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On robust H∞ filter design for uncertain neural systems: LMI optimization approach
Applied Mathematics and Computation 159 (2004) 625–639 www.elsevier.com/locate/amc
On robust H1 filter design for uncertain neural systems: LMI optimization approach Ju H. Park a
a,*
, O. Kwon b, S. Lee b, S. Won
b
School of Electrical Engineering and Computer Science, Yeungnam University, 214-1 Dae-Dong, Kyongsan 712-749, South Korea b Department of Electrical Engineering, Pohang University of Science and Technology, San 31, Hyoja-dong, Pohang 790-784, South Korea
Abstract This article deals with the robust H1 filtering problem for neutral delay differential systems with parametric uncertainties. A linear matrix inequality (LMI) approach is proposed to design the robust H1 filter such that the filtering system remains asymptotically stable and the bound of H1 norm is minimized. The Lyapunov stability theory is used for analysis of the system. A numerical example is provided to illustrate the validity of proposed design approach. Ó 2003 Elsevier Inc. All rights reserved. Keywords: Neutral systems; Uncertainty; H1 filtering; Lyapunov theory; Linear matrix inequalities
1. Introduction State estimation has been one of the fundamental issues in the control area. Recently, there has been a lot of interest on the problem of robust H1 filtering for dynamic systems with parametric uncertainties. In the H1 filtering, the
*
Corresponding author. E-mail address: [email protected] (J.H. Park).
0096-3003/$ - see front matter Ó 2003 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2003.09.025
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Nomenclature Rn n-dimensional real space Rmn set of all real m by n matrices xT (or AT ) transpose of vector x (or matrix A) kk Euclidean vector norm or the induced matrix 2-norm P > 0 (respectively P < 0) matrix P is symmetric positive (respectively negative) definite I identity matrix with appropriate dimension H the elements below the main diagonal of a symmetric block matrix C0 a set of all continuous differentiable function on given interval diag f g block diagonal matrix L2 the space of square vector functions on ½0; 1Þ with R 1 integrable 2 1=2 norm k k2 ð 0 k k dtÞ
exogenous noise signal is assumed to be energy bounded rather than Gaussian, and the problem is to design a filter such that the H1 norm of the system, which reflects the worst-case gain of the system, is minimized [11]. On the other hand, the stability analysis and stabilization problem for delay differential systems has been received considerable attention during the last few decades [1,2]. It is well-known that the delay is often main cause of instability and poor performance of systems. More recent years, the stability analysis for neutral delay-differential equations/systems of various types has been extensively studied [3–9]. Various methods has been introduced to derive less conservative and concise stability criteria for neutral systems [3–8]. A control scheme to guarantee the adequate level of performance has been presented by Park [9], and the design problem on observer-based controller of a class of neutral system is investigated in Park [10]. However, so far, the robust H1 filtering problem for neutral differential systems has not been fully investigated and remains to be important and challenging. In this paper, the problem of robust H1 filtering for neutral delay-differential systems subjected to parameter uncertainties is investigated. The uncertainties are assumed to be bounded. Using the Lyapunov functional technique combined with LMI technique, we develop a robust H1 filter for this system, which makes the closed-loop system asymptotically stable and the bound of H1 norm be minimized. A stability criterion for the existence of the filter is derived in terms of LMIs, and theirs solutions provide a parameterized representation of the filter. The LMIs can be easily solved by various efficient convex optimization algorithms [13].
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2. Problem formulation Consider a class of neutral delay-differential system of the form: d ½xðtÞ A2 xðt sÞ ¼ ðA0 þ DA0 ðtÞÞxðtÞ þ ðA1 þ DA1 ðtÞÞxðt hÞ þ BwðtÞ; dt yðtÞ ¼ ðC þ DCðtÞÞxðtÞ þ DwðtÞ; xðt0 þ hÞ ¼ /ðhÞ;
8h 2 ½ H ; 0 ; ð1Þ
where xðtÞ 2 Rn is the state vector, A0 , A1 , A2 , B, C, and D are known constant real matrices of appropriate dimensions, wðtÞ 2 Rm is the noise signal vector (including process and measurement noises) in L2 , yðtÞ 2 Rl is the measured output, h and s are the positive constant time delays, DA0 ðtÞ, DA1 ðtÞ and DCðtÞ are time-varying parameter uncertainties, H ¼ maxfh; sg, /ðÞ 2 C0 : ½ H ; 0 ! Rn is the initial vector. The admissible uncertainties are assumed to be of the form DA0 ðtÞ ¼ D0 F0 ðtÞE0 ;
DA1 ðtÞ ¼ D1 F1 ðtÞE1 ;
DCðtÞ ¼ D2 F2 ðtÞE2 ; ð2Þ
where D0 , D1 , D2 , E0 , E1 and E2 are real constant matrices with appropriate dimensions, and F0 ðtÞ, F1 ðtÞ and F2 ðtÞ are unknown perturbation matrices with Lebesque measurable elements and satisfies kF0 ðtÞk 6 1;
kF1 ðtÞk 6 1;
kF2 ðtÞk 6 1:
The robust filter design problem tackled in this paper consists of obtaining an estimate nðtÞ of the signal xðtÞ that provides small estimation error eðtÞ ¼ xðtÞ nðtÞ for all w 2 L2 and uncertainties. Attention is focused on the design of a time-invariant stable filter of order n with state-space realization of the form d ½nðtÞ A2 nðt sÞ ¼ GnðtÞ þ A1 nðt hÞ þ KyðtÞ; dt nðt0 þ hÞ ¼ 0; h 2 ½ H ; 0 ;
ð3Þ
where G and K are the filter parameter to be determined later. Then, it follows that the error dynamics is given by _ e_ ðtÞ ¼ x_ ðtÞ nðtÞ ¼ GeðtÞ þ ðA0 þ DA KðC þ DCÞ GÞxðtÞ þ A1 eðt hÞ þ DA1 xðt hÞ þ A2 e_ ðt sÞ þ ðB KDÞwðtÞ
ð4Þ
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and the output of error is defined as ye ðtÞ ¼ LeðtÞ;
ð5Þ
where L is a known constant matrix. Thus, by (1) and (4), it is easy to obtain the augmented system d 2 zðt sÞ ¼ ðA 0 þ DA 0 ÞzðtÞ þ ðA 1 þ DA 1 Þzðt hÞ þ BwðtÞ; ½zðtÞ A dt ye ðtÞ ¼ LzðtÞ; ð6Þ where
A0 A1 0 A2 0 0 ; A2 ¼ ; ; A1 ¼ A0 KC G G 0 A1 0 A2 B xðtÞ ¼ ¼ ½ 0 L ; zðtÞ ¼ B ; L ; B KD eðtÞ DA0 0 0 D0 0 ðtÞ ¼ F0 ðtÞ½ E0 0 ¼ F0 ðtÞ½ E0 0 þ DA D0 DA0 KDC 0 0 0 þ F2 ðtÞ½ E2 0 KD2 0 þ D 02 F0 ðtÞE 0 þ D 2 F2 ðtÞE 2; 01 F0 ðtÞE D 0 0 0 1 ðtÞ ¼ 1 F1 ðtÞE 1: DA F1 ðtÞ½ E1 0 D ¼ DA1 0 D1 ð7Þ 0 ¼ A
Let Tzw denote a stable operator from signal wðtÞ to signal zðtÞ. Then H1 norm of Tzw is defined as kzðtÞk2 kTzw k1 ¼ sup : kwðtÞk wðtÞ2L2 2 Here, we introduce H1 performance measure for system (6) as follows: Z 1 J¼ ½yeT ðtÞye ðtÞ c2 wT ðtÞwðtÞ dt:
ð8Þ
0
Therefore, the problem tackled in this paper is to determine the filter (3) within the upper bound, i.e, kTye w k1 ¼ sup wðtÞ
kye ðtÞk2 < c: kwðtÞk2
In this situation, the filtering error system is said to have a guaranteed c level of noise attenuation. In next section, the criterion for minimization of c level will be derived using LMI convex optimization approach.
