~ Pergamon Copyright t~ 1996 The Franklin …of optimal discrete-time nonlinear analysis and feedback control with nonlinear non- quadratic performance criteria. Specifically, we consider

~ Pergamon J. Franklin Inst. Vol. 333B, No. 6, pp. 849 860, 1996

Copyright t~ 1996 The Franklin Institute P I h S0016--0032(96)00057-9 Published by Elsevier Science Ltd

Printed in Great Britain 0016M)032/96 $15.00 + 0.00

Discrete-time Nonlinear An@sis and Feedback Control with Nonquadratic Performance Criteria

by W A S S I M M. H A D D A D a n d V I J A Y A - S E K H A R C H E L L A B O I N A

School of Aerospace Engineering, Georgia Institute of Technology, Atlanta, GA 30332-0150, U.S.A.

(Received 13 March 1996," accepted 26 April 1996)

ABSTRACT : In this paper we develop a unified framework to address the problem of discrete-time optimal nonlinear analysis and feedback control. Asymptotic stability of the closed-loop nonlinear system is 9uaranteed by means of a Lyapunov function which can clearly be seen to be the solution to the steady-state form of the discrete-time Bellman equation thus 9uaranteein9 both stability and optimality. The overall framework provides the foundation for extendin 9 discrete-time linea~ quadratic controller synthesis to nonlinear-nonquadratic problems. Copyright © 1996 Published by Elsevier Science Ltd

L Introduction

Under certain conditions nonlinear controllers offer significant advantages over linear controllers. In particular, if the plant dynamics and/or system measurements are nonlinear (1~4), the plant/measurement disturbances are either nonadditive or non- Gaussian, the performance measure considered is nonquadratic (5-15), the plant model is uncertain (16-21), or the control signals/state amplitudes are constrained (22-25), nonlinear controllers yield better performance than the best linear controllers. In Ref. (26) a unified framework for continuous-time nonl inea~nonquadrat ic problems was presented in a simplified and tutorial manner. The basic underlying ideas of the results in Ref. (26) are based on the fact that the steady-state solution of the Hamil ton-Jacobi- Bellman equation is a Lyapunov function for the nonlinear system thus guaranteeing both stability and optimality.

In this paper we extend the framework developed in Ref. (26) to address the problem of optimal discrete-time nonlinear analysis and feedback control with nonlinear nonquadratic performance criteria. Specifically, we consider discrete-time autonomous nonlinear regulation in feedback control problems on an infinite horizon involving nonl inea~nonquadrat ic performance functionals. As in the continuous-time case, the performance functional can be evaluated in closed-form as long as the nonl inea~ nonquadratic cost functional considered is related in a specific way to an underlying Lyapunov function that guarantees asymptotic stability of the nonlinear closed-loop system. This Lyapunov function is shown to be the solution of the discrete-time

849

850 W. M. Haddad and V.-S. Chellaboina

steady-state Bellman equation arising from the principle of optimality in dynamic programming and plays a key role in constructing the optimal nonlinear control law. The overall f ramework provides the foundation for extending discrete-time linear- quadratic synthesis to nonlinear-nonquadrat ic problems.

The contents of the paper are as follows. In Section 2 we consider a nonlinear discrete-time system with a nonl inea~nonquadra t ic performance functional evaluated over the infinite horizon. The performance functional is then evaluated in terms of a Lyapunov function that guarantees asymptotic stability of the nonlinear discrete-time system. This result is then specialized to the l inea~quadrat ic case as well as to a multilinear setting. In Section 3 we state a discrete-time nonlinear optimal control problem. Theorem 3.1 presents sufficient conditions that characterize an optimal nonlinear feedback controller that guarantees asymptotic stability of nonlinear discrete- time systems. The results are then specialized to the l inear-quadratic case to draw connections with the discrete-time linear-quadratic regulator problem. In Section 4 we close the paper with conclusions.

In this paper we use the following standard notion. Let N denote real numbers and let ~" × m denote real n x m matrices. Let N" ×" (P" × n) denote n x n non-negative (positive) definite matrices and let JV denote the set of non-negative integers. Furthermore, A >~ 0 (A > 0) denotes the fact that the Hermitian matrix A is non-negative (positive) definite and A/> B (A > B) denotes the fact that A - B >~ 0 (A - B > 0). Finally, A ® B denotes the Kronecker product of matrices A and B.

