On Strong Stability and Robust Strong Stability of Linear Difference Equations with Two Delays
aa r X i v : . [ m a t h . D S ] J un On Strong Stability and Robust Strong Stability of LinearDifference Equations with Two Delays
Bin Zhou ∗ Abstract
This paper provides a necessary and sufficient condition for guaranteeing exponential stability of thelinear difference equation x ( t ) = Ax ( t − a ) + Bx ( t − b ) where a > , b > A, B are n × n square matrices, in terms of a linear matrix inequality (LMI) of size ( k + 1) n × ( k + 1) n where k ≥ A, B ) appear as highlynonlinear functions, the proposed LMI condition involves matrices that are linear functions of (
A, B ) . Such a property is further used to deal with the robust stability problem in case of norm boundeduncertainty and polytopic uncertainty, and the state feedback stabilization problem. Solutions to thesetwo problems are expressed by LMIs. A time domain interpretation of the proposed LMI conditionin terms of Lyapunov-Krasovskii functional is given, which helps to reveal the relationships among theexisting methods. Numerical example demonstrates the effectiveness of the proposed method.
Keywords:
Linear difference equations; Exponential stability; Necessary and sufficient conditions;Linear matrix inequality.
Throughout this paper, we use A ⊗ B to denote the Kronecker product of matrices A and B. For a matrix A, the symbols | A | , k A k , A T , A H , and ρ ( A ) denote respectively its determinant, norm, transpose, conjugatetranspose, and spectral radius. For a square matrix P , P > x ( t ) = N X i =1 A i x ( t − r i ) , (1)where r i > A i are square matrices, is frequently encountered in neutral-type time-delaysystems [11, 18] and coupled differential-functional equations [10, 15]. The stability of system (1) is usuallythe necessary condition for ensuring the asymptotic stability of the above two types of time-delay systems,and thus has attracted considerable attentions in the literature [3, 4, 7, 10, 12, 21, 23].It is known that (1) is stable if and only if its spectral abscissa is less than zero [12]. However, the spectralabscissa of (1) is not continuous in delays and the stability might be destroyed by arbitrarily small changesin the delay [1, 12]. Therefore, the concept of strong stability was introduced by [12] to handle this hy-persensitivity of the stability with respect to delays, which has been generalized in [18]. To go further, weintroduce the following result from Theorem 6.1 (Chapter 9, p. 286) in [12]. Lemma 1
System (1) is strongly stable if and only if max θ i ∈ [0 , π ] ,i =1 , ,...,N ρ N X i =1 A i e j θ i ! < . (2) ∗ The author is with the Center for Control Theory and Guidance Technology, Harbin Institute of Technology, Harbin,150001, China. Email: [email protected], [email protected] . θ i ∈ [0 , π ] , i = 1 , , . . . , N. Strongstability of (1) was tested via deciding positive definiteness of a multivariate trigonometric polynomial matrix,which is then solved as a converging hierarchy of LMIs [13]. The condition in [13] needs to compute thecharacteristic equation of (1), which is not explicitly expressed as functions of the coefficients, and thusseems difficult to be used for robust stability analysis. For a single delay, strong stability can be checkedby computing the generalized eigenvalues of a pair of matrices [16, 17] as well as the matrix pencil basedapproach [19]. The method of cluster treatment of characteristic roots was used in [20] to derive the stabilitymaps of (1) with three delays. For more related work, see [10, 12, 13, 20] and the references therein.In this note, we restrict ourself to a special case of (1) where N = 2, for which we rewrite (1) as x ( t ) = Ax ( t − a ) + Bx ( t − b ) , (3)where a, b are positive constants, and A, B are n × n square matrices. Regarding the existence of a solution,the continuity/discontinuity of the solution, and definitions for stability of the solution, readers are suggestedto refer [3] and [12] for details. Notice that, by Lemma 1, system (3) is strongly stable if and only if ρ (∆ θ ) < , θ ∈ [0 , π ] , ∆ θ = A + B e − j θ . (4)It came to our attention that condition (4) happens to be equivalent to the stability of the 2-D linear systemdescribed by the Fornasini-Marchesini second model x ( i + 1 , j + 1) = Ax ( i, j + 1) + Bx ( i + 1 , j ) , (5)which has been well studied in the literature [6, 9]. For stability analysis of (5), a necessary and sufficientcondition expressed by an LMI of size 3 n × n was established in [6]. Lemma 2
The system (3)/(5) is strongly/exponentially stable if and only if ρ ( A + B ) < , (6) and there exist two symmetric matrices P ∈ R n × n , P ∈ R n × n and a matrix P ∈ R n × n such that − P − P − P P T3 − P T3 P P + P < E T E, (7) where E = [ B T ⊗ A, A T ⊗ B, A T ⊗ A + B T ⊗ B − I n ⊗ I n ] . This result is almost the same as Theorem 1 in [6], where E is replaced by E ∗ = [ B ⊗ A, A ⊗ B, A ⊗ A + B ⊗ B − I n ⊗ I n ] . The proof given in [6] is based on the Guardian map and the positive real lemma. Motivated by [5],we provide in Appendix a simple proof based on the well-known Yakubovich-Kalman-Popov (YKP) lemma.Another necessary and sufficient conditions, which involve the generalized eigenvalues of two matrices withsize 2 n × n , were obtained in [9], which were also established initially for testing stability of the 2-D linearsystem (5).Bliman established in [2] another LMI based necessary and sufficient conditions for testing stability of (5).To introduce this result, for any k ∈ N + , we define A k = B AB · · · A k − B B · · · A k − B . . . . . . ...0 B ∈ R kn × kn , B k = A k − A k − ... AI n ∈ R kn × n , (8) A k = B AB A B · · · A k − BB AB · · · A k − B . . . . . . ... B ABB ∈ R kn × kn , B k = A k A k − ... A A ∈ R kn × n . (9)2or two symmetric matrices P , Q ∈ R kn × kn , we define a linear function Ω k (cid:0) P , Q (cid:1) ∈ R ( k +1) n × ( k +1) n as Ω k (cid:0) P , Q (cid:1) = " A T k P A k − P + A T k Q A k − A T k QA k A T k P B k + A T k Q B k − A T k QB k B T k P A k + B T k Q A k − B T k QA k B T k P B k + B T k Q B k − B T k QB k . (10) Lemma 3 [2] If there exist positive definite matrices P k , Q k ∈ R kn × kn such thatΩ k (cid:0) P k , Q k (cid:1) < , (11) then system (3)/(5) is stable. Moreover, if (3)/(5) is stable, there exists an integer k ∗ ≥ , such that (11) issolvable with P k > , Q k > , ∀ k ≥ k ∗ . Notice that Lemma 3 is slightly different from the original one in [2] where the result is built for a general2-D linear system, and is expressed in a recursive form. Even for k = 2 , the LMI in Lemma 3 is nonlinearin A and B , and thus can not be used for robust stability analysis.In this note, motivated by [2], we will establish a new necessary and sufficient condition for testing strongstability of system (3). Different from Lemmas 2 and 3, the proposed LMI condition involves matrices thatare linear functions of ( A, B ) . With the help of this property, the robust stability problem in case of normbounded uncertainty is investigated, and the results are also expressed by LMIs (see Section 2). We also givetime-domain interpretations of the proposed LMI condition and the Bliman condition, which help to revealthe relationships among them and the other existing methods such as those in [3] and [4] (see Section 3).
