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On direct adaptive control design for nonlinear discrete-time uncertain systems
ARTICLE IN PRESS
Journal of the Franklin Institute 345 (2008) 119–135 www.elsevier.com/locate/jfranklin
On direct adaptive control design for nonlinear discrete-time uncertain systems$ Simon Hsu-Sheng Fu, Chi-Cheng Cheng Department of Mechanical and Electro-Mechanical Engineering, National Sun Yat-Sen University, 70 Lien-hai road, Kaohsiung 804, Taiwan, ROC Received 26 December 2006; received in revised form 14 July 2007; accepted 31 July 2007
Abstract In this paper, we develop a direct adaptive control framework for adaptive stabilization of the MIMO nonlinear uncertain systems, which can be represented as discrete-time normal form with input-to-state zero dynamics. The framework is Lyapunov-based and guarantees partial stability of the closed-loop systems, such that the adaptation of the feedback gains can stabilize the closed-loop system without the knowledge of the system parameters. In addition, our results show that the adaptive feedback laws can be characterized by Kronecker calculus. Two numerical examples are given to demonstrate the efficacy of the proposed framework. r 2007 The Franklin Institute. Published by Elsevier Ltd. All rights reserved. Keywords: Discrete-time normal form systems; Direct adaptive; Partial stability; Lyapunov-based methods; Kronecker calculus; Normal form
1. Introduction For system theory, the development of the discrete-time system is usually parallel with the continuous-time system [1,2]. Although there were many discrete-time stability results have been published [3–6]. However, for adaptive control we still give less understanding on discrete-time world. Especially, the Lyapunov analysis for adaptive control is difficult
$
This research was supported under the Grant National Defense Educational Program, Taiwan, ROC.
Corresponding author.
E-mail addresses: [email protected] (S.H.-S. Fu), [email protected] (C.-C. Cheng). 0016-0032/$32.00 r 2007 The Franklin Institute. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.jfranklin.2007.07.002
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in discrete-time cases, because the Lyapunov candidate cannot usually be chosen such that the negative definiteness of the Lyapunov difference can be easily proved [7–9]. In this paper, we first illustrate the discrete-time partial stability results. Next, by introducing a Lyapunov candidate function, the proposed direct adaptive control framework achieves adaptive stabilization of the MIMO nonlinear uncertain systems, where the nonlinear system can be represented in discrete-time normal form [10–12] with input-to-state stable zero dynamics. Furthermore, the stability criterion in [3] and the partial stability hypothesis is satisfied. Our result is Lyapunov-based and guarantees partial asymptotic stability of the closed-loop system and the nonlinear adaptive controller does not require the knowledge of the system parameters, only the sign definiteness of the input matrix is required. The proof of the nonlinear framework is an extension from the linear solution in [13]. Finally, two examples are presented to demonstrate the efficacy of the proposed direct adaptive controller. 2. Discrete-time partial stability theory If a system is partially asymptotic stable, that is, asymptotic stability with respect to part of the system states associated with the plant [15]. A direct adaptive framework in continuous time has been developed to guarantee partial asymptotic stability for nonlinear uncertain systems [14]. In this section, we first state and prove the discrete-time version of partial stability. Next, the adaptive control framework is proposed based on this stability result. The discrete-time version of barbalat lemma is given next without proof. Theorem 2.1. Let f : R ! R, and let ftn g1 n¼0 be Pnan unbounded sequence. Suppose that the corresponding sequence ffðtn Þg1 n¼0 , and limn!1 i¼1 jfðti Þj exists and finite. Then fðtn Þ ! 0 as tn ! 1. We consider the discrete-time dynamical system is given by x1 ðk þ 1Þ ¼ f 1 ðx1 ðkÞ; x2 ðkÞÞ;
x1 ð0Þ ¼ x10 ,
x2 ðk þ 1Þ ¼ f 2 ðx1 ðkÞ; x2 ðkÞÞ;
x2 ð0Þ ¼ x20 ,
n1
(1) (2) n2
where x1 2 D, D R such that 0 2 D, x2 2 R , f 1 ð0; x2 Þ ¼ 0 and f 1 ð; x2 Þ is globally Lipschitz, and f 2 : D Rn2 ! Rn2 is such that f 2 ðx1 ; Þ is continuous for every x1 2 D. Under the above conditions, we assume that the solution ðx1 ðkÞ; x2 ðkÞÞ, kX0, to Eqs. (1) and (2) exists and unique. The definition of partial stability is given as following; that is, stability with respect to x1 , for the nonlinear dynamical system (1), (2). We denote by x1 ðkÞ ¼ x1 ðk; 0; x10 Þ the solution of the system (1) subject to the initial condition x1 ð0Þ ¼ x1 ð0; 0; x10 Þ. Definition 2.1. The nonlinear dynamical system (1), (2) is Lyapunov stable with respect to x1 if for every 40, and x20 2 Rn2 , there exists d ¼ dð; x20 Þ40 such that kx01 kod implies that kx1 ðk; 0; x01 Þko for all kX0. The nonlinear dynamical system (1), (2) is Lyapunov stable with respect to x1 uniformly in x20 if for every 40, there exists d ¼ dðÞ40 such that kx01 kod implies that kx1 ðk; 0; x01 Þko for all kX0 and for all x20 2 Rn2 . The nonlinear dynamical system (1), (2) is asymptotically stable with respect to x1 if it is Lyapunov stable with respect to x1 and for every x20 2 Rn2 , there exists d ¼ dðx20 Þ40 such that kx01 kod implies that limk!1 kx1 ðk; 0; x01 Þk ¼ 0. The nonlinear dynamical system (1), (2) is asymptotically stable with respect to x1 uniformly in x20 if it is Lyapunov stable
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with respect to x1 uniformly in x20 and there exists d40 such that kx01 kod implies that limk!1 kx1 ðk; 0; x01 Þk ¼ 0 for all x20 2 Rn2 . The nonlinear dynamical system (1), (2) is globally asymptotically stable with respect to x1 uniformly in x20 if it is Lyapunov stable with respect to x1 uniformly in x20 and limk!1 kx1 ðk; 0; x01 Þk ¼ 0 for all x10 2 Rn1 and x20 2 Rn2 . Let V ¼ V ðx1 ; x2 Þ is x1 positive-definite function [15] and the discrete-time nonlinear dynamical system given by (1), (2). The discrete-time partial stability result is as following. Theorem 2.2. Consider the discrete-time nonlinear dynamical system given by Eqs. (1), (2), and assume D Rn2 is a positive invariant set with respect to Eqs. (1), (2). Furthermore, assume there exist functions V : D Rn2 ! R, W 1 ; W 2 ; W 2 D ! R, such that V ð; Þ is C1 , W 1 ðÞ and W 2 ðÞ are smooth and positive definite, W ðÞ is smooth and nonnegative definite, and, for all ðx1 ; x2 Þ 2 D Rn2 , W 1 ðx1 ÞpV ðx1 ; x2 ÞpW 2 ðx1 Þ,
(3)
DV ðx1 ; x2 Þp W ðx1 Þ,
(4)
Then there exists D0 D such that ðx10 ; x20 Þ 2 D0 Rn2 , x1 ðkÞ ! R9fx1 2 D : W ðx1 Þ ¼ 0g as k ! 1. If, in addition, D ¼ Rn1 and W 1 ðÞ is radially unbounded, then for all ðx10 ; x20 Þ 2 Rn1 Rn2 , x1 ðkÞ ! R9fx1 2 Rn1 : W ðx1 Þ ¼ 0g as k ! 1. Proof. Assume that Eqs. (3) and (4) hold. Let 40 be such that B D, and d ¼ dðÞ40 be such that if x10 2 Bd then x1 ðkÞ 2 B , kX0. Now, since V ðx1 ; x2 Þ is monotonically nonincreasing and bounded from below by zero, it follows that limk!1 V ðx1 ðkÞ; x2 ðkÞÞ exists and finite. Hence, for every kX0 the Lyapunov difference DV ðx1 ðkÞ; x2 ðkÞÞ ¼ V ðx1 ðk þ 1Þ; x2 ðk þ 1ÞÞ V ðx1 ðkÞ; x2 ðkÞÞp W ðx1 ðkÞÞ
(5)
then by induction W ðx1 ð0ÞÞpV ðx1 ð0Þ; x2 ð0ÞÞ V ðx1 ð1Þ; x2 ð1ÞÞ, .. . W ðx1 ðkÞÞpV ðx1 ðkÞ; x2 ðkÞÞ V ðx1 ðk þ 1Þ; x2 ðk þ 1ÞÞ. Then k X
W ðx1 ðiÞÞpV ðx10 ; x20 Þ V ðx1 ðk þ 1Þ; x2 ðk þ 1ÞÞ.
(6)
i¼0
P Therefore, ki¼0 W ðx1 ðiÞÞ exists and is finite. Since the system (1), (2) is smooth and W ðÞ is differentiable on a compact set B it follows that W ðx1 ðkÞÞ is smooth at every kX0. Now, By Theorem 2.1, we conclude that W ðx1 ðkÞÞ ! 0 as k ! 1. Finally, if, in addition, D ¼ Rn1 and W 1 ðÞ is radially unbounded, for every x10 2 Rn1 , there exists ; d40 such that x10 2 Bd and x1 ðkÞ 2 B , kX0, then for all ðx10 ; x20 Þ 2 Rn1 Rn2 , x1 ðkÞ ! R9fx1 2 Rn1 : W ðx1 Þ ¼ 0g as k ! 1. & Theorem 2.2 indicates the discrete-time nonlinear dynamical system given by (1), (2) is Lyapunov stable with respect to x1 uniformly in x20 .