A scalar function 0 : ~" --. N is q-multilinear if q is a positive integer and ~b(x) is a • . i n linear combination of terms of the form X',lX~... x , , where iJ is a non-negative integer

for j = 1 . . . . . n, and i~ + i2 + " " + i, = q. Furthermore, a q-multilinear function ~( ' ) is non-negative definite (resp. positive definite) if ~b(x) ~> 0 for all xE R" [resp., O(x) > 0 for all nonzero x s ~"]. Note that if q is odd then O(x) cannot be positive definite. I f ~,(') is a q-multilinear function then ~(-) can be represented by means of Kronecker products, that is, ~b(x) is given by ~b(x) = IJllx[q] where ~ge NI ×,~ and x tqj ~- x ® x ® . . . ® x (q times).

IL Stability Analysis o f Discrete-time Nonlinear Systems

In this section we present sufficient conditions for stability for a nonlinear discrete- time system. In particular we consider the problem of evaluating a nonlinear-nonquadratic performance functional depending upon a nonlinear discrete-time differential equation. It is shown that the cost functional can be evaluated in closed-form as long as the cost functional is related in a specific way to an underlying Lyapunov function that guarantees stability.

In this paper we restrict our attention to time-invariant infinite horizon systems. For the following result, let ~ a R" be an open set, assume 0 e 9 , let L : ~ ~ N, and let f : ~ ~ R" be such that f(0) = 0.

Theorem I Consider the nonlinear discrete-time system

x ( k + l ) = f ( x ( k ) ) , x(O) = x 0 , k~JV', (1)

with performance functional

Discrete-time Nonlinear Analysis and Feedback Control 851

J(xo) & ~ L(x (k ) ) . (2) k=0

Furthermore, assume there exists a function V: ~ --+ ~ such that

V(0) = 0, (3)

V ( x ) > O , x s g , x v ~0, (4)

V ( f ( x ) ) - V(x) < O, x s ~ , x ¢ O, (5)

L(x) + V( f ( x ) ) - V(x) = O, x 6 9 . (6)

Then there exists a neighbourhood of the origin D0 c ~ such that if x0s D0, then x(k ) = O, k s JV, is a locally asymptotically stable solution to Eq. (1), and

Finally, if ~ -- [~" and

J(xo) = V(xo). (7)

V(x) ~ oo as IIxll -~ ~ , (8)

then the solution x(k) , k s JV, of Eq. (1) is globally asymptotically stable.

Proof'. Let x(k) , k s J V ' , satisfy Eq. (1). Then

A V ( x ( k ) ) ~= V ( x ( k + I ) ) - V(x(k) ) = V ( f ( x ( k ) ) - V(x(k) ) , k s J V . (9)

Hence it follows from Eq. (5) that

A V ( x ( k ) ) < O, k s J V , x (k ) ~ O. (10)

Thus, from Eqs (3), (4) and (10) it follows that V(') is a Lyapunov function (27) for Eq. (1), which proves local asymptotic stability of the solution x (k ) = O, k s ~ U . Consequently, x ( k ) - ~ 0 as k ~ ~ for all initial conditions x0E D0 for some neighborhood of the origin D0 c 9 . Now Eq. (9) implies that

o = -AV(x (k ) )+ V( f (x (k ) ) ) - V(x(k)), k s Y ,

and hence, using Eq. (6),

L ( x ( k ) ) = - A V(x(k) ) + L (x (k ) ) + V ( f ( x ( k ) ) ) - V (x (k ) ) = - A V(x (k ) ).

Now, summing over [0, N] yields

N

L(x (k ) ) = - V ( x ( N ) ) + V(xo). k = 0

Letting N --+ ~ and noting that V(x(N)) ~ 0 for all x0 e D0 yields J(xo) = V(xo). Finally, for ~ = ~" global asymptotic stability is a direct consequence of the radially unbounded condition (8) on V(x). •

Rem ark 2.1 The key feature of Theorem I is that it provides sufficient conditions for stability of

discrete-time nonlinear systems. Furthermore, the nonlinear-nonquadratic per-


formance functional is given in terms of an underlying Lyapunov function that guarantees asymptotic stability.

The following corollary specializes Theorem I to discrete-time linear systems.

