical expectation (average), and {•••} averages

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over the distribution of the random initial value EQ satisfying a Gaussian distribution. The correlation time of colored noise is r = \IX. Anticontrol of chaos can be realized under the right colored noise parameters, as thos i shown above. For example, anticontrol of chemical chaos is achieved from a period-1 orbit, shown in Figure 6.9, to chemical chaos shown in Figure 6.10, if D = 10~~> and the correlation time r = 0.05, where the other parameters are the same as before. It is emphasized here that the colored-noise-induced anticontrol of chemical chaos is closely related to the correlation time which leads to non-equilibrium transitions, such as from a single limit cycle to multiple limit cycles (two, three, or more), and then to chaos, and vice versa. In summary, it is found that external noise can actually be quite beneficial in such tasks as control and anticontrol of chaos, phase locking emgancement, chaos synchronization, hyperchaos and spatiotemporal chaos suppression, as well as spatiotemporal stochastic resonance. Anticontrol of Hamiltonian quantum systems Control and anticontrol of quantum chaos in several fields of contemporary physics and chemistry appears to be challenging. These include such as molecular dynamics in laser and quantum optics. A few examples of complete control of the quantum state have been proposed for some particular quantum-system atoms interacting with quantized an electromagnetic field in a single-mode resonator. A switching control method for classical systems has been extended to the control of quantum systems [33]. Complete control is achieved not only over a finite number of quantum states but also over the unitary evolution of a generic Hamiltonian system. In this approach, by switching on and off two distinct time-independent perturbations in an alternating equence, the control effect is obtained. For an Af-level system, the sequence is periodic, and each period consists of JV 2 time intervals: ^,?2> - -,? 2 , which are found by solving a so-called inverse Floquet problem. One finds this solution by linearizing the system in the vicinity of the identity map solution, which gives the intervals for the identity transformation. This nonlinear problem is solved numerically by minimizing the sum of absolute values of the coefficients of the characteristic polynomial of the evolution operator as a function of JV time intervals. An experimental setting is given in [33] for the control method and two examples of control of an atom. In this section, designing a time-delay driving method for anticontrol of temporal chaos of quantum fined system is briefly introduced [34]. These methods for anticontrol of quantum chaos of a quantum network is seen to be highly significant. For example, under non-equilibrium boundary conditions and under optical pumping, photons interacting with two-level impurity atoms in a photonic band-gap material

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233

can assume a new collective steady state, intermediate between that of incoherent and coherent light. Corresponding to this new optical state, the impurity atoms acquire a steady-state polarization whose phase varies randomly from atom to atom and the resulting collective steady state is the optical analog of a quantum spin glass or quantum network. This can be used to construct a neural network or quantum computing system. One of the key questions is how to design a "predictable" quantum chaotic state by applying a driving field on the quantum network. To do so, the design problem of a time-delayed driving field is discussed. The idea of this design is to yield quantum temporal chaos for a quantum system consisting of N two-level atoms confined in a volume V interacting with a single photon mode. The approach taken is based on a spectral statistical analysis using the spectral decomposition of the Hamiltonian (obtained by using a projection approach). This provides a method for generating or constructing temporal chaos in the quantum network through the spectral decomposition of the Hamiltonian, irrespective of whether the system is in equilibrium or not. The Hamiltonian of a collection of TV two-level atoms confined within a cavity (such as a photonic band gap) with a resonant transition frequency COQ and a periodic boundary condition, driven by a time-dependent photonic mode can be written as [33,34]: H = T.-^-crl

+YJ{a+ja + a+a-j)f{t,T)

(6.4.26)

where h is the Plank constant, COQ is a resonant transition frequency. Also, for generality, the external field f(t, r) is given as a function of time t and delay time r . In this equation, a :) and b:) represent the states on which the j th atom is in upper and lower level states, respectively, a* describes atomic excitation of the jth atom; az- describes they'th atomic inversion; a,a+ are the annihilation and creation operator for photons in the resonant dielectric mode, respectively, and the term a+<xy describes the process in which the atom is taken from the upper state into the lower state and a photon of mode is created, while term o+-a describes the opposite process. The exact form of Eq. (6.4.26) is determined by spectral analysis as described below. For simplicity, and to reveal the origins of temporal dynamics owing to the driving field, it is reasonable to neglect the self-interaction terms hva+a

and

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G. Chen et al.

