Robust Stabilization for Singular Systems with ... - Semantic Scholar
FrM13.5
Proceeding of the 2004 American Control Conference Boston, Massachusetts June 30 - July 2, 2004
Robust Stabilization for Singular Systems with Time-Delays and Saturating Controls Wu-Neng Zhou, Ren-Quan Lu, Hong-Ye Su, and Jian Chu, Member, IEEE
Abstract—We studied the delay-dependent stabilization problem for a class of uncertain singular system with time-delays and saturating controls. Theorems derived give sufficient conditions for delay-dependent stabilization of the singular systems with a combination of saturating controls and multiple time-delays in both state and control; we assumed the delays to be constant bounded but unknown, moreover, the uncertainties are also described to be unknown but bounded and the nonlinear terms included in the systems are fallen into a set. Under these sufficient conditions, the solution of the uncertain singular system is regular, impulse free, and locally asymptotically stable for all admissible uncertainties. Furthermore, the results based on several Linear Matrix Inequalities (LMIs) are developed to guarantee stability and be computed effectively. Finally, we advance an example to demonstrate the superiority of this method. I.
INTRODUCTION
C
ONYTOL of singular systems has been extensively studied in the past years due to the fact that singular systems better describe physical systems than regular ones. A great number of results based on the theory of regular systems (or state-space systems) have been extended to the area of singular systems [1]-[2]. Recently, robust stability and robust stabilization for uncertain singular systems with time-delays have been considered [3]-[5]. Moreover, the problem of stabilizing linear systems with saturating controls has been widely studied these last years because of its practical interest [6] and [7]. However, to the best of our knowledge, the problems of robust stabilization for uncertain singular system included both time-delays and Manuscript received September 20, 2003. This work was supported by the National Outstanding Youth Science Foundation of P. R. China (NSFC: 60025308), the Teaching and Research Award Program for Outstanding Young Teachers in Higher Education Institutions of MOE P. R. China and the National Natural Science Foundation of P. R. China (NSFC: 10371106) W. N. Zhou is with the National Laboratory of Industrial Control Technology, Institute of Advanced Process Control, Zhejiang University, Yuquan Campus, Hangzhou 310027, P. R. China and the Mathematics Institute, Zhejiang Normal University, Jinhua 321004, P. R. China (e-mail: [email protected] ). R. Q. Lu, H. Y. Su, and J. Chu are with the National Laboratory of Industrial Control Technology, Institute of Advanced Process Control, Zhejiang University, Yuquan Campus, Hangzhou 310027, P. R. China (e-mail: [email protected], [email protected], [email protected] ).
0-7803-8335-4/04/$17.00 ©2004 AACC
saturating controls have not been fully investigated yet. In this paper, we concerned with the delay-dependent robust stabilization of a continuous-time subject to multiple time-delays in both state and control, saturating controls and nonlinear terms. The synthesis problem addressed is to design a memoryless state feedback control law such that the resulting closed-loop system is regular, impulse free and stable for all admissible uncertainties, and a sufficient condition for the existence of such a control law is presented in terms of several linear matrix inequalities (LMIs). II. SYSTEM DESCRIPTION AND DEFINITIONS Consider the following uncertain singular systems with time-delays and saturating controls described by k
(Σ) : Ex(t ) = ( A0 + ∆A0 ( x, t )) x(t ) + ∑ ( Ai + ∆Ai ( x, t )) i =1
× x(t − hi (t )) + E10 f (σ(t )) + B10 w(t ) + ( B20 + ∆B20 ( x, t ))u '(t ) k
+ ∑ ( B2i + ∆B2i ( x, t ))u '(t − gi (t )),
(1)
i =1
u '(t ) = sat (u (t )), x(t ) = φ (t ), t ∈ [−τ,0], σ(t ) = Cx(t ) sat (u (t )) = [ sat (u1 (t )) sat (u2 (t )) sat (um (t ))], where x(t ) ∈ R n is the state vector, u (t ) ∈ R m is control input vector to the actuator (emitted from the designed controller), u '(t ) ∈ R m is the control input vector to the plant,
w(t ) ∈ R p is the disturbance input vector from L2 [0, ∞ ) . The matrix E ∈ R n×n may be singular, we shall assume that rank E = r ≤ n . The matrices A0 , Ai , E10 , B10 , B20 and B2i are known real constant matrices with appropriate dimensions. The matrices ∆A0 (•), ∆Ai (•), ∆B20 (•) and ∆B2i (•) are time-invariant matrices representing norm-bounded parameter uncertainties, and are assumed to be of the following form: [ ∆A0 (•) ∆Ai (•) ∆B20 (•) ∆B2i (•)] (2) = GF ( x, t ) [ H1 H 2i H 3 H 4i ] where G , H 1 , H 2i , H 3 and H 4i are known real constant matrices with appropriate dimensions. The uncertain matrix F ( x(t ), t ) with Lebesgue measurable elements satisfies
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F T ( x(t ), t ) F ( x(t ), t ) ≤ I . (3) The input vector is assumed to satisfy actuator limitations, i.e. u (t ) ∈ U ⊂ R m with U = {u (t ) ∈ R m ; −u0( i ) ≤ u( i ) (t ) ≤ u0(i ) , u0(i ) > 0, i = 1,
, m} (4)
The actuator is described by the nonlinearity u0( i ) if u( i ) (t ) > u0( i ) sat (u( i ) (t )) = u( i ) (t ) if − u0(i ) ≤ u( i ) (t ) ≤ u0( i ) (5) if u( i ) (t ) < −u0(i ) −u0( i ) hi (t ) and gi (t ) are unknown scalars denoting the delays in the state and control, respectively, and it is assumed that there exist positive numbers h, g and τ such that (6) 0 ≤ hi (t ), gi (t ) ≤ h, g ≤ τ for all t , i = 1, , k . φ (t ) is smooth vector-valued continuous initial function defined in the Banach space Cτ . In this paper, every nonlinear term is assumed to be of the form as follows f j (i) ∈ K j [0, k j ] = { f j (σ j ) f j (0) = 0,0 < σ j f j (σ j ) (7) ≤ k jσ 2j (σ j ≠ 0)}, j = 1,2, , n where k j are positive scalars. The nominal unforced singular delay systems (1) can be written as k
Ex(t ) = A0 x(t ) + ∑ Ai x(t − hi (t )) + E10 f (σ(t )).
