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Stability, l1-gain and l∞-gain analysis for discrete-time positive switched singular delayed systems
Applied Mathematics and Computation 275 (2016) 95–106
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Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Stability, l1 -gain and l∞ -gain analysis for discrete-time positive switched singular delayed systems Shuo Li, Zhengrong Xiang∗ School of Automation, Nanjing University of Science and Technology, Nanjing 210094, People’s Republic of China
a r t i c l e
i n f o
Keywords: Singular systems Positive switched systems l1 -gain performance l∞ -gain performance Time delay Switching Lyapunov function
a b s t r a c t In this paper, the problems of stability, l1 -gain and l∞ -gain analysis for positive switched singular systems with time delay are addressed. Both l1 -gain and l∞ -gain performance are analyzed. Firstly, a necessary and sufficient condition of positivity for the system is established by using the singular value decomposition method. Then by introducing new state vectors, the positive switched singular system is transformed into a standard positive switched system, and sufficient conditions for the positive switched singular delayed system to be asymptotically stable with an l1 -gain performance and an l∞ -gain performance are obtained, respectively. Finally, three numerical examples are presented to demonstrate the proposed results. © 2015 Elsevier Inc. All rights reserved.
1. Introduction Singular systems, which are also referred to as descriptor systems, implicit systems or differential-algebraic systems, have constantly gained a great interest, due to its capacity to describe the dynamic and algebraic relationships between state variables simultaneously [1]. Singular systems are found in a broad spectrum of applications including power systems, economics systems, robotic systems, chemical processes, biological systems, etc. [2–8]. In practice, physical systems involve variables that have nonnegative property, such as population levels, absolute temperature, and so on. Such systems are referred to as positive systems [9,10], whose states and outputs are nonnegative whenever the initial conditions and inputs are nonnegative. Positive systems are frequently encountered in various fields, for instance, pharmacokinetics [11], chemical reaction [12] and internet congestion control [13]. For positive systems, one often uses the linear co-positive Lyapunov functional instead of the traditional quadratic Lyapunov functional for stability analysis, due to that the former will yield less conservative stability conditions [14]. In addition l1 -gain or l∞ -gain rather than l2 -gain is sometimes more appropriate to be adopted as a system performance measure [15,16]. l2 -norm represents the energy, while l1 -norm and l∞ -norm account for the sum of quantities and the maximum of quantities, respectively [17]. Stability and performance analysis for standard positive systems have been studied by many authors [18–20], and some results on positive singular systems have also been obtained [21–23]. On the other hand, many practical systems are always subjected to abrupt variation in their structures and parameters, such as failures and repairs of the components, changes in the interconnections of subsystems, and sudden environment changes. One effective method of characterizing the abrupt changes is by using the switched system approach [24–26]. There are fruitful results on analysis and synthesis of switched systems [27–30]. When positive singular systems experience the switching-type changes in their parameters, it becomes difficult to study this kind of positive switched singular systems due to the coupling ∗
Corresponding author. Tel.: +86 13951012297. E-mail address: [email protected], [email protected] (Z. Xiang).
http://dx.doi.org/10.1016/j.amc.2015.11.053 0096-3003/© 2015 Elsevier Inc. All rights reserved.
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between the switching, the algebraic constraints in singular models and the non-negativity limits of state valuables. Therefore, the positive switched singular system considered here is of practical importance, and meanwhile poses theoretical challenges in its formal analysis. There are only a few recent publications investigating positive switched singular systems [31,32]. Xia et al. [31] dealt with the problem of state feedback controller design for singular positive Markovian jump systems with partly known transition rates. In [32], the problems of stability for a class of switched positive descriptor systems with average dwell time switching were investigated. However, Xia et al. [31] and Qi and Gao [32] only focused on the autonomous case without taking time delay and exogenous disturbance into account. As time delay and exogenous disturbance are frequently encountered in practice, it is necessary and significant to further investigate positive switched singular systems with those two kinds of phenomena. When taking time delays and exogenous disturbance into account, the problem of choosing an appropriate mode-dependent co-positive Lyapunov function and how to analyze the stability and system performance will be more complicated and challenging. To the best of authors knowledge, no relevant work that considers this kind of system has been published. Therefore, the main objective of this work is to investigate the stability, the l1 -gain and l∞ -gain performance for positive switched singular systems with time delay. The main contributions of this paper are as follows: (1) By using the singular value decomposition method, we propose a necessary and sufficient condition of positivity for switched singular delayed systems; (2) By introducing new state vectors, we derive a sufficient delay-dependent criterion with regard to the considered system to be uniformly asymptotically stable; (3) In consideration of the exogenous disturbance, the disturbance attenuation performance (the l1 -gain and l∞ -gain performance in this paper) is analyzed for the positive switched singular system with time delay; (4) All the conditions obtained in this paper are presented in terms of linear programming (LP). The remaining of the paper is as follows. The problem formulation and some necessary lemmas are provided in Section 2. In Section 3, the positivity, stability and performance analysis are developed. Three numerical examples are presented to demonstrate the obtained results in Section 4. In Section 5, concluding remarks are given. Notations: [A]ij denotes the entry of matrix A in the ith row and jth column; A 0( ≺, , ≺) means that all entries of matrix A are nonnegative (non-positive, positive, negative); AB(A B) means that A − B 0(A − B0 ); AT means the transpose of matrix A; det(A ) denotes the determinant of matrix A; deg(· ) denotes the degree of a polynomial; rank(A ) denotes the rank of matrix A; R(R+ ) is the set of all real (positive real) numbers; Rn Rn+ is the n-dimensional real (positive real) vector space; Rm × n is the set of all m × n-dimensional real matrices; Z + refers to the set of all positive integers; 1n ∈ Rn denotes a column vector with each entry equal to 1; The vector 1-norm is denoted by x1 = nk=1 |xk | with xk being the kth element of x ∈ Rn ; Given ν : R → Rn , the ∞ l1 -norm is defined by νl1 = k=k ν (k )1 . We say ν (k) ∈ l1 [k0 , ∞) if νl1 < ∞; The vector ∞-norm is denoted by x∞ = 0 max1≤k≤n |xk | with xk being the kth element of x ∈ Rn ; Given ν : R → Rn , the l∞ -norm is defined by νl∞ = supk≥0 ν (k )∞ . We say ν (k) ∈ l∞ [k0 , ∞) if νl∞ < ∞. Matrices and vectors, if their dimensions are not explicitly stated, are assumed to have compatible dimensions for algebraic operations. 2. Problem formulation We consider the following discrete-time linear switched singular system with time delay,
⎧ ⎨Ex(k + 1 ) = Aσ (k) x(k ) + Adσ (k) x(k − d ) + Fσ (k) w(k ), z(k ) = Cσ (k) x(k ) + Dσ (k) w(k ), ⎩ x(θ ) = ϕ (θ ), θ = −d, −d + 1, . . . , 0,
(1)
where x(k) ∈ Rn is the state vector, w(k ) ∈ Rq is the disturbance input which belongs to l1 (or l∞ ), z(k) ∈ Rm is the controlled output. d is the constant integer time delay and ϕ (θ ) is a given discrete vector-valued initial condition of the system. {σ (k), k ≥ 0} is a switching signal taking values in a finite set N = {1, 2, . . . , N} with N being the number of subsystems, moreover, σ (k ) = i denotes the ith subsystem is activated, i ∈ N . Ai , Adi , Fi , Ci and Di , i ∈ N, are known constant matrices with appropriate dimensions. The matrix E ∈ Rn × n is singular and rankE = r < n. Since rankE = r < n, then there exist non-singular matrices P, Q ∈ Rn × n such that PEQ = [I0r 00]. In this paper, for simplicity, let E = [I0r 00] and other system matrices have the following form, ∀i ∈ N :
Ai1 Ai = Ai3
Ai2 , Ai4
Adi1 Adi = Adi3
Adi2 , Adi4
Fi =
Fi1 , Fi2
Ci = [Ci1 Ci2 ],
(2)
where x(k ) = [xx1 ((kk ))], x1 (k) ∈ Rr and x2 (k ) ∈ Rn−r . 2
Definition 1. [2] Consider system (1), (i) For a given i ∈ N, the pair (E, Ai ) is said to be regular if det(zE − Ai ) = 0; (ii) For a given i ∈ N, the pair (E, Ai ) is said to be causal if it is regular and degree(det(zE − Ai )) = rankE; Remark 1. As in [3], the regularity and causality of the pair (E, Ai ), ∀i ∈ N, in system (1) ensures the existence and uniqueness of the solution to this system with any given initial conditions ϕ (θ ), θ = −d, −d + 1, . . . , 0.
