An efficient method to design robust observer ... - Semantic Scholar
Applied Mathematics and Computation 158 (2004) 29–44 www.elsevier.com/locate/amc
An efficient method to design robust observer-based control of uncertain linear systems Chang-Hua Lien Department of Electrical Engineering, I-Shou University, Kaohsiung 840, Taiwan, ROC
Abstract In this paper, the robust observer-based control for a class of uncertain linear systems is considered. Exponential stabilizability for systems is studied and the convergence rate of system is given. Linear matrix inequality (LMI) approach is used to design the feedback control. The control and observer gains are given from the LMI formulations. A numerical example is given to illustrate our results. Ó 2003 Elsevier Inc. All rights reserved. Keywords: Robust observer-based control; Lyapunov theory; LMI approach; Uncertain linear systems
1. Introduction In many practical systems, the analysis of a mathematical model is usually an important work for a control engineer so as to control a system. However, the mathematical model always contains some uncertain elements; these uncertainties may be due to additive unknown internal or external noise, environmental influence, nonlinearities such as hysteresis or friction, poor plant knowledge, reduced-order models, uncertain or slowly varying parameters. The states of a system are not always measurable in many control systems and applications; such as realization of feedback control, system supervision, gasfired furnace system, and fault diagnosis. Hence the state observer will be used
E-mail address: [email protected] (C.-H. Lien). 0096-3003/$ - see front matter Ó 2003 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2003.08.062
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C.-H. Lien / Appl. Math. Comput. 158 (2004) 29–44
Nomenclature Rn n-dimensional real space Rmn set of all real m by n matrices AT (respectively, xT ) transpose of matrix A (respectively, vector x) kxk Euclidean norm of vector x rankðAÞ rank of matrix A kmin ðP Þ minimal eigenvalue of real symmetric matrix P kmax ðP Þ maximal eigenvalue of real symmetric matrix P P > 0 (respectively, P < 0) P is a positive (respectively, negative) definite symmetric matrix I unit matrix
to reconstruct the states of a dynamic system. The robust observer-based control is probably well suited in such situations for stabilization of uncertain system. Furthermore, an efficient approach to design this robust observerbased control is very important and has became a focus of much research in recent years. These motivate us to consider the observer design and robust observer-based control for uncertain systems. Recently, much effort has been devoted to design the observer or observer-based control of uncertain systems with many approaches. In [9], the spectrum assignment method was introduced to design the observer. Lyapunov stability theory is used to design the state observers for linear time varying or nonlinear systems [2,5]. Linear matrix inequality (LMI) approach is also a powerful tool in the control theory and applications; such as the sliding mode observersÕ design [7], output feedback control [4], and stability analysis of time-delay systems [6]. In this paper, we will adopt these two useful methodologies (i.e. Lyapunov stability theory and LMI approach) to the design of the robust observer-based control for a class of uncertain systems. The control and observer gains could be found from the LMI formulations. A numerical example is given to illustrate the use of our results.
2. Problem formulation and main results Consider the following uncertain system: x_ ðtÞ ¼ ðA þ DAðtÞÞxðtÞ þ ðB þ DBðtÞÞuðtÞ;
ð1aÞ
yðtÞ ¼ ðC þ DCðtÞÞxðtÞ þ ðD þ DDðtÞÞuðtÞ;
ð1bÞ
C.-H. Lien / Appl. Math. Comput. 158 (2004) 29–44
31
where x 2 Rn , u 2 Rm is the input vector, y 2 Rq is the output vector, A 2 Rnn , B 2 Rnm , C 2 Rqn , and D 2 Rqm are known matrices, DAðtÞ, DBðtÞ, DCðtÞ, and DDðtÞ are some perturbed matrices. The following assumptions are made on system (1). Assumption 1. Suppose that the matrix B is full column rank (i.e., rankðBÞ ¼ m). Assumption 2. The perturbed matrices DAðtÞ, DBðtÞ, DCðtÞ, and DDðtÞ satisfy DAðtÞ ¼ M1 F1 ðtÞN1 ;
DBðtÞ ¼ M2 F2 ðtÞN2 ;
DCðtÞ ¼ M3 F3 ðtÞN3 ;
DDðtÞ ¼ M4 F4 ðtÞN4
FiT ðtÞFi ðtÞ 6 I;
i ¼ 1; 2; 3; 4;
where the matrices Mi and Ni , i ¼ 1; 2; 3; 4, are known with appropriate dimension. Remark 1. In Assumption 1, the full column rank of matrix B is a classic assumption in many practical control systems. Note that if this assumption does not hold, we can choose the linear independent columns of matrix B and setting the controls that correspond to those other linear dependent columns are zero; see for example 2 3 2 3 1 0 1 u1 ðtÞ 60 1 27 6 7 6 7 B¼6 7 ¼ ½B1 ; B2 ; B3 ; uðtÞ ¼ 4 u2 ðtÞ 5; 41 0 15 u3 ðtÞ 0 0 0 BuðtÞ ¼ B1 u1 ðtÞ þ B2 u2 ðtÞ þ B3 u3 ðtÞ: Since the matrix B is not full column rank, but the column range space of matrix ½B1 ; B2 is same as matrix B. Hence the control input u3 ðtÞ could be selected as zero and the input space is not be influenced. Furthermore, column u1 ðtÞ b uðtÞ and the matrix B b ¼ ½B1 ; B2 is full we have BuðtÞ ¼ ½B1 ; B2 ¼ B^ u2 ðtÞ column rank. This assumption is very useful to analysis the performance of the control systems. QðyÞ þ q I SðyÞ Lemma 1 [1]. The LMI < 0 is equivalent to SðyÞT RðyÞ RðyÞ < 0; QðyÞ SðyÞRðyÞ1 SðyÞT < q I;
32
C.-H. Lien / Appl. Math. Comput. 158 (2004) 29–44 T
T
where QðyÞ ¼ QðyÞ , RðyÞ ¼ RðyÞ , and SðyÞ depend affinely on y and q is a real number. Proof. This proof is a trivial extension of Schur complement of [1].