Corollary 2.1 Let A ~ ~n×n and R e ~n×n. Consider the linear system

x ( k + l ) = A x ( k ) , x ( 0 ) = X o , k~JV, (11)


J(xo) & ~ x T ( k ) R x ( k ) . (12) k - O

Furthermore, assume there exists P e 0 z" ×" such that

P = A T p A + R . (13)

Then x (k ) = O, k ~ ~/', is a globally asymptotically stable solution to Eq. (11) and

J(xo) = x~Pxo. (14)

Proof'. The result is a direct consequence of Theorem I with f ( x ) = A x , L ( x ) = x T Rx , V(x) = xXPx, and ~ = Rn. Specifically, conditions (3) and (4) are trivially satisfied. Now V ( f ( x ) ) - - V ( x ) = x T ( A T p A - - P ) x and hence it follows from Eq. (13) that L(x ) + V ( f ( x ) ) - V(x) = 0 so that all the conditions of Theorem I are satisfied. Finally, since V(x) is radially unbounded, the solution x (k ) = O, k e Y , of Eq. (11) is globally asymptotically stable. •

Note that if Eq. (6) holds, then Eq. (5) is equivalent to

L(x ) > 0 x e ~ , x # O. (15)

More generally, assume A is asymptotically stable, let P be given by Eq. (13), and consider the case in which L, f a n d V are given by

L ( x ) = x T R x + h (x) ,

f ( x ) = A x + N(x ) ,

V(x) = xT p x + o ( X ) ,

(16)

(17)

(18)

where h : ~ ~ N and 9 : ~ ~ N are nonquadratic and N : ~ ~ Nn is nonlinear. In this case Eq. (6) holds if and only if

0 = x T R x + h (x) + x TAT P A x + N T (x) P N ( x ) + 2x TAr P N ( x ) -- x T P x

+ 9 ( A x + N ( x ) ) - - g ( x ) , x e ~ , (19)

or, equivalently,

0 = xT(ATPA -- P + R ) x + 9 ( A x + N ( x ) )

- - 9 ( x ) + h ( x ) + N T ( x ) P N ( x ) + 2 x T A T P N ( x ) , x ~ . (20)

If A is asymptotically stable, then we can choose P to satisfy Eq. (13) as in the l inea~

D i s c r e t e - t i m e N o n l i n e a r A n a l y s i s and F e e d b a c k C o n t r o l 853

quadratic case. Now, suppose N ( x ) =- 0 and let P satisfy Eq. (13). Then Eq. (20) specializes to

0 = 9 ( A x ) - - 9 ( x ) + h ( x ) , x ~ . (21)

Next, given h('), we determine the existence of a function 9(') satisfying Eq. (21). To this end, we focus our attention on multilinear functions for which Eq. (21) holds with @ = ~ " .

L e m m a 2.1 Let A ~ ~ ' × " be asymptotically stable and let h : ~" ~ ~ be a q-multilinear function.

Then there exists a unique q-multilinear function g ' ~" ---, ~ such that

0 = 9 ( A x ) - g ( x ) + h ( x ) , x ~ ~ ' . (22)

Furthermore, if h ( x ) is non-negative (resp., positive) definite, then g ( x ) is non-negative (resp., positive) definite.

Proof' . Let h ( x ) = ~ x [q] and define g ( x ) & F x [ql, where F & --tIJ(A[q]--I[n ql) 1. Note t h a t A [ql - I ~ q~ is invertible since A, and hence A tqj, is discrete-time asymptotically stable. Now note that for all x e N",

9 ( A x ) - 9 ( x ) = F ( ( A x ) tql - x tql) = F(Atklxfq] _xtql)

---- F ( A tql -- I [ f ) x [ql = -- t~x[q] = -- h ( x ) .

To prove uniqueness, suppose that ~(x) = f 'x Iql also satisfies Eq. (22). Then it follows that

F ( A [q] - - I[nql)x [ql = f ' ( A [q] - - I[nq])x [q], x ~ ~n.

Since A lql is discrete-time asymptotically stable and (A[q]) j = (A J) [q], it follows that, for all x ~ ~",

F x tql = F ( A tql -- I~ql)(A rql - - I f ) - ' x fql

= _ F(A~ql --I~q I) ~ (Alql)Jx Iql j = l

= -- F(A Eql - - I f ) ~ (w)tqlx tql j = l

= - ~ F ( A tqj - Ilql)(AJx)[ql j = l

= _ ~ f f ( A tqj _ I~ql)(Wx)tql j = l

= Fx[q],

which shows that g(x) = 9(x), x ~ R". Finally, if h ( x ) is non-negative definite, then it follows that, for all x ~ IW,


g(x) = -Ud(Atql-II,, ql) lxtqJ = ud ~ (A[ql)Jx [q] = ufl (AJ)[qlx [q] = °fl(AJx)[q] >~ O. j = l j=~ j = l