(8)
i =1
Using the Leibniz-Newton formula [7], then the singular delay system (8) can be written as k
Ex(t ) = ( A0 + ∑ Ai ) x (t ) + E10 f (σ (t ))
π 1 x(t )
k
[A0 x(t + θ ) + ∑ Ai
k
Lemma 3 [8]. Given matrices A, Θ, Ξ, Γ and F (σ ) of
appropriate dimensions and with Θ symmetrical and F (σ ) satisfying F T (σ ) F (σ ) ≤ I . Then we have: a) If the following inequality holds, Θ + ΓF (σ )Ξ + (ΓF (σ )Ξ )T < 0 if and only if there exists a scalar ε > 0 such that Θ + εΓΓT + ε −1ΞT Ξ < 0 b) For any symmetric matrix P > 0 and scalar ε > 0 such that ε I − ΞPΞT > 0 , then ( A + ΓF (σ )Ξ) P ( A + ΓF (σ )Ξ )T
III. ANALYSIS OF ROBUST STABILITY
i =1
A. Analysis of robust Stability of Systems (8) The main result is derived as follows, it gives the sufficient condition of robust stability for the singular delay system (8).The proof of it is similar to [2] and [9] and is omitted. Theorem 1. If there exists a series of positive definite symmetric Q, Q1i , Q2i , Q3i , i = 1 k , a matrix P , and the scalars ε , γ and τ such that
and impulse free . Lemma 1 (Krasovskii theorem [1]). The singular delay systems (8) is said to be locally asymptotically stable if there exists a positive definite symmetric matrix P , positive scalars π 1 , π 2 , π 3 , v and γ , for any initial condition φ (t )
EPT = PE T ≥ 0 W τ N1 τ N 2 τ N 3 τ N T τΩ 0 0 1 M = 1T 0} , whereas, for a continuous function
if
Definition 2. The uncertain singular systems ( Σ ) is said to be robustly stable if the systems ( Σ ) with u (t ) ≡ 0, u (t − g i (t )) ≡ 0, w(t ) ∈ L2 [0, ∞) is regular, impulse free and locally asymptotically stable for all admissible uncertainties. Definition 3. The uncertain singular delay systems ( Σ ) is said to be robustly stabilizable if there exists a linear state feedback control law u (t ) = Λx(t ), Λ ∈ R m×n such that the resultant closed-loop system is robustly stable in the sense of Definition 3. In this case, u (t ) = Λx(t ) is said to be a robust state feedback control law for system ( Σ ). Lemma 2 [8]. Given vector x, y , a positive definite symmetric matrix R with appropriate dimensions, then for any scalar ε > 0 , we have ±2 xT y ≤ ε xT Rx + ε −1 y T R −1 y.
(9)
regular and impulse free if the pair ( E , A + ∑ Ai ) is regular
in
and
V ( x(t ), t ) ≤ −π 3 x(t ) , V ( x(t ), t ) ≤ V (φ (0), 0) .
i =1
x (t ) = φ (t ), t ∈ [−2τ , 0] Throughout this paper, we shall use the following concepts and introduce the following useful lemmas. Definition 1. The singular delay systems (8) is said to be
confined
,
≤ APAT + APΞT (ε I − ΞPΞT ) −1 ΞPAT + εΓΓT
× x(t − hi (t ) + θ ) + E10 f (σ (t + θ ))]dθ
remain
2
k
0
− hi ( t )
i =1
≤ V ( x(t ), t ) ≤ π 2 x(t ) 2
i =1
− ∑ Ai ∫
2
V ( x(t ), t ) : R n × R + → R
such
that
(10a)
(10b)
(10c)
where
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k
k
i =1
i =1
W = ( A0 + Σ Ai ) PT + P ( A0 + Σ Ai )T k
+τ Σ Ai (Q1i + Q2i + Q3i ) AiT + ε E10 E10T + ε −1 PC T K T KCPT i =1i
N1 = [ PA0T
PA0T ], N 2 = [ PA1T PA2T
N 3 = [ PC T K T E10T PC T K T E10T
Ω1 = −diag {Q11 , Q12 , Ω 2 = −diag {Q21 , Q22 ,
PAkT ] ,
PC T K T E10T ],
, Q1k } , K = diag {k1 ,
kn } ,
, Q2 k } , Ω3 = − diag {Q31 , Q32 ,
, Q3k } ,
then the singular delay system (8) is regular, impulse free and locally asymptotically stable for any initial condition belonging to the set Φ 0 = {φ (t ) ||| φ (t ) ||2 ≤ δ } with δ = λmin (Q) / γπ 2 where kτ 2 π 2 = λmax ( EPT ) + max λmax [ PA0T Q1−i 1 A0 PT ] 2
Corollary 1 (Disturbance-free case). If there exist a series of positive definite symmetric Q, Q1i , Q2i , Q3i , Q4i , Q5i , Q6i , R1i , R2 i , R3i , R4i , R5i , R6i , i = 1 k , a matrix P , and the scalars ε 2 , ε 3i , γ , i = 1, , k and τ such that M ∆ < 0, Ξ1 ≥ 0, Ξ 2 ≤ 0 and the expression of EPT = PE T ≥ 0 hold, then the uncertain singular delay system (1) is regular, impulse free and locally asymptotically stable for any initial condition belonging to the set
Φ 0 = {φ (t ) ||| φ (t ) ||2 ≤ δ } with δ = where
kτ 2 max λmax [ P ( A0 ∆ 2 i + B20 ∆ Λ )T (Q1−i 1 + R1−i 1 )( A0 ∆ + B20 ∆ Λ ) PT ]
π 2 = λmax ( EPT ) + kτλmax ( PΛT ΛPT ) +
i
3kτ 2 max λmax [ PAiT∆ (Q2−i1 + R2−i1 ) Ai∆ PT ] i 2 2 kτ + max λmax [PCT K T E10T (Q3−i1 + R3−i1 )E10 KCPT ] 2 i 3kτ 2 + max λmax [P(B2i∆ Λ)T (Q4−i1 + R4−i1 )(B2i∆ Λ)PT ] 2 i kτ 2 + max λmax [P(B20∆ Λ)T (Q5−i1 + R5−i1 ) 2 i 3kτ 2 × (B20∆ Λ)PT ]+ max λmax [P(B2i∆ Λ)T i 2 −1 −1 T × (Q6i + R6i )(B2i∆ Λ)P ] +
3kτ max λmax [ PAiT Q2−i1 Ai PT ] + i 2 kτ 2 max λmax [ PC T K T E10T Q3−i1 E10 KCPT ]) + 2 i 2
B. Disturbance-Free Case (with w(t ) = 0 ) When w(t ) = 0 , for the uncertain singular system ( Σ ), introduce the control law u (t ) = 2Λx(t ) , where the control law gain matrix Λ ∈ R m×n is to be found, and the closed-loop system is k
(Σ′) : Ex(t ) = ( A0 ∆ + B20 ∆ Λ) x(t ) + ∑ Ai ∆ x(t − hi (t ))
and
i =1
k
+ E10 f (σ(t )) + ∑ B2i ∆ Λx(t − gi (t )) i =1
P −1 E ΛTi Ξ1 = ≥0 γ u02i Λi
(11)
k
+ B20 ∆η (t ) + ∑ B2i∆η (t − gi (t )) and M ∆
where [•]∆ = [ •] + ∆ [•] ( [•] denoting the matrix), and
η (t ) = sat (2Λx(t )) − Λx(t ), η (t − gi (t )) = sat (2Λx(t − gi (t )) − Λx(t − gi (t )) Obviously, the vector function η (t ) satisfies the following
inequality
η (t − gi (t )) ≤ xT (t − gi (t ))ΛT Λx(t − gi (t ))
(12)
Ω3 are the same as theorem 1, and k
k
W∆ = ( A0 ∆ + ∑ Ai∆ + B20 ∆ Λ + ∑ B2i∆ Λ )PT + P( A0 ∆ i =1
i =1
k
k
k
i =1
i =1
i =1
+ ∑ Ai∆ + B20∆ Λ + ∑ B2i∆ Λ)T +τ ∑ Ai∆ (Q1i k
+ Q2i + Q3i + Q4i + Q5i + Q6i ) AiT∆ + τ ∑ B2i∆Λ(Q1i i =1
By (4), one has x(t ) ∈ S (u0 ,1m ) , where
+Q2i + Q3i + Q4i + Q5i + Q6i )(B2i∆Λ)
T
S (u0 ,1m ) = {x(t ) ∈ R n |{u (t ) ∈ R m ; −u0(i ) ≤ Λ (i ) x(t ) ≤ u0( i ) , i = 1,
(14a)
Q PT Ξ2 = (14b) ≤0 P I is shown in the next page, where N 3 , Ω1 , Ω 2 and
i =1
x(t ) = φ (t ), t ∈ [−τ, 0]
η T (t )η (t ) ≤ xT (t )ΛT Λx(t ),
λmin (Q) , γπ2
, m}
(13)
Using the same method as theorem 1 and taking into (12), (13) and [7], furthermore introducing the idea of generalized quadratic stability and generalized quadratic stabilization in [3], one can deduce the following corollary.