h
Lemma 2. For any vectors x; y 2 Rn , matrix A 2 Rnp , B 2 Rpn , F ðtÞ 2 Rpp , constant e > 0, and F ðtÞ satisfies F T ðtÞF ðtÞ 6 I, then we have 2xT AF ðtÞBy 6 e1 xT AAT x þ e y T BT By:
Proof. This proof can be done by the following treatments: 2xT AF ðtÞBy 6 e1 xT AAT x þ e y T BT F T ðtÞF ðtÞBy 6 e1 xT AAT x þ e y T BT By: The suitable dynamic observer and observer-based control for the system (1) are given by ^x_ ðtÞ ¼ A^xðtÞ þ BuðtÞ þ L½yðtÞ ^y ðtÞ;
ð2Þ
^y ðtÞ ¼ C^xðtÞ þ DuðtÞ;
ð3Þ
uðtÞ ¼ K^xðtÞ;
ð4Þ
where ^x 2 Rn is the estimation of x, ^y 2 Rq is the observer output, K 2 Rmn is the control gain, L 2 Rnq is the observer gain. By (2)–(4), systems (1) and (2) can be rewritten as " # " A BK þ DAðtÞ DBðtÞK d xðtÞ ¼ dt eðtÞ DAðtÞ DBðtÞK LDCðtÞ þ LDDðtÞK " # xðtÞ ; eðtÞ
A LC þ DBðtÞK LDDðtÞK
where eðtÞ ¼ xðtÞ ^xðtÞ is the estimation error of system. Define T zðtÞ ¼ xðtÞT
eðtÞT
#
BK þ DBðtÞK
ð5Þ
C.-H. Lien / Appl. Math. Comput. 158 (2004) 29–44
33
and V ðzðtÞÞ ¼ zT ðtÞQzðtÞ;
ð6Þ
where matrix Q > 0. The time derivative of V ðzðtÞÞ, along the trajectories of (5) is given by 8" #T < A BK BK T V_ ðzt Þ ¼ z ðtÞ Q : 0 A LC " #9 = A BK BK zðtÞ þ 2zT ðtÞQ þQ 0 A LC ; "
DAðtÞ DBðtÞK
DBðtÞK
DAðtÞ DBðtÞK LDCðtÞ þ LDDðtÞK DBðtÞK LDDðtÞK 8" #T " #9 = < A BK BK A BK BK zðtÞ QþQ ¼ zT ðtÞ : 0 A LC 0 A LC ; " T
þ 2z ðtÞQ
M1 F1 ðtÞN1 0 "
T
þ 2z ðtÞQ
M1 F1 ðtÞN1 0
# zðtÞ
# zðtÞ
M2 F2 ðtÞN2 K M2 F2 ðtÞN2 K
#
zðtÞ M2 F2 ðtÞN2 K M2 F2 ðtÞN2 K " # 0 0 T zðtÞ þ 2z ðtÞQ LM3 F3 ðtÞN3 0 " # 0 0 T zðtÞ: þ 2z ðtÞQ LM4 F4 ðtÞN4 K LM4 F4 ðtÞN4 K
ð7Þ
Remark 2. The observer design in (2) and (3) is a full-order Luenberger observer [8]. The stability of classic Luenberger observer-based control of system (1) would not be guaranteed when some perturbations appear in view of (7). In this paper, we will use the LMI approach to design the suitable robust observer-based control of system (1). P 0 In order to decouple this stability analysis, the matrix Q is setting as 0 R and from (7) and Lemma 2, we have
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C.-H. Lien / Appl. Math. Comput. 158 (2004) 29–44
" V_ ðzt Þ ¼ zT ðtÞ
AT P þ PA K T BT P PBK
#
PBK
zðtÞ AT R þ RA C T LT R RLC (" ) " # PM1 PM2 T þ 2z ðtÞ F1 ðtÞ½ N1 0 þ F2 ðtÞ½ N2 K N2 K zðtÞ RM1 RM2 (" ) # " # 0 0 þ 2zT ðtÞ F3 ðtÞ½ N3 0 þ F4 ðtÞ½ N4 K N4 K zðtÞ RLM3 RLM4 " # PBK AT P þ PA K T BT P PBK T 6 z ðtÞ zðtÞ AT R þ RA C T LT R RLC K T BT P " # # ( " PM1 N1T T 1 T T þ z ðtÞ e1 ½ N1 0 M1 P M1 R þ e1 RM1 0 ( " " # # ) PM2 K T N2T 1 T T T þ e2 ½ N2 K N2 K M2 P M2 R zðtÞ þ z ðtÞ e2 RM2 K T N2T ) " # " # 0 N3T T T þ e1 þ e ½ 0 zðtÞ N 0 M3 L R 3 3 3 RLM3 0 ( " # 0 T 1 þ z ðtÞ e4 0 M4T LT R RLM4 ) " # T T K N4 þ e4 ½ N4 K N4 K zðtÞ ð8aÞ K T N4T K T BT P #
(" V_ ðzt Þ ¼ zT ðtÞ
R11 R12 RT12 R22
#
" þ e2
K T N2T N2 K
K T N2T N2 K
#
K T N2T N2 K K T N2T N2 K #) K T N4T N4 K K T N4T N4 K þ e4 zðtÞ K T N4T N4 K K T N4T N4 K # " # ( " ) PM PM 1 2 þ zT ðtÞ e1 M1T P M1T R þ e1 M2T P M2T R zðtÞ 1 2 RM1 RM2 # ( " 0 þ zT ðtÞ e1 0 M3T LT R 3 RLM3 " # ) 0 ð8bÞ þ e1 0 M4T LT R zðtÞ ¼ zT ðtÞRzðtÞ; 4 RLM4 "
where P and R are positive definite symmetric matrices, e1 , e2 , e3 , and e4 are some positive constants, and
C.-H. Lien / Appl. Math. Comput. 158 (2004) 29–44
35
R11 ¼ AT P þ PA K T BT P PBK þ e1 N1T N1 þ e3 N3T N3 ; R12 ¼ PBK; R22 ¼ AT R þ RA C T LT R RLC; T T R11 R12 K N2 N2 K K T N2T N2 K R¼ þ e2 RT12 R22 K T N2T N2 K K T N2T N2 K T T PM1 T K N4 N4 K K T N4T N4 K 1 T þ e4 þ e M P M R 1 1 1 RM1 K T N4T N4 K K T N4T N4 K PM2 T 0 1 T þ e1 þ e M 0 M3T LT R P M R 2 3 2 2 RLM3 RM2 0 þ e1 0 M4T LT R : 4 RLM4
ð9Þ
First we present an LMI result for the exponential stability of system (5) with no input perturbations (i.e. DBðtÞ ¼ DDðtÞ ¼ 0, M2 ¼ M4 ¼ N2 ¼ N4 ¼ 0). The suitable control gain K 2 Rmn and observer gain L 2 Rnq for system (5) with no input perturbations could be solved by the following results. Theorem 1. System (1) with no input perturbations is exponentially stabilizable by (2)–(4) provided that there exist some positive constants e1 , e3 , q, two positive b 2 Rmn , L ^ 2 Rnq , and definite symmetric matrices P , R 2 Rnn , and matrices K mm b P 2 R , such that 2 3 X11 X12 PM1 0 6 XT ^ 37 X22 RM1 LM 6 12 7 ð10Þ 6 T 7 < 0; 4 M1 P M1T R e1 I 0 5 ^T 0 e3 I 0 M3T L PB ¼ B Pb ;
ð11Þ
where b11 þ q I; X11 ¼ X b; X12 ¼ B K
b11 ¼ AT P þ PA K b T BT B K b þ e1 N T N1 þ e3 N T N3 ; X 1 3
b22 þ q I; X22 ¼ X
^T : b22 ¼ AT R þ RA LC ^ CTL X
b and The stabilizing observer-based control gains are given by K ¼ Pb 1 K 1 ^ L ¼ R L, and the systems convergence rate is ½q=ð2 max ½kmax ðP Þ; kmax ðRÞÞ. Proof. By Lemma 1, the LMI condition (10) implies " # 1 " T b11 X12 0 0 PM1 e1 I M1 P X R1 ¼ T ^ b RM 0 e LM I 0 X 22 X 1 3 3 12
< qI:
M1T R ^T M3T L
#
ð12Þ
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C.-H. Lien / Appl. Math. Comput. 158 (2004) 29–44
b, The matrix R1 is equal to the matrix R in (8) and (9) with PB ¼ B Pb , K ¼ Pb 1 K ^ and M2 ¼ M4 ¼ N2 ¼ N4 ¼ 0. By conditions (6), (8) and (12), we L ¼ R1 L, have min ½kmin ðP Þ; kmin ðRÞ kzðtÞk2 6 V ðzðtÞÞ 6 max ½kmax ðP Þ; kmax ðRÞ kzðtÞk2 and 2 V_ ðzðtÞÞ 6 q kzðtÞk :
We conclude that the system (5) with no input perturbations is exponentially stabilizable by (2)–(4) with the convergence rate ½q=ð2 max ½kmax ðP Þ; kmax ðRÞÞ [3]. h Remark 3. In Theorem 1, since the matrix B 2 Rnm is full column rank, this implies that the columns of matrices B and PB are all linear independent with P > 0. Hence if the equation (11) is satisfied for some P > 0, the matrix Pb must be nonsingular [4]. Remark 4. The conditions (10) and (11) are not the classic LMI solvable form. We can use efficient software Scilab to solve the LMI problem with the equality constraint (11). Note that if the state perturbations of system have the properties of N1 ¼ N3 and F1 ðtÞ ¼ F3 ðtÞ. The following result can be obtained from Theorem 1 in view of (8a) and (8b). Corollary 1. System (1) with N1 ¼ N3 , F1 ðtÞ ¼ F3 ðtÞ, and no input perturbations is exponentially stabilizable by (2)–(4) provided that there exist two positive constants e1 , q, two positive definite symmetric matrices P , R 2 Rnn , and mab 2 Rmn , L ^ 2 Rnq , and Pb 2 Rmm , such that trices K 2 3 X12 PM1 X11 6 T ^ 37 X22 RM1 LM ð13Þ 4 X12 5 < 0; ^T M TP M TR M TL e1 I 1
1
3
PB ¼ B Pb ; where b11 þ q I; X11 ¼ X b; X12 ¼ B K
b11 ¼ AT P þ PA K b T BT B K b þ e1 N T N1 ; X 1
b22 þ q I; X22 ¼ X
^T : b22 ¼ AT R þ RA b X LC C T L b and The stabilizing observer-based control gains are given by K ¼ Pb 1 K 1 b L ¼ R L, and the systems convergence rate is ½q=ð2 max ½kmax ðP Þ; kmax ðRÞÞ.