For any k ∈ N + , we denote A k = (cid:20) I ( k − n (cid:21) ∈ R kn × kn , B k = (cid:20) I n (cid:21) ∈ R kn × n , (12)and L k = (cid:2) I kn kn × n (cid:3) ∈ R kn × ( k +1) n , (13)which are independent of ( A, B ) , and A k = B A · · · B A . . . .... . . . . . 0
B AB ∈ R kn × kn , B k = A ∈ R kn × n , (14)which are linear matrix functions of ( A, B ) . For two symmetric matrices
P, Q ∈ R kn × kn , we define Ω k ( P, Q ) = (cid:2) A k B k (cid:3) T ( P − Q ) (cid:2) A k B k (cid:3) − L T k P L k , Ω k ( P, Q ) = Ω k ( P, Q ) + (cid:2) A k B k (cid:3) T Q (cid:2) A k B k (cid:3) , (15)which are linear functions of P, Q and, moreover, Ω k ( P, Q ) is independent of (
A, B ) . Theorem 1
If there exist positive definite matrices P k , Q k ∈ R kn × kn such thatΩ k ( P k , Q k ) < , (16) then system (3) is strongly stable. Moreover, if system (3) is strongly stable, there exists an integer k ∗ ≥ ,such that (16) is solvable with P k > , Q k > , ∀ k ≥ k ∗ . roof. Let z k = z k,k ... z k, z k, = (cid:0) e j θ I kn − A k (cid:1) − B k , (17)which is equivalent to e j θ I n − I n · · · j θ I n . . . . . . .... . . − I n j θ I n − I n e j θ I n z k,k z k,k − ... z k, z k, = I n . Solving this equation recursively from the bottom to the up gives z k = e − k j θ I n ...e − θ I n e − j θ I n . (18)With this we get from (12), (13) and (14) that (cid:2) A k B k (cid:3) (cid:20) (cid:0) e j θ I kn − A k (cid:1) − B k I n (cid:21) = (cid:2) A k B k (cid:3) (cid:20) z k I n (cid:21) = e − j( k − θ I n ...e − j θ I n I n = e j θ z k , (19) (cid:2) A k B k (cid:3) (cid:20) (cid:0) e j θ I kn − A k (cid:1) − B k I n (cid:21) = (cid:2) A k B k (cid:3) (cid:20) z k I n (cid:21) = e − j( k − θ ∆ θ ...e − j θ ∆ θ ∆ θ = e j θ z k ∆ θ , (20)and L k (cid:20) (cid:0) e j θ I kn − A k (cid:1) − B k I n (cid:21) = (cid:0) e j θ I kn − A k (cid:1) − B k = z k . (21)Therefore, we can obtain (cid:20) z k I n (cid:21) H Ω k ( P k , Q k ) (cid:20) z k I n (cid:21) = (cid:20) z k I n (cid:21) H (cid:16)(cid:2) A k B k (cid:3) T P k (cid:2) A k B k (cid:3) − L T k P k L k (cid:17) (cid:20) z k I n (cid:21) + (cid:20) z k I n (cid:21) H (cid:2) A k B k (cid:3) T Q k (cid:2) A k B k (cid:3) (cid:20) z k I n (cid:21) − (cid:20) z k I n (cid:21) H (cid:2) A k B k (cid:3) T Q k (cid:2) A k B k (cid:3) (cid:20) z k I n (cid:21) = (cid:0) e j θ z k ∆ θ (cid:1) H Q k e j θ z k ∆ θ − (cid:0) e j θ z k (cid:1) H Q k e j θ z k + (cid:0) e j θ z k (cid:1) H P k e j θ z k − z H k P k z k =∆ H θ z H k Q k z k ∆ θ − z H k Q k z k < , (22)which implies (4) since z H k Q k z k > . We next prove the converse. By Lemma 11 in Appendix A2, we know that there exists a k ∗ ≥ (cid:0) ∆ kθ (cid:1) H ∆ kθ < I n , ∀ θ ∈ [0 , π ] , k ≥ k ∗ . (23)4enote Q ∗ k = W T k W k where (see the notation in Appendix A2) W k = B [ k − B [ k − A [1] · · · B [1] A [ k − A [ k − B [ k − B [ k − A [1] . . . A [ k − . . . . . . ... B [1] A [1] I n . (24)It follows that Q ∗ k ≥ Q ∗ k > B is nonsingular. For any integer i ≥ , by the binomialexpansion theorem, we have (cid:0) A + B e − j θ (cid:1) i = A [ i ] + B [1] A [ i − e − j θ + B [2] A [ i − e − θ + · · · + B [ i − A [1] e − ( i − θ + B [ i ] e − i j θ = (cid:2) B [ i ] B [ i − A [1] · · · B [1] A [ i − A [ i ] (cid:3) e − i j θ I n e − ( i − θ I n ...e − j θ I n I n . It follows that W k e j θ z k = W k e − ( k − θ I n e − ( k − θ I n ...e − j θ I n I n = ∆ k − θ ∆ k − θ ...∆ θ I n . Let Θ k ( Q ) = (cid:2) A k B k (cid:3) T Q (cid:2) A k B k (cid:3) − (cid:2) A k B k (cid:3) T Q (cid:2) A k B k (cid:3) . We then have from (23) and equations (19) and (20) that (cid:20) z k I n (cid:21) H Θ k ( Q ∗ k ) (cid:20) z k I n (cid:21) = (cid:0) e j θ z k ∆ θ (cid:1) H Q ∗ k e j θ z k ∆ θ − (cid:0) e j θ z k (cid:1) H Q ∗ k e j θ z k =∆ H θ (cid:16)(cid:0) W k e j θ z k (cid:1) H W k e j θ z k (cid:17) ∆ θ − (cid:0) W k e j θ z k (cid:1) H W k e j θ z k =∆ H θ ∆ k − θ ...∆ θ I n H ∆ k − θ ...∆ θ I n ∆ θ − ∆ k − θ ...∆ θ I n H ∆ k − θ ...∆ θ I n = (cid:0) ∆ kθ (cid:1) H ∆ kθ − I n < . As A k is Schur stable, by the YKP lemma in Appendix A2, the above inequality holds true if and only ifthere exists a symmetric matrix P ∗ k ∈ R kn × kn such that0 > (cid:20) A T k P ∗ k A k − P ∗ k A T k P ∗ k B k B T k P ∗ k A k B T k P ∗ k B k (cid:21) + Θ k ( Q ∗ k )= (cid:2) A k B k (cid:3) T P ∗ k (cid:2) A k B k (cid:3) − L T k P ∗ k L k + (cid:2) A k B k (cid:3) T Q ∗ k (cid:2) A k B k (cid:3) − (cid:2) A k B k (cid:3) T Q ∗ k (cid:2) A k B k (cid:3) = Ω k ( P ∗ k , Q ∗ k ) . (25)By comparing (25) with (15), we know that the LMI in (16) is feasible with ( P k , Q k ) = ( P ∗ k , Q ∗ k ) . In thefollowing, we will show that P ∗ k >