T + ε1E10 E10T + ε1−1PCT K T KCPT + ε 2 B20∆ B20 ∆ k
k
+ ε 2−1PΛT ΛPT + ∑ε3i B2i∆ B2Ti∆ + ∑ε3−i1PΛT ΛPT i =1
N1∆ = [ P ( A1∆ + B20 ∆ Λ )T N 20 = [ PA1T∆ PA1T∆ 2
i =1
P( A1∆ + B20 ∆ Λ )T ],
PAkT∆ ]
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N 4 ∆ = [ P ( B21∆ Λ )T
P ( B2 k ∆ Λ )T ] = N 6 ∆ ,
N 5 ∆ = [ P( B20 ∆ Λ )
P( B20 ∆ Λ ) ],
T
where the matrix N i is the matrix N i∆ (i = 1, uncertainty in corollary 1, and
T
Ω j = − diag {Q j1 , Q j 2 , Ω′j = − diag { R j1 , R j 2 ,
k
, Q jk } , j = 4,5, 6,
, R jk } , j = 1, 2,
i =1
, 6.
T0 , Ti , T ji , Pji such that the following inequalities are satisfied: T T B20 B20 + B20 H 4T ( β 0 I − H 3 H 3T )-1 H 3 B20 + β 0 GG T ≤ T0 (15) B2i B2Ti + B2i H 4Ti ( β i I − H 4 H 4Ti )-1 H 4i B2Ti + β i GG T ≤ Ti
(16)
Ai Q ji A + Ai Q ji H ( β ji I - H 2i Q ji H ) H 2i Q ji A T -1 2i
T i
(17)
+ β ik GG T ≤ T ji B2i Z ji B2Ti + B2i Z ji H 4Ti (δ ji I − H 4i Z ji H 4Ti ) −1
(18)
×H 4i Z ji B2Ti + δ ji GG T ≤ Pji where Z ji ≥ ΛQ ji ΛT , j = 1,
, 6, i = 1,
, k , and
β 0 I − H 3 H > 0, β i I − H 4i H > 0, T 3
T 4i
Using Lemma 3b, we have T T B20 ∆ B20 ∆ ≤ T0 , B2 i ∆ B2 i ∆ ≤ Ti , T 2 i∆
(20)
≤ Pji
By Corollary 1, (2) and (20), it follows that M ∆ = M + Θ1 F ( x(t ), t )Θ2 + (Θ1 F ( x(t ), t )Θ2 )T < 0
(21)
where Θ1 = diag {G , G ,
, G} , Θ 2 = Θ2 0 0 , and T T T Θ 2 = [ P( H1 + ∑ H 2i + H 3 Λ + ∑ H 4i Λ ) , Θ 21 , ΘT21 ], i
i
ΘT21 = [ P ( H1 + H 3 Λ )T ,
T , P( H1 + H 3 Λ)T , PH 21 ,
,0, PΛ H ,
T 2k
T
PH , 0,
T , PΛT H 3T , PΛT H 41 ,
T
T 4k
, PΛT H 4Tk ],
M ′ = W ′ − τ N1 (Ω1−1 + Ω1′−1 ) N1T − τ N 2 (Ω −21 + Ω′2−1 ) N 2T − τ N 3 (Ω + Ω′ ) N − τ N 4 (Ω + Ω′ ) N −1 3
−1 3
T 3
,
, PΛ H , PΛ H 3T ,
T 41
−1 4
i =1
i =1
6
k
6
k
+ τ ∑∑ Pji +τ ∑∑Tji + ε 2T0 + ε 2−1PΛT ΛPT j =1 i =1
j =1 i =1
k
k
i =1
i =1
+ ∑ε3iTi + ∑ε3−i1PΛT ΛPT By Lemma 3a and (21), we can obtain that there exists a scalar α > 0 such that M ′ + αΘ1Θ1T + α −1ΘT2 Θ2 < 0 (22) For simplicity we introduce the matrix ϒ ∈ R n×( n − r ) satisfying Eϒ = 0 and rank ϒ = n − r . It’s easy to see that there exist invertible matrices L1 and L2 ∈ R n× n from the proof of Theorem 1 such that P P P = L1 PL−2T = 11 12 0 P22 T r ×( n − r ) where P11 = P11 ≥ 0, P12 ∈ R , P22 ∈ R ( n − r )×( n − r ) . On the other hand, from Eϒ = 0 and rank ϒ = n − r , it implies that there exists an invertible matrix Γ ∈ R ( n − r )×( n − r ) such that 0 ϒ = L2 Γ I n−r Hence P P I 0 −1 P = L1−1 11 12 LT2 = ( L1−1 r L2 ) 0 0 0 P22 P 0 T −1 P12 − T × ( L2 11 L2 ) + ( L1 Γ ) P22 0 In −r T T × (Γ [ 0 I n − r ] L2 ) EX + Y ϒT where
Ai∆ Q ji A ≤ T ji , B2i∆ ΛQ ji Λ B T
i =1
+ B20 Λ + ∑ B2i Λ)T + ε1 E10 E10T + ε1−1 PC T K T KCPT
(19)
β ij I − H 2i Q ji H 2Ti > 0, δ ji I − H 4i Z ji H 4Ti > 0
T i∆
k
k
(14c) In the following we shall discuss how to solve the control law gain matrix Λ in the following analysis by using LMI technology. Assume that there exist scalars β 0 , β i , β ji , δ ji > 0 and positive definite symmetric matrices
T 2i
k
W ′ = ( A0 + ∑ Ai + B20 Λ + ∑ B2i Λ )PT + P( A0 + ∑ Ai
W∆ τ N1∆ τ N2∆ τ N3 τ N4∆ τ N5∆ τ N6∆ τ N1∆ τ N2∆ τ N3 τ N4∆ τ N5∆ τ N6∆ τ N T τΩ 0 0 0 0 0 0 0 0 0 0 0 1 1∆ τ N2T∆ τΩ2 0 0 0 0 0 0 0 0 0 0 0 T τΩ3 0 0 0 0 0 0 0 0 0 0 0 τ N3 τ N T τΩ4 0 0 0 0 0 0 0 0 0 0 0 4∆ T τΩ5 0 0 0 0 0 0 0 0 0 0 0 τ N5∆ M ∆ = τ N6T∆ τΩ6 0 0 0 0 0 0 0 0 0 0 0 0, Y = L1 Γ 0 I n−r P22 Furthermore EPT = E ( EX + Y ϒT )T = EXE T = ( EX + Y ϒT ) E T = PE T ≥ 0 Ψ = Λ ( EX + Y ϒT )T ΛΖT ( X , Y ) . Define Without loss of generality, we can assume that Ζ( X , Y ) = EX + Y ΦT is invertible. Define matrix M ′′ , as shown at the top of the next page, where k
k
W ′′ = ( A0 + ∑ Ai ) Z T ( X , Y ) + B20 Ψ + ∑ B2i Ψ + Z ( X , Y ) i =1 k
i =1
k
× ( A0 + Σ Ai ) + ( B20 Ψ + Σ B2i Ψ )T +ε1 E10 E10T T
i =1
i =1
6
k
+ ε Z ( X , Y )C K KCZ ( X , Y )T + τ Σ Σ Pji −1 1