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3. Adaptive control for nonlinear discrete-time uncertain systems In this section we begin by considering the problem of characterizing adaptive feedback control laws for nonlinear uncertain discrete-time systems. Specifically, consider the following controlled nonlinear uncertain discrete time system G given by xðk þ 1Þ ¼ f ðxðkÞÞ þ GðxðkÞÞuðxðkÞÞ;
xð0Þ ¼ x0 ,
^ uðxðkÞÞ ¼ GðxðkÞÞKðkÞF ðxðkÞÞ,
(7) (8)
where xðkÞ 2 Rn is the state vector, u : Rn ! Rm is the control vector, K 2 Rms is the feedback gain matrix, f : Rn ! Rn is nonlinear system dynamics, G : Rn ! Rnm is input dynamics, G^ : Rn ! Rmm , and F : Rn ! Rs , with F ð0Þ ¼ 0. Furthermore, for the nonlinear system G we assume that the required properties for the existence and uniqueness of solutions are satisfied and zero state observable. The Lyapunov direct method gives sufficient conditions for Lyapunov stability of a discrete-time dynamical system. Here, we found that the adaptive laws of the feedback gains can be characterized by Kronecker calculus and the solution of Sylvester equation in terms of the Kronecker algebra [16] is given as Proposition 3.1. ^ þ X B^ ¼ C, ^ where A^ 2 Rnn , B^ 2 Rmm , Proposition 3.1. Consider the matrix equation AX nm nm C^ 2 R , and X 2 R . Then there exists a unique matrix X: ^ 1 vec C , ^ X ¼ vec1 ½ðB^ T AÞ
(9)
^ B^ T I n þ I m A^ 2 Rnmnm . If and only if det ðB^ T AÞa0. ^ where B^ T A9 Next, we state and prove the main result of this paper. Theorem 3.1. Consider the nonlinear discrete-time system G given by Eqs. (7), (8) is zerostate observable. Assume 9 a matrix K g 2 Rms such that ^ xðk þ 1Þ ¼ f ðxðkÞÞ þ GðxðkÞÞGðxðkÞÞK g F ðxðkÞÞ9f c ðxðkÞÞ,
(10)
is globally asymptotically stable, where f c : Rn ! Rn . Furthermore, let the output defined as y9‘ðxðkÞÞ, where ‘ : Rn ! Rt , and let V s : Rn ! R be such that V s ðÞ is positive definite, radially unbounded, and V s ð0Þ ¼ 0. In addition, let 0 ¼ V s ðf c ðxðkÞÞÞ V s ðxðkÞÞ þ ‘T ðxðkÞÞ‘ðxðkÞÞ;
8x 2 Rn
(11)
and V s ðf ðxðkÞÞ þ GðxðkÞÞuðxðkÞÞÞ ¼ V s ðf ðxðkÞÞÞ þ P1u ðxðkÞÞuðxðkÞÞ þ uT ðxðkÞÞP2u ðxðkÞÞuðxðkÞÞ,
ð12Þ
T ^ where P1u : Rn ! R1m , P2u : Rn ! Rmm , and P^ : Rn ! Rmm . P2u ðxðkÞÞ9P^ ðxðkÞÞPðxðkÞ. mm ss Finally, let Q 2 R and Y 2 R are positive definite design matrices, where T ^ Y 1 ¼ N^ N40, N^ 2 Rss . If the following is satisfied T ^ QG^ ðxðkÞÞP2u ðxðkÞÞGðxðkÞÞa0.
(13)
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Then the adaptive feedback control (8) with the update law T 1 1 ^ KðkÞ ¼ vec1 f½eak F ðxðkÞÞF T ðxðkÞÞY ðQG^ ðxðkÞÞP2u ðxðkÞÞGðxðkÞÞÞ T
1 ^ vecðQG^ ðxðkÞÞP2u ðxðkÞÞGðxðkÞÞÞ ðKðk 1Þ T
12eak QG^ ðxðkÞÞPT1u ðxðkÞÞF T ðxðkÞÞÞY g,
ð14Þ
guarantees that the solution ðxðkÞ; Kðk 1Þ; Zðk 1ÞÞ ð0; K g ; 0Þ of the closed-loop system given by (7), (8), and (14) is asymptotically stable. Furthermore, if ‘ðxðkÞÞ ! 0 as k ! 1. ‘T ðxðkÞÞ‘ðxðkÞÞ40, xa0, then xðkÞ ! 0 as k ! 1 for all xð0Þ. Proof. Based on the results of linear uncertain system, the proof is an extension of [13]. To show asymptotically stable of the closed-loop system (7), (8), and (14), we consider the Lyapunov candidate function V ðxðkÞ; Kðk 1Þ; Zðk 1ÞÞ ¼ V s ðxðkÞÞ þ trQ1 K g Zðk 1ÞK Tg þ tr Q1 ðKðk 1Þ K g Þ Zðk 1ÞðKðk 1Þ K g ÞT ,
ð15Þ
where ZðkÞ 2 Rss , kX0, is a nonnegative nonincreasing matrix function. Noted that Q is positive definite matrices, V s ðxðkÞÞ40, and V ð0; K g ; 0Þ ¼ 0, V ðxðkÞ; Kðk 1Þ; Zðk 1ÞÞ40 for all ðxðkÞ; Kðk 1Þ; Zðk 1ÞÞað0; K g ; 0Þ. Next, since V ðxðkÞ; Kðk 1Þ; Zðk 1ÞÞ is radially unbounded, that is V ðxðkÞ; Kðk 1Þ; Zðk 1ÞÞ ! 1, as xðkÞ ! 1. Furthermore, by Eq. (15) we found that the solution of two time step ahead xðk 1Þ, xðkÞ are required for obtaining the next prediction xðk þ 1Þ, which may fall into the trajectory-dependent hypothesis. The corresponding Lyapunov difference is given by DV ðkÞ ¼ V ðxðk þ 1Þ; KðkÞ; ZðkÞÞ V ðxðkÞ; Kðk 1Þ; Zðk 1ÞÞ,
(16)
we add and subtract V s ðf c ðxðkÞÞÞ given by Eq. (10), to and from Eq. (16) so that DV ðkÞ ¼ ðV s ðf c ðxðkÞÞÞ V s ðxðkÞÞÞ þ tr Q1 KðkÞZðkÞK T ðkÞ 2 tr Q1 KðkÞZðkÞK Tg þ 2 tr Q1 Kðk 1ÞZðk 1ÞK Tg ^ P1u ðxðkÞÞGðxðkÞÞK g F ðxðkÞÞ þ P1u ðxðkÞÞuðxðkÞÞ T ^ F T ðxðkÞÞK Tg G^ ðxðkÞÞP2u ðxðkÞÞGðxðkÞÞK g F ðxðkÞÞ
þ uT ðxðkÞÞP2u ðxðkÞÞuðxðkÞÞ þ 2 tr Q1 K g ðZðkÞ Zðk 1ÞÞK Tg tr Q1 Kðk 1ÞZðk 1ÞK T ðk 1Þ.
ð17Þ
Next, consider the following update law KðkÞ ¼ Kðxðk 1ÞÞ QF^ ðxðkÞÞF T ðxðkÞÞZ 1 ðk 1Þ,
(18)
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where F^ : Rn ! Rm . Next, let ZðkÞ ¼ eak Y 1 , a be positive real number. Furthermore, by the fact that tr xyT ¼ yT x, if x; y 2 Rn . Then substitute (18) into (17) so that DV ðkÞ ¼ ðV s ðf c ðxðkÞÞÞ V s ðxðkÞÞÞ þ ½eak eakþa tr Q1 K Tg K g T ^ þ ½F T ðxðkÞÞK T ðxðkÞÞG^ ðxðkÞÞP2u ðxðkÞÞGðxðkÞÞ T
^ ^ 2F^ ðxðkÞÞ KðkÞF ðxðkÞÞ ½P1u ðxðkÞÞGðxðkÞÞ þ P1u ðxðkÞÞGðxðkÞÞ T T ^ þ F T ðxðkÞÞK Tg G^ ðxðkÞÞP2u ðxðkÞÞGðxðkÞÞ 2F^ ðxðkÞÞ K g F ðxðkÞÞ T kF T ðxðkÞÞZ 1 ðk 1ÞF ðxðkÞÞkF^ ðxðkÞÞQF^ ðxðkÞÞ
þ ½eak eakþa tr Q1 ½KðkÞ K g T Y 1 ½KðkÞ K g .
ð19Þ
We then consider T T ^ F^ ðxðkÞÞ ¼ 12G^ ðxðkÞÞPT1u ðxðkÞÞ þ G^ ðxðkÞÞP2u ðxðkÞÞGðxðkÞÞKðkÞF ðxðkÞÞ.
(20)
After substituting Eq. (20) to Eq. (19), and dropping the negative definitiveness terms, (16) becomes DV pV s ðf c ðxðkÞÞÞ V s ðxðkÞÞ.
(21)
Since e2ak e2akþ2a o0, for kX0, and by Condition (11), then the resulting Lyapunov difference becomes DV p ‘T ðxðkÞÞ‘ðxðkÞÞ.
(22)
This proves that the solution ðxðkÞ; Kðk 1Þ; Zðk 1ÞÞ ¼ ð0; K g ; 0Þ of the closed-loop system given by Eqs. (7), (8), and (14) is asymptotically stable if ‘ðxðkÞÞa0, kX1, then xðkÞ ! 0 as k ! 1 for all xð0Þ 2 Rn . By Theorem 2.2, the solution is asymptotic x-stability and ‘ðxðkÞÞ ! 0 as k ! 1. Concluding above, by Eqs. (18) and (20), the feedback law is obtained by solving the following Sylvester equation at every time step T
1 ^ ½QG^ ðxðkÞÞP2u ðxðkÞÞGðxðkÞÞ KðkÞ þ KðkÞ½eak F ðxðkÞÞF T ðxðkÞÞ T 1 ^ ¼ ½QG^ ðxðkÞÞP2u ðxðkÞÞGðxðkÞÞ T ½Kðk 1Þ 12eak QG^ ðxðkÞÞPT1u ðxðkÞÞF T ðxðkÞÞ .
Then by Proposition 3.1, (23) becomes (14).
ð23Þ
&
The solution may depends on the choice of P1u ðxðkÞÞ and P2u ðxðkÞÞ, here we provide the extension of Theorem 3.1 as following.