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Proof. Suppose N1 ¼ N3 and F1 ðtÞ ¼ F3 ðtÞ, some terms in (8a) could be combined as PM1 0 2zT ðtÞ F1 ðtÞ½ N1 0 zðtÞ þ 2zT ðtÞ F3 ðtÞ½ N3 0 zðtÞ RLM3 RM1 PM1 ¼ 2zT ðtÞ F ðtÞ½ N1 0 zðtÞ: RM1 RLM3 1 The other formulations are same as (8b) and the proof of Theorem 1. h When the control gain K 2 Rmn has been designed from Theorem 1 or Corollary 1, we may use this control gain to design the suitable robust observer-based control from the following results. Theorem 2. System (1) is exponentially stabilizable by (2)–(4) provided that there exist some positive constants e1 , e2 , e3 , e4 , q, two positive definite symmetric ^ 2 Rnq , such that matrices P , R 2 Rnn , and matrices L 3 2 Y11 Y12 PM1 PM2 0 0 6 T ^ 3 LM ^ 47 7 6 Y12 Y22 RM1 RM2 LM 7 6 7 6 M TP T M R e I 0 0 0 1 7 6 1 1 ð14Þ 7 < 0; 6 T T 6 M2 P M2 R 0 e2 I 0 0 7 7 6 6 0 ^T M3T L 0 0 e3 I 0 7 5 4 T ^T 0 M4 L 0 0 0 e4 I where Y11 ¼ Yb11 þ q I; Yb11 ¼ AT P þ PA K T BT P PBK þ e1 N1T N1 þ e2 K T N2T N2 K þ e3 N3T N3 þ e4 K T N4T N4 K; Y12 ¼ PBK e2 K T N2T N2 K e4 K T N4T N4 K; Y22 ¼ Yb22 þ q I; ^T þ e2 K T N T N2 K þ e4 K T N T N4 K: ^ CTL Yb22 ¼ AT R þ RA LC 2 4 ^ and the The stabilizing observer-based control gains are given by K and L ¼ R1 L, systems convergence rate is ½q=ð2 max ½kmax ðP Þ; kmax ðRÞÞ. ^ in view of (8a) Proof. This is similar to the proof of Theorem 1 with L ¼ R1 L and (8b). h
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C.-H. Lien / Appl. Math. Comput. 158 (2004) 29–44
Some special cases of Theorem 2 are formulate in the following results. (a) Case (1) M1 ¼ M2 and F1 ðtÞ ¼ F2 ðtÞ. Corollary 2. System (1) with M1 ¼ M2 and F1 ðtÞ ¼ F2 ðtÞ is exponentially stabilizable by (2)–(4) provided that there exist some positive constants e1 , e3 , e4 , q, ^ 2 Rnq , two positive definite symmetric matrices P , R 2 Rnn , and matrices L such that 3 2 Y12 PM1 0 0 Y11 7 6 T ^ 3 LM ^ 47 6 Y12 Y22 RM1 LM 7 6 7 6 T M1T R e1 I 0 0 7 < 0; ð15Þ 6 M1 P 7 6 T ^T 7 6 0 M3 L 0 e3 I 0 5 4 ^T 0 M4T L 0 0 e4 I where Y11 ¼ Yb11 þ q I; T Yb11 ¼ AT P þ PA K T BT P PBK þ e1 ðN1 N2 KÞ ðN1 N2 KÞ
þ e3 N3T N3 þ e4 K T N4T N4 K; T
Y12 ¼ PBK þ e1 ðN1 N2 KÞ N2 K e4 K T N4T N4 K; Y22 ¼ Yb22 þ q I; ^T þ e1 K T N T N2 K þ e4 K T N T N4 K: ^ CTL Yb22 ¼ AT R þ RA LC 2 4 ^ and the The stabilizing observer-based control gains are given by K and L ¼ R1 L, systems convergence rate is ½q=ð2 max ½kmax ðP Þ; kmax ðRÞÞ. Proof. This is similar to the proofs of Theorem 1 and Corollary 1 with ^ in view of (8a) and (8b). h L ¼ R1 L (b) Case (2) M3 ¼ M4 and F3 ðtÞ ¼ F4 ðtÞ. Corollary 3. System (1) with M3 ¼ M4 and F3 ðtÞ ¼ F4 ðtÞ is exponentially stabilizable by (2)–(4) provided that there exist some positive constants e1 , e2 , e3 , q, ^ 2 Rnq , two positive definite symmetric matrices P , R 2 Rnn , and matrices L such that
C.-H. Lien / Appl. Math. Comput. 158 (2004) 29–44
2
Y11 6 T 6 Y12 6 6 T 6 M1 P 6 6 M TP 4 2 0
Y12
PM1
PM2
Y22
RM1
RM2
M1T R M2T R ^T M2T L
e1 I
0
0
e2 I
0
0
39
3
0 7 ^ 37 LM 7 7 0 7 < 0; 7 0 7 5 e3 I
ð16Þ
where Y11 ¼ Yb11 þ q I; Yb11 ¼ AT P þ PA K T BT P PBK þ e1 N1T N1 þ e2 K T N2T N2 K T
þ e3 ðN3 N4 KÞ ðN3 N4 KÞ; Y12 ¼ PBK e2 K T N2T N2 K þ e3 ðN3 N4 KÞT N4 K; Y22 ¼ Yb22 þ q I; ^T þ e2 K T N T N2 K þ e3 K T N T N4 K: ^ CTL Yb22 ¼ AT R þ RA LC 2 4 ^ and the The stabilizing observer-based control gains are given by K and L ¼ R1 L, systems convergence rate is ½q=ð2 max ½kmax ðP Þ; kmax ðRÞÞ. (c) Case (3) N1 ¼ N3 and F1 ðtÞ ¼ F3 ðtÞ. Corollary 4. System (1) with N1 ¼ N3 and F1 ðtÞ ¼ F3 ðtÞ is exponentially stabilizable by (2)–(4) provided that there exist some positive constants e1 , e2 , e4 , q, two ^ 2 Rnq , such positive definite symmetric matrices P , R 2 Rnn , and matrices L that 3 2 Y11 Y12 PM1 PM2 0 7 6 T ^ 3 RM2 ^ 47 6 Y12 Y22 RM1 LM LM 7 6 7 6 T ^T e1 I 0 0 7 < 0; ð17Þ 6 M1 P M1T R M3T L 7 6 T 7 6 M TP M R 0 e I 0 2 2 5 4 2 ^T 0 M4T L 0 0 e4 I where Y11 ¼ Yb11 þ q I; Yb11 ¼ AT P þ PA K T BT P PBK þ e1 N1T N1 þ e2 K T N2T N2 K þ e4 K T N4T N4 K; Y12 ¼ PBK e2 K T N2T N2 K e4 K T N4T N4 K;
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C.-H. Lien / Appl. Math. Comput. 158 (2004) 29–44
Y22 ¼ Yb22 þ q I; ^T þ e2 K T N T N2 K þ e4 K T N T N4 K: ^ CTL Yb22 ¼ AT R þ RA LC 2
4
^ and the The stabilizing observer-based control gains are given by K and L ¼ R1 L, systems convergence rate is ½q=ð2 max ½kmax ðP Þ; kmax ðRÞÞ. (d) Case (4) N2 ¼ N4 and F2 ðtÞ ¼ F4 ðtÞ. Corollary 5. System (1) with N2 ¼ N4 and F2 ðtÞ ¼ F4 ðtÞ is exponentially stabilizable by (2)–(4) provided that there exist some positive constants e1 , e2 , e3 , q, two ^ 2 Rnq , such positive definite symmetric matrices P , R 2 Rnn , and matrices L that 3 2 Y12 PM1 PM2 0 Y11 7 6 T ^ 3 7 ^ 4 6 Y12 LM Y22 RM1 RM2 LM 7 6 7 6 T M1T R e1 I 0 0 7 < 0; ð18Þ 6 M1 P 7 6 T T T T 6M P M R M L ^ 0 e2 I 0 7 2 4 5 4 2 T ^T 0 M3 L 0 0 e3 I where Y11 ¼ Yb11 þ q I; Yb11 ¼ AT P þ PA K T BT P PBK þ e1 N1T N1 þ e2 K T N2T N2 K þ e3 N3T N3 ; Y12 ¼ PBK e2 K T N2T N2 K; Y22 ¼ Yb22 þ q I;
^T þ e2 K T N T N2 K: ^ CTL Yb22 ¼ AT R þ RA LC 2
^ and the The stabilizing observer-based control gains are given by K and L ¼ R1 L, systems convergence rate is ½q=ð2 max ½kmax ðP Þ; kmax ðRÞÞ. Remark 5. Notice that the conditions (14)–(18) are affined with all their respective arguments with a fixed K 2 Rmn given in Theorem 1 or Corollary 1. Hence we can use the LMI method to obtain the suitable observer gain L to estimate the states of the original uncertain system. Remark 6. Notice that for any one of perturbed matrices DAðtÞ, DBðtÞ, DCðtÞ, or DDðtÞ is equal to zero, it represents that some one of the terms Mi ¼ Ni ¼ 0 in view of Assumption 2. By the derivation of (8a) and (8b), the corresponding matrices Mi , Ni , and constant ei , could be deleted and LMI conditions (10), (13)–(18) are reduce by deleting their corresponding elements. Suppose that DðtÞ ¼ 0, the corresponding matrices M4 and N4 , and constant e4 could also be deleted in (14), (15) and (17).