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3. Robust stability analysis and H‘ filter design Before proceeding further, we will give two lemmas and a fact. Lemma 1 [14]. For any constant symmetric positive-definite matrix H 2 Rmm , a scalar r > 0, and the vector function x : ½0; r ! Rm such that the integrations in the following are well defined, then Z r T Z r Z r T x ðsÞHxðsÞ ds P xðsÞ ds H xðsÞ ds : r 0
0
0
Lemma 2 [12]. For given positive scalars h and s and any E1 ; E2 2 Rnn , the operator Dðxt Þ : C0 ! Rn defined by Z t Dðxt Þ ¼ xðtÞ þ E1 xðsÞ ds E2 xðt sÞ ð9Þ t h
is stable if there exist a positive definite matrix C and positive scalars a1 and a2 such that a1 þ a2 < 1; T E2 CE2 a1 C H
hE2T CE1 h2 E1T CE1 a2 C
< 0:
ð10Þ
Fact 1 (Schur complement). Given constant symmetric matrices R1 , R2 , R3 where R1 ¼ RT1 and 0 < R2 ¼ RT2 , then R1 þ RT3 R 1 2 R3 < 0 if and only if R2 R3 R1 RT3 < 0; or < 0: RT3 R1 R3 R2 Here, we define an operator Dðzt Þ : C0 ! Rn Z t 2 zðt sÞ; Dðzt Þ ¼ zðtÞ þ A1 zðsÞ ds A t h
zt ¼ zðt þ hÞ;
ð11Þ
8 h 2 ½ H ; 0 :
Remark 1. By Lemma 2, the operator Dðzt Þ is stable if there exist a positive definite matrix C and positive scalars a1 and a2 such that " # T CA 2 a1 C T CA 1 A h A 2 2 ð12Þ 1 a2 C < 0; and a1 þ a2 < 1: T CA H h2 A 1
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Differentiating Dðzt Þ and combining Eq. (6) leads to þ DA 0 ðtÞÞzðtÞ þ DA 1 ðtÞzðt hÞ þ BwðtÞ _ t Þ ¼ ðA Dðz þ DA 0 ðtÞÞDðzt Þ þ DA 1 ðtÞzðt hÞ þ BwðtÞ ¼ ðA Z t þ DA 0 ðtÞÞA 1 þ DA 0 ðtÞÞA 2 zðt þ sÞ; ðA zðsÞ ds ðA
ð13Þ
t h
¼A 0 þ A 1. where A Now, we have the first main result of the paper. Theorem 1. Consider the system (6) with wðtÞ ¼ 0. For given scalars h > 0 and s > 0, suppose that there exist C > 0, a1 > 0, and a2 > 0 satisfying (12). If there exist positive scalars ei for i ¼ 0; 1; . . . ; 9, P > 0, Q1 > 0, Q2 > 0, and Q3 > 0 satisfying the following LMI: 2 3 A 2 þ QA 2 hP A A 1 hQA 1 R1 R2 0 P A 6H R 7 0 0 0 3 6 7 6 H H R4 7 < 0; ð14Þ 0 0 6 7 T 4H H H 5 hA2 Q R5 H H H H R6 where TP þ P A þ Q þ e0 E TE T T R1 ¼ A 0 0 þ e1 E0 E0 þ e2 E2 E2 ; 02 P D 2 PD 1 PD 01 P D 02 P D 2 hP D 01 hP D 02 hP D 2 ; 01 P D R2 ¼ P D R3 ¼ diag f e0 I; e1 I; e2 I; e3 I; e4 I; e5 I; e6 I; he7 I; he8 I; he9 Ig; T TE R4 ¼ Q2 þ e3 E 1 1 þ A2 Q A2 ; TE T T T T T R5 ¼ Q3 þ e4 A 2 0 E0 A2 þ e5 A2 E0 E0 A2 þ e6 A2 E2 E2 A2 ; 2 TE T T T T T R6 ¼ hQ1 þ e7 hA 1 0 E0 A1 þ e8 hA1 E0 E0 A1 þ e9 hA1 E2 E2 A1 þ h Q;
Q ¼ hQ1 þ Q2 þ Q3 ; then, the filtering error system (6) is asymptotically stable. Proof. Consider the following legitimate Lyapunov functional candidate [1]: V ðt; zðtÞÞ ¼ V1 þ V2 þ V3 þ V4 ; where V1 ¼ DT ðzt ÞP Dðzt Þ; V3 ¼
Z
V2 ¼
ð15Þ
Z
t
ðs t þ hÞzT ðsÞQ1 zðsÞ ds;
t h t T
z ðsÞQ2 zðsÞ ds;
V4 ¼
Z
t h
and P > 0, Q1 > 0, Q2 > 0, and Q3 > 0.
t
zT ðsÞQ3 zðsÞ ds; t s
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631
Taking the time derivative of V along the solution of (13) gives that dV1 þ DA 0 ðtÞÞDðzt Þ þ DA 1 ðtÞzðt hÞ ¼ 2 ðA dt T Z t ðA þ DA0 ðtÞÞA1 zðsÞ ds ðA þ DA0 ðtÞÞA2 zðt sÞ P Dðzt Þ; t h
ð16Þ Z t dV2 ¼ hzT ðtÞQ1 zðtÞ zT ðsÞQ1 zðsÞ ds 6 hzT ðtÞQ1 zðtÞ dt t h Z t T Z t 1 1 zðsÞ ds ðhQ1 Þ zðsÞ ds ; h t h h t h
ð17Þ
dV3 ¼ zT ðtÞQ2 zðtÞ zT ðt hÞQ2 zðt hÞ; dt
ð18Þ
dV4 ¼ zT ðtÞQ3 zðtÞ zT ðt sÞQ3 zðt sÞ; dt
ð19Þ
where Lemma 1 is utilized in (17). Here note that 1 zT ðtÞQi zðtÞ ¼ DT ðzt ÞQi Dðzt Þ 2DT ðzt ÞQi A Z 1 þ A
1 2 A
T Z 1 zðsÞ ds Qi A
t t h
Z
t
Z
t
2 zðt sÞ zðsÞ ds þ 2DT ðzt ÞQi A t h
t
zðsÞ ds
t h
T T Qi A 2 zðt sÞ þ zT ðt sÞA 2 zðt sÞ: zðsÞ ds Qi A 2
t h
Then, we have 4 X dV dVi ¼ 6 vT ðtÞXvðtÞ; dt dt i¼1
ð20Þ
Rt T where vT ðtÞ ¼ ½DT ðzt ÞzT ðt hÞzT ðt sÞðh 1 t h zðsÞ dsÞ and 3 20 1 þ DA 0 ðtÞÞT P ðA 7 6@ 1 ðtÞ þ DA 0 ðtÞÞ A X1;2 X1;3 P DA 7 6 þP ðA 7 6 7 6 þQ X¼6 7: T 7 6 H Q þ A QA 0 0 2 2 2 6 TQ 7 5 4 H H Q3 hA 2 H H H hQ1 þ h2 Q ð21Þ þ DA 0 ðtÞÞA 2 þ QA 2 and X1;3 ¼ hP ðA þ DA 0 ðtÞÞA 1 hQA 1. where X1;2 ¼ P ðA
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Therefore, if X < 0, there exists the positive scalar c such that dV 2 6 ckzðtÞk : dt
ð22Þ
Using the well-known inequality 2aT b 6
1 T a a þ ebT b e
ð23Þ
for any a; b 2 Rn and scalar e > 0, one can eliminate the uncertain factors 0 ðtÞ and DA 1 ðtÞ. Thus, we get the following Fi ðtÞ which are included in DA relation: 2
X1 6H X66 4H H
0 X2 H H
A 2 þ QA 2 PA 0 X3 H
3 A 1 hQA 1 hP A 7 0 7 X; 5 TQ hA
ð24Þ
2
X4
where TP þ P A þ Q þ e 1 P D T P þ e0 E TE 01 D X1 ¼ A 0 01 0 0 1 T T 1 T E 0 þ e P D P þ e2 E 2E T þ e 1 P D TP 2D 1D þ e P D02 D P þ e1 E 1
02
0
2
2
2
3
1
1 T 1 T 1 T T þ e 1 4 P D01 D01 P þ e5 P D02 D02 P þ e6 P D2 D2 P þ e7 hP D01 D01 P 1 T T þ e 1 8 hP D02 D02 P þ e9 hP D2 D2 P ;
T TE X2 ¼ Q2 þ e3 E 1 1 þ A2 Q A2 ; TE T T T T T X3 ¼ Q3 þ e4 A 2 0 E 0 A2 þ e 5 A2 E0 E0 A2 þ e 6 A2 E2 E2 A2 ; 2 TE T T T T T X4 ¼ hQ1 þ e7 hA 1 0 E0 A1 þ e8 hA1 E0 E0 A1 þ e9 hA1 E2 E2 A1 þ h Q:
is equivalent to By Fact 1 (Schur complement), the negative-definiteness of X the LMI (14). Thus, the feasibility of the LMI (14) guarantees the negativeness (22) of dV . Therefore, by Theorem 9.8.1 (pp. 292–293) of Hale and Lunel [1] dt with the stable operator Dðzt Þ and (22), we conclude that system (1) and (6) are both asymptotically stable. h Next, the following theorem provides a criterion for the existence of a linear stable filter assuring a robust H1 performance for neutral systems (1). Theorem 2. Let A ¼ A0 þ A1 and q ¼ c2 . For wðtÞ 2 L2 and given delays h and s, consider the system (6). If the following optimization problem:
J.H. Park et al. / Appl. Math. Comput. 159 (2004) 625–639
633
minimize q subject to 2 6 P1 6 6 6 6H 6 6 6 6H 6 6 6 6 6H 6 6 6H 6 6 6H 6 6 6H 6 6 6H 6 6 6H 6 6 6H 4 H
P2
P4
0
0
0
P3
0
0
0
0
P5
P6
0
0
H
H
P7
0
H
H
H
H
H
H
H
!