T
T
T
j =1 i =1
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W ′′ T τ N 1 τ N 2T T τ N 3 τ N T 4 τ N 5T M ′′ = τ N 6T τ N T 1T τ N 2 τ N T 3 τ N 4T T τ N 5 T τ N 6
τ N1 τ N 2 τ N 3 τ N 4 τ N 5 τ N 6 τ N1 τ N 2 τ N 3 τ N 4 τ N 5 τ N 6 0 0 0 0 0 0 0 0 0 0 0 τΩ1 0 0 0 0 0 0 0 0 0 0 0 τΩ 2 0 0 0 0 0 0 0 0 0 0 0 τΩ 3 0 0 0 0 0 0 0 0 0 0 0 τΩ 4 0 0 0 0 0 0 0 0 0 0 0 τΩ 5 0 0 0 0 0 0 0 0 0 0 0 τΩ 6 0 0 0 0 0 0 0 0 0 0 0 τ Ω 1′ 0 0 0 0 0 0 0 0 0 0 0 τΩ ′2 0 0 0 τ Ω ′3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 τΩ ′4 0 0 0 0 0 0 0 0 0 0 0 0 τ Ω ′5 0 0 0 0 0 0 0 0 0 0 0 0 τ Ω ′6
B2i Z ji B2Ti + δ ji GGT − Pji H 4i Z ji B2Ti
(23) k
k
W ′′ = ( A0 + ∑ Ai ) Z T ( X , Y ) + B20 Ψ + ∑ B2i Ψ + Z ( X , Y ) i =1 k
i =1
× ( A0 + ∑ Ai )T + ( B20 Ψ + ∑ B2i Ψ )T +ε1 E10 E10T i =1
6
k
+ ε1−1 Z ( X , Y )C T K T KCZ T ( X , Y )T + τ ∑∑ Pji j =1 i =1
6
k
k
k
i =1
i =1
+τ ∑∑ T ji + ε 2T0 + ε 2−1Ψ T Ψ + ∑ ε 3iTi + ∑ ε 3−i1Ψ T Ψ j =1 i =1
N1 = [ Z ( X , Y ) A +Ψ B T 0
T
N 2 = [Z ( X , Y ) A
T 1
Z ( X , Y ) A +Ψ B
T 20
T 0
T 2
N 3 = [ Z ( X , Y )C K E T
T 10
T 20
],
T k
Z ( X ,Y ) A
T
T
Z ( X , Y ) A ],
Z ( X , Y )C T K T E10T
Z ( X , Y )C T K T E10T ], T N 4 = [Ψ T B21
Ψ T B2Tk ] = N 6 ,
T N 5 = [Ψ T B20
Ω′j = −diag { R j1 , R j 2 ,
, R jk } , j = 1, 2,
, 6.
and the scalars and τ , i = 1, , k ,
EX + Y ϒ ≥ Z ji
(24a)
Z ji ≥ Q ji
(24b)
T
Ψ ≥0 Z ji
(24c)
T B20 B20 + β 0 GG T − T0 B20 H 3T ≤ 0 (24d) T H 3 B20 H 3 H 3T − β 0 I B2i B2Ti + β i GG T − Ti B2i H 4Ti ≤ 0 (24e) H 4i B2Ti H 4i H 4Ti − β i I
Ai Q ji AiT + β ji GG T − T ji H 2i Q ji AiT
(24i)
M ′′ Θ1 Θ′2T T −1 (24j) Θ1 −α I 0 , if there exist a series of positive definite symmetric X , Q ji , R ji , Z ji , Z ji , T0 , Ti , T ji , Pji , V ji , Q , a matrix Y , Ψ , and
uncertain time-delay singular system, it is found, using the software package LMI lab, that this system is regular, impulse free and locally asymptotically stable for any time-delay τ ≤ 0.6684, When τ = 0.5 , the corresponding calculation results are as follows: 6.0657 4.7504 −1.1697 X = , Y = −2.3394 , 4.7504 6.0657
the scalars ε1 , ε 2 , ε 3i , β 0 , β i , satisfying (35a-i) for i = 1,
W11 Θ1T Θ′2 Z ( X , Y ) B10T 0 0
Θ1
−α 0 0 0 0
−1
β ji , δ ji ,α , µ ji , γ ,ν ,υ and τ ,
, k , j = 1,
, 7 and
Θ′2T 0 −α I 0 0 0
B10 Z T ( X , Y ) 0 0 I 0 0 0 0 0 0 0 . n
T
−1
−1
[9]
J.K. Hale, Introduction to Functional Differential Equations. New York: Springer-Verlag, 1993. L. Dai, Singular Control Systems. Berlin: Springer-Verlag; 1989. S. Xu, P.V. Dooren, R. Stefan, and J. Lam, “Robust stability and stabilization for singular systems with state delay and parameter uncertainty”, IEEE Trans. Automat. Control, Vol. 47, pp. 1122-1128, 2002. S. Xu, J. Lam, and L. Zhang, “Robust D-Stability analysis for uncertain discrete singular systems with state delay”, IEEE Trans.Circuits Syst. I, Vol. 49, pp. 551-555, 2002. P.Shi and V.Dragan, “Asymptotic H ∞ Control of Singular Perturbed Systems with Parametic Uncertainties”, IEEE Trans. Automat. Control, Vol. 44, No. 9, pp. 1738-1742, 1999. S. Tarbouriech, P.L.D. Peres, G. Garcia, and I. Queinnec, “Delay-dependent stabilization and disturbance tolerance for time-delay systems subject to actuator saturation”, IEE Proc. –Control Theory Appl., Vol. 149, pp. 387-393, 2002. H. -Y. Su, F. Liu and J. Chu, “Robust stabilization of uncertain time-delay systems containing saturating actuators”, IEE Proc. –Control Theory Appl., Vol. 148, pp. 323-328, 2001. S. Boyd, L. El Ghaoui, E. Feron, and V. Balakrishnan, Linear Matrix Inequalities in Systems and Control Theory. Philadelphia, PA: SIAM, 1994. R. -Q. Lu, H. -Y. Su and J. Chu, “Roust H ∞ Control for A Class of Uncertain Lur’e Singular Systems with Time-Delays”, in Proc. IEEE Conf. Decision & Control Hawaii 2003, pp. 5585-559.