C.-H. Lien / Appl. Math. Comput. 158 (2004) 29–44
41
Remark 7. There are two ways to solve the design of robust observer-based control. In the first way, we can use Theorem 1 or Corollary 1 to design the control gain K and use the control gain K to Theorem 2 or Corollaries 2–5 to design the observer gain L. In the second way, the constant q > 0 in Theorem 1 or Corollary 1 could be seen as a known parameter and use to solve a suitable control gain K, such that Theorem 2 or Corollaries 2–5 is satisfied for this control gain K. If Theorem 2 and Corollaries 2–5 cannot be satisfied for this control gain K, we can reduce or increase the parameter q in Theorem 1 or Corollary 1 until Theorem 2 or Corollaries 2–5 is satisfied.
3. Numerical example Consider the system (1) 2 3 1 0 1 A ¼ 4 0 3 1 5; 1 1 6 2 0 aðtÞ 6 bðtÞ DAðtÞ ¼ 4 0 cðtÞ 0 DCðtÞ ¼ ½ 0
dðtÞ
with the following parameters: 2 3 1 0 B ¼ 4 0 1 5; C ¼ ½ 0 1 1 ; 0 0 3 2 3 0 aðtÞ 0 7 6 7 0 5; DBðtÞ ¼ 4 0 bðtÞ 5; 0 cðtÞ 0
0 ;
D ¼ 0;
DDðtÞ ¼ 0;
ð19Þ
where jaðtÞj 6 0:3, jbðtÞj 6 0:3, jcðtÞj 6 0:3, jdðtÞj 6 0:3. Comparing (19) with Assumption 2, we have M1 ¼ M2 ¼ I;
M3 ¼ 1; M4 ¼ 0; 2 aðtÞ=0:3 0 6 0 bðtÞ=0:3 F1 ðtÞ ¼ F2 ðtÞ ¼ 4 0
0
3
0 0
7 5;
cðtÞ=0:3
F3 ðtÞ ¼ dðtÞ=0:3; F4 ðtÞ ¼ 0; 2 3 2 3 0 0:3 0 0:3 0 N1 ¼ 4 0 0:3 0 5; N2 ¼ 4 0 0:3 5; 0:3 0 0 0:3 0
N3 ¼ ½ 0 0:3 0 ;
N4 ¼ 0:
By Theorem 1 and Remark 7, the conditions (10) and (11) are satisfied with e1 ¼ 28:8546; e3 ¼ 38:3984; q ¼ 7:2327; 2 3 2:2726 0:9181 0 P ¼ 4 0:9181 11:6588 0 5; 0 0 3:8891
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C.-H. Lien / Appl. Math. Comput. 158 (2004) 29–44
2
2:7701
b ¼ K
Pb ¼
3 2:7701 7 2:9058 5;
2:0749 11:9621
4:0744 6 R ¼ 4 2:0749
2:9058
5:0511
15:4519 0:0636
2:4084 1:9156 ; 8:1703 2:7409
2:2726
0:9181
0:9181
11:6588
2
3 22:5171 7 ^¼6 L 4 13:2686 5; 6:5996
:
The system (1) with (19) and no input perturbations (i.e. DBðtÞ ¼ 0, DDðtÞ ¼ 0) would be exponentially stabilized by the system (2)–(4) with convergence rate 0.2602 and the control and observer gains are given by " b ¼ K ¼ Pb 1 K
7:0249
0:8022
0:9687
# and
0:5586
2
0:6376 0:3114 3
7:2865 6 7 7 ^¼6 L ¼ R1 L 6 0:5797 7: 4 5 3:0230
Note that M1 ¼ M2 , F1 ðtÞ ¼ F2 ðtÞ, and DDðtÞ ¼ 0, the condition (15) in Corollary 2 is satisfied with e1 ¼ 111:9672; 2
140:9821
6 P ¼ 4 22:4035 30:4236 2
26:6052
6 R ¼ 4 177:7166
e3 ¼ 2535:3; 22:4035
30:4236
3
165:7687 26:6167
7 26:6167 5; 106:2643
17:7166
29:3685
269:0297
29:3685 137:1184 2
q ¼ 19:8723;
3 807:3646 ^ ¼ 4 52:3038 5; L 213:7169
3
7 137:1184 5; 233:0026
C.-H. Lien / Appl. Math. Comput. 158 (2004) 29–44
43
Fig. 1. Trajectories of the uncontrolled system.
in view of Remark 6. System (1) with (19) would be exponentially stabilized by the system (2)–(4) with convergence rate 0.0255 and the control and observer gains are given by 2 3 46:7563 7:0249 0:8022 0:9687 ^ ¼ 4 8:2988 5: K¼ and L ¼ R1 L 0:5586 0:6376 0:3114 9:8598 ð20Þ With e.g.,
2
3 3 6 7 xð0Þ ¼ 4 3 5; 1
3 4 6 7 ^xð0Þ ¼ 4 1 5; 3
bðtÞ ¼ 0:3 sinð2tÞ;
2
aðtÞ ¼ 0:3 cosð5tÞ;
cðtÞ ¼ 0:3 sinð4tÞ;
dðtÞ ¼ 0:2 sinð3tÞ;
some state trajectories for uncontrolled and feedback-controlled system (1) with (19) and (20) are depicted in Figs. 1 and 2, respectively. 4. Conclusion In this paper, we have studied the observer-based control problem for a class of uncertain systems. LMI approach has been developed to construct linear full-order observer to guarantee the feedback-controlled system is
44
C.-H. Lien / Appl. Math. Comput. 158 (2004) 29–44
Fig. 2. Trajectories of the feedback-controlled system.
exponentially stabilizable. A numerical example has been given to demonstrate the use of our results. Acknowledgements The research reported here was supported by the National Science Council of Taiwan, ROC under grant nos. NSC 90-2213-E-214-062 and NSC 91-2213E-214-016. References [1] S.P. Boyd, L. El Ghaoui, E. Feron, V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory, SIAM, Philadelphia, 1994. [2] K.K. Busawon, M. Saif, A state observer for nonlinear systems, IEEE Transactions on Automatic Control 44 (1999) 2098–2103. [3] F.M. Callier, C.A. Desoer, Linear System Theory, Springer-Verlag, New York, 1991. [4] C.H. Kuo, C.H. Fang, Stabilization of uncertain linear systems via static output feedback control, 2003 Automatic Control Conference, Taiwan, vol. 1, 2003, pp. 1607–1611. [5] D.W. Gu, F.W. Poon, A robust state observer scheme, IEEE Transactions on Automatic Control 46 (2001) 1958–1963. [6] C.H. Lien, J.D. Chen, Discrete-delay-independent and discrete-delay-dependent criteria for a class of neutral systems, ASME Journal of Dynamic Systems, Measurement, and Control 125 (2003) 33–41. [7] C.P. Tan, C. Edwards, An LMI approach for designing sliding mode observers, International Journal of Control 74 (1998) 1559–1568. [8] K. Zhou, J.C. Doyle, Essentials of Robust Control, Prentice-Hall, New Jersey, 1998. [9] P. Zitek, Anisochronic state observers for hereditary systems, International Journal of Control 71 (1998) 581–599.