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Corollary 3.1. Consider the nonlinear discrete-time system G given by Eqs. (7), (8). Assume there exist a matrix K g 2 Rms such that the zero solution xðkÞ ¼ 0 to (10) is globally asymptotically stable. Furthermore, let the output defined as y9‘ðxðkÞÞ, where ‘ : Rn ! Rt , and let V s : Rn ! R be such that V s ðÞ is positive definite, radially unbounded, V s ð0Þ ¼ 0, and Eqs. (11), (12) hold. Then the adaptive feedback control (8) with the update law KðkÞ ¼ vec1 f½eak F ðxðkÞÞF T ðxðkÞÞY PðxðkÞÞ 1 vec½PðxðkÞÞKðk 1Þ g, T
ð24Þ
^ PðxðkÞÞ ¼ ðQG^ ðxðkÞÞð12P1 þ G T ðxðkÞÞPGðxðkÞÞÞGðxðkÞÞÞ , 1
guarantees that the solution ðxðkÞ; Kðk 1Þ; Zðk 1ÞÞ ð0; K g ; 0Þ of the closed-loop system given by Eqs. (7), (8), and (24) is asymptotically stable and ‘ðxðkÞÞ ! 0 as k ! 1. ‘T ðxðkÞÞ‘ðxðkÞÞ40, xa0, then xðkÞ ! 0 as k ! 1 for all xð0Þ. Proof. The result is a direct application of Theorem 3.1 with P1u ðxðkÞÞ ¼ uT ðxðkÞÞP1 , where P1 40, and P2u ðxðkÞÞ ¼ uT ðxðkÞÞPuðxðkÞÞ. & Corollary 3.2. Consider the nonlinear discrete-time system G given by Eqs. (7) and (8). Assume that there exist a matrix K g 2 Rms such that the zero solution xðkÞ ¼ 0 to Eq. (10) is globally asymptotically stable. Furthermore, let the output defined as y9‘ðxðkÞÞ, where ‘ : Rn ! Rt , and let V s : Rn ! R be such that V s ðÞ is positive definite, radially unbounded, V s ð0Þ ¼ 0, and Eqs. (11), (12) hold. Then the adaptive feedback control (8) with the update law KðkÞ ¼ vec1 f½eak F ðxðkÞÞF T ðxðkÞÞY PðxðkÞÞ 1 vec PðxðkÞÞðKðk 1Þ T 12eak QG^ ðxðkÞÞG T ðxðkÞÞPf c ðxðkÞÞF T ðxðkÞÞY Þg, T
ð25Þ
^ PðxðkÞÞ ¼ ðQG^ ðxðkÞÞG ðxðkÞÞPGðxðkÞÞGðxðkÞÞÞ , T
1
guarantees that the solution ðxðkÞ; Kðk 1Þ; Zðk 1ÞÞ ð0; K g ; 0Þ of the closed-loop system given by Eqs. (7), (8), and (24) is asymptotically stable and ‘ðxðkÞÞ ! 0 as k ! 1. ‘T ðxðkÞÞ‘ðxðkÞÞ40, xa0, then xðkÞ ! 0 as k ! 1 for all xð0Þ. Proof. The result is a direct application of Theorem 2f Tc ðxðkÞÞPGðxðkÞÞ, P2u ðxðkÞÞ ¼ uT ðxðkÞÞPuðxðkÞÞ, and V s ðf ðxðkÞÞÞ ¼ f Tc ðxðkÞÞPf c ðxðkÞÞ:
&
3.1
with
P1u ðxðkÞÞ ¼ (26)
Note that the adaptive control laws (14), (24) and (25) do not require explicit knowledge of the gain matrix K g and system dynamics f ðxðkÞÞ, if Eq. (7) is in discrete-time normal form with stable internal dynamics [10,11], then we can always construct a function F ðÞ with F ð0Þ ¼ 0, such that the zero solution is asymptotically stable. 4. Specialization to systems with uncertain dynamics In this section, we consider the case where the system dynamics f ðxðkÞÞ and input matrix GðxðkÞÞ ¼ B are uncertain, where, we assume that the controlled nonlinear uncertain system G is given by xðk þ 1Þ ¼ f ðxðkÞÞ þ BuðxðkÞÞ; uðxðkÞÞ ¼ KðkÞF ðxðkÞÞ,
xð0Þ ¼ x0 ,
(27) (28)
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and " f ðxðkÞÞ ¼
A0 0mn
#
"
# 0ðnmÞ1 , xðkÞ þ Au f u ðxðkÞÞ
(29)
where A0 2 RðnmÞn is a known matrix capturing multivariable canonical form [1], Au 2 Rmq is a matrix of uncertain constant parameters, f u : Rn ! Rq and satisfies f u ð0Þ ¼ 0. In addition, we assume that B ¼ ½0ðnmÞm ; Bs T , where Bs 2 Rmm is an unknown symmetric sign definite matrix; that is, Bs 40 or Bs o0. For the statement of the next result define B0 9½0mðnmÞ ; I m T for Bs 40, and B0 9½0mðnmÞ ; I m T for Bs o0. Corollary 4.1. Consider the controlled nonlinear discrete-time system G given by Eqs. (27) and (28), where Bs is an unknown symmetric matrix and the sign definiteness of Bs is known. Assume there exist a stabilizing gain matrix K g 2 Rms and a function F : Rn ! Rs , with F ð0Þ ¼ 0, such that xðk þ 1Þ ¼ f ðxðkÞÞ þ BK g F ðxðkÞÞ ¼ f c ðxðkÞÞ;
xð0Þ ¼ x0 ,
(30)
is globally asymptotically stable. Furthermore, since Eq. (27) is zero-state observable and output y9‘ðxðkÞÞ, where ‘ : Rn ! Rt , and let V s : Rn ! R be such that V s ðÞ is positive definite, radially unbounded, V s ð0Þ ¼ 0, and Eq. (11) satisfied, where V s ðxðk þ 1ÞÞ ¼ V s ðf ðxðkÞÞÞ þ P1u ðxðkÞÞuðxðkÞÞ þ uT ðxðkÞÞP2u ðxðkÞÞuðxðkÞÞ.
(31)
Then the adaptive feedback control (28) with the update laws KðkÞ ¼ vec1 f½F ðxðkÞÞF T ðxðkÞÞY e2ak ð2BT0 PB0 Þ1 1 vecð2BT0 PB0 Þ1 ðKðk 1Þ 12e2ak BT0 Pf c ðxðkÞÞF T ðxðkÞÞY Þg,
ð32Þ
guarantees that the closed-loop system given by Eqs. (27), (28), and (32) is Lyapunov stable and ‘ðxðkÞÞ ! 0 as k ! 1. If ‘T ðxðkÞÞ‘ðxðkÞÞ40, xa0, then xðkÞ ! 0 as k ! 1 for all xð0Þ 2 Rn . Proof. This result is a direct consequence of Theorem 3.1 with the following: P2u ðxðkÞÞ ¼ 2BT PBQ;
P ¼ N T N;
PT1u ðxðkÞÞ ¼ BT Pf c ðxðkÞÞ;
^ GðxðkÞÞ ¼ I,
(33)
V s ðxðkÞÞ ¼ f Tc ðxðkÞÞPf c ðxðkÞÞ,
(34)
and the update law becomes KðkÞ ¼ Kðk 1Þ 12eak QBT Pf c ðxðkÞÞF T ðxðkÞÞY 2eak QBT PBQKðkÞF ðxðkÞÞF T ðxðkÞÞY .
ð35Þ 1
Since Q is arbitrary positive-definite matrices, Q can be replaced by jBs j and the positivedefinite square root jBs j9ðB2s Þ1=2 . Now, since Bs is symmetric and sign definite it follows from the Schur decomposition that Bs ¼ UDBs U, where U is orthogonal and DBs is real diagonal. Hence, jBs j1 BT ¼ ½0mðnmÞ ; Im ¼ BT0 , where Im ¼ I m for Bs 40 and Im ¼ I m for Bs o0. Then by Proposition 3.1, Eq. (35) implies Eq. (32). & Corollary 4.2. Consider the nonlinear discrete-time system G given by Eqs. (27) and (28), where Bs is an unknown symmetric matrix and the sign definiteness of Bs is known. Assume there exist a stabilizing gain matrix K g 2 Rms and a function F : Rn ! Rs , with F ð0Þ ¼ 0,
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such that Eq. (30) is globally asymptotically stable. Furthermore, assume that (27) is zerostate observable and output y9‘ðxðkÞÞ, where ‘ : Rn ! Rt , and let V s : Rn ! R be such that V s ðÞ is positive definite, radially unbounded, V s ð0Þ ¼ 0, and (11) and (31) hold. Then the adaptive feedback control law (28) with the update law KðkÞ ¼ vec1 f½F ðxðkÞÞF T ðxðkÞÞY e2ak ð12I m þ BT0 PB0 Þ1 1 vecð12I m þ BT0 PB0 Þ1 Kðk 1Þg,
ð36Þ
guarantees that the closed-loop system given by Eqs. (27), (28), and (36) is Lyapunov stable and ‘ðxðkÞÞ ! 0 as k ! 1. If ‘T ðxðkÞÞ‘ðxðkÞÞ40, xa0, then xðkÞ ! 0 as k ! 1 for all xð0Þ 2 Rn . Proof. This result is a direct consequence of Corollary 4.1 with the following: ^ T PBQ; ^ P2u ðxðkÞÞ ¼ QB PT1u ðxðkÞÞ ¼ P1 uT ðxðkÞÞ;
P ¼ N T N;
^ Q40,
V s ðxðkÞÞ40;
P1 40.
(37) (38)
In addition, let Q^ be replaced by qjBs j1 , q is positive real number, and specifically, Q ¼ q2 I m , P1 ¼ q2 I m . &
5. Numerical examples In this section we present two numerical examples to demonstrate the utility of the proposed direct nonlinear adaptive control framework. First, we consider the uncertain controlled nonlinear discrete-time system in normal form given by 3 2 3 2 x2 ðkÞ 0 0 7 6 2 7 6 ax ðkÞ þ bx2 ðkÞ cosðx2 ðkÞÞ 7 þ 6 1 0 7uðxðkÞÞ xðk þ 1Þ ¼ 6 5 4 1 5 4 cx3 ðkÞ þ dx31 ðkÞ 0 1 2 3 0 6 2 7 ax ðkÞ þ bx2 ðkÞ cosðx2 ðkÞÞ 7 þ BuðxðkÞÞ, ¼ A0 xðkÞ þ 6 ð39Þ 4 1 5 3 cx3 ðkÞ þ dx1 ðkÞ where a, b, c, and d are unknown parameters. Next, let f c ðxðkÞÞ to be 2 3 0 6 2 7 ax ðkÞ þ bx2 ðkÞ cosðx2 ðkÞÞ 7 f c ðxðkÞÞ ¼ A0 xðkÞ þ 6 4 1 5 cx3 ðkÞ þ dx31 ðkÞ " # 0 þ jBs j1 ðYn f u ðxðkÞÞ Yf u ðxðkÞÞ þ Fn f^u ðxðkÞÞÞ, Bs
ð40Þ
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2
3
x21
7 6 6 x2 ðkÞ cosðx2 ðkÞÞ 7 7 6 f u ðxðkÞÞ ¼ 6 7; 7 6 x ðkÞ 3 5 4 x31 ðkÞ
" Y¼
a
b
0
0
0
0
c
d
# ;
" f^u ðxðkÞÞ ¼
x1
1 0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 -1
0
10
20 30 Time step
40
50
40
50
Fig. 1. System state X 1 .
x2
1.2 1 0.8 0.6 0.4 0.2 0 -0.2
0
10
20 30 Time step Fig. 2. System state X 2 .
x1 ðkÞ x2 ðkÞ
# .
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Then we can construct F ðxðkÞÞ ¼ ½x21 ðkÞ; x2 ðkÞ cosðx2 ðkÞÞ; x3 ðkÞ; x31 ðkÞ; x1 ðkÞ; x2 ðkÞ T , and Yn and Fn be chosen such that ^ Yn f u ðxðkÞÞ þ Fn f^u ðxðkÞÞ ¼ AxðkÞ,
(41)
where A^ 2 R23 is arbitrary, so that " # A~ 0 f c ðxðkÞÞ ¼ xðkÞ ¼ Ac xðkÞ, A^ specifically, let 2
0 1 6 Ac ¼ 4 0:5 0:4 0:3 0:5
3 0 7 0:1 5; 0:9
(42)
2
3 0 0 6 7 B0 ¼ 4 1 0 5. 0 1
(43)
Next, by using the update law (32) and choosing the proper design matrices Y ¼ 15I 6 , R ¼ 0:75I 3 , and a ¼ 0:025. Let V s ðxÞ ¼ xT Px, where P satisfies the condition P ¼ ATc PAc þ R. The simulation start with xð0Þ ¼ ½1; 1; 1:5 , and let a ¼ 0:5, b ¼ 0:1, c ¼ 0:3, and d ¼ 0:5. At time k ¼ 19, the states are perturbed xð19Þ ¼ ½0:75; 1; 1 , and the system parameters are changed to a ¼ 0:65, b ¼ 0:25, c ¼ 0:45, and d ¼ 0:55. The controller does not have the knowledge of the system parameters, either the perturbation of the states. Figs. 1–3 depict the states versus the time step, Fig. 4 illustrates the control inputs at each time step, and Figs. 5 and 6 show the update gains. The results show that the controller with the update law given by Eq. (32) can stabilize the system when system has uncertain entries in both system parameters
x3
1
0.5
0
-0.5
-1
-1.5
0
10
20
30
Time step Fig. 3. System state X 3 .