4991
Proceeding of the 2004 American Control Conference Boston, Massachusetts June 30 - July 2, 2004
Robust Stabilization for Singular Systems with Time-Delays and Saturating Controls Wu-Neng Zhou, Ren-Quan Lu, Hong-Ye Su, and Jian Chu, Member, IEEE
Abstract—We studied the delay-dependent stabilization problem for a class of uncertain singular system with time-delays and saturating controls. Theorems derived give sufficient conditions for delay-dependent stabilization of the singular systems with a combination of saturating controls and multiple time-delays in both state and control; we assumed the delays to be constant bounded but unknown, moreover, the uncertainties are also described to be unknown but bounded and the nonlinear terms included in the systems are fallen into a set. Under these sufficient conditions, the solution of the uncertain singular system is regular, impulse free, and locally asymptotically stable for all admissible uncertainties. Furthermore, the results based on several Linear Matrix Inequalities (LMIs) are developed to guarantee stability and be computed effectively. Finally, we advance an example to demonstrate the superiority of this method. I.
INTRODUCTION
C
ONYTOL of singular systems has been extensively studied in the past years due to the fact that singular systems better describe physical systems than regular ones. A great number of results based on the theory of regular systems (or state-space systems) have been extended to the area of singular systems [1]-[2]. Recently, robust stability and robust stabilization for uncertain singular systems with time-delays have been considered [3]-[5]. Moreover, the problem of stabilizing linear systems with saturating controls has been widely studied these last years because of its practical interest [6] and [7]. However, to the best of our knowledge, the problems of robust stabilization for uncertain singular system included both time-delays and Manuscript received September 20, 2003. This work was supported by the National Outstanding Youth Science Foundation of P. R. China (NSFC: 60025308), the Teaching and Research Award Program for Outstanding Young Teachers in Higher Education Institutions of MOE P. R. China and the National Natural Science Foundation of P. R. China (NSFC: 10371106) W. N. Zhou is with the National Laboratory of Industrial Control Technology, Institute of Advanced Process Control, Zhejiang University, Yuquan Campus, Hangzhou 310027, P. R. China and the Mathematics Institute, Zhejiang Normal University, Jinhua 321004, P. R. China (e-mail: [email protected] ). R. Q. Lu, H. Y. Su, and J. Chu are with the National Laboratory of Industrial Control Technology, Institute of Advanced Process Control, Zhejiang University, Yuquan Campus, Hangzhou 310027, P. R. China (e-mail: [email protected], [email protected], [email protected] ).
0-7803-8335-4/04/$17.00 ©2004 AACC
saturating controls have not been fully investigated yet. In this paper, we concerned with the delay-dependent robust stabilization of a continuous-time subject to multiple time-delays in both state and control, saturating controls and nonlinear terms. The synthesis problem addressed is to design a memoryless state feedback control law such that the resulting closed-loop system is regular, impulse free and stable for all admissible uncertainties, and a sufficient condition for the existence of such a control law is presented in terms of several linear matrix inequalities (LMIs). II. SYSTEM DESCRIPTION AND DEFINITIONS Consider the following uncertain singular systems with time-delays and saturating controls described by k
(Σ) : Ex(t ) = ( A0 + ∆A0 ( x, t )) x(t ) + ∑ ( Ai + ∆Ai ( x, t )) i =1
× x(t − hi (t )) + E10 f (σ(t )) + B10 w(t ) + ( B20 + ∆B20 ( x, t ))u '(t ) k
+ ∑ ( B2i + ∆B2i ( x, t ))u '(t − gi (t )),
(1)
i =1
u '(t ) = sat (u (t )), x(t ) = φ (t ), t ∈ [−τ,0], σ(t ) = Cx(t ) sat (u (t )) = [ sat (u1 (t )) sat (u2 (t )) sat (um (t ))], where x(t ) ∈ R n is the state vector, u (t ) ∈ R m is control input vector to the actuator (emitted from the designed controller), u '(t ) ∈ R m is the control input vector to the plant,
w(t ) ∈ R p is the disturbance input vector from L2 [0, ∞ ) . The matrix E ∈ R n×n may be singular, we shall assume that rank E = r ≤ n . The matrices A0 , Ai , E10 , B10 , B20 and B2i are known real constant matrices with appropriate dimensions. The matrices ∆A0 (•), ∆Ai (•), ∆B20 (•) and ∆B2i (•) are time-invariant matrices representing norm-bounded parameter uncertainties, and are assumed to be of the following form: [ ∆A0 (•) ∆Ai (•) ∆B20 (•) ∆B2i (•)] (2) = GF ( x, t ) [ H1 H 2i H 3 H 4i ] where G , H 1 , H 2i , H 3 and H 4i are known real constant matrices with appropriate dimensions. The uncertain matrix F ( x(t ), t ) with Lebesgue measurable elements satisfies
4986
F T ( x(t ), t ) F ( x(t ), t ) ≤ I . (3) The input vector is assumed to satisfy actuator limitations, i.e. u (t ) ∈ U ⊂ R m with U = {u (t ) ∈ R m ; −u0( i ) ≤ u( i ) (t ) ≤ u0(i ) , u0(i ) > 0, i = 1,
, m} (4)
The actuator is described by the nonlinearity u0( i ) if u( i ) (t ) > u0( i ) sat (u( i ) (t )) = u( i ) (t ) if − u0(i ) ≤ u( i ) (t ) ≤ u0( i ) (5) if u( i ) (t ) < −u0(i ) −u0( i ) hi (t ) and gi (t ) are unknown scalars denoting the delays in the state and control, respectively, and it is assumed that there exist positive numbers h, g and τ such that (6) 0 ≤ hi (t ), gi (t ) ≤ h, g ≤ τ for all t , i = 1, , k . φ (t ) is smooth vector-valued continuous initial function defined in the Banach space Cτ . In this paper, every nonlinear term is assumed to be of the form as follows f j (i) ∈ K j [0, k j ] = { f j (σ j ) f j (0) = 0,0 < σ j f j (σ j ) (7) ≤ k jσ 2j (σ j ≠ 0)}, j = 1,2, , n where k j are positive scalars. The nominal unforced singular delay systems (1) can be written as k
Ex(t ) = A0 x(t ) + ∑ Ai x(t − hi (t )) + E10 f (σ(t )).
