Pergamon

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,7. ,~)'anklhl hTst. Vol. 335B, No. 5, pp. 827 839. 1998 1998 The Franklin Inslitute

Pergamon

P1 !: SO016-0032(97)000134i

PublishedbyElsevierSci. . . . . Ltd Printed in Great Britain 0Ill 6 003298 S 19.11(1+ 0.00

Optimal Discrete-time Controlfor Non-linear Cascade @stems b ) ' W A S S I M M. H A D D A D ,

V I J A Y A - S E K H A R C H E L L A B O 1 N A , J E R R Y L. F A U S Z

School o[ A erospace Engineering, Georgia Institute q[' Technology, Atlanta, GA 30332-0150, U.S.A. and CHAOUKIABDALLAH Electrical and Computer Engineering Department, University 0[ N e w Mexico, Albuquerque, N M 87131, U.S.A. ( Receired in final f o r m 28 Norember 1996," accep ted 11 February 1997)

ABSTRACT: bl this paper we develop an optbnalio'-based.fi'amework Jbr designing controlh, rs Jbr discrete-l#ne non-linear cascade systems. Slwc(fically, usinq a non-linear non-quadratic opt#hal control Ji'amework we develop a family O[ylohally stabilizing backstepph, q-O'pe controllers parameterized hy the cost .[unctional that is minhnized. Furthermore, it is shown that the control L l ' a p u n o t " fimction guaranteehTg closed-loop stabilio' is a solution to the slea~O'-state Bellman equation.ft," the controlled a3,stem attd thus guarantees both optimali O' aml stabiliO'. 1998 The Franklin Institute. Published by Elsevier Science Ltd Kevwords: nonlinear control: discrete-time; Bellman principal of optimality; control Lyapunov functions; cascade systems.

Introduction

Since most physical processes evolve naturally in continuous time, it is not surprising that the bulk of non-linear control theory has been developed for continuous-time systems. Nevertheless, it is the overwhelming trend to implement controllers digitally. Despite this fact, the development of nonlinear control theory for discrete-time systems has lagged its continuous-time counterpart. This is in part due to the fact that concepts such as zero dynamics, normal forms, and minimum phase are much more intricate for discrete-time systems. For example, in contrast to the continuous-time case, technicalities involving passivity analysis tools needed to prove global stability via smooth feedback controllers (1) as well as system-relative degree requirements (2) are more involved in the discrete-time case. Recent work involving differential geometric methods (3-7) employing concepts of zero dynamics and feedback linearization have been applied to discrete-time systems. In particular, the results in Refs (3-7) parallel continuous-time results on linearization 827

828

W . M . H a d d a d et al.