0. 5traightforward computation gives that W k (cid:2) A k B k (cid:3) = B [ k ] B [ k − A [1] · · · B [1] A [ k − A [ k ] B [ k − B [ k − A [1] . . . A [ k − . . . . . . ... B [2] B [1] A [1] A [2] B [1] A [1] , and W k (cid:2) A k B k (cid:3) = B [ k − B [ k − A [1] · · · B [1] A [ k − A [ k − B [ k − B [ k − A [1] . . . A [ k − B [1] A [1] I n . It follows that we can write W k A k = (cid:20) ( k − n × n U k n × n n × ( k − n (cid:21) , W k B k = A [ k − ... A [1] I n ,W k A k = (cid:20) B [ k ] V k ( k − n × n U k (cid:21) , W k B k = A [ k ] ... A [2] A [1] , where U k = B [ k − B [ k − A [1] · · · B [1] A [ k − . . . . . . ... B [2] B [1] A [1] B [1] ,V k = (cid:2) B [ k − A [1] B [ k − A [2] · · · B [1] A [ k − (cid:3) . We also denote C k = (cid:2) B [ k ] V k (cid:3) = (cid:2) B [ k ] B [ k − A [1] · · · B [1] A [ k − (cid:3) ∈ R n × kn ,D k = A [ k ] ∈ R n × n . Then, by straightforward computations, we obtain A T k W T k W k A k − A T k W T k W k A k = " (cid:0) B [ k ] (cid:1) T B [ k ] (cid:0) B [ k ] (cid:1) T V k V T k B [ k ] V T k V k + U T k U k − (cid:20) n × n n × ( k − n ( k − n × n U T k U k (cid:21) = " (cid:0) B [ k ] (cid:1) T B [ k ] (cid:0) B [ k ] (cid:1) T V k V T k B [ k ] V T k V k = C T k C k . Similarly, we have A T k W T k W k B k − A T k W T k W k B k = (cid:0) B [ k ] (cid:1) T A [ k ] V T k A [ k ] + U T k A [ k − ... A [1] − n × n U T k A [ k − ... A [1] " (cid:0) B [ k ] (cid:1) T A [ k ] V T k A [ k ] = C T k D k , and B T k W T k W k B k − B T k W T k W k B k = (cid:16) A [ k ] (cid:17) T A [ k ] − I n = D T k D k − I n . Therefore, we can get Ω k ( P ∗ k , Q ∗ k ) = (cid:20) A T k P ∗ k A k − P ∗ k + C T k C k A T k P k B k + C T k D k B T k P ∗ k A k + D T k C k B T k P ∗ k B k + D T k D k − I n (cid:21) , (26)which, together with (25), implies that A T k P ∗ k A k − P ∗ k + C T k C k < . As A k is Schur stable, the above equation implies P ∗ k > . By now we have shown that, if B is nonsingular, the LMI in (16) is solvable with positive definite matrices P ∗ k and Q ∗ k = W T k W k . However, if B is singular, the matrix Q ∗ k = W T k W k is only semi-positive definite. Inthe following, we will show that the LMI in (16) is also feasible with ( P k , Q k ) = ( P ∗ k , Q ∗ k + εI kn ) where ε > Ω k ( P ∗ k , Q ∗ k + εI kn ) < . (27)In fact, it follows from (15) that Ω k ( P ∗ k , Q ∗ k + εI kn ) = Ω k ( P ∗ k , Q ∗ k ) + Ω k (0 kn × kn , εI kn )= Ω k ( P ∗ k , Q ∗ k ) + ε (cid:16)(cid:2) A k B k (cid:3) T (cid:2) A k B k (cid:3) − (cid:2) A k B k (cid:3) T (cid:2) A k B k (cid:3)(cid:17) ≤ Ω k ( P ∗ k , Q ∗ k ) + ε (cid:16)(cid:13)(cid:13)(cid:2) A k B k (cid:3)(cid:13)(cid:13) + (cid:13)(cid:13)(cid:2) A k B k (cid:3)(cid:13)(cid:13) (cid:17) . Since Ω k ( P ∗ k , Q ∗ k ) is independent of ε and satisfies (25), there exists a sufficiently small ε > (cid:20) Ω k ( P k , Q k ) [ A k , B k ] T Q k Q k [ A k , B k ] − Q k (cid:21) < , whose left hand side is a linear function of ( A, B ) . Thus, the most important feature of Theorem 1, whencompared with the results in [2] (see Lemma 3), the result in [6] (see Lemma 2) and the method in [13],is that the coefficient (
A, B ) appears as a linear function. Such a property is helpful for solving the robuststability analysis problem, as made clear below.Consider the perturbed system of (3) x ( t ) = ( A + ∆ A ) x ( t − a ) + ( B + ∆ B ) x ( t − b ) , (28)where A ∈ R n × n and B ∈ R n × n are the same as that in (3) and (cid:2) ∆ B ∆ A (cid:3) = E F (cid:2) B A (cid:3) , (29)where E ∈ R n × p , B ∈ R q × n , A ∈ R q × n are known matrices, and F ∈ R p × q denotes the norm boundeduncertainty (which can be time-varying) that satisfies F T F ≤ I q . (30)For further using, we denote (cid:2) A k B k (cid:3) = B A · · ·
0. . . . . . . . . ... B A B A ∈ R kq × ( k +1) n . heorem 2 The uncertain linear difference equation (28) is exponentially stable for any F ∈ R p × q satisfying(30) if there exists an integer k ≥ , positive definite matrices P k , Q k ∈ R kn × kn and a positive definite matrix S k ∈ R k × k such that the following LMI is satisfied: (cid:20) Ω k ( P k , Q k ) + (cid:2) A k B k (cid:3) T ( S k ⊗ I q ) (cid:2) A k B k (cid:3) (cid:2) A k B k (cid:3) T Q k ( I k ⊗ E ) (cid:0) I k ⊗ E T0 (cid:1) Q k (cid:2) A k B k (cid:3) (cid:0) I k ⊗ E T0 (cid:1) Q k ( I k ⊗ E ) − S k ⊗ I p (cid:21) < . (31) Proof.
For notation simplicity, we denote C k = (cid:2) A k B k (cid:3) , C k = (cid:2) A k B k (cid:3) , Ω k = Ω k ( P k , Q k ) , and Ω k = Ω k ( P k , Q k ) . Notice that we can write0 > (cid:20) Ω k + C T k ( S k ⊗ I q ) C k C T k Q k ( I k ⊗ E ) (cid:0) I k ⊗ E T0 (cid:1) Q k C k (cid:0) I k ⊗ E T0 (cid:1) Q k ( I k ⊗ E ) − S k ⊗ I p (cid:21) = (cid:20) Ω k + C T k Q k C k + C T k ( S k ⊗ I q ) C k C T k Q k ( I k ⊗ E ) (cid:0) I k ⊗ E T0 (cid:1) Q k C k (cid:0) I k ⊗ E T0 (cid:1) Q k ( I k ⊗ E ) − S k ⊗ I p (cid:21) = (cid:20) Ω k + C T k ( S k ⊗ I q ) C k ( k +1) n × kp kp × ( k +1) n − S k ⊗ I p (cid:21) + (cid:20) C T k Q k (cid:0) I k ⊗ E T0 (cid:1) Q k (cid:21) Q − k (cid:20) C T k Q k (cid:0) I k ⊗ E T0 (cid:1) Q k (cid:21) T , which, by a Schur complement, is equivalent to Ω k + C T k ( S k ⊗ I q ) C k ( k +1) n × kp C T k Q k kp × ( k +1) n − S k ⊗ I p (cid:0) I k ⊗ E T0 (cid:1) Q k Q k C k Q k ( I k ⊗ E ) − Q k < . By a congruence transformation, this is equivalent to Ω k + C T k ( S k ⊗ I q ) C k C T k Q k ( k +1) n × kp Q k C k − Q k Q k ( I k ⊗ E )0 kp × ( k +1) n (cid:0) I k ⊗ E T0 (cid:1) Q k − S k ⊗ I p < . By a Schur complement, the above inequality holds true if and only if0 > (cid:20) Ω k C T k Q k Q k C k − Q k (cid:21) + (cid:20) ( k +1) n × kp Q k ( I k ⊗ E ) (cid:21) (cid:0) S − k ⊗ I p (cid:1) (cid:20) ( k +1) n × kp Q k ( I k ⊗ E ) (cid:21) T + (cid:2) C k kq × kn (cid:3) T ( S k ⊗ I q ) (cid:2) C k kq × kn (cid:3) . (32)By (29) we have ∆ C k , ∆ B ∆ A · · · B ∆ A . . . ... .... . . . . . 0 0∆ B ∆ A B ∆ A = E F B E F A · · · E F B E F A . . . ... 0. . . . . . 