(8)
i =1
Using the Leibniz-Newton formula [7], then the singular delay system (8) can be written as k
Ex(t ) = ( A0 + ∑ Ai ) x (t ) + E10 f (σ (t ))
π 1 x(t )
k
[A0 x(t + θ ) + ∑ Ai
k
Lemma 3 [8]. Given matrices A, Θ, Ξ, Γ and F (σ ) of
appropriate dimensions and with Θ symmetrical and F (σ ) satisfying F T (σ ) F (σ ) ≤ I . Then we have: a) If the following inequality holds, Θ + ΓF (σ )Ξ + (ΓF (σ )Ξ )T < 0 if and only if there exists a scalar ε > 0 such that Θ + εΓΓT + ε −1ΞT Ξ < 0 b) For any symmetric matrix P > 0 and scalar ε > 0 such that ε I − ΞPΞT > 0 , then ( A + ΓF (σ )Ξ) P ( A + ΓF (σ )Ξ )T
III. ANALYSIS OF ROBUST STABILITY
i =1
A. Analysis of robust Stability of Systems (8) The main result is derived as follows, it gives the sufficient condition of robust stability for the singular delay system (8).The proof of it is similar to [2] and [9] and is omitted. Theorem 1. If there exists a series of positive definite symmetric Q, Q1i , Q2i , Q3i , i = 1 k , a matrix P , and the scalars ε , γ and τ such that
and impulse free . Lemma 1 (Krasovskii theorem [1]). The singular delay systems (8) is said to be locally asymptotically stable if there exists a positive definite symmetric matrix P , positive scalars π 1 , π 2 , π 3 , v and γ , for any initial condition φ (t )
EPT = PE T ≥ 0 W τ N1 τ N 2 τ N 3 τ N T τΩ 0 0 1 M = 1T 0} , whereas, for a continuous function
if
Definition 2. The uncertain singular systems ( Σ ) is said to be robustly stable if the systems ( Σ ) with u (t ) ≡ 0, u (t − g i (t )) ≡ 0, w(t ) ∈ L2 [0, ∞) is regular, impulse free and locally asymptotically stable for all admissible uncertainties. Definition 3. The uncertain singular delay systems ( Σ ) is said to be robustly stabilizable if there exists a linear state feedback control law u (t ) = Λx(t ), Λ ∈ R m×n such that the resultant closed-loop system is robustly stable in the sense of Definition 3. In this case, u (t ) = Λx(t ) is said to be a robust state feedback control law for system ( Σ ). Lemma 2 [8]. Given vector x, y , a positive definite symmetric matrix R with appropriate dimensions, then for any scalar ε > 0 , we have ±2 xT y ≤ ε xT Rx + ε −1 y T R −1 y.
(9)
regular and impulse free if the pair ( E , A + ∑ Ai ) is regular
in
and
V ( x(t ), t ) ≤ −π 3 x(t ) , V ( x(t ), t ) ≤ V (φ (0), 0) .
i =1
x (t ) = φ (t ), t ∈ [−2τ , 0] Throughout this paper, we shall use the following concepts and introduce the following useful lemmas. Definition 1. The singular delay systems (8) is said to be
confined
,
≤ APAT + APΞT (ε I − ΞPΞT ) −1 ΞPAT + εΓΓT
× x(t − hi (t ) + θ ) + E10 f (σ (t + θ ))]dθ
remain
2
k
0
− hi ( t )
i =1
≤ V ( x(t ), t ) ≤ π 2 x(t ) 2
i =1
− ∑ Ai ∫
2
V ( x(t ), t ) : R n × R + → R
such
that
(10a)
(10b)
(10c)
where
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k
k
i =1
i =1
W = ( A0 + Σ Ai ) PT + P ( A0 + Σ Ai )T k
+τ Σ Ai (Q1i + Q2i + Q3i ) AiT + ε E10 E10T + ε −1 PC T K T KCPT i =1i
N1 = [ PA0T
PA0T ], N 2 = [ PA1T PA2T
N 3 = [ PC T K T E10T PC T K T E10T
Ω1 = −diag {Q11 , Q12 , Ω 2 = −diag {Q21 , Q22 ,
PAkT ] ,
PC T K T E10T ],
, Q1k } , K = diag {k1 ,
kn } ,
, Q2 k } , Ω3 = − diag {Q31 , Q32 ,
, Q3k } ,
then the singular delay system (8) is regular, impulse free and locally asymptotically stable for any initial condition belonging to the set Φ 0 = {φ (t ) ||| φ (t ) ||2 ≤ δ } with δ = λmin (Q) / γπ 2 where kτ 2 π 2 = λmax ( EPT ) + max λmax [ PA0T Q1−i 1 A0 PT ] 2
Corollary 1 (Disturbance-free case). If there exist a series of positive definite symmetric Q, Q1i , Q2i , Q3i , Q4i , Q5i , Q6i , R1i , R2 i , R3i , R4i , R5i , R6i , i = 1 k , a matrix P , and the scalars ε 2 , ε 3i , γ , i = 1, , k and τ such that M ∆ < 0, Ξ1 ≥ 0, Ξ 2 ≤ 0 and the expression of EPT = PE T ≥ 0 hold, then the uncertain singular delay system (1) is regular, impulse free and locally asymptotically stable for any initial condition belonging to the set
Φ 0 = {φ (t ) ||| φ (t ) ||2 ≤ δ } with δ = where
kτ 2 max λmax [ P ( A0 ∆ 2 i + B20 ∆ Λ )T (Q1−i 1 + R1−i 1 )( A0 ∆ + B20 ∆ Λ ) PT ]