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Definition 2. [9] System (1) is said to be positive if, for any initial conditions ϕ (θ ) 0, w(k )0 and any switching signals σ (k), we have x(k) 0 and z(k) 0 for all k ≥ 0. Lemma 1. [3] System (1) with (2) is causal if and only if Ai4 is nonsingular, ∀i ∈ N . In the light of Lemma 1, when system (1) with (2) is regular and causal, it can be rewritten as:
⎧ x1 (k + 1 ) = A¯ σ (k)1 x1 (k ) + A¯ dσ (k)1 x1 (k − d ) + A¯ dσ (k)2 x2 (k − d ) + F¯σ (k)1 w(k ), ⎪ ⎪ ⎪ ⎪ ⎪ ⎨x2 (k ) = A¯ σ (k)3 x1 (k ) + A¯ dσ (k)3 x1 (k − d ) + A¯ dσ (k)4 x2 (k − d ) + F¯σ (k)2 w(k ), z(k ) = Cσ (k)1 x1 (k ) + Cσ (k)2 x2 (k ) + Dσ (k) w(k ), ⎪ ⎪ ⎪ θ = −d, −d + 1, . . . , 0, x 1 (θ ) = ϕ1 (θ ), ⎪ ⎪ ⎩ x2 (θ ) = ϕ2 (θ ), θ = −d, −d + 1, . . . , 0,
(3)
where for each i ∈ N,
A¯ i1 = Ai1 − Ai2 A−1 A , i4 i3
A¯ di1 = Adi1 − Ai2 A−1 A , i4 di3
A¯ di2 = Adi2 − Ai2 A−1 A , i4 di4 A¯ i3 = −A−1 A , i4 i3
F¯i1 = Fi1 − Ai2 A−1 F , i4 i2
A¯ di3 = −A−1 A , i4 di3
A¯ di4 = −A−1 A , i4 di4
F¯i2 = −A−1 F . i4 i2
Lemma 2. [23] Assume that (E, Ai ), ∀i ∈ N, is regular and causal. Then system (1) with (2) is positive if and only if system (3) is positive. Definition 3. [28] System (1) with w(k ) = 0 is said to be (i) regular, causal, if the pair (E, Ai ), ∀i ∈ N, is regular, causal; (ii) uniformly asymptotically stable, if for any ε > 0, there is a δ (ε ) > 0 such that for arbitrary switching signal σ (k), sup−d≤θ ≤0 ϕ (θ ) < δ implies x(k) < ε for k ≥ 0 and there is also a δ > 0 such that for arbitrary switching signal σ (k), sup−d≤θ ≤0 ϕ (θ ) < δ implies x(k) → 0 as k → ∞. Definition 4. Assume that system (1) with w(k ) = 0 is regular, causal, uniformly asymptotically stable. For a given scalar γ > 0 and the initial condition ϕ (k ) = 0, k = −d, −d + 1, . . . , 0, system (1) is said to be uniformly asymptotically stable with an l1 -gain performance if the following condition is satisfied: ∞
z ( k ) 1 < γ
k=0
∞
w(k )1 , ∀w(k ) ∈ l1 ;
k=0
system (1) is said to be uniformly asymptotically stable with an l∞ -gain performance if the following condition is satisfied:
sup z(k )∞ < γ sup w(k )∞ , k≥0
k≥0
∀w(k ) ∈ l∞ .
(4)
In this paper, our aim is to investigate the positivity, asymptotical stability, l1 -gain performance, and l∞ -gain performance for the switched singular delayed system (1). 3. Main results This section will focus on the analysis of positivity, stability, l1 -gain and l∞ -gain performance. 3.1. Positivity analysis Before analyzing the stability and system performance, we firstly present a necessary and sufficient condition for the positivity of system (1). Theorem 1. Assume that (E, Ai ), ∀i ∈ N, is regular and causal. System (1) with (2) is positive if and only if A¯ i1 0, A¯ di1 0, A¯ di2 0, F¯i1 0, A¯ i3 0, A¯ di3 0, A¯ di4 0, F¯i2 0, Ci1 0, Ci2 0, Di 0, ∀i ∈ N, where A¯ i1 , A¯ di1 , A¯ di2 , F¯i1 , A¯ i3 , A¯ di3 , A¯ di4 , F¯i2 , Ci1 , Ci2 are defined in (3). Proof. System (1) with (2) is regular and causal due to that (E, Ai ), ∀i ∈ N, is regular and causal. Sufficiency. We will show that system (1) is positive. We first prove that the solution x(k ) = [xT1 (k ) d]. It follows from Lagrange’s formula and the first equation of (3) that
x1 (k ) = A¯ ki1 x1 (0 ) +
k−1
θ =0
xT2 (k )]T is positive on [0,
θ −1 (A¯ x (θ − d ) + A¯ x (θ − d ) + F¯ w(θ )). A¯ k− i1 di1 1 di2 2 i1
Since A¯ i1 0, it follows that A¯ ki1 0, k ≥ 0. From x1 (0) 0, we have A¯ ki1 x1 (0 )0. Moreover, since A¯ di1 0, A¯ di2 0, F¯i1 0, by the same way we have A¯ k−θ −1 A¯ di1 x1 (θ − d )0, A¯ k−θ −1 A¯ di2 x2 (θ − d )0, A¯ k−θ −1 F¯i1 w(θ )0, for all 0 ≤ θ ≤ k − 1 ≤ d. Hence, we have x1 (k) 0, ∀k ∈ [0, d].