An efficient method to design robust observer-based control of uncertain linear systems Chang-Hua Lien Department of Electrical Engineering, I-Shou University, Kaohsiung 840, Taiwan, ROC
Abstract In this paper, the robust observer-based control for a class of uncertain linear systems is considered. Exponential stabilizability for systems is studied and the convergence rate of system is given. Linear matrix inequality (LMI) approach is used to design the feedback control. The control and observer gains are given from the LMI formulations. A numerical example is given to illustrate our results. Ó 2003 Elsevier Inc. All rights reserved. Keywords: Robust observer-based control; Lyapunov theory; LMI approach; Uncertain linear systems
1. Introduction In many practical systems, the analysis of a mathematical model is usually an important work for a control engineer so as to control a system. However, the mathematical model always contains some uncertain elements; these uncertainties may be due to additive unknown internal or external noise, environmental influence, nonlinearities such as hysteresis or friction, poor plant knowledge, reduced-order models, uncertain or slowly varying parameters. The states of a system are not always measurable in many control systems and applications; such as realization of feedback control, system supervision, gasfired furnace system, and fault diagnosis. Hence the state observer will be used
E-mail address: [email protected] (C.-H. Lien). 0096-3003/$ - see front matter Ó 2003 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2003.08.062
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C.-H. Lien / Appl. Math. Comput. 158 (2004) 29–44
Nomenclature Rn n-dimensional real space Rmn set of all real m by n matrices AT (respectively, xT ) transpose of matrix A (respectively, vector x) kxk Euclidean norm of vector x rankðAÞ rank of matrix A kmin ðP Þ minimal eigenvalue of real symmetric matrix P kmax ðP Þ maximal eigenvalue of real symmetric matrix P P > 0 (respectively, P < 0) P is a positive (respectively, negative) definite symmetric matrix I unit matrix
to reconstruct the states of a dynamic system. The robust observer-based control is probably well suited in such situations for stabilization of uncertain system. Furthermore, an efficient approach to design this robust observerbased control is very important and has became a focus of much research in recent years. These motivate us to consider the observer design and robust observer-based control for uncertain systems. Recently, much effort has been devoted to design the observer or observer-based control of uncertain systems with many approaches. In [9], the spectrum assignment method was introduced to design the observer. Lyapunov stability theory is used to design the state observers for linear time varying or nonlinear systems [2,5]. Linear matrix inequality (LMI) approach is also a powerful tool in the control theory and applications; such as the sliding mode observersÕ design [7], output feedback control [4], and stability analysis of time-delay systems [6]. In this paper, we will adopt these two useful methodologies (i.e. Lyapunov stability theory and LMI approach) to the design of the robust observer-based control for a class of uncertain systems. The control and observer gains could be found from the LMI formulations. A numerical example is given to illustrate the use of our results.
2. Problem formulation and main results Consider the following uncertain system: x_ ðtÞ ¼ ðA þ DAðtÞÞxðtÞ þ ðB þ DBðtÞÞuðtÞ;
ð1aÞ
yðtÞ ¼ ðC þ DCðtÞÞxðtÞ þ ðD þ DDðtÞÞuðtÞ;
ð1bÞ
C.-H. Lien / Appl. Math. Comput. 158 (2004) 29–44
31
where x 2 Rn , u 2 Rm is the input vector, y 2 Rq is the output vector, A 2 Rnn , B 2 Rnm , C 2 Rqn , and D 2 Rqm are known matrices, DAðtÞ, DBðtÞ, DCðtÞ, and DDðtÞ are some perturbed matrices. The following assumptions are made on system (1). Assumption 1. Suppose that the matrix B is full column rank (i.e., rankðBÞ ¼ m). Assumption 2. The perturbed matrices DAðtÞ, DBðtÞ, DCðtÞ, and DDðtÞ satisfy DAðtÞ ¼ M1 F1 ðtÞN1 ;
DBðtÞ ¼ M2 F2 ðtÞN2 ;
DCðtÞ ¼ M3 F3 ðtÞN3 ;
DDðtÞ ¼ M4 F4 ðtÞN4
FiT ðtÞFi ðtÞ 6 I;
i ¼ 1; 2; 3; 4;
where the matrices Mi and Ni , i ¼ 1; 2; 3; 4, are known with appropriate dimension. Remark 1. In Assumption 1, the full column rank of matrix B is a classic assumption in many practical control systems. Note that if this assumption does not hold, we can choose the linear independent columns of matrix B and setting the controls that correspond to those other linear dependent columns are zero; see for example 2 3 2 3 1 0 1 u1 ðtÞ 60 1 27 6 7 6 7 B¼6 7 ¼ ½B1 ; B2 ; B3 ; uðtÞ ¼ 4 u2 ðtÞ 5; 41 0 15 u3 ðtÞ 0 0 0 BuðtÞ ¼ B1 u1 ðtÞ þ B2 u2 ðtÞ þ B3 u3 ðtÞ: Since the matrix B is not full column rank, but the column range space of matrix ½B1 ; B2 is same as matrix B. Hence the control input u3 ðtÞ could be selected as zero and the input space is not be influenced. Furthermore, column u1 ðtÞ b uðtÞ and the matrix B b ¼ ½B1 ; B2 is full we have BuðtÞ ¼ ½B1 ; B2 ¼ B^ u2 ðtÞ column rank. This assumption is very useful to analysis the performance of the control systems. QðyÞ þ q I SðyÞ Lemma 1 [1]. The LMI < 0 is equivalent to SðyÞT RðyÞ RðyÞ < 0; QðyÞ SðyÞRðyÞ1 SðyÞT < q I;
32
C.-H. Lien / Appl. Math. Comput. 158 (2004) 29–44 T
T
where QðyÞ ¼ QðyÞ , RðyÞ ¼ RðyÞ , and SðyÞ depend affinely on y and q is a real number. Proof. This proof is a trivial extension of Schur complement of [1].