40
50
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0.02 0 -0.02 -0.04 -0.06
0
5
10
15
20
25
30
35
40
45
50
35
40
45
50
35
40
45
50
Time step Control U2
0.06 0.04 0.02 0 -0.02
0
5
10
15
20 25 30 Time step
Fig. 4. Control input U.
Gain
0.02 K(1,1) K(1,2) K(1,3) K(1,4) K(1,5) K(1,6)
0.015 0.01 0.005 0 -0.005 -0.01 -0.015 -0.02
0
5
10
15
20
25
30
Time step
Fig. 5. Adaptive gains.
and input matrix. Furthermore, an immediate application for system perturbed, the simulation shows that the controller can readapt and drive the system and stabilize the system.
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Gain
0.04 K(2,1) K(2,2) K(2,3) K(2,4) K(2,5) K(2,6)
0.03 0.02 0.01 0 -0.01 -0.02 -0.03
0
5
10
15
20
25
30
35
40
45
50
Time step Fig. 6. Adaptive gains.
Next, we consider an uncertain controlled nonlinear discrete-time system given by [11] 3 2 x2 ðkÞ 2 3 7 6 0 ðx2 ðkÞ x3 ðkÞÞ 7 6 þ bx1 ðkÞ ax1 ðkÞ 7 6 6 27 ðx1 ðkÞ þ 1Þ xðk þ 1Þ ¼ 6 7 þ 4 ðx2 ðkÞ þ 1Þ 5uðxðkÞÞ, 7 6 x2 ðkÞ x3 ðkÞ 5 4 0 þ dx1 ðkÞ cðx1 ðkÞ x2 ðkÞ 1Þ x1 ðkÞ þ 1 (44) where a, b, and c are unknown system parameters, and denote xðkÞ ¼ ½x1 ðkÞ; x2 ðkÞ; x3 ðkÞ T be state vector. For simplicity and control purpose, Eq. (44) can be rewritten as 2 3 " # " # x2 ðkÞ 0 x1 ðk þ 1Þ 6 7 ¼ 4 x1 ðkÞðx2 ðkÞ x3 ðkÞÞ uðxðkÞÞ, 5þ þ bx1 ðkÞ a ðx2 ðkÞ þ 1Þ2 x2 ðk þ 1Þ ðx1 ðkÞ þ 1Þ x3 ðk þ 1Þ ¼ cðx1 ðkÞ x2 ðkÞ 1Þ
x2 ðkÞ x3 ðkÞ þ dx1 ðkÞ, x1 ðkÞ þ 1
where jcjo1. Next, we apply the Corollary 3.1 to Eq. (45), and let f c ðxðkÞÞ to be 2 3 " # 0 x1 ðkÞ 6 7 ðx2 ðkÞ x3 ðkÞÞ f c ðxðkÞÞ ¼ A0 þ4 5 þ bx ax ðkÞ ðkÞ 1 1 x2 ðkÞ ðx1 ðkÞ þ 1Þ " # 0 þ jBs j1 ðYn f u ðxðkÞÞ Yf u ðxðkÞÞ þ Fn f^u ðxðkÞÞÞ, Bs
ð45Þ
ð46Þ
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2
3
x1
6 f u ðxðkÞÞ ¼ 4
7 ðx2 ðkÞ x3 ðkÞÞ 5; Y ¼ ½b; a , x1 ðkÞ ðx1 ðkÞ þ 1Þ " # f u ðxðkÞÞ f^u ðxðkÞÞ ¼ x2 ðkÞ; F ðxðkÞÞ ¼ . f^u ðxðkÞÞ
Furthermore, Yn and Fn be chosen so that " f c ðxðkÞÞ ¼ Ac
# x1 ðkÞ , x2 ðkÞ
(47)
x1
1 0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0
-0.2
-0.2
-0.4
0
10
20
30
x2
1
40
50
60
70
80
-0.4
0
10
20
Time step
30
x3
0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 -1
0
40
50
Time step
10
20
30
40
50
60
Time step Fig. 7. (a) X 1 , (b) X 2 , and (c) X 3 .
70
80
60
70
80
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where Ac is asymptotically stable. Since the sign definiteness of the input matrix in Eq. (45) is known. Specifically, we select Ac ¼
0
1
0:1
0:8
0.5
;
B0 ¼
0 1
.
(48)
x 10-3
Control U
0 -0.5 -1 -1.5 -2 -2.5 -3 -3.5 -4
0
10
20
30
40
50
60
70
80
Time step
Fig. 8. Adaptive gains.
1
x 10-3
Gain
0 K(1,1) K(1,2) K(1,3)
-1 -2 -3 -4 -5 -6
0
10
20
30
40
50
Time step Fig. 9. Adaptive gains.
60
70
80
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Next, by using the update law (32) and choosing Y ¼ 10I 3 , R ¼ 0:02I 3 , and a ¼ 0:01. Let V s ðxÞ ¼ xT Px, where P satisfies the condition P ¼ ATc PAc þ R. The simulation start with xð0Þ ¼ ½1; 0:8; 1 , and let a ¼ 0:2, b ¼ 0:4, c ¼ 0:5, and d ¼ 0:5. At time k ¼ 39, the states are perturbed xð39Þ ¼ ½0:5; 0:3; 0:35 , and the system parameters are changed to a ¼ 0:25, b ¼ 0:42, c ¼ 0:45, and d ¼ 0:55. The controller does not have the knowledge of the system parameters, either the perturbations of the states and the parameters. Fig. 7 depict the state versus time, Fig. 8 shows the control inputs at each time step, and Fig. 9 illustrates the update gains. The results show that the controller with the update law (32) can stabilize the system, while there are uncertain entries in system parameters and input matrix. The controller readapts the system when unknown perturbation in states and system parameters exist. 6. Conclusion In this paper, we developed direct adaptive discrete-time nonlinear control frameworks for uncertain nonlinear discrete-time systems. Our approach is Lyapunov-based and we proved that the closed-loop system is partially asymptotic stability, that is, asymptotic stability with respect to part of the closed-loop system states. Furthermore, we assumed that the nonlinear system can be represented in normal form, and the sign definiteness of input matrix is known. The nonlinear adaptive controller was constructed without knowledge of the system dynamics. Our future research will emphasize on the system with exogenous disturbances and release the trajectory dependent hypothesis. References [1] W.J. Rugh, Linear System Theory, Prentice-Hall, Englewood Cliffs, NJ, 1996, pp. 2362–2365. [2] K. Premaratne, E.I. Jury, Discrete-time positive-real lemma revisited: the discrete-time counterpart of Kalman–Yakubovitch lemma, IEEE Trans. on Circuits Systems-I Fundamental Theory Appl. 41 (11) (1994) 747–750. [3] L. Hitz, B.D.O Anderson, Discrete positive-real functions and their application to system stability, Proc. IEE 116 (1) (1969) 153–155. [4] K.M. Passino, A.N. Michael, P.J. Antsaklis, Lyapunov stability of a class of discrete events systems, IEEE Trans. Automatic Control 39 (2) (1994) 269–279. [5] G.J. Silva, A. Datta, Adaptive internal model control: the discrete-time case, Int. J. Adaptive Control Signal Processing 15 (2001) 15–36. [6] I. Bar-Kana, Absolute stability and robust discrete adaptive control of multivariable systems, Control and Dynamic Systems 31 (1989) 157–183. [7] R. Venugopal, V.G. Rao, D.S. Bernstein, Optimal Lyapunov-based backward horizon adaptive stabilization, in: Proceedings of the American Control Conference, Chicago, IL, 2000, pp. 1654–1658. [8] I. Kanellakopoulos, A discrete-time adaptive nonlinear systems, IEEE Trans. Automatic Control 39 (1994) 2362–2365. [9] R. Venugopal, D.S. Bernstein, Adaptive disturbance rejection using ARMARKOV system representations, in: Proceedings of the 36th IEEE CDC, San Diego, CA, 1999, pp. 1654–1658. [10] B. Hamzi, J.P. Barbot, W. Kang, Normal form for discrete time parameterized systems with uncontrollable linearization, in: Proceedings of the 38th Conference on Decision and Control, Phoenix, AZ, 1999, pp. 2035–2038. [11] C. Califano, S. Monaco, D. Normand-Cryot, On the discrete-time normal form, IEEE Trans. Automatic Control 43 (11) (1998) 1654–1658. [12] A. Madani, S. Monaco, D. Normand-Cryot, Adaptive control of discrete-time dynamics in parametric strictfeedback form, in: Proceedings of the 35th Conference on Decision and Control, 1996, pp. 2659–2664.
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[13] S.H. Fu, C.C. Cheng, Direct adaptive feedback design for linear discrete-time uncertain systems, Asia J. Control (2004) 421–427. [14] W.M. Haddad, T. Hayakawa, Direct adaptive control for nonlinear uncertain systems with exogenous disturbances, J. Signal Processing Adaptive Control (2002) 151–172. [15] V.I. Vorotnikov, Partial Stability and Control, Birkhauser, Basel, 1998, pp. 26–27. [16] D.S. Bernstein, W.M. Haddad, Control System Synthesis: The Fixed Structure Approach, Georgia Tech Bookstore, 1997.