J jfca^afc and to focus on the interaction terms for atoms, photons and external fields. Here, J -^ is the process coefficient for the j -th atom excitation and the k th atomic inversion, and v is a jump frequency of the photon (together, h,v is the energy of one photon jump). Denote _z _ a a

b

jKbj

j\ j

(6.4.27) a b

jl j

*)-

a

=

•j

b

a

jK J

which satisfy the spin-— algebra of the Pauli matrices. Substituting the definitions of Eq. (6.4.27) into the Hamiltonian of Eq. (6.4.26), the Hamiltonian operator is written as H(t,r) = H0+V(t,T) where HQ is the Hamiltonian for a quantum confined system in cavity, given by b :,n + \)(b.-,« + !

a n

7=1

2

V

j' Kaj'n

and V is the interaction and time-dependent driving field f(t,z) temporal chaos, expressed by N

v{t,r)= x s aj,n\(bj,n

+

\

bj,n

+

used to generate

\\Uj,n ]f(t, T).

j=\n The spectral decomposition of Hamiltonian without driving field f{t,r) calculated as 1

N-\ ho0

CD

(2) (1)

— + 2gcos 8 j cos 6^

j=0k=0 where the eigenvalues are

(2)\/(l)

(2)

can be

Introduction

to Chaos Control and Anti-Control

(1) 2n 9 J-^J>

7=0,1,

(2)

j=

2

235

0,l,-,N-\,

0j=—j,

and the eigenvectors are

!

(D\

(1)

X

TT2

\

e

i9 J

J

or

(2)

(2)\

e

,

z m

') H

i6

J (2)

id e J (2)

i(N-\)d 2

J

or (2)\

(2)

1

i(N-l)0n

X

Hence, 0) (2) tf(?,r) Xj®Xk®fm

hco

^

V + 2f(z)cOS

& (1) (2) Xj®Xk®fm )

\

6 ; COS 9fc

Using the projection operator method, it is proved that the eigenvalue can be solved as

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G. Chen et al.

f{t,r)\fm)=m\fm) The eigenvectors are in a Rigged Hilbert Space having the following structure:

| / / » ) e * (fm\ex,eHs0*, where is a dense subspace of the Hilbert space H . This shows that the right eigenvectors of f(t,r) are as an infinite combination of polynomials in the test space <j>, while the left eigenvectors are a finite combination of the delta function and its derivatives lying in the dual space of . Thus, f(t,r) is an important factor that determines the convergence of the series expansion of the right and left eigenvectors. It also determines if the evolution operator of the system is non-unitary, and if the system evolution is irreversible. The existence of generalized eigenvectors (with respect to the time parameter /) beyond the Hilbert space for the driving field f(t,r) can imply that the system is in a temporally chaotic state. The spectral statistics given below will clarify this point. The imposed quantum temporal chaos can be studied from the evolution of the statistical properties of the spectrum for the total Hamiltonian. This is because the transition to quantum chaos from a regular quantum system is related to the occurrence of level repulsion and attraction in the spectrum. One type of spectrum that reflects quantum chaos is the nearest neighbor spacing between eigenvalues appearing to the Wigner-like distribution rather than the Poisson-like distribution [35,36]. This claim is supported by both experimental analysis and theoretical calculations, and is based on the statistical properties of the spectrum for quantum chaotic systems. In this approach, the evidence for the existence of quantum temporal chaos in the system under consideration is the neighbor spacing between corresponding eigenvalues, as they evolve non-monotonically with change in a delay time parameter, T . Then, using this non-monotonical distribution one can determine / ( r ) for anti-controlling quantum temporal chaotic states of the system. For example, based on a Wigner-type distribution, PW(T) = T exp(-r), and fix(1) (2) ing c o s ^ . c o s ^ , one can design the / ( r ) such that Pw{r) =

{f(z)-f{T0))exp[-{f(T)-f{TQ))2)

and, hence, determine its explicit form: f(T)x±lnl^Ql+f(

PW(T)

) , if _Z
JK0

'

PW(T)

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237

This is a necessary condition, which enables an arbitrary spacing between the corresponding eigenvalues to evolve with a Wigner-type of distribution and indices for quantum temporal chaos in this time-delayed quantum system. The above provides a method to design a time-delayed external field for driving the quantum system into quantum temporal chaos. The delayed time parameter r acts as a key control parameter for inducing quantum temporal chaos in the system. The eigenvectors of the driving field are obtained using projection methodology. It reveals that they evolve in a dual pair of spaces beyond the Hilbert space. One benefit of this approach is in providing a definite form of eigenvalue which is only a function of the driving field, / ( r ) . This enables a simple design of a driving field suitable for inducing temporal chaos, based on Wigner-like distributions of the spectral spacing (with respect to r ). The anticontrol method discussed in this section suggests a general approach to anticontrolling temporal chaos of quantum systems, which may be easily tested by experiment, e.g., by using an external optical field with a flexible or random delayed time.