If, in addition, h(x) is positive definite, then g(x) is positive definite. • To illustrate Lemma 2.1, consider the linear system (11) and let h(x) be a positive-

definite q-multilinear function, where q is necessarily even. Furthermore, let g(x) be the positive-definite q-multilinear function given by Lemma 2.1. Then, since g(Ax) --g(x) < O, x~ ~", x ~ O, it follows that g(x) is a Lyapunov function for Eq. (11). Hence Lemma 2.1 can be used to generate Lyapunov functions of specific structure.

Suppose now that h(x) in Eq. (16) is of the more general form

h(x) = ~ h2v(x), (23) v ~ 2

where, for v = 2 . . . . . r, h2~." R n ~ ~ is a non-negative-definite 2v-multilinear function. Now, using Lemma 2.1, let g2v '~n~ ~ be the non-negative-definite 2v-multilinear functions satisfying

and define

0 =gzv(Ax)-g~,,(x)+hz,~(x), x 6 ~ " , v = 2 . . . . . r, (24)

g(x) ~= ~ gz,~(x). (25) v--2

Now summing eq. (24) over v yields Eq. (22). Since Eq. (6) is satisfied with L(x) and V(x) given by Eqs (16) and (18), respectively, Eq. (7) implies that

J(xo) = x~Pxo +g(xo). (26)

As another illustration of condition (22), suppose that V(x) is constrained to be of the form

V(x) = xr px + (xT Mx) 2, (27)

where P satisfies Eq. (13) and M is an n×n symmetric matrix. In this case 9(x) = (xTMx) 2 is a non-negative-definite 4-multilinear function. Then Eq. (22) yields

h(x) = - (x v (AT MA + M)x) (x r (A'r MA - M)x). (28)

If/~ is an n × n symmetric matrix and M is chosen to satisfy

M = ATMA + 1~, (29)

then Eq. (28) implies that h(x) satisfying Eq. (22) is of the form

h(x) = (x T (ATMA + M)x) (xV/~x). (30)

Note that if/~ is non-negative definite, then M is also non-negative definite and thus h(x) is a non-negative-definite 4-multilinear function. Thus, if V(x) is of the form (27), and L(x) is given by


L(x) = xT Rx + (xT (A~ MA + M)x)(xT ~qx), (31)

where M and R satisfy Eq. (29), then condition (22), and hence Eq. (6), is satisfied. The following lemma generalizes the above results to general polyquadratic functionals.

Lemma 2.2 Let A ~ ~" ×" be asymptotically stable, R 6 P" ×", and/~q E: ~ n × n q : 2 . . . . . r. Consider

the linear system (11) with performance functional

J(xo) a= xV(k)Rx(k) 4- x'r(k)Rqx(k)) k=0

q 1 × E ( x T ( k ) M q x ( k ) ) J - I ( x T ( k ) A T M q A x ( k ) ) q - j • (32)

j=l

Furthermore, assume there exist P ~ [P" ×" and Mq ~ I~" ×", q = 2 . . . . . r, such that

P = A T p A + R , (33)

Mq = ATMqA +l~q, q = 2 . . . . . r. (34)

Then x(k) = O, k E JV, is a globally asymptotically stable solution to Eq. (11) and

J(xo) = xT exo 4- ~ (XT MqXo) q. (35) q-2

Proof'. The result is a direct consequence of Theorem I with

f (x) = Ax, L(x) = xTRx4- X T I ~ q X X T M q X ) J - i ( x T A T M q A X ) q j , q=2 L j=l

r V(x) = xT Px + ~ (XT MqX) q,

q=2

and ~ = R". Specifically, conditions (3) and (4) are trivially satisfied. Now

V(f(x) ) - V(x) = xT (AT pA -- P)x

4- q=2L~ IxT(ATMqA--Mq)x j = l ~" (XTMqx)J-I(xTATMqAX)q J]

and hence it follows from Eqs (33) and (34) that L(x) + V(f(x)) -- V(x) = 0 so that all the conditions of Theorem I are satisfied. Finally, since V(x) is radially unbounded, Eq. (11) is globally asymptotically stable.