P1 AA2
ðR2 þ R5 þ R8 ÞA2
þðR1 þ R4 þ R7 ÞA2 0 ðP2 A0 M2 C M1 ÞA2
!
0
!
ðM1 þ P2 A1 ÞA2
þAT2 ðR2 þ R5 þ R8 ÞT
þðR3 þ R6 þ R9 ÞA2
0
0
0
P8
P9
0
0
H
H
P10
0
0
H
H
H
H
P11
R8
H
H
H
H
H
H
R9
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
!
hP1 AA1
3 hðR2 þ R5 þ R8 ÞA1
hðR1 þ R4 þ R7 ÞA1 0 ! hðP2 A0 M2 C M1 ÞA1 hAT1 ðR2 þ R5 þ R8 ÞT 0 0 0 hAT2 ðR1
þ R4 þ R7 Þ
hðR2 þ R5 þ R8 ÞT A2 P12 H
7 7 7 7 0 0 7 7 ! 7 hðM1 þ P2 A1 ÞA1 7 P2 B M2 D 7 7 hðR3 þ R6 þ R9 ÞA1 7 7 7 0 0 7 7 7<0 0 0 7 7 7 0 0 7 7 T hA2 ðR2 þ R5 þ R8 Þ 0 7 7 7 7 hAT2 ðR3 þ R6 þ R9 Þ 0 7 7 2 7 hR2 þ h ðR2 þ R5 þ R8 Þ 0 7 7 2 hR3 þ h ðR3 þ R6 þ R9 Þ 0 5
H
R1 H
R2 R3
> 0;
c2 I
H
R4 H
R5 R6
P1 B
> 0;
R7 H
R8 R9
ð25Þ
>0
ð26Þ
has the positive definite solutions, P1 , P2 , R1 , R3 , R4 , R6 , R7 , R9 , positive scalars e0 ; e1 ; . . . ; e9 ; q, and matrices M1 , M2 , R2 , R5 , R8 , then the filtering error system has a guaranteed c level of noise attenuation by robust H1 filter (3).
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J.H. Park et al. / Appl. Math. Comput. 159 (2004) 625–639
Here, the following notations are defined: P1 ¼ AT P1 þ P1 A þ R1 þ R4 þ R7 þ e0 E0T E0 þ e1 E0T E0 þ e2 E2T E2 ; P2 ¼ ½ P1 D0 P1 D0 hP1 D0 ; P3 ¼ diag f e0 I; e4 I; he7 Ig; P4 ¼ AT0 P2 C T M2T M1T þ R2 þ R5 þ R8 ; P5 ¼ M1 þ M1T þ P2 A1 þ AT1 P2 þ R3 þ R6 þ R9 þ LT L; P6 ¼ ½ P2 D0 M2 D2 P2 D1 P2 D0 M2 D2 hP2 D0 hM2 D2 ; P7 ¼ diag f e1 I; e2 I; e3 I; e5 I; e6 I; he8 I; he9 Ig; P8 ¼ R4 þ e3 E1T E1 þ AT2 ðR1 þ R4 þ R7 ÞA2 ; P9 ¼ R5 þ AT2 ðR2 þ R5 þ R8 ÞA2 ; P10 ¼ R6 þ AT2 ðR3 þ R6 þ R9 ÞA2 ; P11 ¼ R7 þ e4 AT2 E0T E0 A2 þ e5 AT2 E0T E0 A2 þ e6 AT2 E2T E2 A2 ; P12 ¼ hR1 þ h2 ðR1 þ R4 þ R7 Þ þ e7 hAT1 E0T E0 A1 þ e8 hAT1 E0T E0 A1 þ e9 hAT1 E2T E2 A1 :
ð27Þ
Proof. To establish the H1 performance for the filtering error system, first notice that since the system (6) is asymptotically stable, then zðtÞ tends to zero as t ! 1. Next, assuming zero initial conditions for the filtering error system, the performance index is Z 1 J¼ ½yeT ðtÞye ðtÞ c2 wT ðtÞwðtÞ dt 0 Z 1 ¼ ½yeT ðtÞye ðtÞ c2 wT ðtÞwðtÞ þ V_ ðt; zðtÞÞ dt 0
þ V ðt; zðtÞÞjt¼0 V ðt; zðtÞÞjt!1 ;
ð28Þ
where V ðt; zðtÞÞ is as in (15). Considering that V ðt; zðtÞÞjt¼0 ¼ 0 and V ðt; zðtÞÞjt!1 ! 0, one can easily infer from (28) that Z 1 J6 fT ðtÞW fðtÞ dt; ð29Þ h i 0 R t where fT ðtÞ ¼ DT ðzt Þ zT ðt hÞ zT ðt sÞ ðð1=hÞ t h zðsÞ dsÞT wT ðtÞ and 2 3 1 ðtÞ þ DA 0 ðtÞÞA 1 hQA 1 P B P DA W2 hP ðA W1 6 H Q2 þ A T QA 2 0 0 0 7 2 6 7 6 W ¼6H H Q3 0 0 7 7; 4H H H hQ1 þ h2 Q 0 5 H H H H c2 I ð30Þ T
þ DA 0 ðtÞÞ P þ P ðA þ DA 0 ðtÞÞ þ Q þ L T L where W1 ¼ ðA DA0 ðtÞÞA2 þ QA2 .