π 2 = λmax ( EPT ) + kτλmax ( PΛT ΛPT ) +
i
3kτ 2 max λmax [ PAiT∆ (Q2−i1 + R2−i1 ) Ai∆ PT ] i 2 2 kτ + max λmax [PCT K T E10T (Q3−i1 + R3−i1 )E10 KCPT ] 2 i 3kτ 2 + max λmax [P(B2i∆ Λ)T (Q4−i1 + R4−i1 )(B2i∆ Λ)PT ] 2 i kτ 2 + max λmax [P(B20∆ Λ)T (Q5−i1 + R5−i1 ) 2 i 3kτ 2 × (B20∆ Λ)PT ]+ max λmax [P(B2i∆ Λ)T i 2 −1 −1 T × (Q6i + R6i )(B2i∆ Λ)P ] +
3kτ max λmax [ PAiT Q2−i1 Ai PT ] + i 2 kτ 2 max λmax [ PC T K T E10T Q3−i1 E10 KCPT ]) + 2 i 2
B. Disturbance-Free Case (with w(t ) = 0 ) When w(t ) = 0 , for the uncertain singular system ( Σ ), introduce the control law u (t ) = 2Λx(t ) , where the control law gain matrix Λ ∈ R m×n is to be found, and the closed-loop system is k
(Σ′) : Ex(t ) = ( A0 ∆ + B20 ∆ Λ) x(t ) + ∑ Ai ∆ x(t − hi (t ))
and
i =1
k
+ E10 f (σ(t )) + ∑ B2i ∆ Λx(t − gi (t )) i =1
P −1 E ΛTi Ξ1 = ≥0 γ u02i Λi
(11)
k
+ B20 ∆η (t ) + ∑ B2i∆η (t − gi (t )) and M ∆
where [•]∆ = [ •] + ∆ [•] ( [•] denoting the matrix), and
η (t ) = sat (2Λx(t )) − Λx(t ), η (t − gi (t )) = sat (2Λx(t − gi (t )) − Λx(t − gi (t )) Obviously, the vector function η (t ) satisfies the following
inequality
η (t − gi (t )) ≤ xT (t − gi (t ))ΛT Λx(t − gi (t ))
(12)
Ω3 are the same as theorem 1, and k
k
W∆ = ( A0 ∆ + ∑ Ai∆ + B20 ∆ Λ + ∑ B2i∆ Λ )PT + P( A0 ∆ i =1
i =1
k
k
k
i =1
i =1
i =1
+ ∑ Ai∆ + B20∆ Λ + ∑ B2i∆ Λ)T +τ ∑ Ai∆ (Q1i k
+ Q2i + Q3i + Q4i + Q5i + Q6i ) AiT∆ + τ ∑ B2i∆Λ(Q1i i =1
By (4), one has x(t ) ∈ S (u0 ,1m ) , where
+Q2i + Q3i + Q4i + Q5i + Q6i )(B2i∆Λ)
T
S (u0 ,1m ) = {x(t ) ∈ R n |{u (t ) ∈ R m ; −u0(i ) ≤ Λ (i ) x(t ) ≤ u0( i ) , i = 1,
(14a)
Q PT Ξ2 = (14b) ≤0 P I is shown in the next page, where N 3 , Ω1 , Ω 2 and
i =1
x(t ) = φ (t ), t ∈ [−τ, 0]
η T (t )η (t ) ≤ xT (t )ΛT Λx(t ),
λmin (Q) , γπ2
, m}
(13)
Using the same method as theorem 1 and taking into (12), (13) and [7], furthermore introducing the idea of generalized quadratic stability and generalized quadratic stabilization in [3], one can deduce the following corollary.
T + ε1E10 E10T + ε1−1PCT K T KCPT + ε 2 B20∆ B20 ∆ k
k
+ ε 2−1PΛT ΛPT + ∑ε3i B2i∆ B2Ti∆ + ∑ε3−i1PΛT ΛPT i =1
N1∆ = [ P ( A1∆ + B20 ∆ Λ )T N 20 = [ PA1T∆ PA1T∆ 2
i =1
P( A1∆ + B20 ∆ Λ )T ],
PAkT∆ ]
4988
N 4 ∆ = [ P ( B21∆ Λ )T
P ( B2 k ∆ Λ )T ] = N 6 ∆ ,
N 5 ∆ = [ P( B20 ∆ Λ )
P( B20 ∆ Λ ) ],
T
where the matrix N i is the matrix N i∆ (i = 1, uncertainty in corollary 1, and
T
Ω j = − diag {Q j1 , Q j 2 , Ω′j = − diag { R j1 , R j 2 ,
k
, Q jk } , j = 4,5, 6,
, R jk } , j = 1, 2,
i =1
, 6.
T0 , Ti , T ji , Pji such that the following inequalities are satisfied: T T B20 B20 + B20 H 4T ( β 0 I − H 3 H 3T )-1 H 3 B20 + β 0 GG T ≤ T0 (15) B2i B2Ti + B2i H 4Ti ( β i I − H 4 H 4Ti )-1 H 4i B2Ti + β i GG T ≤ Ti
(16)
Ai Q ji A + Ai Q ji H ( β ji I - H 2i Q ji H ) H 2i Q ji A T -1 2i
T i
(17)
+ β ik GG T ≤ T ji B2i Z ji B2Ti + B2i Z ji H 4Ti (δ ji I − H 4i Z ji H 4Ti ) −1
(18)
×H 4i Z ji B2Ti + δ ji GG T ≤ Pji where Z ji ≥ ΛQ ji ΛT , j = 1,
, 6, i = 1,
, k , and
β 0 I − H 3 H > 0, β i I − H 4i H > 0, T 3
T 4i
Using Lemma 3b, we have T T B20 ∆ B20 ∆ ≤ T0 , B2 i ∆ B2 i ∆ ≤ Ti , T 2 i∆
(20)
≤ Pji
By Corollary 1, (2) and (20), it follows that M ∆ = M + Θ1 F ( x(t ), t )Θ2 + (Θ1 F ( x(t ), t )Θ2 )T < 0
(21)
where Θ1 = diag {G , G ,
, G} , Θ 2 = Θ2 0 0 , and T T T Θ 2 = [ P( H1 + ∑ H 2i + H 3 Λ + ∑ H 4i Λ ) , Θ 21 , ΘT21 ], i
i
ΘT21 = [ P ( H1 + H 3 Λ )T ,
T , P( H1 + H 3 Λ)T , PH 21 ,
,0, PΛ H ,
T 2k
T
PH , 0,
T , PΛT H 3T , PΛT H 41 ,
T
T 4k
, PΛT H 4Tk ],
M ′ = W ′ − τ N1 (Ω1−1 + Ω1′−1 ) N1T − τ N 2 (Ω −21 + Ω′2−1 ) N 2T − τ N 3 (Ω + Ω′ ) N − τ N 4 (Ω + Ω′ ) N −1 3
−1 3
T 3
,
, PΛ H , PΛ H 3T ,
T 41
−1 4
i =1
i =1
6
k
6
k
+ τ ∑∑ Pji +τ ∑∑Tji + ε 2T0 + ε 2−1PΛT ΛPT j =1 i =1
j =1 i =1
k
k
i =1
i =1