Journal of the Franklin Institute 345 (2008) 119–135 www.elsevier.com/locate/jfranklin
On direct adaptive control design for nonlinear discrete-time uncertain systems$ Simon Hsu-Sheng Fu, Chi-Cheng Cheng Department of Mechanical and Electro-Mechanical Engineering, National Sun Yat-Sen University, 70 Lien-hai road, Kaohsiung 804, Taiwan, ROC Received 26 December 2006; received in revised form 14 July 2007; accepted 31 July 2007
Abstract In this paper, we develop a direct adaptive control framework for adaptive stabilization of the MIMO nonlinear uncertain systems, which can be represented as discrete-time normal form with input-to-state zero dynamics. The framework is Lyapunov-based and guarantees partial stability of the closed-loop systems, such that the adaptation of the feedback gains can stabilize the closed-loop system without the knowledge of the system parameters. In addition, our results show that the adaptive feedback laws can be characterized by Kronecker calculus. Two numerical examples are given to demonstrate the efficacy of the proposed framework. r 2007 The Franklin Institute. Published by Elsevier Ltd. All rights reserved. Keywords: Discrete-time normal form systems; Direct adaptive; Partial stability; Lyapunov-based methods; Kronecker calculus; Normal form
1. Introduction For system theory, the development of the discrete-time system is usually parallel with the continuous-time system [1,2]. Although there were many discrete-time stability results have been published [3–6]. However, for adaptive control we still give less understanding on discrete-time world. Especially, the Lyapunov analysis for adaptive control is difficult
$
This research was supported under the Grant National Defense Educational Program, Taiwan, ROC.
Corresponding author.
E-mail addresses: [email protected] (S.H.-S. Fu), [email protected] (C.-C. Cheng). 0016-0032/$32.00 r 2007 The Franklin Institute. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.jfranklin.2007.07.002
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in discrete-time cases, because the Lyapunov candidate cannot usually be chosen such that the negative definiteness of the Lyapunov difference can be easily proved [7–9]. In this paper, we first illustrate the discrete-time partial stability results. Next, by introducing a Lyapunov candidate function, the proposed direct adaptive control framework achieves adaptive stabilization of the MIMO nonlinear uncertain systems, where the nonlinear system can be represented in discrete-time normal form [10–12] with input-to-state stable zero dynamics. Furthermore, the stability criterion in [3] and the partial stability hypothesis is satisfied. Our result is Lyapunov-based and guarantees partial asymptotic stability of the closed-loop system and the nonlinear adaptive controller does not require the knowledge of the system parameters, only the sign definiteness of the input matrix is required. The proof of the nonlinear framework is an extension from the linear solution in [13]. Finally, two examples are presented to demonstrate the efficacy of the proposed direct adaptive controller. 2. Discrete-time partial stability theory If a system is partially asymptotic stable, that is, asymptotic stability with respect to part of the system states associated with the plant [15]. A direct adaptive framework in continuous time has been developed to guarantee partial asymptotic stability for nonlinear uncertain systems [14]. In this section, we first state and prove the discrete-time version of partial stability. Next, the adaptive control framework is proposed based on this stability result. The discrete-time version of barbalat lemma is given next without proof. Theorem 2.1. Let f : R ! R, and let ftn g1 n¼0 be Pnan unbounded sequence. Suppose that the corresponding sequence ffðtn Þg1 n¼0 , and limn!1 i¼1 jfðti Þj exists and finite. Then fðtn Þ ! 0 as tn ! 1. We consider the discrete-time dynamical system is given by x1 ðk þ 1Þ ¼ f 1 ðx1 ðkÞ; x2 ðkÞÞ;
x1 ð0Þ ¼ x10 ,
x2 ðk þ 1Þ ¼ f 2 ðx1 ðkÞ; x2 ðkÞÞ;
x2 ð0Þ ¼ x20 ,
n1
(1) (2) n2
where x1 2 D, D R such that 0 2 D, x2 2 R , f 1 ð0; x2 Þ ¼ 0 and f 1 ð; x2 Þ is globally Lipschitz, and f 2 : D Rn2 ! Rn2 is such that f 2 ðx1 ; Þ is continuous for every x1 2 D. Under the above conditions, we assume that the solution ðx1 ðkÞ; x2 ðkÞÞ, kX0, to Eqs. (1) and (2) exists and unique. The definition of partial stability is given as following; that is, stability with respect to x1 , for the nonlinear dynamical system (1), (2). We denote by x1 ðkÞ ¼ x1 ðk; 0; x10 Þ the solution of the system (1) subject to the initial condition x1 ð0Þ ¼ x1 ð0; 0; x10 Þ. Definition 2.1. The nonlinear dynamical system (1), (2) is Lyapunov stable with respect to x1 if for every 40, and x20 2 Rn2 , there exists d ¼ dð; x20 Þ40 such that kx01 kod implies that kx1 ðk; 0; x01 Þko for all kX0. The nonlinear dynamical system (1), (2) is Lyapunov stable with respect to x1 uniformly in x20 if for every 40, there exists d ¼ dðÞ40 such that kx01 kod implies that kx1 ðk; 0; x01 Þko for all kX0 and for all x20 2 Rn2 . The nonlinear dynamical system (1), (2) is asymptotically stable with respect to x1 if it is Lyapunov stable with respect to x1 and for every x20 2 Rn2 , there exists d ¼ dðx20 Þ40 such that kx01 kod implies that limk!1 kx1 ðk; 0; x01 Þk ¼ 0. The nonlinear dynamical system (1), (2) is asymptotically stable with respect to x1 uniformly in x20 if it is Lyapunov stable
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with respect to x1 uniformly in x20 and there exists d40 such that kx01 kod implies that limk!1 kx1 ðk; 0; x01 Þk ¼ 0 for all x20 2 Rn2 . The nonlinear dynamical system (1), (2) is globally asymptotically stable with respect to x1 uniformly in x20 if it is Lyapunov stable with respect to x1 uniformly in x20 and limk!1 kx1 ðk; 0; x01 Þk ¼ 0 for all x10 2 Rn1 and x20 2 Rn2 . Let V ¼ V ðx1 ; x2 Þ is x1 positive-definite function [15] and the discrete-time nonlinear dynamical system given by (1), (2). The discrete-time partial stability result is as following. Theorem 2.2. Consider the discrete-time nonlinear dynamical system given by Eqs. (1), (2), and assume D Rn2 is a positive invariant set with respect to Eqs. (1), (2). Furthermore, assume there exist functions V : D Rn2 ! R, W 1 ; W 2 ; W 2 D ! R, such that V ð; Þ is C1 , W 1 ðÞ and W 2 ðÞ are smooth and positive definite, W ðÞ is smooth and nonnegative definite, and, for all ðx1 ; x2 Þ 2 D Rn2 , W 1 ðx1 ÞpV ðx1 ; x2 ÞpW 2 ðx1 Þ,
(3)
DV ðx1 ; x2 Þp W ðx1 Þ,
(4)
Then there exists D0 D such that ðx10 ; x20 Þ 2 D0 Rn2 , x1 ðkÞ ! R9fx1 2 D : W ðx1 Þ ¼ 0g as k ! 1. If, in addition, D ¼ Rn1 and W 1 ðÞ is radially unbounded, then for all ðx10 ; x20 Þ 2 Rn1 Rn2 , x1 ðkÞ ! R9fx1 2 Rn1 : W ðx1 Þ ¼ 0g as k ! 1. Proof. Assume that Eqs. (3) and (4) hold. Let 40 be such that B D, and d ¼ dðÞ40 be such that if x10 2 Bd then x1 ðkÞ 2 B , kX0. Now, since V ðx1 ; x2 Þ is monotonically nonincreasing and bounded from below by zero, it follows that limk!1 V ðx1 ðkÞ; x2 ðkÞÞ exists and finite. Hence, for every kX0 the Lyapunov difference DV ðx1 ðkÞ; x2 ðkÞÞ ¼ V ðx1 ðk þ 1Þ; x2 ðk þ 1ÞÞ V ðx1 ðkÞ; x2 ðkÞÞp W ðx1 ðkÞÞ
(5)
then by induction W ðx1 ð0ÞÞpV ðx1 ð0Þ; x2 ð0ÞÞ V ðx1 ð1Þ; x2 ð1ÞÞ, .. . W ðx1 ðkÞÞpV ðx1 ðkÞ; x2 ðkÞÞ V ðx1 ðk þ 1Þ; x2 ðk þ 1ÞÞ. Then k X
W ðx1 ðiÞÞpV ðx10 ; x20 Þ V ðx1 ðk þ 1Þ; x2 ðk þ 1ÞÞ.
(6)
i¼0
P Therefore, ki¼0 W ðx1 ðiÞÞ exists and is finite. Since the system (1), (2) is smooth and W ðÞ is differentiable on a compact set B it follows that W ðx1 ðkÞÞ is smooth at every kX0. Now, By Theorem 2.1, we conclude that W ðx1 ðkÞÞ ! 0 as k ! 1. Finally, if, in addition, D ¼ Rn1 and W 1 ðÞ is radially unbounded, for every x10 2 Rn1 , there exists ; d40 such that x10 2 Bd and x1 ðkÞ 2 B , kX0, then for all ðx10 ; x20 Þ 2 Rn1 Rn2 , x1 ðkÞ ! R9fx1 2 Rn1 : W ðx1 Þ ¼ 0g as k ! 1. & Theorem 2.2 indicates the discrete-time nonlinear dynamical system given by (1), (2) is Lyapunov stable with respect to x1 uniformly in x20 .