0 ... E F B E F A E F B E F A = ( I k ⊗ E ) ( I k ⊗ F ) C k . By using (30) we can compute (cid:0) I k ⊗ F T (cid:1) ( S k ⊗ I p ) ( I k ⊗ F ) = S k ⊗ F T F ≤ S k ⊗ I q . (cid:20) Ω k ( C k + ∆ C k ) T Q k Q k ( C k + ∆ C k ) − Q k (cid:21) = (cid:20) Ω k C T k Q k Q k C k − Q k (cid:21) + (cid:20) ( k +1) n × ( k +1) n ∆ C T k Q k Q k ∆ C k kn × kn (cid:21) = (cid:20) Ω k C T k Q k Q k C k − Q k (cid:21) + (cid:20) ( k +1) n × kp Q k ( I k ⊗ E ) (cid:21) ( I k ⊗ F ) (cid:2) C k kq × kn (cid:3) + (cid:2) C k kq × kn (cid:3) T (cid:0) I k ⊗ F T (cid:1) (cid:20) ( k +1) n × kp Q k ( I k ⊗ E ) (cid:21) T ≤ (cid:20) Ω k C T k Q k Q k C k − Q k (cid:21) + (cid:20) ( k +1) n × kp Q k ( I k ⊗ E ) (cid:21) (cid:0) S − k ⊗ I p (cid:1) (cid:20) ( k +1) n × kp Q k ( I k ⊗ E ) (cid:21) T + (cid:2) C k kq × kn (cid:3) T (cid:0) I k ⊗ F T (cid:1) ( S k ⊗ I p ) ( I k ⊗ F ) (cid:2) C k kq × kn (cid:3) ≤ (cid:20) Ω k C T k Q k Q k C k − Q k (cid:21) + (cid:20) ( k +1) n × kp Q k ( I k ⊗ E ) (cid:21) (cid:0) S − k ⊗ I p (cid:1) (cid:20) ( k +1) n × kp Q k ( I k ⊗ E ) (cid:21) T + (cid:2) C k kq × kn (cid:3) T ( S k ⊗ I q ) (cid:2) C k kq × kn (cid:3) < . (33)By a Schur complement, the above inequality is equivalent to0 > Ω k + ( C k + ∆ C k ) T Q k ( C k + ∆ C k )= (cid:2) A k B k (cid:3) T ( P k − Q k ) (cid:2) A k B k (cid:3) − L T k P k L k + (cid:2) A k + ∆ A k B k + ∆ B k (cid:3) T Q k (cid:2) A k + ∆ A k B k + ∆ B k (cid:3) . By Theorem 1, we know that system (28) is exponentially stable. The proof is finished.The merit of the proof of Theorem 2 is that we have utilized the fact that (
A, B ) appears as a linear functionin the LMIs, which helps to eliminate the uncertain matrix F in the LMI (16). This can not be achieved forthe LMI in Lemmas 2 and 3. Moreover, from the proof we can see that the only conservatism comes fromthe usage of the inequality in Lemma 12. Thus the condition in Theorem 2 is considered to be quite tight.By using again the property that ( A, B ) appears in the matrix Ω k ( P k , Q k ) as a quadratic function, we canextend easily the results in Theorem 2 to the case of polytopic type uncertainty, say, (cid:2) ∆ A ∆ B (cid:3) ∈ co (cid:8)(cid:2) A ( i ) B ( i ) (cid:3) , i = 1 , , . . . , N (cid:9) , (34)where A ( i ) , B ( i ) , i = 1 , , . . . , N are given matrices. Denote h A ( i ) k B ( i ) k i = B + B ( i ) A + A ( i ) · · · B + B ( i ) A + A ( i ) B + B ( i ) A + A ( i ) ∈ R kn × ( k +1) n . Then we obtain immediately the following theorem.
Theorem 3
The uncertain linear difference equation (28), where ∆ A and ∆ B satisfy (34), is exponentiallystable if there exists positive definite matrices P k , Q k ∈ R kn × kn such thatΩ ( i ) k ( P k , Q k ) = Ω k ( P k , Q k ) + h A ( i ) k B ( i ) k i T Q k h A ( i ) k B ( i ) k i < , (35) are satisfied for i = 1 , , . . . , N. roof. Notice that (35) implies Ω k ( P k , Q k ) h A ( i ) k B ( i ) k i T Q k Q k h A ( i ) k B ( i ) k i − Q k < , where i = 1 , , . . . , N . It follows that, for any α i ≥ , i = 1 , , . . . , N with α + α + · · · + α N = 1 , and (cid:2) ∆ A ∆ B (cid:3) = N X i =1 α i (cid:2) A ( i ) B ( i ) (cid:3) , we have 0 > N P i =1 α i Ω k ( P k , Q k ) N P i =1 α i h A ( i ) k B ( i ) k i T Q k Q k N P i =1 α i h A ( i ) k B ( i ) k i − N P i =1 α i Q k = (cid:20) Ω k ( P k , Q k ) ( C k + ∆ C k ) T Q k Q k ( C k + ∆ C k ) − Q k (cid:21) , which is exactly in the form of (33). The remaining of the proof is similar to that of Theorem 2 and isomitted. We first provide time-domain interpretations of Theorem 1 and Lemma 3 by establishing LKFs.
Lemma 4
For any integer k ≥ , there holds x ( t ) = k X i =0 A [ i ] B [ k − i ] x ( t − ia − ( k − i ) b ) . (36) Proof.
Clearly, it follows from (3) that (36) holds true with k = 1 . Assume that (36) is true with k = m, namely, x ( t ) = m X i =0 A [ i ] B [ m − i ] x ( t − ia − ( m − i ) b ) . (37)Then, by inserting (3) into (37), we have x ( t ) = m X i =0 A [ i ] B [ m − i ] ( Ax ( t − ( i + 1) a − ( m − i ) b ) + Bx ( t − ia − ( m + 1 − i ) b ))= m X i =0 A [ i ] B [ m − i ] Bx ( t − ia − ( m + 1 − i ) b ) + m X i =0 A [ i ] B [ m − i ] Ax ( t − ( i + 1) a − ( m − i ) b )= m X i =0 A [ i ] B [ m − i ] Bx ( t − ia − ( m + 1 − i ) b ) + m +1 X j =1 A [ j − B [ m +1 − j ] Ax ( t − ja − ( m + 1 − j ) b )= A [0] B [ m ] Bx ( t − ( m + 1) b ) + m X i =1 A [ i ] B [ m − i ] Bx ( t − ia − ( m + 1 − i ) b )+ m X j =1 A [ j − B [ m +1 − j ] Ax ( t − ja − ( m + 1 − j ) b ) + A [ m ] B [0] Ax ( t − ( m + 1) a )= B [ m +1] x ( t − ( m + 1) b ) + A [ m +1] x ( t − ( m + 1) a )10 m X i =1 (cid:16) A [ i ] B [ m − i ] B + A [ i − B [ m +1 − i ] A (cid:17) x ( t − ia − ( m + 1 − i ) b ) . (38)Notice that (see (80) in Appendix A2) A [ i ] B [ m − i ] B + A [ i − B [ m +1 − i ] A = A [ i ] B [ m +1 − i ] , i = 1 , , . . . , m, substitution of which into (38) gives x ( t ) = B [ m +1] x ( t − ( m + 1) b ) + m X i =1 A [ i ] B [ m +1 − i ] x ( t − ia − ( m + 1 − i ) b ) + A [ m +1] x ( t − ( m + 1) a )= m +1 X i =0 A [ i ] B [ m +1 − i ] x ( t − ia − ( m + 1 − i ) b ) . Therefore, (36) holds with k = m + 1 . The proof is finished by mathematical induction.In the following, we assume, without loss of generality, that b > a since otherwise we can change the roles of a and b. Lemma 5
For any integer k ≥ , let X k ( t ) = x ( t − kb ) x ( t − ( k − b − a ) ... x ( t − b − ( k − a ) x ( t − b − ( k − a ) ∈ R kn ,U k ( t ) = x ( t − ka ) ∈ R n ,Y k ( t ) = x ( t ) ∈ R n . (39) Then ( U k ( t ) , X k ( t ) , Y k ( t )) satisfies X k ( t + b − a ) = A k X k ( t ) + B k U k ( t ) ,Y k ( t ) = C k X k ( t ) + D k U k ( t ) . (cid:27) (40) Proof.