h
Lemma 2. For any vectors x; y 2 Rn , matrix A 2 Rnp , B 2 Rpn , F ðtÞ 2 Rpp , constant e > 0, and F ðtÞ satisfies F T ðtÞF ðtÞ 6 I, then we have 2xT AF ðtÞBy 6 e1 xT AAT x þ e y T BT By:
Proof. This proof can be done by the following treatments: 2xT AF ðtÞBy 6 e1 xT AAT x þ e y T BT F T ðtÞF ðtÞBy 6 e1 xT AAT x þ e y T BT By: The suitable dynamic observer and observer-based control for the system (1) are given by ^x_ ðtÞ ¼ A^xðtÞ þ BuðtÞ þ L½yðtÞ ^y ðtÞ;
ð2Þ
^y ðtÞ ¼ C^xðtÞ þ DuðtÞ;
ð3Þ
uðtÞ ¼ K^xðtÞ;
ð4Þ
where ^x 2 Rn is the estimation of x, ^y 2 Rq is the observer output, K 2 Rmn is the control gain, L 2 Rnq is the observer gain. By (2)–(4), systems (1) and (2) can be rewritten as " # " A BK þ DAðtÞ DBðtÞK d xðtÞ ¼ dt eðtÞ DAðtÞ DBðtÞK LDCðtÞ þ LDDðtÞK " # xðtÞ ; eðtÞ
A LC þ DBðtÞK LDDðtÞK
where eðtÞ ¼ xðtÞ ^xðtÞ is the estimation error of system. Define T zðtÞ ¼ xðtÞT
eðtÞT
#
BK þ DBðtÞK
ð5Þ
C.-H. Lien / Appl. Math. Comput. 158 (2004) 29–44
33
and V ðzðtÞÞ ¼ zT ðtÞQzðtÞ;
ð6Þ
where matrix Q > 0. The time derivative of V ðzðtÞÞ, along the trajectories of (5) is given by 8" #T < A BK BK T V_ ðzt Þ ¼ z ðtÞ Q : 0 A LC " #9 = A BK BK zðtÞ þ 2zT ðtÞQ þQ 0 A LC ; "
DAðtÞ DBðtÞK
DBðtÞK
DAðtÞ DBðtÞK LDCðtÞ þ LDDðtÞK DBðtÞK LDDðtÞK 8" #T " #9 = < A BK BK A BK BK zðtÞ QþQ ¼ zT ðtÞ : 0 A LC 0 A LC ; " T
þ 2z ðtÞQ
M1 F1 ðtÞN1 0 "
T
þ 2z ðtÞQ
M1 F1 ðtÞN1 0
# zðtÞ
# zðtÞ
M2 F2 ðtÞN2 K M2 F2 ðtÞN2 K
#
zðtÞ M2 F2 ðtÞN2 K M2 F2 ðtÞN2 K " # 0 0 T zðtÞ þ 2z ðtÞQ LM3 F3 ðtÞN3 0 " # 0 0 T zðtÞ: þ 2z ðtÞQ LM4 F4 ðtÞN4 K LM4 F4 ðtÞN4 K
ð7Þ
Remark 2. The observer design in (2) and (3) is a full-order Luenberger observer [8]. The stability of classic Luenberger observer-based control of system (1) would not be guaranteed when some perturbations appear in view of (7). In this paper, we will use the LMI approach to design the suitable robust observer-based control of system (1). P 0 In order to decouple this stability analysis, the matrix Q is setting as 0 R and from (7) and Lemma 2, we have
34
C.-H. Lien / Appl. Math. Comput. 158 (2004) 29–44
" V_ ðzt Þ ¼ zT ðtÞ
AT P þ PA K T BT P PBK
#
PBK
zðtÞ AT R þ RA C T LT R RLC (" ) " # PM1 PM2 T þ 2z ðtÞ F1 ðtÞ½ N1 0 þ F2 ðtÞ½ N2 K N2 K zðtÞ RM1 RM2 (" ) # " # 0 0 þ 2zT ðtÞ F3 ðtÞ½ N3 0 þ F4 ðtÞ½ N4 K N4 K zðtÞ RLM3 RLM4 " # PBK AT P þ PA K T BT P PBK T 6 z ðtÞ zðtÞ AT R þ RA C T LT R RLC K T BT P " # # ( " PM1 N1T T 1 T T þ z ðtÞ e1 ½ N1 0 M1 P M1 R þ e1 RM1 0 ( " " # # ) PM2 K T N2T 1 T T T þ e2 ½ N2 K N2 K M2 P M2 R zðtÞ þ z ðtÞ e2 RM2 K T N2T ) " # " # 0 N3T T T þ e1 þ e ½ 0 zðtÞ N 0 M3 L R 3 3 3 RLM3 0 ( " # 0 T 1 þ z ðtÞ e4 0 M4T LT R RLM4 ) " # T T K N4 þ e4 ½ N4 K N4 K zðtÞ ð8aÞ K T N4T K T BT P #
(" V_ ðzt Þ ¼ zT ðtÞ
R11 R12 RT12 R22
#
" þ e2
K T N2T N2 K
K T N2T N2 K
#
K T N2T N2 K K T N2T N2 K #) K T N4T N4 K K T N4T N4 K þ e4 zðtÞ K T N4T N4 K K T N4T N4 K # " # ( " ) PM PM 1 2 þ zT ðtÞ e1 M1T P M1T R þ e1 M2T P M2T R zðtÞ 1 2 RM1 RM2 # ( " 0 þ zT ðtÞ e1 0 M3T LT R 3 RLM3 " # ) 0 ð8bÞ þ e1 0 M4T LT R zðtÞ ¼ zT ðtÞRzðtÞ; 4 RLM4 "
where P and R are positive definite symmetric matrices, e1 , e2 , e3 , and e4 are some positive constants, and
C.-H. Lien / Appl. Math. Comput. 158 (2004) 29–44
35
R11 ¼ AT P þ PA K T BT P PBK þ e1 N1T N1 þ e3 N3T N3 ; R12 ¼ PBK; R22 ¼ AT R þ RA C T LT R RLC; T T R11 R12 K N2 N2 K K T N2T N2 K R¼ þ e2 RT12 R22 K T N2T N2 K K T N2T N2 K T T PM1 T K N4 N4 K K T N4T N4 K 1 T þ e4 þ e M P M R 1 1 1 RM1 K T N4T N4 K K T N4T N4 K PM2 T 0 1 T þ e1 þ e M 0 M3T LT R P M R 2 3 2 2 RLM3 RM2 0 þ e1 0 M4T LT R : 4 RLM4
ð9Þ
First we present an LMI result for the exponential stability of system (5) with no input perturbations (i.e. DBðtÞ ¼ DDðtÞ ¼ 0, M2 ¼ M4 ¼ N2 ¼ N4 ¼ 0). The suitable control gain K 2 Rmn and observer gain L 2 Rnq for system (5) with no input perturbations could be solved by the following results. Theorem 1. System (1) with no input perturbations is exponentially stabilizable by (2)–(4) provided that there exist some positive constants e1 , e3 , q, two positive b 2 Rmn , L ^ 2 Rnq , and definite symmetric matrices P , R 2 Rnn , and matrices K mm b P 2 R , such that 2 3 X11 X12 PM1 0 6 XT ^ 37 X22 RM1 LM 6 12 7 ð10Þ 6 T 7 < 0; 4 M1 P M1T R e1 I 0 5 ^T 0 e3 I 0 M3T L PB ¼ B Pb ;
ð11Þ
where b11 þ q I; X11 ¼ X b; X12 ¼ B K
b11 ¼ AT P þ PA K b T BT B K b þ e1 N T N1 þ e3 N T N3 ; X 1 3
b22 þ q I; X22 ¼ X
^T : b22 ¼ AT R þ RA LC ^ CTL X
b and The stabilizing observer-based control gains are given by K ¼ Pb 1 K 1 ^ L ¼ R L, and the systems convergence rate is ½q=ð2 max ½kmax ðP Þ; kmax ðRÞÞ. Proof. By Lemma 1, the LMI condition (10) implies " # 1 " T b11 X12 0 0 PM1 e1 I M1 P X R1 ¼ T ^ b RM 0 e LM I 0 X 22 X 1 3 3 12
< qI:
M1T R ^T M3T L
#
ð12Þ
36
C.-H. Lien / Appl. Math. Comput. 158 (2004) 29–44
b, The matrix R1 is equal to the matrix R in (8) and (9) with PB ¼ B Pb , K ¼ Pb 1 K ^ and M2 ¼ M4 ¼ N2 ¼ N4 ¼ 0. By conditions (6), (8) and (12), we L ¼ R1 L, have min ½kmin ðP Þ; kmin ðRÞ kzðtÞk2 6 V ðzðtÞÞ 6 max ½kmax ðP Þ; kmax ðRÞ kzðtÞk2 and 2 V_ ðzðtÞÞ 6 q kzðtÞk :
We conclude that the system (5) with no input perturbations is exponentially stabilizable by (2)–(4) with the convergence rate ½q=ð2 max ½kmax ðP Þ; kmax ðRÞÞ [3]. h Remark 3. In Theorem 1, since the matrix B 2 Rnm is full column rank, this implies that the columns of matrices B and PB are all linear independent with P > 0. Hence if the equation (11) is satisfied for some P > 0, the matrix Pb must be nonsingular [4]. Remark 4. The conditions (10) and (11) are not the classic LMI solvable form. We can use efficient software Scilab to solve the LMI problem with the equality constraint (11). Note that if the state perturbations of system have the properties of N1 ¼ N3 and F1 ðtÞ ¼ F3 ðtÞ. The following result can be obtained from Theorem 1 in view of (8a) and (8b). Corollary 1. System (1) with N1 ¼ N3 , F1 ðtÞ ¼ F3 ðtÞ, and no input perturbations is exponentially stabilizable by (2)–(4) provided that there exist two positive constants e1 , q, two positive definite symmetric matrices P , R 2 Rnn , and mab 2 Rmn , L ^ 2 Rnq , and Pb 2 Rmm , such that trices K 2 3 X12 PM1 X11 6 T ^ 37 X22 RM1 LM ð13Þ 4 X12 5 < 0; ^T M TP M TR M TL e1 I 1
1
3
PB ¼ B Pb ; where b11 þ q I; X11 ¼ X b; X12 ¼ B K
b11 ¼ AT P þ PA K b T BT B K b þ e1 N T N1 ; X 1
b22 þ q I; X22 ¼ X
^T : b22 ¼ AT R þ RA b X LC C T L b and The stabilizing observer-based control gains are given by K ¼ Pb 1 K 1 b L ¼ R L, and the systems convergence rate is ½q=ð2 max ½kmax ðP Þ; kmax ðRÞÞ.