i1
i1
i1
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On the other hand, according to A¯ i3 0, A¯ di3 0, A¯ di4 0, F¯i2 0, x1 (k) 0, x1 (k − d )0, x2 (k − d )0, k ∈ [0, d] and from the second equation of (3) we have x2 (k) 0, ∀k ∈ [0, d]. Thus, the solution x(k) of system (1) is positive on [0, d]. Moreover, from Ci1 0, Ci2 0, Di 0 and the third equation of (3), we can directly get z(k) 0 on [0, d]. Using the step method, we can extend the consideration for the intervals [d, 2d], [2d, 3d], etc. Repeating this procedure iteratively, it follows that x(k) 0, z(k) 0, ∀k ≥ 0. Necessity. Suppose that system (1) is positive. Consider the first equation of (3) on [0, d]. We first prove A¯ i1 0. ad absurdum, A¯ i1 is not nonnegative. That is, suppose there exist some indices p, q ∈ {1, 2, . . . , r} such that [A¯ i1 ] pq < 0. Take the initial condition:
ϕ1 (0 ) = eq , ϕ1 (θ ) = 0, θ ∈ [−d, 0 ); ϕ2 (θ ) = 0, w(θ ) = 0, θ ∈ [−d, 0], where eq , q = 1, 2, . . . , r denotes the qth vector of canonical basis in Rr , and the solution on [0, d] is x1 (1 ) = A¯ i1 eq , which implies that [x1 (1 )] p = [A¯ i1 ] pq < 0, this contradicts the fact that x1 (1) 0. Therefore, we conclude that A¯ i1 0. We now prove A¯ di1 0, ad absurdum, suppose there exist some indices p, q ∈ {1, 2, . . . , r} such that [A¯ di1 ] pq < 0. Take the initial condition:
ϕ1 (θ ) = 0, θ ∈ (−d, 0], ϕ1 (−d ) = eq ; ϕ2 (θ ) = 0, w(θ ) = 0, θ ∈ [−d, 0], where eq , q = 1, 2, . . . , r denotes the qth vector of canonical basis in Rr , and the solution is x1 (1 ) = A¯ di1 eq , which implies that [x1 (1 )] p = [A¯ di1 ] pq < 0, this is a contradiction to the fact that x1 (1) 0. Therefore, A¯ di1 0. To prove A¯ di2 0, we choose the initial condition:
ϕ2 (θ ) = 0, θ ∈ (−d, 0], ϕ2 (−d ) = eq ; ϕ1 (θ ) = 0, w(θ ) = 0, θ ∈ [−d, 0], where eq , q = r + 1, r + 2, . . . , n denotes the qth vector of canonical basis in Rn−r , then by the same argument we obtain A¯ di2 0. To prove F¯i1 0, we choose the initial condition:
ϕ1 (θ ) = 0, ϕ2 (θ ) = 0, w(θ ) = eq , θ ∈ [−d, 0], where eq , q = 1, 2, . . . , n denotes the qth vector of canonical basis in Rn , then by the same argument we obtain F¯i1 0. We now show that A¯ i3 0, A¯ di3 0, A¯ di4 0, F¯i2 0. To this end, consider the second equation of (3) on [0, d], choosing the initial condition:
ϕ1 (0 ) = eq , ϕ1 (θ ) = 0, ϕ2 (θ ) = 0, θ ∈ [−d, 0 ); w(θ ) = 0, θ ∈ [−d, 0], we have x2 (0 ) = A¯ i3 eq 0, q = 1, 2, . . . , r, then it gives A¯ i3 0. Choosing the initial condition:
ϕ1 (−d ) = eq , ϕ1 (θ ) = 0, θ ∈ (−d, 0]; ϕ2 (θ ) = 0, θ ∈ [−d, 0 ); w(θ ) = 0, θ ∈ [−d, 0], the solution x2 (0 ) = A¯ di3 eq 0, q = 1, 2, . . . , r is obtained, then it gives A¯ di3 0. Choosing the initial condition:
ϕ2 (−d ) = eq , ϕ2 (θ ) = 0, θ ∈ (−d, 0 ); ϕ1 (θ ) = 0, w(θ ) = 0, θ ∈ [−d, 0], we have x2 (0 ) = A¯ di4 eq 0, q = r + 1, r + 2, . . . , n, then it gives A¯ di4 0. Choosing the initial condition:
ϕ2 (θ ) = 0, θ ∈ [−d, 0 ); ϕ1 (θ ) = 0, w(θ ) = eq , θ ∈ [−d, 0], we may get x2 (0 ) = F¯i2 eq 0, q = 1, 2, . . . , n, then it gives F¯i2 0. Furthermore, we show that Ci1 0, Ci2 0, Di 0. Considering the third equation of (3) on [0, d], choosing the initial condition:
ϕ1 (0 ) = eq , ϕ1 (θ ) = 0, θ ∈ [−d, 0 ); ϕ2 (θ ) = 0, w(θ ) = 0, θ ∈ [−d, 0], we can obtain z(0 ) = Ci1 eq , q = 1, 2, . . . , r, then it gives Ci1 0. Choosing the initial condition:
ϕ2 (0 ) = eq , ϕ2 (θ ) = 0, θ ∈ [−d, 0 ); ϕ1 (θ ) = 0, w(θ ) = 0, θ ∈ [−d, 0], we can get z(0 ) = Ci2 eq 0, q = r + 1, r + 2, . . . , n, then it gives Ci2 0. Choosing the initial condition:
ϕ1 (θ ) = 0, ϕ2 (θ ) = 0, w(θ ) = eq , θ ∈ [−d, 0], the solution z(0 ) = Di eq 0, q = 1, 2, . . . , n is obtained, then it gives Di 0. This completes the proof. In the sequence, 3.2. Stability analysis In this section, based on Theorem 1, some sufficient conditions will be provided to ensure the asymptotical stability of the positive system (1) (w(k ) = 0), namely, the following system (5):
Ex(k + 1 ) = Aσ (k) x(k ) + Adσ (k) x(k − d ), x(θ ) = ϕ (θ ), θ = −d, −d + 1, . . . , 0,
(5)
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where E, Ai , and Adi , i ∈ N, satisfy (2). It is easy to see that system (5) can be written as:
x1 ( k + 1 ) = Aσ ( k ) 1 x1 ( k ) + A σ ( k ) 2 x2 ( k ) + A d σ ( k ) 1 x1 ( k − d ) + A d σ ( k ) 2 x2 ( k − d ) , 0 = Aσ ( k ) 3 x 1 ( k ) + A σ ( k ) 4 x 2 ( k ) + A d σ ( k ) 3 x 1 ( k − d ) + A d σ ( k ) 4 x 2 ( k − d ) .