and
þ W2 ¼ P ðA
J.H. Park et al. / Appl. Math. Comput. 159 (2004) 625–639
Again, using the inequality (23), we have 2 A 2 þ QA2 0 PA T L X1 þ L 6 0 H X2 6 ¼6 W 6W H H X3 6 4 H H H H H H Here, define the matrices Q1 , Q2 , Q3 and P as R4 R5 R1 R2 Q1 ¼ ð1=hÞ T ; Q2 ¼ ; R2 R3 RT5 R6 P1 0 P¼ ; 0 P2
635
A 1 hQA 1 hP A 0 1 hQA X4 H
3 PB 0 7 7 0 7 7: 0 5 c2 I ð31Þ
Q3 ¼
R7 RT8
R8 ; R9 ð32Þ
where P1 , P2 , R1 , R3 , R4 , R6 , R7 and R9 are the positive definite matrix defined in (26). < 0 is modified to Utilizing (7) and (32), the inequality W 2 U1 U2 0 0 P1 AA2 þ ðR1 þ R4 þ R7 ÞA2 ðR2 þ R5 þ R8 ÞA2 6 P2 ðA0 KC GÞA2 P2 ðG þ A1 ÞA2 6 0 6 H U3 0 T 6 þAT2 ðR2 þ R5 þ R8 Þ þðR3 þ R6 þ R9 ÞA2 6 6 H H P8 P9 0 0 6 6 0 0 6 H H H P10 6 6H H H H P R 11 8 6 6 6H H H H H R9 6 6H H H H H H 6 6 4H H H H H H H H H H hP1 AA1 hðR1 þ R4 þ R7 ÞA1 hP2 ðA0 KC GÞA1 hAT1 ðR2 þ R5 þ R8 Þ 0 0
T
H hðR2 þ R5 þ R8 ÞA1 hP2 ðG þ A1 ÞA1 hðR3 þ R6 þ R9 ÞA1 0 0
hAT2 ðR1 þ R4 þ R7 Þ
hAT2 ðR2 þ R5 þ R8 Þ
hðR2 þ R5 þ R8 ÞT A2
hAT2 ðR3 þ R6 þ R9 Þ
P12
hR2 þ h2 ðR2 þ R5 þ R8 Þ
H H
hR3 þ h2 ðR3 þ R6 þ R9 Þ H
H P1 B
3
7 P2 B P2 KD 7 7 7 7 7 0 7 7 0 7 7 < 0; 7 0 7 7 7 0 7 7 0 7 7 5 0 c2 I ð33Þ
636
J.H. Park et al. / Appl. Math. Comput. 159 (2004) 625–639
where T 1 T 1 T U1 ¼ P1 þ e 1 0 P1 D0 D0 P1 þ e4 P1 D0 D0 P1 þ e7 hP1 D0 D0 P1 ; T
U2 ¼ ðA0 KC GÞ P2 þ R2 þ R5 þ R8 ; T U3 ¼ ðG þ A1 ÞT P2 þ P2 ðG þ A1 Þ þ R3 þ R6 þ R9 þ e 1 1 P2 D0 D0 P2 T 1 1 T þ e 1 2 P2 KD2 D2 K P2 þ e3 P2 D1 D1 P2 þ e5 P2 D0 D0 P2 T 1 T 1 T T þ e 1 6 P2 KD2 D2 K P2 þ e8 hP2 D0 D0 P2 þ e9 hP2 KD2 D2 K P2 þ L L
and P1 , P8 , and P9 are defined in (27). Using some changes of variables, M1 ¼ P2 G and M2 ¼ P2 K, the inequality (33) is changed to the LMI (25). Therefore, the inequalities (25) and (26) imply that J < 0 for any nonzero wðtÞ 2 L2 , i.e., the filtering error system has a guaranteed c level of noise attenuation, which concludes the proof. h Remark 2. Given any solutions of the LMIs (25) and (26) in Theorem 2, the corresponding robust H1 filter of the form (3) will be constructed from the 1 relations G ¼ P 1 2 M1 and K ¼ P2 M2 . Remark 3. In this article, in order to solve the LMIs, we utilize MatlabÕs LMI Control Toolbox [15], which implements state-of-the-art interior-point algorithms, which is significantly faster than classical convex optimization algorithms [13]. Example. To illustrate the proposed method, consider the neural system 0 3 0:2 0:1 d xðtÞ xðt 1Þ ¼ dt 0:1 0:2 4 5 1 0:3 0:2 þ F0 ðtÞ½ 0 0:1 xðtÞ þ 1 0:2 0:3 1 1 þ F1 ðtÞ½ 0:1 0:1 xðt 0:5Þ þ wðtÞ; 0 0:5 yðtÞ ¼ ð½ 1
0 þ F2 ðtÞ½ 0:1
0:1 ÞxðtÞ þ wðtÞ;
where Fi ðtÞT Fi ðtÞ 6 I for i ¼ 0; 1; 2. The output of error system is defined as ye ðtÞ ¼ ½ 1
1 eðtÞ:
The aim is to design a robust H1 filter (3) for above system. First, checking the stability condition (12) for operator Dðzt Þ gives the solutions:
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2
0:7490 6 0:0044 C¼6 4 0 0
0:0044 0 0:7490 0 0 0:7490 0 0:0044
637
3
0 0 7 7; 0:0044 5 0:7490
a1 ¼ 0:3333;
a2 ¼ 0:3333:
Next, by applying the minimization problem of Theorem 2 to above system using LMI toolbox [15], the solutions of optimization problem in the inequalities (25) and (26) are as follows: 0:1701 0:1101 0:6795 0:3221 P1 ¼ ; P2 ¼ ; 0:1101 0:1615 0:3221 0:9955 0:1324 0:0867 R1 ¼ ; 0:0867 0:1478 0:0255 0:0144 0:4043 0:2017 R2 ¼ ; R3 ¼ ; 0:0210 0:0131 0:2017 0:5774 0:0476 0:0473 0:0049 0:0052 R4 ¼ ; R5 ¼ ; 0:0473 0:0485 0:0045 0:0051 0:1096 0:1060 0:1394 0:1291 R6 ¼ ; R7 ¼ ; 0:1060 0:1115 0:1291 0:1539 0:0275 0:0403 ; R8 ¼ 0:0119 0:0463 0:8812 0:6713 2:5064 0:1475 R9 ¼ ; M1 ¼ ; 0:6713 0:7702 4:6942 4:0887 0:9731 ; M2 ¼ 0:7325 q ¼ 0:1812; e3 ¼ 2:0789; e5 ¼ 29:7619; e9 ¼ 14:5728:
e0 ¼ 2:3617; e4 ¼ 10
4
e1 ¼ 5:1754;
e2 ¼ 4:463;
4:8687;
e6 ¼ 10:8645;
e7 ¼ 2:0256;
e8 ¼ 4:439;
Therefore, in light of Remark 2, the parameters G and K of robust H1 filter are 1:7168 2:5557 1:2795 G¼ ; K¼ : 4:1602 4:9341 0:3219 Therefore, the obtained robust H1 filter stabilizes the filtering error system and minimizes H1 norm within c ¼ q1=2 ¼ 0:4257 in the presence of delay and uncertainties.
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4. Conclusions This paper investigated the design of robust H1 filter for neutral delaydifferential system with parametric uncertainties in the system state-space model. A methodology for the design of a stable linear filter that assures asymptotic stability and minimization of H1 performance for the filtering error system is studied. The filter is obtained from a convex optimization problem described in terms of LMIs. The validity of proposed filter design algorithm has been checked through a numerical example.
Acknowledgements J.H. Park is grateful to H.Y. Jung, H.J. Baek, W.C. Shim, and S.G. Kim for their good advice about my research, and to my students, I.G. Jung, Y.I. Ban, and S.D. Lee for their devoted contribution in my laboratory. This work was supported by grant No. R05-2003-000-10173-0 from Korea Science & Engineering Foundation.
References [1] J. Hale, S.M. Verduyn Lunel, Introduction to Functional Differential Equations, SpringerVerlag, New York, 1993. [2] V. Kolmanovskii, A. Myshkis, Applied Theory of Functional Differential Equations, Kluwer Academic Publishers, Dordrecht, 1992. [3] J.X. Kuang, J.X. Xiang, H.J. Tian, The asymptotic stability of one-parameter methods for neutral differential equations, BIT 34 (1994) 400–408. [4] D.Y. Khusainov, E.V. YunÕkova, Investigation of the stability of linear systems of neutral type by the Lyapunov function method, Differentsialnye Uravneniya 24 (1988) 613–621. [5] G.D. Hu, G.D. Hu, Some simple stability criteria of neutral delay-differential systems, Applied Mathematics and Computation 80 (1996) 257–271. [6] J.H. Park, S. Won, A note on stability of neutral delay-differential systems, Journal of The Franklin Institute 336 (1999) 543–548. [7] J.H. Park, S. Won, Asymptotic stability of neutral systems with multiple delays, Journal of Optimization Theory and Applications 103 (1999) 187–200. [8] J.H. Park, Delay-dependent criterion for asymptotic stability of a class of neutral equations, Applied Mathematics Letters, in press. [9] J.H. Park, Robust guaranteed cost control for uncertain linear differential systems of neutral type, Applied Mathematics and Computations 140 (2003) 523–535. [10] J.H. Park, On the design of observer-based controller of linear neutral delay-differential systems, Applied Mathematics and Computations, in press. [11] R.M. Palhares, C.E. de Souza, P.L.D. Peres, Robust H1 filtering for uncertain discrete-time state-delayed systems, IEEE Transactions on Signal Processing 49 (2001) 1696–1703. [12] D. Yue, S. Won, O. Kwon, Delay-dependent stability of neutral systems with time delay: an LMI approach, IEE Proceedings––Control Theory and Applications 150 (2003) 23–27.