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3. Adaptive control for nonlinear discrete-time uncertain systems In this section we begin by considering the problem of characterizing adaptive feedback control laws for nonlinear uncertain discrete-time systems. Specifically, consider the following controlled nonlinear uncertain discrete time system G given by xðk þ 1Þ ¼ f ðxðkÞÞ þ GðxðkÞÞuðxðkÞÞ;
xð0Þ ¼ x0 ,
^ uðxðkÞÞ ¼ GðxðkÞÞKðkÞF ðxðkÞÞ,
(7) (8)
where xðkÞ 2 Rn is the state vector, u : Rn ! Rm is the control vector, K 2 Rms is the feedback gain matrix, f : Rn ! Rn is nonlinear system dynamics, G : Rn ! Rnm is input dynamics, G^ : Rn ! Rmm , and F : Rn ! Rs , with F ð0Þ ¼ 0. Furthermore, for the nonlinear system G we assume that the required properties for the existence and uniqueness of solutions are satisfied and zero state observable. The Lyapunov direct method gives sufficient conditions for Lyapunov stability of a discrete-time dynamical system. Here, we found that the adaptive laws of the feedback gains can be characterized by Kronecker calculus and the solution of Sylvester equation in terms of the Kronecker algebra [16] is given as Proposition 3.1. ^ þ X B^ ¼ C, ^ where A^ 2 Rnn , B^ 2 Rmm , Proposition 3.1. Consider the matrix equation AX nm nm C^ 2 R , and X 2 R . Then there exists a unique matrix X: ^ 1 vec C , ^ X ¼ vec1 ½ðB^ T AÞ
(9)
^ B^ T I n þ I m A^ 2 Rnmnm . If and only if det ðB^ T AÞa0. ^ where B^ T A9 Next, we state and prove the main result of this paper. Theorem 3.1. Consider the nonlinear discrete-time system G given by Eqs. (7), (8) is zerostate observable. Assume 9 a matrix K g 2 Rms such that ^ xðk þ 1Þ ¼ f ðxðkÞÞ þ GðxðkÞÞGðxðkÞÞK g F ðxðkÞÞ9f c ðxðkÞÞ,
(10)
is globally asymptotically stable, where f c : Rn ! Rn . Furthermore, let the output defined as y9‘ðxðkÞÞ, where ‘ : Rn ! Rt , and let V s : Rn ! R be such that V s ðÞ is positive definite, radially unbounded, and V s ð0Þ ¼ 0. In addition, let 0 ¼ V s ðf c ðxðkÞÞÞ V s ðxðkÞÞ þ ‘T ðxðkÞÞ‘ðxðkÞÞ;
8x 2 Rn
(11)
and V s ðf ðxðkÞÞ þ GðxðkÞÞuðxðkÞÞÞ ¼ V s ðf ðxðkÞÞÞ þ P1u ðxðkÞÞuðxðkÞÞ þ uT ðxðkÞÞP2u ðxðkÞÞuðxðkÞÞ,
ð12Þ
T ^ where P1u : Rn ! R1m , P2u : Rn ! Rmm , and P^ : Rn ! Rmm . P2u ðxðkÞÞ9P^ ðxðkÞÞPðxðkÞ. mm ss Finally, let Q 2 R and Y 2 R are positive definite design matrices, where T ^ Y 1 ¼ N^ N40, N^ 2 Rss . If the following is satisfied T ^ QG^ ðxðkÞÞP2u ðxðkÞÞGðxðkÞÞa0.
(13)
ARTICLE IN PRESS S.H.-S. Fu, C.-C. Cheng / Journal of the Franklin Institute 345 (2008) 119–135
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Then the adaptive feedback control (8) with the update law T 1 1 ^ KðkÞ ¼ vec1 f½eak F ðxðkÞÞF T ðxðkÞÞY ðQG^ ðxðkÞÞP2u ðxðkÞÞGðxðkÞÞÞ T
1 ^ vecðQG^ ðxðkÞÞP2u ðxðkÞÞGðxðkÞÞÞ ðKðk 1Þ T
12eak QG^ ðxðkÞÞPT1u ðxðkÞÞF T ðxðkÞÞÞY g,
ð14Þ
guarantees that the solution ðxðkÞ; Kðk 1Þ; Zðk 1ÞÞ ð0; K g ; 0Þ of the closed-loop system given by (7), (8), and (14) is asymptotically stable. Furthermore, if ‘ðxðkÞÞ ! 0 as k ! 1. ‘T ðxðkÞÞ‘ðxðkÞÞ40, xa0, then xðkÞ ! 0 as k ! 1 for all xð0Þ. Proof. Based on the results of linear uncertain system, the proof is an extension of [13]. To show asymptotically stable of the closed-loop system (7), (8), and (14), we consider the Lyapunov candidate function V ðxðkÞ; Kðk 1Þ; Zðk 1ÞÞ ¼ V s ðxðkÞÞ þ trQ1 K g Zðk 1ÞK Tg þ tr Q1 ðKðk 1Þ K g Þ Zðk 1ÞðKðk 1Þ K g ÞT ,
ð15Þ
where ZðkÞ 2 Rss , kX0, is a nonnegative nonincreasing matrix function. Noted that Q is positive definite matrices, V s ðxðkÞÞ40, and V ð0; K g ; 0Þ ¼ 0, V ðxðkÞ; Kðk 1Þ; Zðk 1ÞÞ40 for all ðxðkÞ; Kðk 1Þ; Zðk 1ÞÞað0; K g ; 0Þ. Next, since V ðxðkÞ; Kðk 1Þ; Zðk 1ÞÞ is radially unbounded, that is V ðxðkÞ; Kðk 1Þ; Zðk 1ÞÞ ! 1, as xðkÞ ! 1. Furthermore, by Eq. (15) we found that the solution of two time step ahead xðk 1Þ, xðkÞ are required for obtaining the next prediction xðk þ 1Þ, which may fall into the trajectory-dependent hypothesis. The corresponding Lyapunov difference is given by DV ðkÞ ¼ V ðxðk þ 1Þ; KðkÞ; ZðkÞÞ V ðxðkÞ; Kðk 1Þ; Zðk 1ÞÞ,
(16)
we add and subtract V s ðf c ðxðkÞÞÞ given by Eq. (10), to and from Eq. (16) so that DV ðkÞ ¼ ðV s ðf c ðxðkÞÞÞ V s ðxðkÞÞÞ þ tr Q1 KðkÞZðkÞK T ðkÞ 2 tr Q1 KðkÞZðkÞK Tg þ 2 tr Q1 Kðk 1ÞZðk 1ÞK Tg ^ P1u ðxðkÞÞGðxðkÞÞK g F ðxðkÞÞ þ P1u ðxðkÞÞuðxðkÞÞ T ^ F T ðxðkÞÞK Tg G^ ðxðkÞÞP2u ðxðkÞÞGðxðkÞÞK g F ðxðkÞÞ
þ uT ðxðkÞÞP2u ðxðkÞÞuðxðkÞÞ þ 2 tr Q1 K g ðZðkÞ Zðk 1ÞÞK Tg tr Q1 Kðk 1ÞZðk 1ÞK T ðk 1Þ.
ð17Þ
Next, consider the following update law KðkÞ ¼ Kðxðk 1ÞÞ QF^ ðxðkÞÞF T ðxðkÞÞZ 1 ðk 1Þ,
(18)
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where F^ : Rn ! Rm . Next, let ZðkÞ ¼ eak Y 1 , a be positive real number. Furthermore, by the fact that tr xyT ¼ yT x, if x; y 2 Rn . Then substitute (18) into (17) so that DV ðkÞ ¼ ðV s ðf c ðxðkÞÞÞ V s ðxðkÞÞÞ þ ½eak eakþa tr Q1 K Tg K g T ^ þ ½F T ðxðkÞÞK T ðxðkÞÞG^ ðxðkÞÞP2u ðxðkÞÞGðxðkÞÞ T
^ ^ 2F^ ðxðkÞÞ KðkÞF ðxðkÞÞ ½P1u ðxðkÞÞGðxðkÞÞ þ P1u ðxðkÞÞGðxðkÞÞ T T ^ þ F T ðxðkÞÞK Tg G^ ðxðkÞÞP2u ðxðkÞÞGðxðkÞÞ 2F^ ðxðkÞÞ K g F ðxðkÞÞ T kF T ðxðkÞÞZ 1 ðk 1ÞF ðxðkÞÞkF^ ðxðkÞÞQF^ ðxðkÞÞ
þ ½eak eakþa tr Q1 ½KðkÞ K g T Y 1 ½KðkÞ K g .
ð19Þ
We then consider T T ^ F^ ðxðkÞÞ ¼ 12G^ ðxðkÞÞPT1u ðxðkÞÞ þ G^ ðxðkÞÞP2u ðxðkÞÞGðxðkÞÞKðkÞF ðxðkÞÞ.
(20)
After substituting Eq. (20) to Eq. (19), and dropping the negative definitiveness terms, (16) becomes DV pV s ðf c ðxðkÞÞÞ V s ðxðkÞÞ.
(21)
Since e2ak e2akþ2a o0, for kX0, and by Condition (11), then the resulting Lyapunov difference becomes DV p ‘T ðxðkÞÞ‘ðxðkÞÞ.
(22)
This proves that the solution ðxðkÞ; Kðk 1Þ; Zðk 1ÞÞ ¼ ð0; K g ; 0Þ of the closed-loop system given by Eqs. (7), (8), and (14) is asymptotically stable if ‘ðxðkÞÞa0, kX1, then xðkÞ ! 0 as k ! 1 for all xð0Þ 2 Rn . By Theorem 2.2, the solution is asymptotic x-stability and ‘ðxðkÞÞ ! 0 as k ! 1. Concluding above, by Eqs. (18) and (20), the feedback law is obtained by solving the following Sylvester equation at every time step T
1 ^ ½QG^ ðxðkÞÞP2u ðxðkÞÞGðxðkÞÞ KðkÞ þ KðkÞ½eak F ðxðkÞÞF T ðxðkÞÞ T 1 ^ ¼ ½QG^ ðxðkÞÞP2u ðxðkÞÞGðxðkÞÞ T ½Kðk 1Þ 12eak QG^ ðxðkÞÞPT1u ðxðkÞÞF T ðxðkÞÞ .
Then by Proposition 3.1, (23) becomes (14).
ð23Þ
&
The solution may depends on the choice of P1u ðxðkÞÞ and P2u ðxðkÞÞ, here we provide the extension of Theorem 3.1 as following.