This can be verified by direct computation. In fact, by definition, we have X k ( t + b − a ) = x ( t − ( k − b − a ) x ( t − ( k − b − a )... x ( t − b − ( k − a ) x ( t − ka ) = A k X k ( t ) + B k U ( t ) , and it follows from Lemma 4 that Y k ( t ) = k X i =0 A [ i ] B [ k − i ] x ( t − ia − ( k − i ) b )= (cid:2) B [ k ] A [1] B [ k − · · · A [ k − B [1] (cid:3) X k ( t ) + A [ k ] x ( t − ka )= C k X k ( t ) + D k U k ( t ) . The proof is finished.We next provide a time-domain interpretation of Theorem 1 by establishing an LKF for the system.11 roposition 1
For any integer k ≥ , let Ω k ( P, Q ) be defined by (15) where ( A k , B k , A k , B k ) is defined by(12)-(14). Consider the following LKF V k ( x t ) = Z t − at − b X T k ( s ) P k X k ( s ) d s + Z tt − a X T k ( s ) Q k X k ( s ) d s, (41) where P k = P T k ∈ R kn × kn and Q k = Q T k ∈ R kn × kn . Then ˙ V k ( x t ) = (cid:20) X k ( t − b ) x ( t − b − ka ) (cid:21) T Ω k ( P k , Q k ) (cid:20) X k ( t − b ) x ( t − b − ka ) (cid:21) . (42) Proof.
From (39) and (40) we know U k ( t + a − b ) = x ( t − ( k − a − b ) and X k ( t − a ) = A k X k ( t − b ) + B k x ( t − ka − b ) . By using (3) and noting the structures of A k and B k , we have X k ( t ) = x ( t − kb ) x ( t − ( k − b − a )... x ( t − b − ( k − a ) x ( t − b − ( k − a ) = B A · · · B A . . . ... 0. . . . . . 0 ...
B A B A x ( t − ( k + 1) b ) x ( t − kb − a )... x ( t − b − ( k − a ) x ( t − b − ka ) = (cid:2) A k B k (cid:3) (cid:20) X k ( t − b ) x ( t − b − ka ) (cid:21) . Therefore, it follows from (47) that˙ V k ( x t ) = X T k ( t − a ) P k X k ( t − a ) − X T k ( t − b ) P k X k ( t − b )+ X T k ( t ) Q k X k ( t ) − X T k ( t − a ) Q k X k ( t − a )= ( A k X k ( t − b ) + B k x ( t − b − ka )) T ( P k − Q k ) ( A k X k ( t − b ) + B k x ( t − b − ka ))+ X T k ( t ) Q k X k ( t ) − X T k ( t − b ) P k X k ( t − b )= (cid:20) X k ( t − b ) x ( t − b − ka ) (cid:21) T (cid:2) A k B k (cid:3) T ( P k − Q k ) (cid:2) A k B k (cid:3) (cid:20) X k ( t − b ) x ( t − b − ka ) (cid:21) − (cid:20) X k ( t − b ) x ( t − b − ka ) (cid:21) T (cid:2) I kn kn × n (cid:3) T P k (cid:2) I kn kn × n (cid:3) (cid:20) X k ( t − b ) x ( t − b − ka ) (cid:21) + (cid:20) X k ( t − b ) x ( t − b − ka ) (cid:21) T (cid:2) A k B k (cid:3) T Q k (cid:2) A k B k (cid:3) (cid:20) X k ( t − b ) x ( t − b − ka ) (cid:21) = (cid:20) X k ( t − b ) x ( t − b − ka ) (cid:21) T Ω k ( P k , Q k ) (cid:20) X k ( t − b ) x ( t − b − ka ) (cid:21) . The proof is finished.Similar to Lemma 5, we can present the following lemma.
Lemma 6
For any integer k ≥ , let A k , B k be defined in (8) and C k = (cid:2) B AB · · · A k − B (cid:3) ∈ R n × kn , D k = A k ∈ R n × n . (43)12 et X k ( t ) = x ( t ) x ( t − a ) ... x ( t − ( k − a ) ∈ R kn ,U k ( t ) = x ( t + b − ka ) ∈ R n ,Y k ( t ) = x ( t + b ) ∈ R n . (44) Then (cid:0) U k ( t ) , X k ( t ) , Y k ( t ) (cid:1) satisfies X k ( t + b − a ) = A k X k ( t ) + B k U k ( t ) ,Y k ( t ) = C k X k ( t ) + D k U k ( t ) . (cid:27) (45) Proof.
It is straightforward to see that, for any i = 0 , , . . . , k − ,x ( t − ia ) = Bx ( t − ia − b ) + Ax ( t − ( i + 1) a )= Bx ( t − ia − b ) + A ( Bx ( t − ( i + 1) a − b ) + Ax ( t − ( i + 2) a ))= Bx ( t − ia − b ) + ABx ( t − ( i + 1) a − b ) + A x ( t − ( i + 2) a )= · · · = Bx ( t − ia − b ) + ABx ( t − ( i + 1) a − b ) + · · · + A k − i − Bx ( t − ( k − a − b ) + A k − i x ( t − ka ) . (46)For i = 1 , , . . . , k − , we write the above k − x ( t − a ) x ( t − a )... x ( t − ( k − a ) x ( t − ka ) = B AB · · · A k − B B · · · A k − B . . . . . . ...0 B x ( t − b ) x ( t − a − b )... x ( t − ( k − a − b ) x ( t − ( k − a − b ) + A k − A k − ... AI n x ( t − ka ) , which can be written as X k ( t − a ) = A k X k ( t − b ) + B k x ( t − ka ) , (47)which is just the first equation (45). On the other hand, with i = 0 in (46), we have x ( t ) = Bx ( t − b ) + ABx ( t − a − b ) + · · · + A k − Bx ( t − ( k − a − b ) + A k x ( t − ka )= (cid:2) B AB · · · A k − B (cid:3) X k ( t − b ) + A k x ( t − ka )= C k X k ( t − b ) + D k x ( t − ka ) , which is just the second equation in (45). The proof is finished.We then can present for Lemma 3 a time-domain interpretation, which parallels Proposition 1. Proposition 2
For any integer k ≥ , let Ω k be defined in (10). Consider the following LKF V k ( x t ) = Z tt − a X T k ( s ) Q k X k ( s ) d s + Z t − at − b X T k ( s ) P k X k ( s ) d s, (48) where P k = P T k ∈ R kn × kn and Q k = Q T k ∈ R kn × kn . Then ˙ V k ( x t ) = (cid:20) X k ( t − b ) x ( t − ka ) (cid:21) T Ω k (cid:0) P k , Q k (cid:1) (cid:20) X k ( t − b ) x ( t − ka ) (cid:21) . (49) Proof.