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[13] B. Boyd, L.E. Ghaoui, E. Feron, V. Balakrishnan, Linear Matrix Inequalities in Systems and Control Theory, SIAM, Philadelphia, 1994. [14] K. Gu, An integral inequality in the stability problem of time-delay systems, in: Proceedings of 39th IEEE CDC Sydney, Australia, 2000, pp. 2805–2810. [15] P. Gahinet, A. Nemirovski, A. Laub, M. Chialali, LMI Control Toolbox UserÕs Guide, The Mathworks, Natick, Massachusetts, 1995.
On robust H1 filter design for uncertain neural systems: LMI optimization approach Ju H. Park a
a,*
, O. Kwon b, S. Lee b, S. Won
b
School of Electrical Engineering and Computer Science, Yeungnam University, 214-1 Dae-Dong, Kyongsan 712-749, South Korea b Department of Electrical Engineering, Pohang University of Science and Technology, San 31, Hyoja-dong, Pohang 790-784, South Korea
Abstract This article deals with the robust H1 filtering problem for neutral delay differential systems with parametric uncertainties. A linear matrix inequality (LMI) approach is proposed to design the robust H1 filter such that the filtering system remains asymptotically stable and the bound of H1 norm is minimized. The Lyapunov stability theory is used for analysis of the system. A numerical example is provided to illustrate the validity of proposed design approach. Ó 2003 Elsevier Inc. All rights reserved. Keywords: Neutral systems; Uncertainty; H1 filtering; Lyapunov theory; Linear matrix inequalities
1. Introduction State estimation has been one of the fundamental issues in the control area. Recently, there has been a lot of interest on the problem of robust H1 filtering for dynamic systems with parametric uncertainties. In the H1 filtering, the
*
Corresponding author. E-mail address: [email protected] (J.H. Park).
0096-3003/$ - see front matter Ó 2003 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2003.09.025
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Nomenclature Rn n-dimensional real space Rmn set of all real m by n matrices xT (or AT ) transpose of vector x (or matrix A) kk Euclidean vector norm or the induced matrix 2-norm P > 0 (respectively P < 0) matrix P is symmetric positive (respectively negative) definite I identity matrix with appropriate dimension H the elements below the main diagonal of a symmetric block matrix C0 a set of all continuous differentiable function on given interval diag f g block diagonal matrix L2 the space of square vector functions on ½0; 1Þ with R 1 integrable 2 1=2 norm k k2 ð 0 k k dtÞ
exogenous noise signal is assumed to be energy bounded rather than Gaussian, and the problem is to design a filter such that the H1 norm of the system, which reflects the worst-case gain of the system, is minimized [11]. On the other hand, the stability analysis and stabilization problem for delay differential systems has been received considerable attention during the last few decades [1,2]. It is well-known that the delay is often main cause of instability and poor performance of systems. More recent years, the stability analysis for neutral delay-differential equations/systems of various types has been extensively studied [3–9]. Various methods has been introduced to derive less conservative and concise stability criteria for neutral systems [3–8]. A control scheme to guarantee the adequate level of performance has been presented by Park [9], and the design problem on observer-based controller of a class of neutral system is investigated in Park [10]. However, so far, the robust H1 filtering problem for neutral differential systems has not been fully investigated and remains to be important and challenging. In this paper, the problem of robust H1 filtering for neutral delay-differential systems subjected to parameter uncertainties is investigated. The uncertainties are assumed to be bounded. Using the Lyapunov functional technique combined with LMI technique, we develop a robust H1 filter for this system, which makes the closed-loop system asymptotically stable and the bound of H1 norm be minimized. A stability criterion for the existence of the filter is derived in terms of LMIs, and theirs solutions provide a parameterized representation of the filter. The LMIs can be easily solved by various efficient convex optimization algorithms [13].
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2. Problem formulation Consider a class of neutral delay-differential system of the form: d ½xðtÞ A2 xðt sÞ ¼ ðA0 þ DA0 ðtÞÞxðtÞ þ ðA1 þ DA1 ðtÞÞxðt hÞ þ BwðtÞ; dt yðtÞ ¼ ðC þ DCðtÞÞxðtÞ þ DwðtÞ; xðt0 þ hÞ ¼ /ðhÞ;
8h 2 ½ H ; 0 ; ð1Þ
where xðtÞ 2 Rn is the state vector, A0 , A1 , A2 , B, C, and D are known constant real matrices of appropriate dimensions, wðtÞ 2 Rm is the noise signal vector (including process and measurement noises) in L2 , yðtÞ 2 Rl is the measured output, h and s are the positive constant time delays, DA0 ðtÞ, DA1 ðtÞ and DCðtÞ are time-varying parameter uncertainties, H ¼ maxfh; sg, /ðÞ 2 C0 : ½ H ; 0 ! Rn is the initial vector. The admissible uncertainties are assumed to be of the form DA0 ðtÞ ¼ D0 F0 ðtÞE0 ;
DA1 ðtÞ ¼ D1 F1 ðtÞE1 ;
DCðtÞ ¼ D2 F2 ðtÞE2 ; ð2Þ
where D0 , D1 , D2 , E0 , E1 and E2 are real constant matrices with appropriate dimensions, and F0 ðtÞ, F1 ðtÞ and F2 ðtÞ are unknown perturbation matrices with Lebesque measurable elements and satisfies kF0 ðtÞk 6 1;
kF1 ðtÞk 6 1;
kF2 ðtÞk 6 1:
The robust filter design problem tackled in this paper consists of obtaining an estimate nðtÞ of the signal xðtÞ that provides small estimation error eðtÞ ¼ xðtÞ nðtÞ for all w 2 L2 and uncertainties. Attention is focused on the design of a time-invariant stable filter of order n with state-space realization of the form d ½nðtÞ A2 nðt sÞ ¼ GnðtÞ þ A1 nðt hÞ þ KyðtÞ; dt nðt0 þ hÞ ¼ 0; h 2 ½ H ; 0 ;
ð3Þ
where G and K are the filter parameter to be determined later. Then, it follows that the error dynamics is given by _ e_ ðtÞ ¼ x_ ðtÞ nðtÞ ¼ GeðtÞ þ ðA0 þ DA KðC þ DCÞ GÞxðtÞ þ A1 eðt hÞ þ DA1 xðt hÞ þ A2 e_ ðt sÞ þ ðB KDÞwðtÞ
ð4Þ
628
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and the output of error is defined as ye ðtÞ ¼ LeðtÞ;
ð5Þ
where L is a known constant matrix. Thus, by (1) and (4), it is easy to obtain the augmented system d 2 zðt sÞ ¼ ðA 0 þ DA 0 ÞzðtÞ þ ðA 1 þ DA 1 Þzðt hÞ þ BwðtÞ; ½zðtÞ A dt ye ðtÞ ¼ LzðtÞ; ð6Þ where
A0 A1 0 A2 0 0 ; A2 ¼ ; ; A1 ¼ A0 KC G G 0 A1 0 A2 B xðtÞ ¼ ¼ ½ 0 L ; zðtÞ ¼ B ; L ; B KD eðtÞ DA0 0 0 D0 0 ðtÞ ¼ F0 ðtÞ½ E0 0 ¼ F0 ðtÞ½ E0 0 þ DA D0 DA0 KDC 0 0 0 þ F2 ðtÞ½ E2 0 KD2 0 þ D 02 F0 ðtÞE 0 þ D 2 F2 ðtÞE 2; 01 F0 ðtÞE D 0 0 0 1 ðtÞ ¼ 1 F1 ðtÞE 1: DA F1 ðtÞ½ E1 0 D ¼ DA1 0 D1 ð7Þ 0 ¼ A
Let Tzw denote a stable operator from signal wðtÞ to signal zðtÞ. Then H1 norm of Tzw is defined as kzðtÞk2 kTzw k1 ¼ sup : kwðtÞk wðtÞ2L2 2 Here, we introduce H1 performance measure for system (6) as follows: Z 1 J¼ ½yeT ðtÞye ðtÞ c2 wT ðtÞwðtÞ dt:
ð8Þ
0
Therefore, the problem tackled in this paper is to determine the filter (3) within the upper bound, i.e, kTye w k1 ¼ sup wðtÞ
kye ðtÞk2 < c: kwðtÞk2
In this situation, the filtering error system is said to have a guaranteed c level of noise attenuation. In next section, the criterion for minimization of c level will be derived using LMI convex optimization approach.