of non-linear systems via state and output feedback. However, as in the continuoustime case, these techniques cancel out system non-linearities and may therefore lead to inefficient designs since the resulting feedback linearizing controller may generate a large control effort to cancel beneficial non-linearities. Backstepping control for continuous-time systems has recently received a great deal of attention in the non-linear control literature (8-12). The popularity of this control methodology can be explained in a large part due to the fact that it provides a framework for designing stabilizing non-linear controllers for a large class of nonlinear cascade systems. Specifically, this framework guarantees stability by providing a systematic procedure for finding a control Lyapunov function for the closed-loop system and choosing the control such that the time derivative of the control Lyapunov function along the trajectories of the closed-loop dynamic system is negative. Furthermore, the controller is obtained in such a way that system non-linearities, which may be useful in attaining performance objectives, need not be canceled as in state or output feedback linearization techniques. Even though discrete-time recursive backstepping techniques have not been developed, the closest discrete-time analog to backstepping is given in Refs (2, 13, 14). Specifically, in Refs (2, 13, 14) discrete-time passivity analysis tools are used to construct control Lyapunov functions guaranteeing global asymptotic stability for block cascade discrete-time systems. However, as in the continuous-time case, no analytical measure of performance or notions of optimality have been shown to exist for the controllers derived in Refs (2, 13, 14). In this paper we develop an optimality-based control design theory for non-linear discrete-time cascade systems. The key motivation for developing an optimal nonlinear control theory framework for discrete-time cascade systems is that it provides a family of candidate controllers parameterized by the cost functional that is minimized. In order to address the optimality-based non-linear control problem we use the nonlinear-non-quadratic optimal control framework developed in Ref. (15). The basic underlying ideas of the results in Ref. (15) rely on the fact that the steady-state solution of the discrete-time Bellman equation is a control Lyapunov function for the nonlinear controlled system, thus guaranteeing both optimality and stability (16). To address the optimal control problem for non-linear cascade systems, we specialize the non-linea~non-quadratic performance criterion given in Ref. (15) to a non-linear non-quadratic function of the state plus a linear and quadratic function of the feedback control. The non-linear feedback control law is then chosen so that the Bellman optimality conditions are satisfied. This is accomplished by choosing the controller such that the control Lyapunov difference is negative along the closed-loop system trajectories while providing sufficient conditions for the existence of asymptotically stabilizing solutions to the Bellman equation. Thus, our results allow us to derive globally asymptotically stabilizing backstepping-type controllers for discrete-time nonlinear systems that minimize a corresponding non-linea~non-quadratic performance functional. In this paper we use the following standard notation. Let iR denote the set of real numbers, let R ...... denote the set of real n x m matrices, let ~ ..... (resp., iP..... ) denote the set of n x n non-negative (resp., positive) definite matrices, and let ,,.t" denote the set of non-negative integers. Furthermore, A > 0 (resp., A > 0) denotes the fact that the Hermitian matrix A is non-negative (resp., positive) definite and A_> B (resp., A > B)

Optimal Controljor Cascade Systems denotes the fact that A - B > 0 matrix.

829

(resp., A - B > 0 ) . Finally, let I,, denote the n x n identity

Optimal Control for Discrete-time Nonlinear Systems In this section we restate a key theorem from Ref. (15) involving a non-linear nonquadratic optimal control framework for addressing optimal controllers of discretetime non-linear systems. In addition, we specialize this result to affine systems with performance criteria involving a linear and quadratic function of the feedback control. We begin by considering the problem of characterizing feedback control laws that minimize a given non-linear-non-quadratic performance functional. For this problem we consider the controlled discrete-time system

x(k+ 1) = F(x(k),u(k)), x(0) = x0, ke, ~,

(1)

where x ( k ) e ~ c R" is the state vector, ~ is an open set with 0 e ~ , u ( k ) e ~ c R"' is the control input, c6 is an arbitrary set with 0 e c6, and F : ~ x ~6'~N" satisfies F(0,0) = 0. The control input u(-) in Eq. (1) is restricted to the class of admissible controls consisting of measurable functions u ( ' ) such that u(k)e4g, for all ke~,f', where the control constraint set 5# c ~. is given with 0e~//. A measurable mapping ~ b : ~ J / ! satisfying 4~(0) = 0 is called a control law. Furthermore if u(k) = ~b(x(k)), where q5 is a control law and x(k), ke~.~', satisfies Eq. (1), then u(. ) is called aJeedback control law. With the mapping ~b:~--*~! the closed-loop system is given by

x(k+ 1) = F(x(k),d?(x(k))), x(O) = xo, ke,.t'.

(2)

For the non-linear controlled system (1) we seek a control law u(k) = (o(x(k)) such that for the closed-loop system (2) the non-linear-non-quadratic performance functional

J(xo,u("))& ~ L(x(k),u(k)), k

(3)

0

is minimized. Sufficient conditions for optimality can be characterized by the steadystate Bellman equation. The performance integrand L is not prescribed a priori, but rather it interacts with the non-linear system, the control Lyapunov function of the closed-loop system, and the stabilizing control law via the Bellman equation (15). In particular, as shown in Ref. (16) and further clarified in Ref. (15), asymptotic stability of the closed-loop system is guaranteed by means of a control Lyapunov function which can be seen to be the solution to the steady-state discrete-time Bellman equation. The control Lyapunov function also plays a central role in the structure of the optimal non-linear feedback control law which in turn minimizes a non-linear-non-quadratic functional. To address the problem of characterizing optimal non-linear discrete-time feedback controllers, let L : ~ x'~#--*R and, for each initial condition x0e~, define the set of asymptotically stabilizing controllers for the non-linear system F ( . , • ) by