By using (46) and noting the structures of A k and B k in (9), we have X k ( t ) = B AB A B · · · A k − B A k B AB . . . A k − B A k − . . . . . . ... ... B AB A B A x ( t − b ) x ( t − a − b )... x ( t − ( k − a − b ) x ( t − ka ) (cid:2) A k B k (cid:3) (cid:20) X k ( t − b ) x ( t − ka ) (cid:21) . Therefore, it follows from (47) that˙ V k ( x t ) = X T k ( t − a ) P k X k ( t − a ) − X T k ( t − b ) P k X k ( t − b )+ X T k ( t ) Q k X k ( t ) − X T k ( t − a ) Q k X k ( t − a )= (cid:0) A k X k ( t − b ) + B k x ( t − ka ) (cid:1) T (cid:0) P k − Q k (cid:1) (cid:0) A k X k ( t − b ) + B k x ( t − ka ) (cid:1) + X T k ( t ) Q k X k ( t ) − X T k ( t − b ) P k X k ( t − b )= (cid:20) X k ( t − b ) x ( t − ka ) (cid:21) T (cid:2) A k B k (cid:3) T (cid:0) P k − Q k (cid:1) (cid:2) A k B k (cid:3) (cid:20) X k ( t − b ) x ( t − ka ) (cid:21) − (cid:20) X k ( t − b ) x ( t − ka ) (cid:21) T (cid:2) I kn kn × n (cid:3) T P k (cid:2) I kn kn × n (cid:3) (cid:20) X k ( t − b ) x ( t − ka ) (cid:21) + (cid:20) X k ( t − b ) x ( t − ka ) (cid:21) T (cid:2) A k B k (cid:3) T Q k (cid:2) A k B k (cid:3) (cid:20) X k ( t − b ) x ( t − ka ) (cid:21) = (cid:20) X k ( t − b ) x ( t − ka ) (cid:21) T Ω k (cid:0) P k , Q k (cid:1) (cid:20) X k ( t − b ) x ( t − ka ) (cid:21) , which completes the proof.One may wonder the relationship between Theorem 1 and Lemma 3. Such a relationship should be revealedfrom the time-domain interpretations of these two LMIs. To investigate this problem, we need to find therelationship between Ω k and Ω k . Such a relationship should be revealed from the time-domain interpretationsof these two LMIs, say, the relationship between X k ( t ) and X k ( t ) , and the relationship between (cid:20) X k ( t − b ) x ( t − b − ka ) (cid:21) and (cid:20) X k ( t − b ) x ( t − ka ) (cid:21) . To this end, we denote, for any integer k ≥ ,W k = B [ k − B [ k − A [1] · · · A [ k − . . . . . . ... B [1] A [1] I n , T k = (cid:20) W k I n (cid:21) . Then we have the following result.
Proposition 3
Let Ω k ( P k , Q k ) and Ω k (cid:0) P k , Q k (cid:1) be defined respectively in (15) and (10). Let P k = W T k P k W k , Q k = W T k Q k W k . (50) Then there holds Ω k ( P k , Q k ) = T T k Ω k (cid:0) P k , Q k (cid:1) T k . (51) Therefore, the LMI in (11) is feasible if and only if the LMI in (16) is feasible.
Proof.
By using Lemma 6 we have X k ( t − b ) = x ( t − b ) x ( t − a − b )... x ( t − ( k − a − b ) = B [ k − B [ k − A [1] · · · A [ k − . . . . . . ... B [1] A [1] I n x ( t − kb ) x ( t − ( k − b − a )... x ( t − b − ( k − a ) x ( t − b − ( k − a ) W k X k ( t ) , from which we get (cid:20) X k ( t − b ) x ( t − b − ka ) (cid:21) = (cid:20) W k X k ( t − b ) x ( t − b − ka ) (cid:21) = (cid:20) W k I n (cid:21) (cid:20) X k ( t − b ) x ( t − b − ka ) (cid:21) = T k (cid:20) X k ( t − b ) x ( t − b − ka ) (cid:21) . Therefore, we have from (48) that V k ( x t − b ) = Z tt − a X T k ( s − b ) Q k X k ( s − b ) d s + Z t − at − b X T k ( s − b ) P k X k ( s − b ) d s, = Z tt − a X T k ( s ) W T k Q k W k X k ( s ) d s + Z t − at − b X T k ( s ) W T k P k W k X k ( s ) d s, (52)and from (49) that ˙ V k ( x t − b ) = (cid:20) X k ( t − b ) x ( t − b − ka ) (cid:21) T Ω k (cid:0) P k , Q k (cid:1) (cid:20) X k ( t − b ) x ( t − b − ka ) (cid:21) = (cid:20) X k ( t − b ) x ( t − b − ka ) (cid:21) T T T k Ω k (cid:0) P k , Q k (cid:1) T k (cid:20) X k ( t − b ) x ( t − b − ka ) (cid:21) . (53)By comparing (52) and (53) with (41) and (42) we know that, if ( P k , Q k ) satisfies (50), then Ω k and Ω k satisfies (51). The proof is finished.It follows that Theorem 1 is equivalent to Lemma 3. Even so, Theorem 1 possesses great advantage overLemma 3 since the system parameters appear linearly (quadratically) in the LMIs (16), which has been veryimportant in the robust stability analysis. We next show the connection to the Carvalho Condition. Lemma 7 [3] The linear difference equation (3) is exponentially stable if there exist two positive definitematrices X ∈ R n × n and Y ∈ R n × n such that the following LMI is satisfiedΦ ( X , Y ) = (cid:20) A BI n (cid:21) T (cid:20) X Y (cid:21) (cid:20) A BI n (cid:21) − (cid:20) X Y (cid:21) < . (54) Proof.
For future use, we give a simple proof here. Choose the following LK functional W ( x t ) = Z tt − a x T ( s ) X x ( s ) d s + Z t − at − b x T ( s ) Y x ( s ) d s, (55)which is such that˙ W ( x t ) = x T ( t ) X x ( t ) − x T ( t − a ) X x ( t − a ) + x T ( t − a ) Y x ( t − a ) − x T ( t − b ) Y x ( t − b )= (cid:20) x ( t − a ) x ( t − b ) (cid:21) T Φ ( X , Y ) (cid:20) x ( t − a ) x ( t − b ) (cid:21) . (56)Since Φ ( X , Y ) < , the stability follows from the Lyapunov stability theorem [3].If we set k = 1 in Theorem 1 and denote E = (cid:20) I n I n (cid:21) , E = I n I n I n , we obtain the following result. 15 emma 8 Let Ω k be defined in (15), Ω k be defined in (10) and Φ be defined in (54). Then, for k = 1 ,there holds Ω ( P , Q ) = E T2 Φ ( Q , P ) E , (57) Ω (cid:0) P , Q (cid:1) = E T2 Φ (cid:0) Q , P (cid:1) E . (58) Thus the result in Lemma 7 [3] is a special case of Lemma 3 and Theorem 1.
Proof.