Remark 2.2 Lemma 2.2 requires the solutions of r - 1 Lyapunov equations in Eq. (34) to obtain

the nonquadratic cost. However, if/~q --/~2, q = 3 . . . . . r, then Mq = M2, q = 3 . . . . . r, satisfies Eq. (34). In this case we require the solution of one Lyapunov equation in Eq. (34).


II. Optimal Control for Discrete-time Nonlinear Systems

In this section we consider a control problem involving a notion of optimality with respect to a nonlinear-nonquadrat ic cost criterion. The optimal feedback controllers are derived as a direct consequence of Theorem I and provide a transparent gen- eralization of the continuous-time Hamil ton-Jacobi-Bel lman conditions for time- invariant, infinite horizon problems for addressing optimal controllers of nonlinear discrete-time systems. To address the discrete-time optimal control problem let ~ c ~ be an open set and let c£ c ~", where 0 • ~ and 0 • ~, and let f : ~ x c~ ~ ~n be such thatJ~(0, 0) = 0. Next, consider the controlled system

x ( k + l ) = f ( x ( k ) , u ( k ) ) , x ( 0 ) = X o , k • J ~', (36)

where the control u(-) is restricted to the class of admissible controls consisting of measurable functions u(') such that u(k) • ~ll for all k e ~ where the control constraint set ~ / /c c£ is given. We assume 0e q/. A measurable mapping q~:~ ~ 9/ satisfying 4~(0) = 0 is called a control law. I f u(k) = O(x(k)), where ~b(-) is a control law and x(k) satisfies Eq. (36), then u(') is called a feedback control. Given a control law qS(.) and a feedback control u(k) = O(x(k)), the closed-loop system has the form

x ( k + l ) =f (x(k) ,O(x(k) ) ) , x(O) = Xo, k e X . (37)

Next we present our main theorem for characterizing discrete-time feedback controllers that guarantee stability for a nonlinear discrete-time system and minimize a nonl inea~nonquadra t ic performance functional. For the statement of this result let /S: ~ x ~k' ~ ~ and define the set of asymptotically stabilizing controllers for the nonlinear system f( . , . ) by 5e(x0) ~ {u( ' ) :u( ' ) is admissible and x(') given by Eq. (36) satisfies x(k) ~ 0 as k ~ Go}.

Theorem H Consider the controlled system (36) with performance functional

J(Xo, u(.)) ~= ~ £(x(k), .(k)), (38) k 0

where u(') is an admissible control. Assume there exists a function V: ~ ~ [R and control law ~b : @ ~ ~/! such that

where

v(0) = 0,

V ( x ) > O , x ~ , x ¢ O ,

q~(0) = o,

v(f(x, ~(x) ) ) - V(x) < O, x • ~, x ~ O,

H(x,O(x)) = O, x • ~ ,

H(x,u) >~ O, x • ~ , uedll,

(39)

(4o)

(41)

(42)

(43)

(44)

H(x, u) ~- if(x, u) + V(f (x , u)) - V(x). (45)


Then, with the feedback control u(') = ~b(x(')), there exists a neighborhood of the origin ~0 c ~ such that if Xo e ~0, the solution x(k) = O, k e ~ , of the closed-loop system (37) is locally asymptotically stable. Furthermore,

J(xo, q~(x('))) = V(xo). (46)

In addition, if Xo e ~o then the feedback control u(') = ~b(x(')) minimizes J(xo, u(')) in the sense that

J(xo, q~(x('))) = min J(xo, u(')). (47) u(-)~,,r(x o)

Finally, if ~ = Nn and

V(x) ~ ac as IIxll ~ ~ , (48)

then the solution x(k) = O, k e Jff, of the closed-loop system (37) is globally asymptotically stable.

Proof'. Local and global asymptotic stability is a direct consequence of Eqs (39)-(42) and (48) by applying Theorem ! to the closed-loop system (37). Furthermore, using Eq. (43), condition (46) is a restatement of Eq. (7) as applied to the closed-loop system. Next, let u(-) e 5~(x0) and let x(') be the solution of Eq. (36). Then it follows that

0 = - A V(x(k))+ V(f(x(k), u (k ) ) ) - V(x(k)).

Hence

ff~(x(k ), u(k) ) = - A V(x(k) ) + ff~(x(k), u(k ) ) + V(ff(x(k ), u(k ) ) - V(x(k ) )

= - A V(x(k)) + H(x(k), u(k)).