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3. Robust stability analysis and H‘ filter design Before proceeding further, we will give two lemmas and a fact. Lemma 1 [14]. For any constant symmetric positive-definite matrix H 2 Rmm , a scalar r > 0, and the vector function x : ½0; r ! Rm such that the integrations in the following are well defined, then Z r T Z r Z r T x ðsÞHxðsÞ ds P xðsÞ ds H xðsÞ ds : r 0
0
0
Lemma 2 [12]. For given positive scalars h and s and any E1 ; E2 2 Rnn , the operator Dðxt Þ : C0 ! Rn defined by Z t Dðxt Þ ¼ xðtÞ þ E1 xðsÞ ds E2 xðt sÞ ð9Þ t h
is stable if there exist a positive definite matrix C and positive scalars a1 and a2 such that a1 þ a2 < 1; T E2 CE2 a1 C H
hE2T CE1 h2 E1T CE1 a2 C
< 0:
ð10Þ
Fact 1 (Schur complement). Given constant symmetric matrices R1 , R2 , R3 where R1 ¼ RT1 and 0 < R2 ¼ RT2 , then R1 þ RT3 R 1 2 R3 < 0 if and only if R2 R3 R1 RT3 < 0; or < 0: RT3 R1 R3 R2 Here, we define an operator Dðzt Þ : C0 ! Rn Z t 2 zðt sÞ; Dðzt Þ ¼ zðtÞ þ A1 zðsÞ ds A t h
zt ¼ zðt þ hÞ;
ð11Þ
8 h 2 ½ H ; 0 :
Remark 1. By Lemma 2, the operator Dðzt Þ is stable if there exist a positive definite matrix C and positive scalars a1 and a2 such that " # T CA 2 a1 C T CA 1 A h A 2 2 ð12Þ 1 a2 C < 0; and a1 þ a2 < 1: T CA H h2 A 1
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Differentiating Dðzt Þ and combining Eq. (6) leads to þ DA 0 ðtÞÞzðtÞ þ DA 1 ðtÞzðt hÞ þ BwðtÞ _ t Þ ¼ ðA Dðz þ DA 0 ðtÞÞDðzt Þ þ DA 1 ðtÞzðt hÞ þ BwðtÞ ¼ ðA Z t þ DA 0 ðtÞÞA 1 þ DA 0 ðtÞÞA 2 zðt þ sÞ; ðA zðsÞ ds ðA
ð13Þ
t h
¼A 0 þ A 1. where A Now, we have the first main result of the paper. Theorem 1. Consider the system (6) with wðtÞ ¼ 0. For given scalars h > 0 and s > 0, suppose that there exist C > 0, a1 > 0, and a2 > 0 satisfying (12). If there exist positive scalars ei for i ¼ 0; 1; . . . ; 9, P > 0, Q1 > 0, Q2 > 0, and Q3 > 0 satisfying the following LMI: 2 3 A 2 þ QA 2 hP A A 1 hQA 1 R1 R2 0 P A 6H R 7 0 0 0 3 6 7 6 H H R4 7 < 0; ð14Þ 0 0 6 7 T 4H H H 5 hA2 Q R5 H H H H R6 where TP þ P A þ Q þ e0 E TE T T R1 ¼ A 0 0 þ e1 E0 E0 þ e2 E2 E2 ; 02 P D 2 PD 1 PD 01 P D 02 P D 2 hP D 01 hP D 02 hP D 2 ; 01 P D R2 ¼ P D R3 ¼ diag f e0 I; e1 I; e2 I; e3 I; e4 I; e5 I; e6 I; he7 I; he8 I; he9 Ig; T TE R4 ¼ Q2 þ e3 E 1 1 þ A2 Q A2 ; TE T T T T T R5 ¼ Q3 þ e4 A 2 0 E0 A2 þ e5 A2 E0 E0 A2 þ e6 A2 E2 E2 A2 ; 2 TE T T T T T R6 ¼ hQ1 þ e7 hA 1 0 E0 A1 þ e8 hA1 E0 E0 A1 þ e9 hA1 E2 E2 A1 þ h Q;
Q ¼ hQ1 þ Q2 þ Q3 ; then, the filtering error system (6) is asymptotically stable. Proof. Consider the following legitimate Lyapunov functional candidate [1]: V ðt; zðtÞÞ ¼ V1 þ V2 þ V3 þ V4 ; where V1 ¼ DT ðzt ÞP Dðzt Þ; V3 ¼
Z
V2 ¼
ð15Þ
Z
t
ðs t þ hÞzT ðsÞQ1 zðsÞ ds;
t h t T
z ðsÞQ2 zðsÞ ds;
V4 ¼
Z
t h
and P > 0, Q1 > 0, Q2 > 0, and Q3 > 0.