6.5 Some concluding remarks Chaos control and anticontrol, as a challenging subject born out of complex dynamics and control systems has triggered many aesthetic as well as intellectual responses from communities of engineering, science, and mathematics. It has led to many important and interesting scientific findings which demonstrates that crossing the traditional boundaries of scientific disciplines is not only important but has become necessary. Notwithstanding the existing accomplishment, chaos control and anticontrol as a profound research subject is far from being complete. In fact, it emerged only about a decade ago and is merely a small territory of the intriguing field of modern nonlinear science. In addition to pursuing deeper understanding and further amelioration of the topics that have been briefly addressed in this chapter, real-world applications are especially significant for assembling the theory. The impact of thorough studies and successful applications of many nontraditional and nontrivial topics in this exciting field will be enormous and far-reaching. Chaos control calls for new efforts and endeavors in the new millennium. In this new era, today's contemporary concepts of nonlinear dynamics and controls may undergo another cycle of rethinking and reorganizing; the chaos control and anticontrol theories, methodologies, and also perspectives may unleash some other new ideas and valuable engineering applications; real breakthroughs may begin to take place, bringing enhancement, improvement, and sustainability to the complex living nature. An unexpected yet exciting new world is yet to come.

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Acknowledgement Some material used are taken and modified from works with Drs. Dejian Lai, Xiaofan Wang, and Xinghuo Yu, and particularly some overview material and description are taken and modified from Referernce [1] coauthored by Dr. Xiaoning Dong. These colleagues and the World Scientific Publishing Company are acknowledged, for their cooperation's and courtesy.

References [I] G. Chen and X. Dong: From Chaos to Order: Perspectives, Methodologies, and Applications, World Scientific Pub. Co., Singapore, 1998. [2] G. Chen: Chaos, Bifurcation, and Their Control, in the Wiley Encyclopedia of Electrical and Electronics Engineering, 1998. [3] G. Chen and D. Lai, "Feedback control of Lyapunov exponents for discrete-time dynamical systems", Int. J. ofBifur. Chaos, vol. 6, pp. 1341-1349, 1996. [4] G. Chen and D. Lai, "Feedback anticontrol of discrete chaos", Int. J. ofBifur. Chaos, vol. 8, pp. 1585-1590, 1998. [5] R. L. Devaney, An Introduction to Chaotic Dynamical Systems, Addison-Wesley, New York, 1987. [6] R. W. Easton, Geometric Methods for Discrete Dynamical Systems, Oxford University Press, New York, 1998. [7] A. L. Fradkov, and A. Yu. Pogromsky, Introduction to Control of Oscillations and Chaos. Singapore: World Scientific, 1999. [8] P. Glendinning, Stability, Instability and Chaos, Cambridge university Press, New York, 1994. [9] F. C. Hoppensteadt, Analysis and Simulation of Chaotic Systems, SpringerVerlag, New york, 1993. [10] H. K. Khalil, Nonlinear Systems, 2nd Ed., Prentice-Hall, Upper Saddle River, NJ 1996. [II] D. W. Jordan and P. Smith, Nonlinear Ordinary Differential Equations, Oxford University Press, Oxford, 1987. [12] V. Laskmikantham, S. Leela, and M. N. Oguztoreti, "Quasi-solutions, vector Lyapunov functions and monotone methods", IEEE Trans, on Auto. Contr., Vol. 26, 1981,pp. 1149-1153. [13] T. Y. Li and J. A. Jorke, "Period three implies chaos", American Mathematical Monthly, Vol. 82, 1975, pp.481-485. [14] I. G. Malkin, Theorie der Stabilitat einer Bewegung, Oldenbourg, Munich, 1959. [15] F. R. Marotto, "Snap-back repellers imply chaos in R"", J. of Mathematical Analysis and Applications, Vol.63, 1978, pp. 199-223. [16] E. Ott, C. Grebogi, and J. A. Yorke, "Controlling chaos", Phys. Rev. Lett., vol. 64, pp. 1196-1199, 1990.