(6)
By introducing the following vector
xˆ(k ) = [xT1 (k ), xT2 (k − 1 ), xT1 (k − 1 ), xT2 (k − 2 )]T ,
(7)
system (6) is rewritten as
Eˆσ (k) xˆ(k + 1 ) = Aˆ σ (k) xˆ(k ) + Aˆ dσ (k) xˆ(k − d + 1 ), where for each i ∈ N,
⎡
Ir 0 ⎢ Eˆi = ⎣ 0 0
−Ai2 −Ai4 0 0
0 0 Ir 0
0 0 0
⎤ ⎥ ⎦,
In−r
⎡
Ai1 A ⎢ Aˆ i = ⎣ i3 Ir 0
0 0 0 In−r
(8)
0 0 0 0
⎤
0 0⎥ , 0⎦ 0
⎡
0 0 ⎢ Aˆ di = ⎣ 0 0
Adi2 Adi4 0 0
Adi1 Adi3 0 0
⎤
0 0⎥ . 0⎦ 0
(9)
Remark 2. By Definitions 1 and 3, one can see that the regularity, causality and uniform asymptotic stability of system (5) are equivalent to those of system (6). Lemma 3. System (5) is regular, causal and uniformly asymptotically stable, if and only if system (8) is a switched standard linear system (i.e. the matrix Eˆi is nonsingular for each i ∈ N ) and uniformly asymptotically stable. Proof. First, from Remark 2, that system (5) is regular, causal and uniformly asymptotically stable is equivalent to that system (6) is regular, causal and uniformly asymptotically stable. From Definition 1 and Lemma 1, that system (6) is regular, causal is equivalent to that matrix Ai4 is nonsingular for each i ∈ N; then from (9), it follows that the matrix Ai4 is nonsingular is equivalent to that the matrix Eˆi in system (8) is nonsingular. From Definition 3, that system (6) is uniformly asymptotically stable is equivalent to that x1 (k) → 0, x2 (k) → 0 as k → ∞; then from (7), x1 (k) → 0, x2 (k) → 0 as k → ∞, it is shown that system (8) is uniformly asymptotically stable. Thus, the conclusion is obtained. The proof is completed. Remark 3. Based on (2) and (7), system (5) is transformed into system (8) with delay d − 1. Based on Lemma 3, the regularity, causality and uniform asymptotical stability for system (5) can be analyzed by investigating the uniform asymptotic stability for the standard linear switched system (8). Theorem 2. Assume that (E, Ai ), ∀i ∈ N, is regular and causal. System (5) with (2) is positive and uniformly asymptotically stable, if 2n conditions in Theorem 1 are satisfied, and for each i ∈ N, there exist vectors νi ∈ R2n + and υ ∈ R+ satisfying the following LPs:
Aˆ Ti Eˆi−T ν j − νi + υ ≺ 0,
(10)
Aˆ Tdi Eˆi−T ν j − υ ≺ 0.
(11)
Proof. By Theorem 1, system (5) with (2) is regular, causal and positive. Based on Lemma 3, to prove that system (5) is uniformly asymptotically stable, it only required to prove that system (8) is uniformly asymptotically stable. We construct the following Lyapunov functional candidate
V (k ) = xˆT (k )νσ (k) +
k−1
xˆT (l )υ ,
(12)
l=k−d+1
where vectors ν σ (k) 0, υ 0. Let the mode at time k and k + 1 be i and j, respectively, that is σ (k ) = i and σ (k + 1 ) = j for any i, j ∈ N. One can obtain that
V (k ) = V (k + 1 ) − V (k ) = xˆT (k + 1 )ν j +
k
xˆT (l )υ − xˆT (k )νi −
l=k−d+2
k−1
xˆT (l )υ
l=k−d+1
= (xˆT (k )Aˆ Ti Eˆi−T + xˆT (k − d + 1 )Aˆ Tdi Eˆi−T )ν j − xˆT (k )νi + xˆT (k )υ − xˆT (k − d + 1 )υ = xˆT (k )[Aˆ Ti Eˆi−T ν j − νi + υ ] + xˆT (k − d + 1 )[Aˆ Tdi Eˆi−T ν j − υ ]. From (10) and (11), it is obtained that V(k) ≤ 0. Then system (8) is uniformly asymptotically stable. The proof is completed.
(13)
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3.3. l1 -gain and l∞ -gain analysis In consideration of the exogenous disturbance, to analyze the disturbance attenuation performance, we consider the l1 -gain performance and l∞ -gain performance for positive system (1) in this section, respectively. From (2) and (7), positive system (1) is rewritten as
Eˆσ (k) xˆ(k + 1 ) = Aˆ σ (k) xˆ(k ) + Aˆ dσ (k) xˆ(k − d + 1 ) + Fˆσ (k) w(k ), z(k ) = Cˆ1σ (k) xˆ(k ) + Cˆ2σ (k) xˆ(k + 1 ) + Dσ (k) w(k ),
where for each i ∈ N,
Fˆi = Fi1T
Fi2T
T
0 0 ,
Cˆ1i = [Ci1 0 0 0],
(14)
Cˆ2i = [0 Ci2 0 0].
(15)
A sufficient condition for the uniformly asymptotical stability with an l1 -gain performance of positive system (1) is proposed in the following theorem. Theorem 3. Assume that (E, Ai ), ∀i ∈ N, is regular and causal. For a given scalar γ > 0, system (1) with (2) is positive and uniformly asymptotically stable with an l1 -gain performance, if conditions in Theorem 1 are satisfied, and for each i ∈ N, there exist vectors 2n νi ∈ R2n + and υ ∈ R+ satisfying the following LPs: T T Aˆ Ti Eˆi−T ν j − νi + υ + Cˆ1i 1m + Aˆ Ti Eˆi−T Cˆ2i 1m ≺ 0,
(16)
T Aˆ Tdi Eˆi−T ν j − υ + Aˆ Tdi Eˆi−T Cˆ2i 1m ≺ 0,
(17)
T 1m + DTi 1m − γ 1q ≺ 0. FˆiT Eˆi−T ν j + FˆiT Eˆi−T Cˆ2i
(18)
Proof. First, note that (16) and (17) implies (10) and (11). When w(k ) = 0, by Theorem 2, system (1) with (2) is regular, causal, positive and uniformly asymptotically stable. Next, to prove that positive system (1) satisfies the l1 -gain performance for any w(k ) ∈ l1 , it is only required to prove system (14) satisfies the l1 -gain performance. Construct Lyapunov functional (12), and let the mode at time k and k + 1 be i and j, that is σ (k ) = i and σ (k + 1 ) = j. Similar to the proof in Theorem 2, it follows that
V (k ) = V (k + 1 ) − V (k ) = (xˆT (k )Aˆ Ti + xˆT (k − d + 1 )Aˆ Tdi + wT (k )FˆiT )Eˆi−T ν j − xˆT (k )νi + xˆT (k )υ − xˆT (k − d + 1 )υ = xˆT (k )[Aˆ Ti Eˆi−T ν j − νi + υ ] + xˆT (k − d + 1 )[Aˆ Tdi Eˆi−T ν j − υ ] + wT (k )[FˆiT Eˆi−T ν j ].