t
zT ðsÞQ3 zðsÞ ds; t s
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631
Taking the time derivative of V along the solution of (13) gives that dV1 þ DA 0 ðtÞÞDðzt Þ þ DA 1 ðtÞzðt hÞ ¼ 2 ðA dt T Z t ðA þ DA0 ðtÞÞA1 zðsÞ ds ðA þ DA0 ðtÞÞA2 zðt sÞ P Dðzt Þ; t h
ð16Þ Z t dV2 ¼ hzT ðtÞQ1 zðtÞ zT ðsÞQ1 zðsÞ ds 6 hzT ðtÞQ1 zðtÞ dt t h Z t T Z t 1 1 zðsÞ ds ðhQ1 Þ zðsÞ ds ; h t h h t h
ð17Þ
dV3 ¼ zT ðtÞQ2 zðtÞ zT ðt hÞQ2 zðt hÞ; dt
ð18Þ
dV4 ¼ zT ðtÞQ3 zðtÞ zT ðt sÞQ3 zðt sÞ; dt
ð19Þ
where Lemma 1 is utilized in (17). Here note that 1 zT ðtÞQi zðtÞ ¼ DT ðzt ÞQi Dðzt Þ 2DT ðzt ÞQi A Z 1 þ A
1 2 A
T Z 1 zðsÞ ds Qi A
t t h
Z
t
Z
t
2 zðt sÞ zðsÞ ds þ 2DT ðzt ÞQi A t h
t
zðsÞ ds
t h
T T Qi A 2 zðt sÞ þ zT ðt sÞA 2 zðt sÞ: zðsÞ ds Qi A 2
t h
Then, we have 4 X dV dVi ¼ 6 vT ðtÞXvðtÞ; dt dt i¼1
ð20Þ
Rt T where vT ðtÞ ¼ ½DT ðzt ÞzT ðt hÞzT ðt sÞðh 1 t h zðsÞ dsÞ and 3 20 1 þ DA 0 ðtÞÞT P ðA 7 6@ 1 ðtÞ þ DA 0 ðtÞÞ A X1;2 X1;3 P DA 7 6 þP ðA 7 6 7 6 þQ X¼6 7: T 7 6 H Q þ A QA 0 0 2 2 2 6 TQ 7 5 4 H H Q3 hA 2 H H H hQ1 þ h2 Q ð21Þ þ DA 0 ðtÞÞA 2 þ QA 2 and X1;3 ¼ hP ðA þ DA 0 ðtÞÞA 1 hQA 1. where X1;2 ¼ P ðA
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Therefore, if X < 0, there exists the positive scalar c such that dV 2 6 ckzðtÞk : dt
ð22Þ
Using the well-known inequality 2aT b 6
1 T a a þ ebT b e
ð23Þ
for any a; b 2 Rn and scalar e > 0, one can eliminate the uncertain factors 0 ðtÞ and DA 1 ðtÞ. Thus, we get the following Fi ðtÞ which are included in DA relation: 2
X1 6H X66 4H H
0 X2 H H
A 2 þ QA 2 PA 0 X3 H
3 A 1 hQA 1 hP A 7 0 7 X; 5 TQ hA
ð24Þ
2
X4
where TP þ P A þ Q þ e 1 P D T P þ e0 E TE 01 D X1 ¼ A 0 01 0 0 1 T T 1 T E 0 þ e P D P þ e2 E 2E T þ e 1 P D TP 2D 1D þ e P D02 D P þ e1 E 1
02
0
2
2
2
3
1
1 T 1 T 1 T T þ e 1 4 P D01 D01 P þ e5 P D02 D02 P þ e6 P D2 D2 P þ e7 hP D01 D01 P 1 T T þ e 1 8 hP D02 D02 P þ e9 hP D2 D2 P ;
T TE X2 ¼ Q2 þ e3 E 1 1 þ A2 Q A2 ; TE T T T T T X3 ¼ Q3 þ e4 A 2 0 E 0 A2 þ e 5 A2 E0 E0 A2 þ e 6 A2 E2 E2 A2 ; 2 TE T T T T T X4 ¼ hQ1 þ e7 hA 1 0 E0 A1 þ e8 hA1 E0 E0 A1 þ e9 hA1 E2 E2 A1 þ h Q:
is equivalent to By Fact 1 (Schur complement), the negative-definiteness of X the LMI (14). Thus, the feasibility of the LMI (14) guarantees the negativeness (22) of dV . Therefore, by Theorem 9.8.1 (pp. 292–293) of Hale and Lunel [1] dt with the stable operator Dðzt Þ and (22), we conclude that system (1) and (6) are both asymptotically stable. h Next, the following theorem provides a criterion for the existence of a linear stable filter assuring a robust H1 performance for neutral systems (1). Theorem 2. Let A ¼ A0 þ A1 and q ¼ c2 . For wðtÞ 2 L2 and given delays h and s, consider the system (6). If the following optimization problem:
J.H. Park et al. / Appl. Math. Comput. 159 (2004) 625–639
633
minimize q subject to 2 6 P1 6 6 6 6H 6 6 6 6H 6 6 6 6 6H 6 6 6H 6 6 6H 6 6 6H 6 6 6H 6 6 6H 6 6 6H 4 H
P2
P4
0
0
0
P3
0
0
0
0
P5
P6
0
0
H
H
P7
0
H
H
H
H
H
H
H
!
P1 AA2
ðR2 þ R5 þ R8 ÞA2
þðR1 þ R4 þ R7 ÞA2 0 ðP2 A0 M2 C M1 ÞA2
!
0
!
ðM1 þ P2 A1 ÞA2
þAT2 ðR2 þ R5 þ R8 ÞT
þðR3 þ R6 þ R9 ÞA2
0
0
0
P8
P9
0
0
H
H
P10
0
0
H
H
H
H
P11
R8
H
H
H
H
H
H
R9
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
!
hP1 AA1
3 hðR2 þ R5 þ R8 ÞA1
hðR1 þ R4 þ R7 ÞA1 0 ! hðP2 A0 M2 C M1 ÞA1 hAT1 ðR2 þ R5 þ R8 ÞT 0 0 0 hAT2 ðR1
þ R4 þ R7 Þ
hðR2 þ R5 þ R8 ÞT A2 P12 H
7 7 7 7 0 0 7 7 ! 7 hðM1 þ P2 A1 ÞA1 7 P2 B M2 D 7 7 hðR3 þ R6 þ R9 ÞA1 7 7 7 0 0 7 7 7<0 0 0 7 7 7 0 0 7 7 T hA2 ðR2 þ R5 þ R8 Þ 0 7 7 7 7 hAT2 ðR3 þ R6 þ R9 Þ 0 7 7 2 7 hR2 þ h ðR2 þ R5 þ R8 Þ 0 7 7 2 hR3 þ h ðR3 þ R6 þ R9 Þ 0 5
H
R1 H
R2 R3
> 0;
c2 I
H
R4 H
R5 R6
P1 B
> 0;
R7 H
R8 R9
ð25Þ
>0
ð26Þ
has the positive definite solutions, P1 , P2 , R1 , R3 , R4 , R6 , R7 , R9 , positive scalars e0 ; e1 ; . . . ; e9 ; q, and matrices M1 , M2 , R2 , R5 , R8 , then the filtering error system has a guaranteed c level of noise attenuation by robust H1 filter (3).
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Here, the following notations are defined: P1 ¼ AT P1 þ P1 A þ R1 þ R4 þ R7 þ e0 E0T E0 þ e1 E0T E0 þ e2 E2T E2 ; P2 ¼ ½ P1 D0 P1 D0 hP1 D0 ; P3 ¼ diag f e0 I; e4 I; he7 Ig; P4 ¼ AT0 P2 C T M2T M1T þ R2 þ R5 þ R8 ; P5 ¼ M1 þ M1T þ P2 A1 þ AT1 P2 þ R3 þ R6 þ R9 þ LT L; P6 ¼ ½ P2 D0 M2 D2 P2 D1 P2 D0 M2 D2 hP2 D0 hM2 D2 ; P7 ¼ diag f e1 I; e2 I; e3 I; e5 I; e6 I; he8 I; he9 Ig; P8 ¼ R4 þ e3 E1T E1 þ AT2 ðR1 þ R4 þ R7 ÞA2 ; P9 ¼ R5 þ AT2 ðR2 þ R5 þ R8 ÞA2 ; P10 ¼ R6 þ AT2 ðR3 þ R6 þ R9 ÞA2 ; P11 ¼ R7 þ e4 AT2 E0T E0 A2 þ e5 AT2 E0T E0 A2 þ e6 AT2 E2T E2 A2 ; P12 ¼ hR1 þ h2 ðR1 þ R4 þ R7 Þ þ e7 hAT1 E0T E0 A1 þ e8 hAT1 E0T E0 A1 þ e9 hAT1 E2T E2 A1 :
ð27Þ
Proof. To establish the H1 performance for the filtering error system, first notice that since the system (6) is asymptotically stable, then zðtÞ tends to zero as t ! 1. Next, assuming zero initial conditions for the filtering error system, the performance index is Z 1 J¼ ½yeT ðtÞye ðtÞ c2 wT ðtÞwðtÞ dt 0 Z 1 ¼ ½yeT ðtÞye ðtÞ c2 wT ðtÞwðtÞ þ V_ ðt; zðtÞÞ dt 0
þ V ðt; zðtÞÞjt¼0 V ðt; zðtÞÞjt!1 ;
ð28Þ
where V ðt; zðtÞÞ is as in (15). Considering that V ðt; zðtÞÞjt¼0 ¼ 0 and V ðt; zðtÞÞjt!1 ! 0, one can easily infer from (28) that Z 1 J6 fT ðtÞW fðtÞ dt; ð29Þ h i 0 R t where fT ðtÞ ¼ DT ðzt Þ zT ðt hÞ zT ðt sÞ ðð1=hÞ t h zðsÞ dsÞT wT ðtÞ and 2 3 1 ðtÞ þ DA 0 ðtÞÞA 1 hQA 1 P B P DA W2 hP ðA W1 6 H Q2 þ A T QA 2 0 0 0 7 2 6 7 6 W ¼6H H Q3 0 0 7 7; 4H H H hQ1 þ h2 Q 0 5 H H H H c2 I ð30Þ T
þ DA 0 ðtÞÞ P þ P ðA þ DA 0 ðtÞÞ þ Q þ L T L where W1 ¼ ðA DA0 ðtÞÞA2 þ QA2 .