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Corollary 3.1. Consider the nonlinear discrete-time system G given by Eqs. (7), (8). Assume there exist a matrix K g 2 Rms such that the zero solution xðkÞ ¼ 0 to (10) is globally asymptotically stable. Furthermore, let the output defined as y9‘ðxðkÞÞ, where ‘ : Rn ! Rt , and let V s : Rn ! R be such that V s ðÞ is positive definite, radially unbounded, V s ð0Þ ¼ 0, and Eqs. (11), (12) hold. Then the adaptive feedback control (8) with the update law KðkÞ ¼ vec1 f½eak F ðxðkÞÞF T ðxðkÞÞY PðxðkÞÞ 1 vec½PðxðkÞÞKðk 1Þ g, T
ð24Þ
^ PðxðkÞÞ ¼ ðQG^ ðxðkÞÞð12P1 þ G T ðxðkÞÞPGðxðkÞÞÞGðxðkÞÞÞ , 1
guarantees that the solution ðxðkÞ; Kðk 1Þ; Zðk 1ÞÞ ð0; K g ; 0Þ of the closed-loop system given by Eqs. (7), (8), and (24) is asymptotically stable and ‘ðxðkÞÞ ! 0 as k ! 1. ‘T ðxðkÞÞ‘ðxðkÞÞ40, xa0, then xðkÞ ! 0 as k ! 1 for all xð0Þ. Proof. The result is a direct application of Theorem 3.1 with P1u ðxðkÞÞ ¼ uT ðxðkÞÞP1 , where P1 40, and P2u ðxðkÞÞ ¼ uT ðxðkÞÞPuðxðkÞÞ. & Corollary 3.2. Consider the nonlinear discrete-time system G given by Eqs. (7) and (8). Assume that there exist a matrix K g 2 Rms such that the zero solution xðkÞ ¼ 0 to Eq. (10) is globally asymptotically stable. Furthermore, let the output defined as y9‘ðxðkÞÞ, where ‘ : Rn ! Rt , and let V s : Rn ! R be such that V s ðÞ is positive definite, radially unbounded, V s ð0Þ ¼ 0, and Eqs. (11), (12) hold. Then the adaptive feedback control (8) with the update law KðkÞ ¼ vec1 f½eak F ðxðkÞÞF T ðxðkÞÞY PðxðkÞÞ 1 vec PðxðkÞÞðKðk 1Þ T 12eak QG^ ðxðkÞÞG T ðxðkÞÞPf c ðxðkÞÞF T ðxðkÞÞY Þg, T
ð25Þ
^ PðxðkÞÞ ¼ ðQG^ ðxðkÞÞG ðxðkÞÞPGðxðkÞÞGðxðkÞÞÞ , T
1
guarantees that the solution ðxðkÞ; Kðk 1Þ; Zðk 1ÞÞ ð0; K g ; 0Þ of the closed-loop system given by Eqs. (7), (8), and (24) is asymptotically stable and ‘ðxðkÞÞ ! 0 as k ! 1. ‘T ðxðkÞÞ‘ðxðkÞÞ40, xa0, then xðkÞ ! 0 as k ! 1 for all xð0Þ. Proof. The result is a direct application of Theorem 2f Tc ðxðkÞÞPGðxðkÞÞ, P2u ðxðkÞÞ ¼ uT ðxðkÞÞPuðxðkÞÞ, and V s ðf ðxðkÞÞÞ ¼ f Tc ðxðkÞÞPf c ðxðkÞÞ:
&
3.1
with
P1u ðxðkÞÞ ¼ (26)
Note that the adaptive control laws (14), (24) and (25) do not require explicit knowledge of the gain matrix K g and system dynamics f ðxðkÞÞ, if Eq. (7) is in discrete-time normal form with stable internal dynamics [10,11], then we can always construct a function F ðÞ with F ð0Þ ¼ 0, such that the zero solution is asymptotically stable. 4. Specialization to systems with uncertain dynamics In this section, we consider the case where the system dynamics f ðxðkÞÞ and input matrix GðxðkÞÞ ¼ B are uncertain, where, we assume that the controlled nonlinear uncertain system G is given by xðk þ 1Þ ¼ f ðxðkÞÞ þ BuðxðkÞÞ; uðxðkÞÞ ¼ KðkÞF ðxðkÞÞ,
xð0Þ ¼ x0 ,
(27) (28)
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and " f ðxðkÞÞ ¼
A0 0mn
#
"
# 0ðnmÞ1 , xðkÞ þ Au f u ðxðkÞÞ
(29)
where A0 2 RðnmÞn is a known matrix capturing multivariable canonical form [1], Au 2 Rmq is a matrix of uncertain constant parameters, f u : Rn ! Rq and satisfies f u ð0Þ ¼ 0. In addition, we assume that B ¼ ½0ðnmÞm ; Bs T , where Bs 2 Rmm is an unknown symmetric sign definite matrix; that is, Bs 40 or Bs o0. For the statement of the next result define B0 9½0mðnmÞ ; I m T for Bs 40, and B0 9½0mðnmÞ ; I m T for Bs o0. Corollary 4.1. Consider the controlled nonlinear discrete-time system G given by Eqs. (27) and (28), where Bs is an unknown symmetric matrix and the sign definiteness of Bs is known. Assume there exist a stabilizing gain matrix K g 2 Rms and a function F : Rn ! Rs , with F ð0Þ ¼ 0, such that xðk þ 1Þ ¼ f ðxðkÞÞ þ BK g F ðxðkÞÞ ¼ f c ðxðkÞÞ;
xð0Þ ¼ x0 ,
(30)
is globally asymptotically stable. Furthermore, since Eq. (27) is zero-state observable and output y9‘ðxðkÞÞ, where ‘ : Rn ! Rt , and let V s : Rn ! R be such that V s ðÞ is positive definite, radially unbounded, V s ð0Þ ¼ 0, and Eq. (11) satisfied, where V s ðxðk þ 1ÞÞ ¼ V s ðf ðxðkÞÞÞ þ P1u ðxðkÞÞuðxðkÞÞ þ uT ðxðkÞÞP2u ðxðkÞÞuðxðkÞÞ.
(31)
Then the adaptive feedback control (28) with the update laws KðkÞ ¼ vec1 f½F ðxðkÞÞF T ðxðkÞÞY e2ak ð2BT0 PB0 Þ1 1 vecð2BT0 PB0 Þ1 ðKðk 1Þ 12e2ak BT0 Pf c ðxðkÞÞF T ðxðkÞÞY Þg,
ð32Þ
guarantees that the closed-loop system given by Eqs. (27), (28), and (32) is Lyapunov stable and ‘ðxðkÞÞ ! 0 as k ! 1. If ‘T ðxðkÞÞ‘ðxðkÞÞ40, xa0, then xðkÞ ! 0 as k ! 1 for all xð0Þ 2 Rn . Proof. This result is a direct consequence of Theorem 3.1 with the following: P2u ðxðkÞÞ ¼ 2BT PBQ;
P ¼ N T N;
PT1u ðxðkÞÞ ¼ BT Pf c ðxðkÞÞ;
^ GðxðkÞÞ ¼ I,
(33)
V s ðxðkÞÞ ¼ f Tc ðxðkÞÞPf c ðxðkÞÞ,
(34)
and the update law becomes KðkÞ ¼ Kðk 1Þ 12eak QBT Pf c ðxðkÞÞF T ðxðkÞÞY 2eak QBT PBQKðkÞF ðxðkÞÞF T ðxðkÞÞY .
ð35Þ 1
Since Q is arbitrary positive-definite matrices, Q can be replaced by jBs j and the positivedefinite square root jBs j9ðB2s Þ1=2 . Now, since Bs is symmetric and sign definite it follows from the Schur decomposition that Bs ¼ UDBs U, where U is orthogonal and DBs is real diagonal. Hence, jBs j1 BT ¼ ½0mðnmÞ ; Im ¼ BT0 , where Im ¼ I m for Bs 40 and Im ¼ I m for Bs o0. Then by Proposition 3.1, Eq. (35) implies Eq. (32). & Corollary 4.2. Consider the nonlinear discrete-time system G given by Eqs. (27) and (28), where Bs is an unknown symmetric matrix and the sign definiteness of Bs is known. Assume there exist a stabilizing gain matrix K g 2 Rms and a function F : Rn ! Rs , with F ð0Þ ¼ 0,
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such that Eq. (30) is globally asymptotically stable. Furthermore, assume that (27) is zerostate observable and output y9‘ðxðkÞÞ, where ‘ : Rn ! Rt , and let V s : Rn ! R be such that V s ðÞ is positive definite, radially unbounded, V s ð0Þ ¼ 0, and (11) and (31) hold. Then the adaptive feedback control law (28) with the update law KðkÞ ¼ vec1 f½F ðxðkÞÞF T ðxðkÞÞY e2ak ð12I m þ BT0 PB0 Þ1 1 vecð12I m þ BT0 PB0 Þ1 Kðk 1Þg,
ð36Þ
guarantees that the closed-loop system given by Eqs. (27), (28), and (36) is Lyapunov stable and ‘ðxðkÞÞ ! 0 as k ! 1. If ‘T ðxðkÞÞ‘ðxðkÞÞ40, xa0, then xðkÞ ! 0 as k ! 1 for all xð0Þ 2 Rn . Proof. This result is a direct consequence of Corollary 4.1 with the following: ^ T PBQ; ^ P2u ðxðkÞÞ ¼ QB PT1u ðxðkÞÞ ¼ P1 uT ðxðkÞÞ;
P ¼ N T N;
^ Q40,
V s ðxðkÞÞ40;
P1 40.
(37) (38)
In addition, let Q^ be replaced by qjBs j1 , q is positive real number, and specifically, Q ¼ q2 I m , P1 ¼ q2 I m . &
5. Numerical examples In this section we present two numerical examples to demonstrate the utility of the proposed direct nonlinear adaptive control framework. First, we consider the uncertain controlled nonlinear discrete-time system in normal form given by 3 2 3 2 x2 ðkÞ 0 0 7 6 2 7 6 ax ðkÞ þ bx2 ðkÞ cosðx2 ðkÞÞ 7 þ 6 1 0 7uðxðkÞÞ xðk þ 1Þ ¼ 6 5 4 1 5 4 cx3 ðkÞ þ dx31 ðkÞ 0 1 2 3 0 6 2 7 ax ðkÞ þ bx2 ðkÞ cosðx2 ðkÞÞ 7 þ BuðxðkÞÞ, ¼ A0 xðkÞ þ 6 ð39Þ 4 1 5 3 cx3 ðkÞ þ dx1 ðkÞ where a, b, c, and d are unknown parameters. Next, let f c ðxðkÞÞ to be 2 3 0 6 2 7 ax ðkÞ þ bx2 ðkÞ cosðx2 ðkÞÞ 7 f c ðxðkÞÞ ¼ A0 xðkÞ þ 6 4 1 5 cx3 ðkÞ þ dx31 ðkÞ " # 0 þ jBs j1 ðYn f u ðxðkÞÞ Yf u ðxðkÞÞ þ Fn f^u ðxðkÞÞÞ, Bs
ð40Þ
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2
3
x21
7 6 6 x2 ðkÞ cosðx2 ðkÞÞ 7 7 6 f u ðxðkÞÞ ¼ 6 7; 7 6 x ðkÞ 3 5 4 x31 ðkÞ
" Y¼
a
b
0
0
0
0
c
d
# ;
" f^u ðxðkÞÞ ¼
x1
1 0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 -1
0
10
20 30 Time step
40
50
40
50
Fig. 1. System state X 1 .
x2
1.2 1 0.8 0.6 0.4 0.2 0 -0.2
0
10
20 30 Time step Fig. 2. System state X 2 .
x1 ðkÞ x2 ðkÞ
# .
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Then we can construct F ðxðkÞÞ ¼ ½x21 ðkÞ; x2 ðkÞ cosðx2 ðkÞÞ; x3 ðkÞ; x31 ðkÞ; x1 ðkÞ; x2 ðkÞ T , and Yn and Fn be chosen such that ^ Yn f u ðxðkÞÞ þ Fn f^u ðxðkÞÞ ¼ AxðkÞ,
(41)
where A^ 2 R23 is arbitrary, so that " # A~ 0 f c ðxðkÞÞ ¼ xðkÞ ¼ Ac xðkÞ, A^ specifically, let 2
0 1 6 Ac ¼ 4 0:5 0:4 0:3 0:5
3 0 7 0:1 5; 0:9
(42)
2
3 0 0 6 7 B0 ¼ 4 1 0 5. 0 1
(43)
Next, by using the update law (32) and choosing the proper design matrices Y ¼ 15I 6 , R ¼ 0:75I 3 , and a ¼ 0:025. Let V s ðxÞ ¼ xT Px, where P satisfies the condition P ¼ ATc PAc þ R. The simulation start with xð0Þ ¼ ½1; 1; 1:5 , and let a ¼ 0:5, b ¼ 0:1, c ¼ 0:3, and d ¼ 0:5. At time k ¼ 19, the states are perturbed xð19Þ ¼ ½0:75; 1; 1 , and the system parameters are changed to a ¼ 0:65, b ¼ 0:25, c ¼ 0:45, and d ¼ 0:55. The controller does not have the knowledge of the system parameters, either the perturbation of the states. Figs. 1–3 depict the states versus the time step, Fig. 4 illustrates the control inputs at each time step, and Figs. 5 and 6 show the update gains. The results show that the controller with the update law given by Eq. (32) can stabilize the system when system has uncertain entries in both system parameters
x3
1
0.5
0
-0.5
-1
-1.5
0
10
20
30
Time step Fig. 3. System state X 3 .