(19)
When the initial condition xˆ(k ) = 0, k = −d + 1, . . . , −1, 0, which yields V (0 ) = 0, then from (19), it follows that T
(z(k )1 − γ w(k )1 ) =
k=0
T
(z(k )1 − γ w(k )1 + V (k )) −
k=0
=
T
V (k )
k=0
T
(z(k )1 − γ w(k )1 + V (k )) − V (T + 1 ) + V (0 )
k=0
≤
T [z(k )1 − γ w(k )1 + V (k )] k=0
≤
T T T [xˆT (k )Cˆ1i 1m + (xˆT (k )Aˆ Ti + xˆT (k − d + 1 )Aˆ Tdi + wT (k )FˆiT )Eˆi−T Cˆ2i 1m + wT (k )DTi 1m k=0
− γ wT (k )1q + V (k )] ≤
T
{xˆT (k )[Aˆ Ti Eˆi−T ν j − νi + υ + Cˆ1iT 1m + Aˆ Ti Eˆi−T Cˆ2iT 1m ]
k=0 T + xˆT (k − d + 1 )[Aˆ Tdi Eˆi−T ν j − υ + Aˆ Tdi Eˆi−T Cˆ2i 1m ] T 1m + DTi 1m − γ 1q ]}. + wT (k )[FˆiT Eˆi−T ν j + FˆiT Eˆi−T Cˆ2i
Therefore, it is obtained from (16)–(18) that for any w(k ) ∈ l1 , The proof is completed.
∞
k=0
z(k )1 < γ
∞
k=0
(20)
w(k )1 .
In the following, a sufficient condition for the uniformly asymptotical stability with an l∞ -gain performance of positive system (1) is proposed.
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Theorem 4. Assume that (E, Ai ), ∀i ∈ N, is regular and causal. For a given scalar γ > 0, system (1) with (2) is positive and uniformly asymptotically stable with an l∞ -gain performance, if conditions in Theorem 1 are satisfied, and for each i ∈ N, there exist vectors 2n νi ∈ R2n + and υ ∈ R+ such that (16) and (17) and the following LP holds: T 1Tq FˆiT Eˆi−T ν j + 1Tq FˆiT Eˆi−T Cˆ2i 1m + 1Tq DTi 1m − γ ≺ 0.
(21)
Proof. First, note that (16) and (17) implies (10) and (11). When w(k ) = 0, by Theorem 2, system (1) with (2) is regular, causal, positive and uniformly asymptotically stable. Next, to prove that positive system (1) satisfies an l∞ -gain performance for any w(k ) ∈ l∞ , it is only need to prove system (14) satisfies an l∞ -gain performance. From (14), we have
z(k )∞ ≤ Cˆ1i xˆ(k )∞ + Cˆ2i Eˆi−1 Aˆ i xˆ(k )∞ + Cˆ2i Eˆi−1 Aˆ di xˆ(k − d + 1 )∞ + Cˆ2i Eˆi−1 Fˆi w(k )∞ + Di w(k )∞ ≤ Cˆ1i xˆ(k ) + Cˆ2i Eˆ −1 Aˆ i xˆ(k ) + Cˆ2i Eˆ −1 Aˆ di xˆ(k − d + 1 ) + Cˆ2i Eˆ −1 Fˆi w(k ) + Di w(k )1 1
i
1
i
1
i
1
T T T ≤ xˆT (k )Cˆ1i 1m + xˆT (k )Aˆ Ti Eˆi−T Cˆ2i 1m + xˆT (k − d + 1 )Aˆ Tdi Eˆi−T Cˆ2i 1m T + w(k )∞ 1Tq FˆiT Eˆi−T Cˆ2i 1m + w(k )∞ 1Tq DTi 1m .
(22)
Construct Lyapunov functional (12), and let σ (k ) = i and σ (k + 1 ) = j. Similar to the proof in Theorem 3, it follows that,
V (k ) + z(k )∞ − γ w(k )∞ ≤ xˆT (k )[Aˆ Ti Eˆi−T ν j − νi + υ + Cˆ1iT 1m + Aˆ Ti Eˆi−T Cˆ2iT 1m ] T + xˆT (k − d + 1 )[Aˆ Tdi Eˆi−T ν j − υ + Aˆ Tdi Eˆi−T Cˆ2i 1m ] T 1m + 1Tq DTi 1m − γ ] + w(k )∞ [1Tq FˆiT Eˆi−T ν j + 1Tq FˆiT Eˆi−T Cˆ2i
≤ 0.
(23)
Then it is derived that
V (k ) ≤ V (0 ) −
k−1
(z(s )∞ − γ w(s )∞ ).
(24)
s=0
Notice that
w(k )∞ ≤ sup w(k )∞ . k≥0
Under zero initial conditions, from (24), we have
z(k )∞ ≤ γ k sup w(k )∞ , k≥0
which implies that,
sup z(k )∞ ≤ γ k sup w(k )∞ . k≥0
k≥0
Then according to Definition 4, positive switched singular system (1) is uniformly asymptotically stable with an l∞ -gain performance. The proof is completed. Remark 4. In LPs (16)–(18), it is easy to design the state feedback controller for the switched singular system (1) by using LP technique, this is the advantage of transforming system (1) into system (14) equivalently in this paper. Remark 5. The stability for system (8) and performance analysis for system (14) can also be solved by transforming systems (8) and (14) into delay-free discrete-time standard positive switched systems by state-augmentation approach, respectively. But if the time delay is large or the order of system is high, the order of the transformed system will be very high, which will lead to lots of computations. The method proposed in this paper avoids the deficiency caused by the method of state-augmentation, and the orders of systems (8) and (14) are independent of the time delay. Remark 6. As stated in [28], the derivative matrix E is assumed to be switch-mode-independent in this paper. If E is also switchmode-dependent, then E is changed to Ei , ∀i ∈ N, and the transformation matrices P and Q should become Pi and Qi so that I 0 ]. 0 0
Pi Ei Qi = [ ri
In this case, the state of the transformed system becomes x(k ) = [xTi1 (k )
xTi2 (k )]T with xTi1 (k ) ∈ Rri and xTi2 (k ) ∈
Rn−ri , which means that there does not exist one common state space coordinate basis for different subsystems; thus it is rather complicated to discuss the transformed system. Hence, some assumptions for Ei (for example, Ei , i ∈ N, have the same right zero subspace) should be given so that Qi remains the same; in this case, the method presented in this paper is also valid. Nonetheless, the general case with E being switch-mode-dependent is an interesting problem for future investigation via other methods.
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Fig. 1. Switching signal.