Let k = 1 . Then it follows from (41) that V ( x t + b ) = Z t − at − b X T1 ( s + b ) P X ( s + b ) d s + Z tt − a X T1 ( s + b ) Q X ( s + b ) d s = Z t − at − b x T ( s ) P x ( s ) d s + Z tt − a x T ( s ) Q x ( s ) d s, (59)and from (42) that ˙ V ( x t + b ) = (cid:20) x ( t − b ) x ( t − a ) (cid:21) T Ω ( P , Q ) (cid:20) x ( t − b ) x ( t − a ) (cid:21) = (cid:20) x ( t − a ) x ( t − b ) (cid:21) T E T2 Ω ( P , Q ) E (cid:20) x ( t − a ) x ( t − b ) (cid:21) . (60)By comparing (59) and (60) with (55) and (56), respectively, we get (57).Similarly, we have from (48) that V ( x t ) = Z tt − a X T1 ( s ) Q X ( s ) d s + Z t − at − b X T1 ( s ) P X ( s ) d s = Z tt − a x T ( s ) Q x ( s ) d s + Z t − at − b x T ( s ) P x ( s ) d s, (61)and from (49) that ˙ V ( x t ) = (cid:20) x ( t − b ) x ( t − a ) (cid:21) T Ω (cid:0) P , Q (cid:1) (cid:20) x ( t − b ) x ( t − a ) (cid:21) = (cid:20) x ( t − a ) x ( t − b ) (cid:21) T E T2 Ω (cid:0) P , Q (cid:1) E (cid:20) x ( t − a ) x ( t − b ) (cid:21) . (62)By comparing (61) and (62) with (55) and (56), respectively, we get (57). The proof is finished.We next investigate the relationship between Theorem 1 and a result in [4]. To this end, we denote N = (cid:20) A B I n (cid:21) , N = (cid:20) I n I n (cid:21) ,M = (cid:20) A B
00 0 I n (cid:21) , M = (cid:20) I n I n (cid:21) . Lemma 9 [4] The linear difference equation (3) is exponentially stable if there exist four positive definitematrices X , Y ∈ R n × n , U , V ∈ R n × n , such that the following LMI is satisfiedΦ ( X ∗ , Y ∗ ) = N T21 X ∗ N − N T22 X ∗ N + M T21 Y ∗ M − M T22 Y ∗ M < , (63) where Y ∗ = Y + (cid:20) U + V
00 0 n × n (cid:21) > , (64) X ∗ = X + (cid:20) n × n V (cid:21) > . (65)16 roof. This lemma is a little different from the original result in [4] and thus a simple proof will be providedfor completeness (also for the purpose of further using). Choose a more general LKF candidate as [4] (wherewe have assumed without loss of generality that µ = 0) W ( x t ) = Z tt − a x T ( s ) U x ( s ) d s + Z tt − b x T ( s ) V x ( s ) d s, + Z tt − c (cid:20) x ( s ) x ( s − a ) (cid:21) T X (cid:20) x ( s ) x ( s − a ) (cid:21) d s, + Z t − ct − b (cid:20) x ( s + c ) x ( s ) (cid:21) T Y (cid:20) x ( s + c ) x ( s ) (cid:21) d s, where c = b − a . It can be verified that Z tt − a x T ( s ) U x ( s ) d s + Z tt − b x T ( s ) V x ( s ) d s = Z tt − a x T ( s ) ( U + V ) x ( s ) d s + Z t − at − b x T ( s ) V x ( s ) d s = Z t − ct − b x T ( s + c ) ( U + V ) x ( s + c ) d s + Z tt − c x T ( s − a ) V x ( s − a ) d s, from which it follows that W ( x t ) = Z tt − c (cid:20) x ( s ) x ( s − a ) (cid:21) T X ∗ (cid:20) x ( s ) x ( s − a ) (cid:21) d s + Z t − ct − b (cid:20) x ( s + c ) x ( s ) (cid:21) T Y ∗ (cid:20) x ( s + c ) x ( s ) (cid:21) d s, (66)whose time-derivative can be evaluated as˙ W ( x t ) = (cid:20) x ( t ) x ( t − a ) (cid:21) T X ∗ (cid:20) x ( t ) x ( t − a ) (cid:21) − (cid:20) x ( t − c ) x ( t − b ) (cid:21) T X ∗ (cid:20) x ( t − c ) x ( t − b ) (cid:21) + (cid:20) x ( t ) x ( t − c ) (cid:21) T Y ∗ (cid:20) x ( t ) x ( t − c ) (cid:21) − (cid:20) x ( t − a ) x ( t − b ) (cid:21) T Y ∗ (cid:20) x ( t − a ) x ( t − b ) (cid:21) = ξ T2 ( t ) Φ ξ ( t ) , (67)where ξ ( t ) = [ x T ( t − a ) , x T ( t − b ) , x T ( t − c )] T . The result then follows again from the Lyapunov stabilitytheorem [3].The decision matrices U and V in (63) are in fact redundant, as shown in the following corollary. Corollary 1
There exist four positive definite matrices X , Y ∈ R n × n , U , V ∈ R n × n such that (63) issatisfied if and only if there exist two positive definite matrices X ∗ , Y ∗ ∈ R n × n such that (63) is satisfied. Proof. If X > , Y > , U > , V > , then it follows from (64)-(65) that X ∗ > , Y ∗ > . On theother hand, if X ∗ > , Y ∗ >
0, we can always find X > , Y > , U > , V > , satisfying (64)-(65), forexample, U = V = εI n , where ε > Proposition 4
Let ( P , Q ) , (cid:0) P , Q (cid:1) and ( X ∗ , Y ∗ ) be related with P = W T2 X ∗ W , Q = E T2 Y ∗ E , (68) X ∗ = P , Y ∗ = E T2 W T2 Q W E . (69) Then Ω ( P , Q ) and Φ ( X ∗ , Y ∗ ) satisfyΩ ( P , Q ) = T T3 E T3 Φ ( X ∗ , Y ∗ ) E T , (70)17 ( X ∗ , Y ∗ ) = E T3 Ω (cid:0) P , Q (cid:1) E . (71) Thus the result in Lemma 9 [4] is a special case of Lemma 3 and Theorem 1.
Proof.
Notice from (41) that V ( x t + b ) = Z t − at − b X T2 ( s + b ) P X ( s + b ) d s + Z tt − a X T2 ( s + b ) Q X ( s + b ) d s = Z t − at − b (cid:20) x ( s − b ) x ( s − a ) (cid:21) T P (cid:20) x ( s − b ) x ( s − a ) (cid:21) d s + Z tt − a (cid:20) x ( s − b ) x ( s − a ) (cid:21) T Q (cid:20) x ( s − b ) x ( s − a ) (cid:21) d s, (72)and from (42) that ˙ V ( x t + b ) = (cid:20) X ( t ) x ( t − a ) (cid:21) T Ω ( P , Q ) (cid:20) X ( t ) x ( t − a ) (cid:21) = x ( t − b ) x ( t − a − b ) x ( t − a ) T Ω ( P , Q ) x ( t − b ) x ( t − a − b ) x ( t − a ) . (73)On the other hand, we get from (66) that W ( x t ) = Z tt − ( b − a ) (cid:20) x ( s ) x ( s − a ) (cid:21) T X ∗ (cid:20) x ( s ) x ( s − a ) (cid:21) d s + Z t − ( b − a ) t − b (cid:20) x ( s + b − a ) x ( s ) (cid:21) T Y ∗ (cid:20) x ( s + b − a ) x ( s ) (cid:21) d s = Z t − at − b (cid:20) x ( s + a ) x ( s ) (cid:21) T X ∗ (cid:20) x ( s + a ) x ( s ) (cid:21) d s + Z tt − a (cid:20) x ( s ) x ( s − b + a ) (cid:21) T Y ∗ (cid:20) x ( s ) x ( s − b + a ) (cid:21) d s, from which we have W ( x t − a ) = Z t − at − b (cid:20) x ( s ) x ( s − a ) (cid:21) T X ∗ (cid:20) x ( s ) x ( s − a ) (cid:21) d s + Z tt − a (cid:20) x ( s − a ) x ( s − b ) (cid:21) T Y ∗ (cid:20) x ( s − a ) x ( s − b ) (cid:21) d s = Z t − at − b (cid:20) x ( s − b ) x ( s − a ) (cid:21) T W T2 X ∗ W (cid:20) x ( s − b ) x ( s − a ) (cid:21) d s + Z tt − a (cid:20) x ( s − b ) x ( s − a ) (cid:21) T E T2 Y ∗ E (cid:20) x ( s − b ) x ( s − a ) (cid:21) d s. (74)Moreover, from (67) we obtain˙ W ( x t − a ) = x ( t − a ) x ( t − a − b ) x ( t − b ) T Φ ( X ∗ , Y ∗ ) x ( t − a ) x ( t − a − b ) x ( t − b ) = x ( t − b ) x ( t − a − b ) x ( t − a ) T T T3 E T3 Φ ( X ∗ , Y ∗ ) E T x ( t − b ) x ( t − a − b ) x ( t − a ) . (75)Thus, by comparing (74) and (75) with (72) and (73) respectively, if (68) is satisfied, we obtain (70). Therelation (69) and (71) can be proven in a similar way. We consider the linear difference equation (3) with A ( α ) = (cid:20) − . − . . α . (cid:21) , B ( β ) = (cid:20) . . − . − . β (cid:21) , where α, β ∈ R are free parameters [23]. We look for the pair ( α, β ) such that system (3) is strongly stable.By a linear search technique, the regions of ( α, β ) obtained by different methods are plotted in Fig. 1.18 -2 -1.5 -1 -0.5 0 0.5 1 1.5 β -1-0.8-0.6-0.4-0.200.20.40.60.81 Figure 1: Pairs ( α, β ) where the conditions in Lemma 7 (marked by ‘+’), Lemma 2 (marked by ‘.’), andTheorem 1 with k = 2 (which is equivalent to Lemma 3 with k = 2, Lemma 9, and Corollary 1) (marked by‘o’) are satisfied, respectively. The square in blue color denotes (cid:3) . One can verify that the obtained region ( α, β ) by Theorem 1 with k = 2 coincides with the exact regionof stability obtained in [23]. This indicates that k = 2 is already very efficient. Actually, thousands ofnumerical examples show that k = 2 in Theorem 1 can lead to necessary and sufficient stability conditions.Thus, the advantage of Theorem 1 over Lemma 2 is that the size of the LMI has been reduced significantly,especially, for large n .We now treat α and β as uncertainties (which might be time-varying) and solve the robust stability problem,particularly, we want to find the maximal value of r > r ∗ ) such that the system (3) is stronglystable for all α ∈ [ − r, r ] and β ∈ [ − r, r ] . To this end, we rewrite A ( α ) = A + ∆ A and B ( β ) = B + ∆ B, where A = (cid:20) − . − . . . (cid:21) , ∆ A = (cid:20) α (cid:21) ,B = (cid:20) . . − . − . (cid:21) , ∆ B = (cid:20) β (cid:21) . It can be verified that (∆ B, ∆ A ) satisfies (29) where F = [ βr , αr ] and E = (cid:20) (cid:21) , B = (cid:20) r (cid:21) , A = (cid:20) r (cid:21) . We clearly have F T F ≤ I . Then, by applying Theorem 2 for different k and applying a linear searchtechnique on r , we can get r ∗ ( k ) . It is found that r ∗ (1) = 0 . r ∗ (2) = r ∗ (3) = 0 . . Denote thesquare (cid:3) k = { ( α, β ) : α ∈ [ − r ∗ ( k ) , r ∗ ( k )] , β ∈ [ − r ∗ ( k ) , r ∗ ( k )] } . It follows that (cid:3) is very close to (cid:3) whichis recorded in Fig. 1. We can see that the square (cid:3) turns to be the maximal square that can be includedin the region where the system is strongly stable for fixed ( α, β ) . This indicates that Theorem 2 can evenprovide necessary and sufficient conditions for robust strong stability for this example.