40
50
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0.02 0 -0.02 -0.04 -0.06
0
5
10
15
20
25
30
35
40
45
50
35
40
45
50
35
40
45
50
Time step Control U2
0.06 0.04 0.02 0 -0.02
0
5
10
15
20 25 30 Time step
Fig. 4. Control input U.
Gain
0.02 K(1,1) K(1,2) K(1,3) K(1,4) K(1,5) K(1,6)
0.015 0.01 0.005 0 -0.005 -0.01 -0.015 -0.02
0
5
10
15
20
25
30
Time step
Fig. 5. Adaptive gains.
and input matrix. Furthermore, an immediate application for system perturbed, the simulation shows that the controller can readapt and drive the system and stabilize the system.
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Gain
0.04 K(2,1) K(2,2) K(2,3) K(2,4) K(2,5) K(2,6)
0.03 0.02 0.01 0 -0.01 -0.02 -0.03
0
5
10
15
20
25
30
35
40
45
50
Time step Fig. 6. Adaptive gains.
Next, we consider an uncertain controlled nonlinear discrete-time system given by [11] 3 2 x2 ðkÞ 2 3 7 6 0 ðx2 ðkÞ x3 ðkÞÞ 7 6 þ bx1 ðkÞ ax1 ðkÞ 7 6 6 27 ðx1 ðkÞ þ 1Þ xðk þ 1Þ ¼ 6 7 þ 4 ðx2 ðkÞ þ 1Þ 5uðxðkÞÞ, 7 6 x2 ðkÞ x3 ðkÞ 5 4 0 þ dx1 ðkÞ cðx1 ðkÞ x2 ðkÞ 1Þ x1 ðkÞ þ 1 (44) where a, b, and c are unknown system parameters, and denote xðkÞ ¼ ½x1 ðkÞ; x2 ðkÞ; x3 ðkÞ T be state vector. For simplicity and control purpose, Eq. (44) can be rewritten as 2 3 " # " # x2 ðkÞ 0 x1 ðk þ 1Þ 6 7 ¼ 4 x1 ðkÞðx2 ðkÞ x3 ðkÞÞ uðxðkÞÞ, 5þ þ bx1 ðkÞ a ðx2 ðkÞ þ 1Þ2 x2 ðk þ 1Þ ðx1 ðkÞ þ 1Þ x3 ðk þ 1Þ ¼ cðx1 ðkÞ x2 ðkÞ 1Þ
x2 ðkÞ x3 ðkÞ þ dx1 ðkÞ, x1 ðkÞ þ 1
where jcjo1. Next, we apply the Corollary 3.1 to Eq. (45), and let f c ðxðkÞÞ to be 2 3 " # 0 x1 ðkÞ 6 7 ðx2 ðkÞ x3 ðkÞÞ f c ðxðkÞÞ ¼ A0 þ4 5 þ bx ax ðkÞ ðkÞ 1 1 x2 ðkÞ ðx1 ðkÞ þ 1Þ " # 0 þ jBs j1 ðYn f u ðxðkÞÞ Yf u ðxðkÞÞ þ Fn f^u ðxðkÞÞÞ, Bs
ð45Þ
ð46Þ
ARTICLE IN PRESS S.H.-S. Fu, C.-C. Cheng / Journal of the Franklin Institute 345 (2008) 119–135
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2
3
x1
6 f u ðxðkÞÞ ¼ 4
7 ðx2 ðkÞ x3 ðkÞÞ 5; Y ¼ ½b; a , x1 ðkÞ ðx1 ðkÞ þ 1Þ " # f u ðxðkÞÞ f^u ðxðkÞÞ ¼ x2 ðkÞ; F ðxðkÞÞ ¼ . f^u ðxðkÞÞ
Furthermore, Yn and Fn be chosen so that " f c ðxðkÞÞ ¼ Ac
# x1 ðkÞ , x2 ðkÞ
(47)
x1
1 0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0
-0.2
-0.2
-0.4
0
10
20
30
x2
1
40
50
60
70
80
-0.4
0
10
20
Time step
30
x3
0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 -1
0
40
50
Time step
10
20
30
40
50
60
Time step Fig. 7. (a) X 1 , (b) X 2 , and (c) X 3 .
70
80
60
70
80
ARTICLE IN PRESS S.H.-S. Fu, C.-C. Cheng / Journal of the Franklin Institute 345 (2008) 119–135
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where Ac is asymptotically stable. Since the sign definiteness of the input matrix in Eq. (45) is known. Specifically, we select Ac ¼
0
1
0:1
0:8
0.5
;
B0 ¼
0 1
.
(48)
x 10-3
Control U
0 -0.5 -1 -1.5 -2 -2.5 -3 -3.5 -4
0
10
20
30
40
50
60
70
80
Time step
Fig. 8. Adaptive gains.
1
x 10-3
Gain
0 K(1,1) K(1,2) K(1,3)
-1 -2 -3 -4 -5 -6
0
10
20
30
40
50
Time step Fig. 9. Adaptive gains.
60
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Next, by using the update law (32) and choosing Y ¼ 10I 3 , R ¼ 0:02I 3 , and a ¼ 0:01. Let V s ðxÞ ¼ xT Px, where P satisfies the condition P ¼ ATc PAc þ R. The simulation start with xð0Þ ¼ ½1; 0:8; 1 , and let a ¼ 0:2, b ¼ 0:4, c ¼ 0:5, and d ¼ 0:5. At time k ¼ 39, the states are perturbed xð39Þ ¼ ½0:5; 0:3; 0:35 , and the system parameters are changed to a ¼ 0:25, b ¼ 0:42, c ¼ 0:45, and d ¼ 0:55. The controller does not have the knowledge of the system parameters, either the perturbations of the states and the parameters. Fig. 7 depict the state versus time, Fig. 8 shows the control inputs at each time step, and Fig. 9 illustrates the update gains. The results show that the controller with the update law (32) can stabilize the system, while there are uncertain entries in system parameters and input matrix. The controller readapts the system when unknown perturbation in states and system parameters exist. 6. Conclusion In this paper, we developed direct adaptive discrete-time nonlinear control frameworks for uncertain nonlinear discrete-time systems. Our approach is Lyapunov-based and we proved that the closed-loop system is partially asymptotic stability, that is, asymptotic stability with respect to part of the closed-loop system states. Furthermore, we assumed that the nonlinear system can be represented in normal form, and the sign definiteness of input matrix is known. The nonlinear adaptive controller was constructed without knowledge of the system dynamics. Our future research will emphasize on the system with exogenous disturbances and release the trajectory dependent hypothesis. References [1] W.J. Rugh, Linear System Theory, Prentice-Hall, Englewood Cliffs, NJ, 1996, pp. 2362–2365. [2] K. Premaratne, E.I. Jury, Discrete-time positive-real lemma revisited: the discrete-time counterpart of Kalman–Yakubovitch lemma, IEEE Trans. on Circuits Systems-I Fundamental Theory Appl. 41 (11) (1994) 747–750. [3] L. Hitz, B.D.O Anderson, Discrete positive-real functions and their application to system stability, Proc. IEE 116 (1) (1969) 153–155. [4] K.M. Passino, A.N. Michael, P.J. Antsaklis, Lyapunov stability of a class of discrete events systems, IEEE Trans. Automatic Control 39 (2) (1994) 269–279. [5] G.J. Silva, A. Datta, Adaptive internal model control: the discrete-time case, Int. J. Adaptive Control Signal Processing 15 (2001) 15–36. [6] I. Bar-Kana, Absolute stability and robust discrete adaptive control of multivariable systems, Control and Dynamic Systems 31 (1989) 157–183. [7] R. Venugopal, V.G. Rao, D.S. Bernstein, Optimal Lyapunov-based backward horizon adaptive stabilization, in: Proceedings of the American Control Conference, Chicago, IL, 2000, pp. 1654–1658. [8] I. Kanellakopoulos, A discrete-time adaptive nonlinear systems, IEEE Trans. Automatic Control 39 (1994) 2362–2365. [9] R. Venugopal, D.S. Bernstein, Adaptive disturbance rejection using ARMARKOV system representations, in: Proceedings of the 36th IEEE CDC, San Diego, CA, 1999, pp. 1654–1658. [10] B. Hamzi, J.P. Barbot, W. Kang, Normal form for discrete time parameterized systems with uncontrollable linearization, in: Proceedings of the 38th Conference on Decision and Control, Phoenix, AZ, 1999, pp. 2035–2038. [11] C. Califano, S. Monaco, D. Normand-Cryot, On the discrete-time normal form, IEEE Trans. Automatic Control 43 (11) (1998) 1654–1658. [12] A. Madani, S. Monaco, D. Normand-Cryot, Adaptive control of discrete-time dynamics in parametric strictfeedback form, in: Proceedings of the 35th Conference on Decision and Control, 1996, pp. 2659–2664.
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[13] S.H. Fu, C.C. Cheng, Direct adaptive feedback design for linear discrete-time uncertain systems, Asia J. Control (2004) 421–427. [14] W.M. Haddad, T. Hayakawa, Direct adaptive control for nonlinear uncertain systems with exogenous disturbances, J. Signal Processing Adaptive Control (2002) 151–172. [15] V.I. Vorotnikov, Partial Stability and Control, Birkhauser, Basel, 1998, pp. 26–27. [16] D.S. Bernstein, W.M. Haddad, Control System Synthesis: The Fixed Structure Approach, Georgia Tech Bookstore, 1997.