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[17] V. M. Popov, Hyperstability of Automatic Control Systems, Springer-Verlag, Berlin, 1973. [18] C. .Robinson, Dynamical Systems: Stability, Symbolic Dynamics, and Chaos, CRC Press, Boca Raton,FL, USA, 1995. [19] P. Touhey, "Yet another definition of chaos", American Mathematical Monthly, May 1997, pp. 411-414. [20] F. Verhulst, Nonlinear Differential Equations and Dynamical Systems, Springer-Verlag, New York, 1990. [21] X- F. Wang and G. Chen, "On feedback anticontrol of discrete chaos", Int. J. of Bifur. Chaos, vol. 9, pp. 1435-1442, 1999. [22] X. F. Wang and G. Chen, "Chaotifying a stable LTI system by tiny feedback control", IEEE Trans. Circ. Syst, vol.47,pp.410415, 2000. [23] X. F. Wang and G. Chen, "Chaotification via arbitrarily small feedback controls: Theory, method and applications", Int. J. of Bifur. Chaos, March 2000, in press. [24] J. Q. Fang, "The effects of white noise on complexity in a two-dimensinal driven damped dynamical dystem", Phys. Lett. A, vol. 142, pp. 344-348, 1989. [25] J. Q. Fang, "The effects of external noise on complexity in a two-dimensinal driven damped dynamical system", in Measures of Complexity and Chaos, ed. by N. B. Abraham, A.M. Albano, A. Passamante and P. E. Rapp, Plenum Press, New York, 1989, pp. 229-234, [26] J. Q. Fang, "Generalization farey organization and generalized winding number in a 2-D DDDS", Phys. Lett. A, Vol. 146, pp. 35-44, 1990. [27] J. Q. Fang, "Colored noise induced nonequlibrium transition in a new chemical oscillator", Commun. Thoer. Phys., Vol.17, pp. 39-48, 1992. [28] J. Q. Fang and S. Y. Xu, "Synchronizing hyperchaos by white noise", Chinese J. ofNucl. Phys., Vol.18, pp. 244-246, 1996. [29] J. F. Lindner, B. S. Prusha, K. E. Clay, "Optimal disorder for taming spatiotemporal chaos", Phys. Lett. A, Vol. 231, pp. 164-172, 1997. [30] A. Maritan and J. R. Banavar, "Chaos, noise, and sychronization", Phys. Rev. Lett, Vol. 72, pp. 1451-1454, 1994. [31] J. Q. Fang, "A physical model for describing the characteristics of BPD and RFPD (I)", Chinese J. ofNucl. Phys., vol. 12, pp. 163-172, 1990. [32] J. Q. Fang, "A physical model for describing the characteristics of BPD and RFPD (II)", Chinese J. ofNucl. Phys., Vol. 13, pp. 71-80, 1991. [33] G. Harel and V. M. Akulin, "Complete control of Hamiltonian quantum systems: Engineering of Floquet evolution", Phys. Rev. Lett., Vol. 82, pp. 1-4, 1999. [34] B. Qiao, H. E. Ruda and J. Q. Fang, "Time delay anticontrol of temporal chaos in a quantum confined system", submitted, 2000. [35] L. E. Reichl, The Transition to Chaos: In Conservative Classical Systems: Quantum Manifestations, Springer-Verlag, New York, 1992. [36] R. Bluel and W. P. Reinhart, Chaos in Atomic Physics, Cambridge Univ. Press, New York, 1997.

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>«;«*>.<• rijui: fci.t

Fig.6.1 The chaotic attractor of a power system I

•0\

C*0t

Cm 0.2

C»ft?

5< / i

1/

C-03

C » 0.-9

£.' -• Q «

\:

I X"

\ \

ii

JL _ _J7

/

v

_J

Fig.6.2 A broad variety of dynamics can be created and modified when chaos is under control

Introduction

to Chaos Control and Anti-Control

4• ^

r

.,

Fig.6.3 A simple electric power system

position o l «He mqtlibffum point
wi!f»ur perluf btation

Fig.6.4 Schematic diagram for the parametric variation control method

241

242

G. Chen et al.

-v-

R ^ V,;_

Fig.6.5 Chua's circuit

Fig.6.6 The double scroll chaotic attractor of Chua's circuit

x2

1.5

— 0.5

-2.0

*1

—1 .O

O.O

X1

Fig.6.7 Stabilized an induced period-l stochastic resonance (a) D^23.98, period-5 (b) ZV=24.03, chaos

1 O

2.0

Introduction 1 .80

-i

0.80

-

-0.20

-

•1.20

-

to Chaos Control and Anti-Control

X-

-2.20

2.20

2.5

*2

1 — 1 .20

1 —0.20 Xi

1— O.80

1. 1.80

1

1 -5

O 5

-

— o.s —

1 .5

-2.5 -2.5

I -1.5

I -0.5

1 0.5

1 1.5

Xi

Fig.6.8 An example of anti-control of chaos (a) Dr=23.98, period-5 (b) ZV=24.03, chaos

1 2.5

243

244

G. Chen et al.

x2 0.0895

4.1

0.0875

0.0855

0.0835 0.2980

0.3070

0.3160Xl

Fig.6.9 Periodic oscillation of X]

X2 0.085000

4.2(a)

0.084000

0.083000

0.082000 0.286000

0.289500

, X

0.293000

Introduction

to Chaos Control and Anti-Control

x2 0.0890

4.2(b)

0.0880

0.0870

0.0860

0.08501. 0.29700

0.30400

0.31100

Q.ZOE + 00

X

4.2(c)

0.20E + 00

0.10.E + 00

0.20.E - 02

0

0.002

0.004

0.006

0.008 '

Fig.6.10 Anticontrol of chemical chaos by using colored noise input

245