and
þ W2 ¼ P ðA
J.H. Park et al. / Appl. Math. Comput. 159 (2004) 625–639
Again, using the inequality (23), we have 2 A 2 þ QA2 0 PA T L X1 þ L 6 0 H X2 6 ¼6 W 6W H H X3 6 4 H H H H H H Here, define the matrices Q1 , Q2 , Q3 and P as R4 R5 R1 R2 Q1 ¼ ð1=hÞ T ; Q2 ¼ ; R2 R3 RT5 R6 P1 0 P¼ ; 0 P2
635
A 1 hQA 1 hP A 0 1 hQA X4 H
3 PB 0 7 7 0 7 7: 0 5 c2 I ð31Þ
Q3 ¼
R7 RT8
R8 ; R9 ð32Þ
where P1 , P2 , R1 , R3 , R4 , R6 , R7 and R9 are the positive definite matrix defined in (26). < 0 is modified to Utilizing (7) and (32), the inequality W 2 U1 U2 0 0 P1 AA2 þ ðR1 þ R4 þ R7 ÞA2 ðR2 þ R5 þ R8 ÞA2 6 P2 ðA0 KC GÞA2 P2 ðG þ A1 ÞA2 6 0 6 H U3 0 T 6 þAT2 ðR2 þ R5 þ R8 Þ þðR3 þ R6 þ R9 ÞA2 6 6 H H P8 P9 0 0 6 6 0 0 6 H H H P10 6 6H H H H P R 11 8 6 6 6H H H H H R9 6 6H H H H H H 6 6 4H H H H H H H H H H hP1 AA1 hðR1 þ R4 þ R7 ÞA1 hP2 ðA0 KC GÞA1 hAT1 ðR2 þ R5 þ R8 Þ 0 0
T
H hðR2 þ R5 þ R8 ÞA1 hP2 ðG þ A1 ÞA1 hðR3 þ R6 þ R9 ÞA1 0 0
hAT2 ðR1 þ R4 þ R7 Þ
hAT2 ðR2 þ R5 þ R8 Þ
hðR2 þ R5 þ R8 ÞT A2
hAT2 ðR3 þ R6 þ R9 Þ
P12
hR2 þ h2 ðR2 þ R5 þ R8 Þ
H H
hR3 þ h2 ðR3 þ R6 þ R9 Þ H
H P1 B
3
7 P2 B P2 KD 7 7 7 7 7 0 7 7 0 7 7 < 0; 7 0 7 7 7 0 7 7 0 7 7 5 0 c2 I ð33Þ
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where T 1 T 1 T U1 ¼ P1 þ e 1 0 P1 D0 D0 P1 þ e4 P1 D0 D0 P1 þ e7 hP1 D0 D0 P1 ; T
U2 ¼ ðA0 KC GÞ P2 þ R2 þ R5 þ R8 ; T U3 ¼ ðG þ A1 ÞT P2 þ P2 ðG þ A1 Þ þ R3 þ R6 þ R9 þ e 1 1 P2 D0 D0 P2 T 1 1 T þ e 1 2 P2 KD2 D2 K P2 þ e3 P2 D1 D1 P2 þ e5 P2 D0 D0 P2 T 1 T 1 T T þ e 1 6 P2 KD2 D2 K P2 þ e8 hP2 D0 D0 P2 þ e9 hP2 KD2 D2 K P2 þ L L
and P1 , P8 , and P9 are defined in (27). Using some changes of variables, M1 ¼ P2 G and M2 ¼ P2 K, the inequality (33) is changed to the LMI (25). Therefore, the inequalities (25) and (26) imply that J < 0 for any nonzero wðtÞ 2 L2 , i.e., the filtering error system has a guaranteed c level of noise attenuation, which concludes the proof. h Remark 2. Given any solutions of the LMIs (25) and (26) in Theorem 2, the corresponding robust H1 filter of the form (3) will be constructed from the 1 relations G ¼ P 1 2 M1 and K ¼ P2 M2 . Remark 3. In this article, in order to solve the LMIs, we utilize MatlabÕs LMI Control Toolbox [15], which implements state-of-the-art interior-point algorithms, which is significantly faster than classical convex optimization algorithms [13]. Example. To illustrate the proposed method, consider the neural system 0 3 0:2 0:1 d xðtÞ xðt 1Þ ¼ dt 0:1 0:2 4 5 1 0:3 0:2 þ F0 ðtÞ½ 0 0:1 xðtÞ þ 1 0:2 0:3 1 1 þ F1 ðtÞ½ 0:1 0:1 xðt 0:5Þ þ wðtÞ; 0 0:5 yðtÞ ¼ ð½ 1
0 þ F2 ðtÞ½ 0:1
0:1 ÞxðtÞ þ wðtÞ;
where Fi ðtÞT Fi ðtÞ 6 I for i ¼ 0; 1; 2. The output of error system is defined as ye ðtÞ ¼ ½ 1
1 eðtÞ:
The aim is to design a robust H1 filter (3) for above system. First, checking the stability condition (12) for operator Dðzt Þ gives the solutions:
J.H. Park et al. / Appl. Math. Comput. 159 (2004) 625–639
2
0:7490 6 0:0044 C¼6 4 0 0
0:0044 0 0:7490 0 0 0:7490 0 0:0044
637
3
0 0 7 7; 0:0044 5 0:7490
a1 ¼ 0:3333;
a2 ¼ 0:3333:
Next, by applying the minimization problem of Theorem 2 to above system using LMI toolbox [15], the solutions of optimization problem in the inequalities (25) and (26) are as follows: 0:1701 0:1101 0:6795 0:3221 P1 ¼ ; P2 ¼ ; 0:1101 0:1615 0:3221 0:9955 0:1324 0:0867 R1 ¼ ; 0:0867 0:1478 0:0255 0:0144 0:4043 0:2017 R2 ¼ ; R3 ¼ ; 0:0210 0:0131 0:2017 0:5774 0:0476 0:0473 0:0049 0:0052 R4 ¼ ; R5 ¼ ; 0:0473 0:0485 0:0045 0:0051 0:1096 0:1060 0:1394 0:1291 R6 ¼ ; R7 ¼ ; 0:1060 0:1115 0:1291 0:1539 0:0275 0:0403 ; R8 ¼ 0:0119 0:0463 0:8812 0:6713 2:5064 0:1475 R9 ¼ ; M1 ¼ ; 0:6713 0:7702 4:6942 4:0887 0:9731 ; M2 ¼ 0:7325 q ¼ 0:1812; e3 ¼ 2:0789; e5 ¼ 29:7619; e9 ¼ 14:5728:
e0 ¼ 2:3617; e4 ¼ 10
4
e1 ¼ 5:1754;
e2 ¼ 4:463;
4:8687;
e6 ¼ 10:8645;
e7 ¼ 2:0256;
e8 ¼ 4:439;
Therefore, in light of Remark 2, the parameters G and K of robust H1 filter are 1:7168 2:5557 1:2795 G¼ ; K¼ : 4:1602 4:9341 0:3219 Therefore, the obtained robust H1 filter stabilizes the filtering error system and minimizes H1 norm within c ¼ q1=2 ¼ 0:4257 in the presence of delay and uncertainties.
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4. Conclusions This paper investigated the design of robust H1 filter for neutral delaydifferential system with parametric uncertainties in the system state-space model. A methodology for the design of a stable linear filter that assures asymptotic stability and minimization of H1 performance for the filtering error system is studied. The filter is obtained from a convex optimization problem described in terms of LMIs. The validity of proposed filter design algorithm has been checked through a numerical example.
Acknowledgements J.H. Park is grateful to H.Y. Jung, H.J. Baek, W.C. Shim, and S.G. Kim for their good advice about my research, and to my students, I.G. Jung, Y.I. Ban, and S.D. Lee for their devoted contribution in my laboratory. This work was supported by grant No. R05-2003-000-10173-0 from Korea Science & Engineering Foundation.
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