This note established a necessary and sufficient condition for guaranteeing strong stability of linear differenceequations with two delays. The most important advantage of the proposed method is that the coefficientsof the linear difference equation appear as linear functions in the proposed conditions, which helps to dealthe robust stability analysis problem. The relationships among the proposed condition and the existing oneswere revealed by establishing a time-domain interpretation of the proposed LMI condition.19 ppendix
A1: A Proof of Lemma 2
Notice that ρ (∆ θ ) < , ∀ θ ∈ R , is equivalent to that ∆ is Schur stable and0 = (cid:12)(cid:12) ∆ H θ ⊗ ∆ θ − I n ⊗ I n (cid:12)(cid:12) = (cid:12)(cid:12) A T ⊗ B e − j θ + B T ⊗ A e j θ + (cid:0) A T ⊗ A + B T ⊗ B − I n ⊗ I n (cid:1)(cid:12)(cid:12) = e − n j θ (cid:12)(cid:12) A T ⊗ B + B T ⊗ A e − θ + (cid:0) A T ⊗ A + B T ⊗ B − I n ⊗ I n (cid:1) e − j θ (cid:12)(cid:12) = e − n j θ (cid:12)(cid:12)(cid:12) C (cid:0) e j θ I n − A (cid:1) − B + D (cid:12)(cid:12)(cid:12) = e − n j θ (cid:12)(cid:12) G (cid:0) e j θ (cid:1)(cid:12)(cid:12) , ∀ θ ∈ R , (76)where G ( s ) = C ( sI n − A ) − B + D with A = (cid:20) n × n I n n × n n × n (cid:21) , B = (cid:20) n × n I n (cid:21) , C = (cid:2) B T ⊗ A A T ⊗ A + B T ⊗ B − I n ⊗ I n (cid:3) , D = A T ⊗ B. The condition (76) is also equivalent to0 > − G H0 (cid:0) e j θ (cid:1) G (cid:0) e j θ (cid:1) = (cid:20) (cid:0) e j θ I n − A (cid:1) − B I n (cid:21) H M (cid:20) (cid:0) e j θ I n − A (cid:1) − B I n (cid:21) where θ ∈ R and M = − (cid:2) C D (cid:3) T (cid:2) C D (cid:3) . Thus, by the YKP lemma (Lemma 10), this is equivalent to the existence of a symmetric matrix P ∈ R n × n such that (cid:20) A T0 P A − P A T0 P B B T0 P A B T0 P B (cid:21) + M < . (77)Let P be partitioned as P = (cid:20) P P P T3 − P (cid:21) , (78)where P i , i = 1 , , , are n × n matrices with P i , i = 1 , A , B ) , (77) is equivalent to the LMI − P − P − P T3 P + P P P T3 − P + M < . (79)Applying the congruent transformation T = I n I n I n , on the LMI (79) gives (7). The proof is finished by noting that ∆ is Schur stable if and only if (6) issatisfied. 20
2: Some Technical Notations and Lemmas
For two matrices A ∈ R n × n , B ∈ R n × n , the shuffle product (power) is defined as [8] A [ i ] B [ j ] = X i + i + ··· + i s = ij + j + ··· + j s = j A i B j A i B i · · · A i s B j s , where ( i, j ) is a pair of nonnegative integers, and i k , j k ≥ , k = 1 , , . . . , s. For example, A [1] B [2] = AB + BAB + B A. There are several simple properties of the shuffle product. For example, [8] A [ i ] B [ j ] = B [ j ] A [ i ] ,A [ i ] B [0] = A i , A [0] B [ j ] = B j ,A [ i ] B [ j ] = A (cid:16) A [ i − B [ j ] (cid:17) + B (cid:16) A [ i ] B [ j − (cid:17) = (cid:16) A [ i − B [ j ] (cid:17) A + (cid:16) A [ i ] B [ j − (cid:17) B. (80)We next recall the so-called Yakubovich-Kalman-Popov (YKP) Lemma. This lemma in the discrete-timesetting is also known as the Szego-Kalman-Popov (SKP) Lemma [14, 22, 25]. Lemma 10 (YKP Lemma) Given A ∈ R n × n , B ∈ R n × m and M ∈ R ( n + m ) × ( n + m ) with (cid:12)(cid:12) e j θ I n − A (cid:12)(cid:12) =0 , ∀ θ ∈ R . Then (cid:20) (cid:0) e j θ I n − A (cid:1) − BI m (cid:21) H M (cid:20) (cid:0) e j θ I n − A (cid:1) − BI m (cid:21) < , holds for all θ ∈ R if and only if there exists a symmetric matrix P ∈ R n × n such that (cid:20) A T P A − P A T P BB T P A B T P B (cid:21) + M < . The next lemma is adopted from [2].
Lemma 11 [2] If the inequality (4) is satisfied, namely, sup θ ∈ [0 , π ] { ρ (∆ θ ) } < , (81) then there exists a k ∗ ∈ N + such that sup θ ∈ [0 , π ] (cid:8)(cid:13)(cid:13) ∆ kθ (cid:13)(cid:13)(cid:9) < , ∀ k ≥ k ∗ . (82)We finally recall a well-known result that was frequently used in robust control literature. Lemma 12 [7] Let X and Y be real matrices of appropriate dimensions. For Q > the following inequalityis satisfied XY + Y T X T ≤ XQX T + Y T Q − Y. eferences [1] Avellar CE, Hale JK. On the zeros of exponential polynomials. Journal of Mathematical Analysis andApplications . 1980, 73(2): 434-452.[2] Bliman PA. Lyapunov equation for the stability of 2-D systems.
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