C.-H. Lien / Appl. Math. Comput. 158 (2004) 29–44
37
Proof. Suppose N1 ¼ N3 and F1 ðtÞ ¼ F3 ðtÞ, some terms in (8a) could be combined as PM1 0 2zT ðtÞ F1 ðtÞ½ N1 0 zðtÞ þ 2zT ðtÞ F3 ðtÞ½ N3 0 zðtÞ RLM3 RM1 PM1 ¼ 2zT ðtÞ F ðtÞ½ N1 0 zðtÞ: RM1 RLM3 1 The other formulations are same as (8b) and the proof of Theorem 1. h When the control gain K 2 Rmn has been designed from Theorem 1 or Corollary 1, we may use this control gain to design the suitable robust observer-based control from the following results. Theorem 2. System (1) is exponentially stabilizable by (2)–(4) provided that there exist some positive constants e1 , e2 , e3 , e4 , q, two positive definite symmetric ^ 2 Rnq , such that matrices P , R 2 Rnn , and matrices L 3 2 Y11 Y12 PM1 PM2 0 0 6 T ^ 3 LM ^ 47 7 6 Y12 Y22 RM1 RM2 LM 7 6 7 6 M TP T M R e I 0 0 0 1 7 6 1 1 ð14Þ 7 < 0; 6 T T 6 M2 P M2 R 0 e2 I 0 0 7 7 6 6 0 ^T M3T L 0 0 e3 I 0 7 5 4 T ^T 0 M4 L 0 0 0 e4 I where Y11 ¼ Yb11 þ q I; Yb11 ¼ AT P þ PA K T BT P PBK þ e1 N1T N1 þ e2 K T N2T N2 K þ e3 N3T N3 þ e4 K T N4T N4 K; Y12 ¼ PBK e2 K T N2T N2 K e4 K T N4T N4 K; Y22 ¼ Yb22 þ q I; ^T þ e2 K T N T N2 K þ e4 K T N T N4 K: ^ CTL Yb22 ¼ AT R þ RA LC 2 4 ^ and the The stabilizing observer-based control gains are given by K and L ¼ R1 L, systems convergence rate is ½q=ð2 max ½kmax ðP Þ; kmax ðRÞÞ. ^ in view of (8a) Proof. This is similar to the proof of Theorem 1 with L ¼ R1 L and (8b). h
38
C.-H. Lien / Appl. Math. Comput. 158 (2004) 29–44
Some special cases of Theorem 2 are formulate in the following results. (a) Case (1) M1 ¼ M2 and F1 ðtÞ ¼ F2 ðtÞ. Corollary 2. System (1) with M1 ¼ M2 and F1 ðtÞ ¼ F2 ðtÞ is exponentially stabilizable by (2)–(4) provided that there exist some positive constants e1 , e3 , e4 , q, ^ 2 Rnq , two positive definite symmetric matrices P , R 2 Rnn , and matrices L such that 3 2 Y12 PM1 0 0 Y11 7 6 T ^ 3 LM ^ 47 6 Y12 Y22 RM1 LM 7 6 7 6 T M1T R e1 I 0 0 7 < 0; ð15Þ 6 M1 P 7 6 T ^T 7 6 0 M3 L 0 e3 I 0 5 4 ^T 0 M4T L 0 0 e4 I where Y11 ¼ Yb11 þ q I; T Yb11 ¼ AT P þ PA K T BT P PBK þ e1 ðN1 N2 KÞ ðN1 N2 KÞ
þ e3 N3T N3 þ e4 K T N4T N4 K; T
Y12 ¼ PBK þ e1 ðN1 N2 KÞ N2 K e4 K T N4T N4 K; Y22 ¼ Yb22 þ q I; ^T þ e1 K T N T N2 K þ e4 K T N T N4 K: ^ CTL Yb22 ¼ AT R þ RA LC 2 4 ^ and the The stabilizing observer-based control gains are given by K and L ¼ R1 L, systems convergence rate is ½q=ð2 max ½kmax ðP Þ; kmax ðRÞÞ. Proof. This is similar to the proofs of Theorem 1 and Corollary 1 with ^ in view of (8a) and (8b). h L ¼ R1 L (b) Case (2) M3 ¼ M4 and F3 ðtÞ ¼ F4 ðtÞ. Corollary 3. System (1) with M3 ¼ M4 and F3 ðtÞ ¼ F4 ðtÞ is exponentially stabilizable by (2)–(4) provided that there exist some positive constants e1 , e2 , e3 , q, ^ 2 Rnq , two positive definite symmetric matrices P , R 2 Rnn , and matrices L such that
C.-H. Lien / Appl. Math. Comput. 158 (2004) 29–44
2
Y11 6 T 6 Y12 6 6 T 6 M1 P 6 6 M TP 4 2 0
Y12
PM1
PM2
Y22
RM1
RM2
M1T R M2T R ^T M2T L
e1 I
0
0
e2 I
0
0
39
3
0 7 ^ 37 LM 7 7 0 7 < 0; 7 0 7 5 e3 I
ð16Þ
where Y11 ¼ Yb11 þ q I; Yb11 ¼ AT P þ PA K T BT P PBK þ e1 N1T N1 þ e2 K T N2T N2 K T
þ e3 ðN3 N4 KÞ ðN3 N4 KÞ; Y12 ¼ PBK e2 K T N2T N2 K þ e3 ðN3 N4 KÞT N4 K; Y22 ¼ Yb22 þ q I; ^T þ e2 K T N T N2 K þ e3 K T N T N4 K: ^ CTL Yb22 ¼ AT R þ RA LC 2 4 ^ and the The stabilizing observer-based control gains are given by K and L ¼ R1 L, systems convergence rate is ½q=ð2 max ½kmax ðP Þ; kmax ðRÞÞ. (c) Case (3) N1 ¼ N3 and F1 ðtÞ ¼ F3 ðtÞ. Corollary 4. System (1) with N1 ¼ N3 and F1 ðtÞ ¼ F3 ðtÞ is exponentially stabilizable by (2)–(4) provided that there exist some positive constants e1 , e2 , e4 , q, two ^ 2 Rnq , such positive definite symmetric matrices P , R 2 Rnn , and matrices L that 3 2 Y11 Y12 PM1 PM2 0 7 6 T ^ 3 RM2 ^ 47 6 Y12 Y22 RM1 LM LM 7 6 7 6 T ^T e1 I 0 0 7 < 0; ð17Þ 6 M1 P M1T R M3T L 7 6 T 7 6 M TP M R 0 e I 0 2 2 5 4 2 ^T 0 M4T L 0 0 e4 I where Y11 ¼ Yb11 þ q I; Yb11 ¼ AT P þ PA K T BT P PBK þ e1 N1T N1 þ e2 K T N2T N2 K þ e4 K T N4T N4 K; Y12 ¼ PBK e2 K T N2T N2 K e4 K T N4T N4 K;
40
C.-H. Lien / Appl. Math. Comput. 158 (2004) 29–44
Y22 ¼ Yb22 þ q I; ^T þ e2 K T N T N2 K þ e4 K T N T N4 K: ^ CTL Yb22 ¼ AT R þ RA LC 2
4
^ and the The stabilizing observer-based control gains are given by K and L ¼ R1 L, systems convergence rate is ½q=ð2 max ½kmax ðP Þ; kmax ðRÞÞ. (d) Case (4) N2 ¼ N4 and F2 ðtÞ ¼ F4 ðtÞ. Corollary 5. System (1) with N2 ¼ N4 and F2 ðtÞ ¼ F4 ðtÞ is exponentially stabilizable by (2)–(4) provided that there exist some positive constants e1 , e2 , e3 , q, two ^ 2 Rnq , such positive definite symmetric matrices P , R 2 Rnn , and matrices L that 3 2 Y12 PM1 PM2 0 Y11 7 6 T ^ 3 7 ^ 4 6 Y12 LM Y22 RM1 RM2 LM 7 6 7 6 T M1T R e1 I 0 0 7 < 0; ð18Þ 6 M1 P 7 6 T T T T 6M P M R M L ^ 0 e2 I 0 7 2 4 5 4 2 T ^T 0 M3 L 0 0 e3 I where Y11 ¼ Yb11 þ q I; Yb11 ¼ AT P þ PA K T BT P PBK þ e1 N1T N1 þ e2 K T N2T N2 K þ e3 N3T N3 ; Y12 ¼ PBK e2 K T N2T N2 K; Y22 ¼ Yb22 þ q I;
^T þ e2 K T N T N2 K: ^ CTL Yb22 ¼ AT R þ RA LC 2
^ and the The stabilizing observer-based control gains are given by K and L ¼ R1 L, systems convergence rate is ½q=ð2 max ½kmax ðP Þ; kmax ðRÞÞ. Remark 5. Notice that the conditions (14)–(18) are affined with all their respective arguments with a fixed K 2 Rmn given in Theorem 1 or Corollary 1. Hence we can use the LMI method to obtain the suitable observer gain L to estimate the states of the original uncertain system. Remark 6. Notice that for any one of perturbed matrices DAðtÞ, DBðtÞ, DCðtÞ, or DDðtÞ is equal to zero, it represents that some one of the terms Mi ¼ Ni ¼ 0 in view of Assumption 2. By the derivation of (8a) and (8b), the corresponding matrices Mi , Ni , and constant ei , could be deleted and LMI conditions (10), (13)–(18) are reduce by deleting their corresponding elements. Suppose that DðtÞ ¼ 0, the corresponding matrices M4 and N4 , and constant e4 could also be deleted in (14), (15) and (17).