W. M, Haddad et al.

830

5~(x0) A {u(" ): u(" ) is admissible and x(. ) given by Eq. (1) satisfies x(k)--+O as k ~ ov }.

(4) Theorem 2.1. (15) Consider the controlled system (1) with performance functional ~ and a control law ~ b : ~ ° k ' such that

(3). Assume there exists a function V : ~

V(0) = 0,

V(x) > O, x ~ ,

(5)

x # O,

(6)

q~(O) = O,

(7)

V(F(x,~(x)))--V(x)_O, x ~ ,

uE°g,

(10)

where

H(x,u) & L(x,u) + V(F(x,u)) -- V(x).

(11)

Then, with the feedback control law u(. ) = ~b(x( • )), there exists a neighborhood D0 c of the origin such that if x 0 ~ 0 the solution x(k) = O, k~A ~, of the closed-loop system (2) is locally asymptotically stable. Furthermore,

J(xo,O(x(" ))) = V(xo),

(12)

and, ifx0e~0, the feedback control law u(.) = q~(x(- )) minimizes J(xo,u(" )) in the sense that

J(xo,O(x(')))=

min

u( • )~.0,

(20)

x e ~ n, x ¢ 0 ,

( x ) - - ~ g ( x ) ( R 2 + P z ( x ) ) '[LT(x)+PJ2(X)] - - V ( x ) < 0 ,

(21) xe~", xv~0,

v q ( x ) +g(x)u) = v(f(x)) + P,2(x)u + #P~(x)u,

(22)

(23)

where u is admissible, and

V ( x ) ~ oc as [Ixl[ ~ oo.

(24)

Then the solution x(k) = 0, k e A ' , o f the closed-loop system

x ( k + l) = f(x(k)) +g(x(k))cb(x(k)), x(0) = x0, ke,/V,

(25)

is globally asymptotically stable with the feedback control law 1

q~(x) = - ~ (R2 + P2(x))

l

[L,(x) + P;2(x)] T,

(26)

and the performance functional (17), with

Lffx) = ~br(x)(R2 + P2(x))(a(x) - V(f(x)) + V(x),

(27)

is minimized in the sense that

J(xo,(a(x(" ))) =

min

u( • )~5t'(x o )

J(xo,u(" )), Xoe~".

(28)

Finally,

J(xo,C~(x('))) = V(xo), x0e~".

(29)

Proof'. The result is a direct consequence o f T h e o r e m 2.1 with F(x,u) = J(x) +g(x)u,

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L(x,u) = L~(x)+L2(x)u+uTR2u, 6j = ~,,, and °k' = ~m. Specifically, with Eqs (15), (16) and (23) the Hamiltonian given by Eq. (11) has the form

H(x,u) = L,(x) + V(f(x)) + [L2(x) + P,2(x)lu + uT(R2 + P2(x))u-- V(x).

(30)

Now, the feedback control law (26) is obtained by setting OH(x,u)/c~u = 0. With Eq. (26), it follows that Eqs (18)-(22) imply Eqs (5) (8). Next, with L 6 x ) given by Eq. (27) and ~b(x) given by Eq. (26), Eq. (9) holds. Finally, since H(x,u) = H(x,u) - H(x,O(x)) = ( u - O(x))T(R2 + P2(x))(u -- c~(x)),

(31)

and R2+P2(x)>O, xe~", condition (10) holds. The result now follows as a direct consequence of Theorem 2.1. Remark 2.3. Note that Eq. (22) is equivalent to AV(x)& V(f(x)+g(x)~a(x))-- V(x)