+ ∑ε3iTi + ∑ε3−i1PΛT ΛPT By Lemma 3a and (21), we can obtain that there exists a scalar α > 0 such that M ′ + αΘ1Θ1T + α −1ΘT2 Θ2 < 0 (22) For simplicity we introduce the matrix ϒ ∈ R n×( n − r ) satisfying Eϒ = 0 and rank ϒ = n − r . It’s easy to see that there exist invertible matrices L1 and L2 ∈ R n× n from the proof of Theorem 1 such that P P P = L1 PL−2T = 11 12 0 P22 T r ×( n − r ) where P11 = P11 ≥ 0, P12 ∈ R , P22 ∈ R ( n − r )×( n − r ) . On the other hand, from Eϒ = 0 and rank ϒ = n − r , it implies that there exists an invertible matrix Γ ∈ R ( n − r )×( n − r ) such that 0 ϒ = L2 Γ I n−r Hence P P I 0 −1 P = L1−1 11 12 LT2 = ( L1−1 r L2 ) 0 0 0 P22 P 0 T −1 P12 − T × ( L2 11 L2 ) + ( L1 Γ ) P22 0 In −r T T × (Γ [ 0 I n − r ] L2 ) EX + Y ϒT where
Ai∆ Q ji A ≤ T ji , B2i∆ ΛQ ji Λ B T
i =1
+ B20 Λ + ∑ B2i Λ)T + ε1 E10 E10T + ε1−1 PC T K T KCPT
(19)
β ij I − H 2i Q ji H 2Ti > 0, δ ji I − H 4i Z ji H 4Ti > 0
T i∆
k
k
(14c) In the following we shall discuss how to solve the control law gain matrix Λ in the following analysis by using LMI technology. Assume that there exist scalars β 0 , β i , β ji , δ ji > 0 and positive definite symmetric matrices
T 2i
k
W ′ = ( A0 + ∑ Ai + B20 Λ + ∑ B2i Λ )PT + P( A0 + ∑ Ai
W∆ τ N1∆ τ N2∆ τ N3 τ N4∆ τ N5∆ τ N6∆ τ N1∆ τ N2∆ τ N3 τ N4∆ τ N5∆ τ N6∆ τ N T τΩ 0 0 0 0 0 0 0 0 0 0 0 1 1∆ τ N2T∆ τΩ2 0 0 0 0 0 0 0 0 0 0 0 T τΩ3 0 0 0 0 0 0 0 0 0 0 0 τ N3 τ N T τΩ4 0 0 0 0 0 0 0 0 0 0 0 4∆ T τΩ5 0 0 0 0 0 0 0 0 0 0 0 τ N5∆ M ∆ = τ N6T∆ τΩ6 0 0 0 0 0 0 0 0 0 0 0 0, Y = L1 Γ 0 I n−r P22 Furthermore EPT = E ( EX + Y ϒT )T = EXE T = ( EX + Y ϒT ) E T = PE T ≥ 0 Ψ = Λ ( EX + Y ϒT )T ΛΖT ( X , Y ) . Define Without loss of generality, we can assume that Ζ( X , Y ) = EX + Y ΦT is invertible. Define matrix M ′′ , as shown at the top of the next page, where k
k
W ′′ = ( A0 + ∑ Ai ) Z T ( X , Y ) + B20 Ψ + ∑ B2i Ψ + Z ( X , Y ) i =1 k
i =1
k
× ( A0 + Σ Ai ) + ( B20 Ψ + Σ B2i Ψ )T +ε1 E10 E10T T
i =1
i =1
6
k
+ ε Z ( X , Y )C K KCZ ( X , Y )T + τ Σ Σ Pji −1 1
T
T
T
j =1 i =1
4989
W ′′ T τ N 1 τ N 2T T τ N 3 τ N T 4 τ N 5T M ′′ = τ N 6T τ N T 1T τ N 2 τ N T 3 τ N 4T T τ N 5 T τ N 6
τ N1 τ N 2 τ N 3 τ N 4 τ N 5 τ N 6 τ N1 τ N 2 τ N 3 τ N 4 τ N 5 τ N 6 0 0 0 0 0 0 0 0 0 0 0 τΩ1 0 0 0 0 0 0 0 0 0 0 0 τΩ 2 0 0 0 0 0 0 0 0 0 0 0 τΩ 3 0 0 0 0 0 0 0 0 0 0 0 τΩ 4 0 0 0 0 0 0 0 0 0 0 0 τΩ 5 0 0 0 0 0 0 0 0 0 0 0 τΩ 6 0 0 0 0 0 0 0 0 0 0 0 τ Ω 1′ 0 0 0 0 0 0 0 0 0 0 0 τΩ ′2 0 0 0 τ Ω ′3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 τΩ ′4 0 0 0 0 0 0 0 0 0 0 0 0 τ Ω ′5 0 0 0 0 0 0 0 0 0 0 0 0 τ Ω ′6
B2i Z ji B2Ti + δ ji GGT − Pji H 4i Z ji B2Ti
(23) k
k
W ′′ = ( A0 + ∑ Ai ) Z T ( X , Y ) + B20 Ψ + ∑ B2i Ψ + Z ( X , Y ) i =1 k
i =1
× ( A0 + ∑ Ai )T + ( B20 Ψ + ∑ B2i Ψ )T +ε1 E10 E10T i =1
6
k
+ ε1−1 Z ( X , Y )C T K T KCZ T ( X , Y )T + τ ∑∑ Pji j =1 i =1
6
k
k
k
i =1
i =1
+τ ∑∑ T ji + ε 2T0 + ε 2−1Ψ T Ψ + ∑ ε 3iTi + ∑ ε 3−i1Ψ T Ψ j =1 i =1
N1 = [ Z ( X , Y ) A +Ψ B T 0
T
N 2 = [Z ( X , Y ) A
T 1
Z ( X , Y ) A +Ψ B
T 20
T 0
T 2
N 3 = [ Z ( X , Y )C K E T
T 10
T 20
],
T k
Z ( X ,Y ) A
T
T
Z ( X , Y ) A ],
Z ( X , Y )C T K T E10T
Z ( X , Y )C T K T E10T ], T N 4 = [Ψ T B21
Ψ T B2Tk ] = N 6 ,
T N 5 = [Ψ T B20
Ω′j = −diag { R j1 , R j 2 ,
, R jk } , j = 1, 2,
, 6.
and the scalars and τ , i = 1, , k ,
EX + Y ϒ ≥ Z ji
(24a)
Z ji ≥ Q ji
(24b)
T
Ψ ≥0 Z ji
(24c)
T B20 B20 + β 0 GG T − T0 B20 H 3T ≤ 0 (24d) T H 3 B20 H 3 H 3T − β 0 I B2i B2Ti + β i GG T − Ti B2i H 4Ti ≤ 0 (24e) H 4i B2Ti H 4i H 4Ti − β i I
Ai Q ji AiT + β ji GG T − T ji H 2i Q ji AiT
(24i)
M ′′ Θ1 Θ′2T T −1 (24j) Θ1 −α I 0 , if there exist a series of positive definite symmetric X , Q ji , R ji , Z ji , Z ji , T0 , Ti , T ji , Pji , V ji , Q , a matrix Y , Ψ , and
uncertain time-delay singular system, it is found, using the software package LMI lab, that this system is regular, impulse free and locally asymptotically stable for any time-delay τ ≤ 0.6684, When τ = 0.5 , the corresponding calculation results are as follows: 6.0657 4.7504 −1.1697 X = , Y = −2.3394 , 4.7504 6.0657
the scalars ε1 , ε 2 , ε 3i , β 0 , β i , satisfying (35a-i) for i = 1,
W11 Θ1T Θ′2 Z ( X , Y ) B10T 0 0
Θ1
−α 0 0 0 0
−1
β ji , δ ji ,α , µ ji , γ ,ν ,υ and τ ,
, k , j = 1,
, 7 and
Θ′2T 0 −α I 0 0 0
B10 Z T ( X , Y ) 0 0 I 0 0 0 0 0 0 0 . n
T
−1
−1
[9]
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