C.-H. Lien / Appl. Math. Comput. 158 (2004) 29–44
41
Remark 7. There are two ways to solve the design of robust observer-based control. In the first way, we can use Theorem 1 or Corollary 1 to design the control gain K and use the control gain K to Theorem 2 or Corollaries 2–5 to design the observer gain L. In the second way, the constant q > 0 in Theorem 1 or Corollary 1 could be seen as a known parameter and use to solve a suitable control gain K, such that Theorem 2 or Corollaries 2–5 is satisfied for this control gain K. If Theorem 2 and Corollaries 2–5 cannot be satisfied for this control gain K, we can reduce or increase the parameter q in Theorem 1 or Corollary 1 until Theorem 2 or Corollaries 2–5 is satisfied.
3. Numerical example Consider the system (1) 2 3 1 0 1 A ¼ 4 0 3 1 5; 1 1 6 2 0 aðtÞ 6 bðtÞ DAðtÞ ¼ 4 0 cðtÞ 0 DCðtÞ ¼ ½ 0
dðtÞ
with the following parameters: 2 3 1 0 B ¼ 4 0 1 5; C ¼ ½ 0 1 1 ; 0 0 3 2 3 0 aðtÞ 0 7 6 7 0 5; DBðtÞ ¼ 4 0 bðtÞ 5; 0 cðtÞ 0
0 ;
D ¼ 0;
DDðtÞ ¼ 0;
ð19Þ
where jaðtÞj 6 0:3, jbðtÞj 6 0:3, jcðtÞj 6 0:3, jdðtÞj 6 0:3. Comparing (19) with Assumption 2, we have M1 ¼ M2 ¼ I;
M3 ¼ 1; M4 ¼ 0; 2 aðtÞ=0:3 0 6 0 bðtÞ=0:3 F1 ðtÞ ¼ F2 ðtÞ ¼ 4 0
0
3
0 0
7 5;
cðtÞ=0:3
F3 ðtÞ ¼ dðtÞ=0:3; F4 ðtÞ ¼ 0; 2 3 2 3 0 0:3 0 0:3 0 N1 ¼ 4 0 0:3 0 5; N2 ¼ 4 0 0:3 5; 0:3 0 0 0:3 0
N3 ¼ ½ 0 0:3 0 ;
N4 ¼ 0:
By Theorem 1 and Remark 7, the conditions (10) and (11) are satisfied with e1 ¼ 28:8546; e3 ¼ 38:3984; q ¼ 7:2327; 2 3 2:2726 0:9181 0 P ¼ 4 0:9181 11:6588 0 5; 0 0 3:8891
42
C.-H. Lien / Appl. Math. Comput. 158 (2004) 29–44
2
2:7701
b ¼ K
Pb ¼
3 2:7701 7 2:9058 5;
2:0749 11:9621
4:0744 6 R ¼ 4 2:0749
2:9058
5:0511
15:4519 0:0636
2:4084 1:9156 ; 8:1703 2:7409
2:2726
0:9181
0:9181
11:6588
2
3 22:5171 7 ^¼6 L 4 13:2686 5; 6:5996
:
The system (1) with (19) and no input perturbations (i.e. DBðtÞ ¼ 0, DDðtÞ ¼ 0) would be exponentially stabilized by the system (2)–(4) with convergence rate 0.2602 and the control and observer gains are given by " b ¼ K ¼ Pb 1 K
7:0249
0:8022
0:9687
# and
0:5586
2
0:6376 0:3114 3
7:2865 6 7 7 ^¼6 L ¼ R1 L 6 0:5797 7: 4 5 3:0230
Note that M1 ¼ M2 , F1 ðtÞ ¼ F2 ðtÞ, and DDðtÞ ¼ 0, the condition (15) in Corollary 2 is satisfied with e1 ¼ 111:9672; 2
140:9821
6 P ¼ 4 22:4035 30:4236 2
26:6052
6 R ¼ 4 177:7166
e3 ¼ 2535:3; 22:4035
30:4236
3
165:7687 26:6167
7 26:6167 5; 106:2643
17:7166
29:3685
269:0297
29:3685 137:1184 2
q ¼ 19:8723;
3 807:3646 ^ ¼ 4 52:3038 5; L 213:7169
3
7 137:1184 5; 233:0026
C.-H. Lien / Appl. Math. Comput. 158 (2004) 29–44
43
Fig. 1. Trajectories of the uncontrolled system.
in view of Remark 6. System (1) with (19) would be exponentially stabilized by the system (2)–(4) with convergence rate 0.0255 and the control and observer gains are given by 2 3 46:7563 7:0249 0:8022 0:9687 ^ ¼ 4 8:2988 5: K¼ and L ¼ R1 L 0:5586 0:6376 0:3114 9:8598 ð20Þ With e.g.,
2
3 3 6 7 xð0Þ ¼ 4 3 5; 1
3 4 6 7 ^xð0Þ ¼ 4 1 5; 3
bðtÞ ¼ 0:3 sinð2tÞ;
2
aðtÞ ¼ 0:3 cosð5tÞ;
cðtÞ ¼ 0:3 sinð4tÞ;
dðtÞ ¼ 0:2 sinð3tÞ;
some state trajectories for uncontrolled and feedback-controlled system (1) with (19) and (20) are depicted in Figs. 1 and 2, respectively. 4. Conclusion In this paper, we have studied the observer-based control problem for a class of uncertain systems. LMI approach has been developed to construct linear full-order observer to guarantee the feedback-controlled system is
44
C.-H. Lien / Appl. Math. Comput. 158 (2004) 29–44
Fig. 2. Trajectories of the feedback-controlled system.
exponentially stabilizable. A numerical example has been given to demonstrate the use of our results. Acknowledgements The research reported here was supported by the National Science Council of Taiwan, ROC under grant nos. NSC 90-2213-E-214-062 and NSC 91-2213E-214-016. References [1] S.P. Boyd, L. El Ghaoui, E. Feron, V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory, SIAM, Philadelphia, 1994. [2] K.K. Busawon, M. Saif, A state observer for nonlinear systems, IEEE Transactions on Automatic Control 44 (1999) 2098–2103. [3] F.M. Callier, C.A. Desoer, Linear System Theory, Springer-Verlag, New York, 1991. [4] C.H. Kuo, C.H. Fang, Stabilization of uncertain linear systems via static output feedback control, 2003 Automatic Control Conference, Taiwan, vol. 1, 2003, pp. 1607–1611. [5] D.W. Gu, F.W. Poon, A robust state observer scheme, IEEE Transactions on Automatic Control 46 (2001) 1958–1963. [6] C.H. Lien, J.D. Chen, Discrete-delay-independent and discrete-delay-dependent criteria for a class of neutral systems, ASME Journal of Dynamic Systems, Measurement, and Control 125 (2003) 33–41. [7] C.P. Tan, C. Edwards, An LMI approach for designing sliding mode observers, International Journal of Control 74 (1998) 1559–1568. [8] K. Zhou, J.C. Doyle, Essentials of Robust Control, Prentice-Hall, New Jersey, 1998. [9] P. Zitek, Anisochronic state observers for hereditary systems, International Journal of Control 71 (1998) 581–599.