4. Numerical examples In this section, three examples are provided to illustrate the results. Example 1. First, we consider the asymptotical stability with an l1 -gain performance for system (1). We consider system (1) with two modes, i.e., N = {1, 2}, and E = [10 00],
0.2 0.1
−6 , −6
0.3 A2 = 0.1
−3 , −2
A1 =
0.2 , 0.1
F1 =
0.1 , 0.1
C1 = [0.1 0.2],
D1 = 0.1,
0.2 = 0.0
0.3 , 0.2
0.2 F2 = , 0.1
C2 = [0.2 0.1],
D2 = 0.1.
Ad2
0.1 0.0
Ad1 =
Direct calculation shows that
det(sE − A1 ) = 6s − 0.6 = 0,
for
s = 0,
deg(det(sE − A1 )) = deg(6s − 0.6 ) = rankE = 1, det(sE − A2 ) = 2s − 0.3 = 0,
for
s = 0,
deg(det(sE − A2 )) = deg(2s − 0.3 ) = rankE = 1. Thus the switched singular system (1) is regular and causal. Furthermore, by simple computation, we obtain
A¯ 11 = 0.1000, A¯ 13 = 0.0167, A¯ 21 = 0.1500, A¯ 23 = 0.0500,
A¯ d11 = 0.1000, A¯ d13 = 0.0000,
A¯ d12 = 0.1000, F¯11 = 0.0000, A¯ d14 = 0.0167, F¯12 = 0.0167, A¯ d21 = 0.2000, A¯ d22 = 0.0000, F¯21 = 0.0500, A¯ d23 = 0.0000, A¯ d24 = 0.1000, F¯22 = 0.0500.
Therefore, by Theorem 1 system (1) is positive. Let d = 2, γ = 0.5, then solving the LPs (16)–(18) in Theorem 3, we can obtain the following feasible solutions
⎡
ν1
⎤
2.7231 ⎢2.8391⎥ =⎣ , 1.4550⎦ 1.5281
⎡
ν2
⎤
2.9665 ⎢2.9542⎥ =⎣ , 1.4759 ⎦ 1.4898
⎡
υ
⎤
0.6529 ⎢1.0258⎥ =⎣ . 1.0965⎦ 0.9456
According to Theorem 3, we can conclude that positive system (1) is asymptotically stable with an l1 -gain performance. The simulation results are shown in Figs. 1 and 2, where the initial condition is x(θ ) = [2 3]T , θ = −2, −1, 0, and the external disturbance is w(k ) = 0.5e−0.5k . Fig. 1. shows the switching signal σ (k) and Fig. 2 shows the state trajectory over 0–40. It is easy to see that the singular system is positive and asymptotically stable.
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103
Fig. 2. State trajectory of the given system.
Example 2. To illustrate better the effectiveness of the method in Example 1, we provide a numerical example, where the order and number of A are equal to three. We consider the asymptotical stability with an l1 -gain performance for system (1) with three 1 0 0 1 0], 0 0 0
modes, i.e., N = {1, 2, 3}, and E = [0
A1 =
A2 =
A3 =
0.2 0.16 0.1
0.16 0.2 0.1
−4.7 −4.7 , −3
0.3 0.15 0.1
0.15 0.3 0.1
−1 −1 , −2
0.3 0.15 0.1
0.15 0.3 0.1
−2.8 −2.8 , −2
Ad1 =
Ad2 =
C1 = [0.1 0.1 0.2],
0.1 0 0
0.1 0 0
0 0.1 0
0.1 0 0
Ad3 =
C2 = [0.2 0.2 0.1],
0 0.1 0
0.2 0.2 , 0.1
F1 =
0.3 0.3 , 0.2
F2 =
0 0.1 0
0.3 0.3 , 0.2
D1 = 0.1,
0.2 0.2 , 0.1
F3 =
0.2 0.1 , 0.1
D2 = 0.1,
0.2 0.2 , 0.1
D3 = 0.1,
C3 = [0.1 0.2 0.1].
Direct calculation shows that
det(sE − A1 ) = 3s2 − 0.26s − 1.6864 = 0,
for s = 0,
deg(det(sE − A1 )) = deg(3s − 0.26s − 1.6864 ) = rankE = 2, 2
det(sE − A2 ) = 2s2 − s + 0.105 = 0,
for s = 0,
deg(det(sE − A2 )) = deg(2s − s + 0.105 ) = rankE = 2, 2
det(sE − A3 ) = 2s2 − 0.64s + 0.051 = 0,
s = 0,
for
deg(det(sE − A3 )) = deg(2s2 − 0.64s + 0.051 ) = rankE = 2, Thus the switched singular system (1) is regular and causal. Furthermore, by simple computation, we obtain
0.0433 A¯ 11 = 0.0033
0.0033 , 0.0433
A¯ 13 = [0.0333 0.0333],
0.25 A¯ 21 = 0.1
0.1 , 0.25
A¯ 23 = [0.05 0.05],
0.1 A¯ d11 = 0
A¯ d13 = [0 0],
0.1 A¯ d21 = 0
A¯ d23 = [0 0],
0 , 0.1
A¯ d14 = 0.0333,
0 , 0.1
F¯12 = 0.0333,
0.2 A¯ d22 = , 0.2
A¯ d24 = 0.1,
0.0433 A¯ d12 = , 0.0433
F¯22 = 0.05,
0.0433 F¯11 = , 0.0433
0.15 F¯21 = , 0.15
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Fig. 3. Switching signal.
0.16 A¯ 31 = 0.01
0.01 , 0.16
0.1 A¯ d31 = 0
A¯ 33 = [0.05 0.05, A¯ d33 = [0 0],
0 , 0.1
A¯ d34 = 0.1,
0.02 A¯ d32 = , 0.02
0.06 F¯31 = , 0.06
F¯32 = 0.05.
Therefore, by Theorem 1 system (1) is positive. Let d = 2, γ = 0.5, then solving the LPs (16)–(18) in Theorem 3, we can obtain the following feasible solutions
⎡
ν1
⎤
⎡
0.8736 ⎢ 0.8751 ⎥ ⎢1.2044⎥ ⎥ =⎢ ⎢0.3869⎥, ⎣ ⎦ 0.3890 0.3819
ν2
⎤
1.3099 ⎢ 1.3183 ⎥ ⎢1.2247⎥ ⎥ =⎢ ⎢ 0.3518 ⎥, ⎣ ⎦ 0.3532 0.3795
⎡
ν3
⎤
0.9605 ⎢1.0594⎥ ⎢ 1.1421 ⎥ ⎥ =⎢ ⎢0.3940⎥, ⎣ ⎦ 0.3938 0.3994
⎡
υ
⎤
0.1445 ⎢ 0.1434 ⎥ ⎢0.6077⎥ ⎥ =⎢ ⎢0.2424⎥. ⎣ ⎦ 0.2445 0.1994
According to Theorem 3, we can conclude that positive system (1) is asymptotically stable with an l1 -gain performance. The simulation results are shown in Figs. 3 and 4, where the initial condition is x(θ ) = [0.5 3 4]T , θ = −2, −1, 0, and the external disturbance is w(k ) = 0.5e−0.5k . Fig. 3. shows the switching signal σ (k) and Fig. 4 shows the state trajectory over 0–40. It is easy to see that the singular system is positive and asymptotically stable. Example 3. We consider the asymptotical stability with an l∞ -gain performance for system (1). Consider system (1) with two modes, i.e., N = {1, 2}, and E = [10 00],
0.2 A1 = 0.1
−6 , −7
Ad1
0.3 0.1
−5 , −4
Ad2 =
A2 =
0.1 = 0.0
0.2 , 0.1
0.1 F1 = , 0.1
C1 = [0.1 0.2],
D1 = 0.1,
0.2 0.0
0.3 , 0.2
F2 =
0.2 , 0.1
C2 = [0.2 0.1],
D2 = 0.1.