Now using Eq. (45) and the fact that u(-) e 5t(Xo), it follows that

J(xo, u(')) = ~ [ - AV( x ( k ) ) + H(x(k), u(k))] k = O

= - lira V(x(k)) + V(xo) + ~ H(x(k), u(k)) k = O

= V(xo)+ ~ H(x(k), u(k)) k = 0

>1 V(xo)

= a~(Xo, q~(x( ' ) ) ,

which yields Eq. (47). •

Remark 3.1 In order to relate Theorem II to the Bellman equation arising from the principle of

optimality in dynamic programming, recall that the principle of optimality has the form


VN k,N(X(N--k))

= rain {E,(x(N--k), u(N--k)) + VN-<k 1).y(f(x(N-k), u(N--k)))}, ke JV',

which characterizes the optimal control for discrete-time time-varying systems on a finite or infinite discrete-time interval. In the time-invariant infinite-horizon case, the Bellman equation reduces to the algebraic relations (43) and (44).

Remark 3.2 Note that Theorem II guarantees optimality with respect to the set of admissible

stabilizing controllers 5P(x0). However, it is important to note that an explicit charac- terization of 5P(x0) is not required.

Remark 3.3 Note that Theorem II yields an optimal stabilizing feedback control law u = ~b(x)

that is independent of the initial condition x0. Next, we specialize Theorem II to discrete-time linear systems and provide con-

nections to the discrete-time l inea~quadratic regulator problem. For the following result let A • ~"×", Be ~,×m, R~ • P"×", and R2• pm×m be given.

Corollary 3.1 Consider the discrete-time linear system

x ( k + l ) = A x ( k ) + B u ( k ) , x (0 )=X o , ke~A/', (49)


J(x0, u(')) & ~ [xT(k)R,x(k) + uT(k)R2u(k)], (50) k--O

where u(') is an admissible control. Furthermore, assume there exists P • P"×" such that

P =- A-rPA + R~ - ATpB(R2 + BTpB)- l BrpA. (51)

Then Eq. (49) is globally asymptotically stable for all x0 ~ [~", with the feedback control u = ~b(x) & --(R2 +BTpB) IBTpAx, and

J(xo, q~(x('))) = XToPXo. (52)

Furthermore,

J(xo,~(x('))) = min J(x0,u( ')), (53) u(')~J(x o)

where 5Z(Xo) is the set of asymptotically stabilizing controllers for Eq. (49) and Xo E ~".

Proof'. The result is a direct consequence of Theorem II with f ( x , u ) = Ax+Bu, L(x, u) = xTR~x+uTR2u, V(x) = xTpx, ~ = ~" and ~//= ~". Specifically, conditions (39) and (40) are trivially satisfied. Next, it follows from Eq. (51) that H(x,c~(x)) = 0 and hence V ( f ( x , C ( x ) ) - V ( x ) < 0 for all x 4: 0. Thus, H(x,u) = H ( x , u ) - H ( x , ck(x)) = [u--49(x)]T(R2 +BTpB)[u-(~(x)] >, 0 so that all of the conditions of Theorem II are satisfied. Finally, since V(x) is radially un-


bounded, the solution x ( k ) = 0 , k ~ J ~ , of Eq. (49) with u ( k ) = ( a ( x ( k ) ) = - (R2 + BTpB) 1BTpAx(k) , is globally asymptotically stable. •

Remark 3.4 The optimal feedback control law qS(x) in Corollary 3.1 is derived using the properties

of H(x, u) as defined in Theorem II. Specifically, since H(x, u) = xV R1 x + uX R2 u + (Ax + B u ) T p ( A x + B u ) - - x T p x it follows that ~2H/~u2 = R2 + B T p B > 0. Now, ~H/~u = 2(R 2 + B T p B ) u + 2 B T P A x = 0 gives the unique global minimum of H(x, u). Hence, since qS(x) minimizes H(x, u) it follows that ~b(x) satisfies ~H/6u = 0 or, equivalently, qS(x) = --(R2 + B T p B ) IBTpAx.

IV. Conclusion

A unified framework was developed to address the problem of discrete-time optimal nonlinear analysis and feedback with nonlinear-quadratic performance criteria. The overall f ramework provides the foundation for generalizing discrete-time l inear~luad- ratic controller synthesis to nonlinear-nonquadrat ic problems.

Acknowledgements This research was supported in part by the National Science Foundation under Grant ECS-

9496249.

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~ Pergamon Copyright t~ 1996 The Franklin …of optimal discrete-time nonlinear analysis and feedback control with nonlinear non- quadratic performance criteria. Specifically, we consider