Direct calculation shows that
det(sE − A1 ) = 7s − 0.8 = 0,
for
s = 0,
deg(det(sE − A1 )) = deg(7s − 0.8 ) = rankE = 1, det(sE − A2 ) = 4s − 0.7 = 0,
for
s = 0,
deg(det(sE − A2 )) = deg(4s − 0.7 ) = rankE = 1. Thus the switched singular system (1) is regular and causal. Furthermore, by simple computation, we obtain
A¯ 11 = 0.1143, A¯ 13 = 0.0143,
A¯ d11 = 0.1000, A¯ d12 = 0.1143, F¯11 = 0.0143, A¯ d13 = 0.0000, A¯ d14 = 0.0143, F¯12 = 0.0143,
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105
Fig. 4. State trajectory of the given system.
A¯ 21 = 0.1750, A¯ 23 = 0.0250,
A¯ d21 = 0.2000, A¯ d22 = 0.0500, F¯21 = 0.0750, A¯ d23 = 0.0000, A¯ d24 = 0.0500, F¯22 = 0.0250.
Therefore, by Theorem 1 system (1) is positive. Let d = 2, γ = 0.5, then solving the LPs (16) and (17), and (21) in Theorem 4, we can obtain the following feasible solutions
⎡
ν1
⎤
1.7497 ⎢ 1.9216 ⎥ =⎣ , 0.9340⎦ 1.0021
⎡
ν2
⎤
1.9330 ⎢1.9484⎥ =⎣ , 0.9412 ⎦ 0.9932
⎡
υ
⎤
0.4042 ⎢0.6874⎥ =⎣ . 0.7081⎦ 0.6234
According to Theorem 4, we can conclude that positive system (1) is asymptotically stable with an l∞ -gain performance. 5. Conclusions In this paper, we have investigated the problems of asymptotical stability, l1 -gain and l∞ -gain analysis for positive switched singular systems with time delay. Firstly, based on the singular value decomposition method, a necessary and sufficient condition for the positivity of the system has been derived. Then by introducing new state vectors, the positive switched singular system is equivalently transformed into a standard positive switched system, and sufficient conditions of asymptotical stability with an l1 -gain performance and an l∞ -gain performance for the time-delay positive switched singular system is obtained, respectively. Finally, three numerical examples are presented to demonstrate the effectiveness of the proposed results. Our future work will be focused on generalizing the obtained results to more general positive switched singular systems where the matrix E is switchmode-dependent. Acknowledgments This work was supported by the National Natural Science Foundation of China under Grant no. 61273120 and the Postgraduate Innovation Project of Jiangsu Province under Grant no. KYLX-0371. References [1] [2] [3] [4]
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Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Stability, l1 -gain and l∞ -gain analysis for discrete-time positive switched singular delayed systems Shuo Li, Zhengrong Xiang∗ School of Automation, Nanjing University of Science and Technology, Nanjing 210094, People’s Republic of China
a r t i c l e
i n f o
Keywords: Singular systems Positive switched systems l1 -gain performance l∞ -gain performance Time delay Switching Lyapunov function
a b s t r a c t In this paper, the problems of stability, l1 -gain and l∞ -gain analysis for positive switched singular systems with time delay are addressed. Both l1 -gain and l∞ -gain performance are analyzed. Firstly, a necessary and sufficient condition of positivity for the system is established by using the singular value decomposition method. Then by introducing new state vectors, the positive switched singular system is transformed into a standard positive switched system, and sufficient conditions for the positive switched singular delayed system to be asymptotically stable with an l1 -gain performance and an l∞ -gain performance are obtained, respectively. Finally, three numerical examples are presented to demonstrate the proposed results. © 2015 Elsevier Inc. All rights reserved.
1. Introduction Singular systems, which are also referred to as descriptor systems, implicit systems or differential-algebraic systems, have constantly gained a great interest, due to its capacity to describe the dynamic and algebraic relationships between state variables simultaneously [1]. Singular systems are found in a broad spectrum of applications including power systems, economics systems, robotic systems, chemical processes, biological systems, etc. [2–8]. In practice, physical systems involve variables that have nonnegative property, such as population levels, absolute temperature, and so on. Such systems are referred to as positive systems [9,10], whose states and outputs are nonnegative whenever the initial conditions and inputs are nonnegative. Positive systems are frequently encountered in various fields, for instance, pharmacokinetics [11], chemical reaction [12] and internet congestion control [13]. For positive systems, one often uses the linear co-positive Lyapunov functional instead of the traditional quadratic Lyapunov functional for stability analysis, due to that the former will yield less conservative stability conditions [14]. In addition l1 -gain or l∞ -gain rather than l2 -gain is sometimes more appropriate to be adopted as a system performance measure [15,16]. l2 -norm represents the energy, while l1 -norm and l∞ -norm account for the sum of quantities and the maximum of quantities, respectively [17]. Stability and performance analysis for standard positive systems have been studied by many authors [18–20], and some results on positive singular systems have also been obtained [21–23]. On the other hand, many practical systems are always subjected to abrupt variation in their structures and parameters, such as failures and repairs of the components, changes in the interconnections of subsystems, and sudden environment changes. One effective method of characterizing the abrupt changes is by using the switched system approach [24–26]. There are fruitful results on analysis and synthesis of switched systems [27–30]. When positive singular systems experience the switching-type changes in their parameters, it becomes difficult to study this kind of positive switched singular systems due to the coupling ∗
Corresponding author. Tel.: +86 13951012297. E-mail address: [email protected], [email protected] (Z. Xiang).
http://dx.doi.org/10.1016/j.amc.2015.11.053 0096-3003/© 2015 Elsevier Inc. All rights reserved.
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between the switching, the algebraic constraints in singular models and the non-negativity limits of state valuables. Therefore, the positive switched singular system considered here is of practical importance, and meanwhile poses theoretical challenges in its formal analysis. There are only a few recent publications investigating positive switched singular systems [31,32]. Xia et al. [31] dealt with the problem of state feedback controller design for singular positive Markovian jump systems with partly known transition rates. In [32], the problems of stability for a class of switched positive descriptor systems with average dwell time switching were investigated. However, Xia et al. [31] and Qi and Gao [32] only focused on the autonomous case without taking time delay and exogenous disturbance into account. As time delay and exogenous disturbance are frequently encountered in practice, it is necessary and significant to further investigate positive switched singular systems with those two kinds of phenomena. When taking time delays and exogenous disturbance into account, the problem of choosing an appropriate mode-dependent co-positive Lyapunov function and how to analyze the stability and system performance will be more complicated and challenging. To the best of authors knowledge, no relevant work that considers this kind of system has been published. Therefore, the main objective of this work is to investigate the stability, the l1 -gain and l∞ -gain performance for positive switched singular systems with time delay. The main contributions of this paper are as follows: (1) By using the singular value decomposition method, we propose a necessary and sufficient condition of positivity for switched singular delayed systems; (2) By introducing new state vectors, we derive a sufficient delay-dependent criterion with regard to the considered system to be uniformly asymptotically stable; (3) In consideration of the exogenous disturbance, the disturbance attenuation performance (the l1 -gain and l∞ -gain performance in this paper) is analyzed for the positive switched singular system with time delay; (4) All the conditions obtained in this paper are presented in terms of linear programming (LP). The remaining of the paper is as follows. The problem formulation and some necessary lemmas are provided in Section 2. In Section 3, the positivity, stability and performance analysis are developed. Three numerical examples are presented to demonstrate the obtained results in Section 4. In Section 5, concluding remarks are given. Notations: [A]ij denotes the entry of matrix A in the ith row and jth column; A 0( ≺, , ≺) means that all entries of matrix A are nonnegative (non-positive, positive, negative); AB(A B) means that A − B 0(A − B0 ); AT means the transpose of matrix A; det(A ) denotes the determinant of matrix A; deg(· ) denotes the degree of a polynomial; rank(A ) denotes the rank of matrix A; R(R+ ) is the set of all real (positive real) numbers; Rn Rn+ is the n-dimensional real (positive real) vector space; Rm × n is the set of all m × n-dimensional real matrices; Z + refers to the set of all positive integers; 1n ∈ Rn denotes a column vector with each entry equal to 1; The vector 1-norm is denoted by x1 = nk=1 |xk | with xk being the kth element of x ∈ Rn ; Given ν : R → Rn , the ∞ l1 -norm is defined by νl1 = k=k ν (k )1 . We say ν (k) ∈ l1 [k0 , ∞) if νl1 < ∞; The vector ∞-norm is denoted by x∞ = 0 max1≤k≤n |xk | with xk being the kth element of x ∈ Rn ; Given ν : R → Rn , the l∞ -norm is defined by νl∞ = supk≥0 ν (k )∞ . We say ν (k) ∈ l∞ [k0 , ∞) if νl∞ < ∞. Matrices and vectors, if their dimensions are not explicitly stated, are assumed to have compatible dimensions for algebraic operations. 2. Problem formulation We consider the following discrete-time linear switched singular system with time delay,
⎧ ⎨Ex(k + 1 ) = Aσ (k) x(k ) + Adσ (k) x(k − d ) + Fσ (k) w(k ), z(k ) = Cσ (k) x(k ) + Dσ (k) w(k ), ⎩ x(θ ) = ϕ (θ ), θ = −d, −d + 1, . . . , 0,
(1)
where x(k) ∈ Rn is the state vector, w(k ) ∈ Rq is the disturbance input which belongs to l1 (or l∞ ), z(k) ∈ Rm is the controlled output. d is the constant integer time delay and ϕ (θ ) is a given discrete vector-valued initial condition of the system. {σ (k), k ≥ 0} is a switching signal taking values in a finite set N = {1, 2, . . . , N} with N being the number of subsystems, moreover, σ (k ) = i denotes the ith subsystem is activated, i ∈ N . Ai , Adi , Fi , Ci and Di , i ∈ N, are known constant matrices with appropriate dimensions. The matrix E ∈ Rn × n is singular and rankE = r < n. Since rankE = r < n, then there exist non-singular matrices P, Q ∈ Rn × n such that PEQ = [I0r 00]. In this paper, for simplicity, let E = [I0r 00] and other system matrices have the following form, ∀i ∈ N :
Ai1 Ai = Ai3
Ai2 , Ai4
Adi1 Adi = Adi3
Adi2 , Adi4
Fi =
Fi1 , Fi2
Ci = [Ci1 Ci2 ],
(2)
where x(k ) = [xx1 ((kk ))], x1 (k) ∈ Rr and x2 (k ) ∈ Rn−r . 2
Definition 1. [2] Consider system (1), (i) For a given i ∈ N, the pair (E, Ai ) is said to be regular if det(zE − Ai ) = 0; (ii) For a given i ∈ N, the pair (E, Ai ) is said to be causal if it is regular and degree(det(zE − Ai )) = rankE; Remark 1. As in [3], the regularity and causality of the pair (E, Ai ), ∀i ∈ N, in system (1) ensures the existence and uniqueness of the solution to this system with any given initial conditions ϕ (θ ), θ = −d, −d + 1, . . . , 0.
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Definition 2. [9] System (1) is said to be positive if, for any initial conditions ϕ (θ ) 0, w(k )0 and any switching signals σ (k), we have x(k) 0 and z(k) 0 for all k ≥ 0. Lemma 1. [3] System (1) with (2) is causal if and only if Ai4 is nonsingular, ∀i ∈ N . In the light of Lemma 1, when system (1) with (2) is regular and causal, it can be rewritten as:
⎧ x1 (k + 1 ) = A¯ σ (k)1 x1 (k ) + A¯ dσ (k)1 x1 (k − d ) + A¯ dσ (k)2 x2 (k − d ) + F¯σ (k)1 w(k ), ⎪ ⎪ ⎪ ⎪ ⎪ ⎨x2 (k ) = A¯ σ (k)3 x1 (k ) + A¯ dσ (k)3 x1 (k − d ) + A¯ dσ (k)4 x2 (k − d ) + F¯σ (k)2 w(k ), z(k ) = Cσ (k)1 x1 (k ) + Cσ (k)2 x2 (k ) + Dσ (k) w(k ), ⎪ ⎪ ⎪ θ = −d, −d + 1, . . . , 0, x 1 (θ ) = ϕ1 (θ ), ⎪ ⎪ ⎩ x2 (θ ) = ϕ2 (θ ), θ = −d, −d + 1, . . . , 0,
(3)
where for each i ∈ N,
A¯ i1 = Ai1 − Ai2 A−1 A , i4 i3
A¯ di1 = Adi1 − Ai2 A−1 A , i4 di3
A¯ di2 = Adi2 − Ai2 A−1 A , i4 di4 A¯ i3 = −A−1 A , i4 i3
F¯i1 = Fi1 − Ai2 A−1 F , i4 i2
A¯ di3 = −A−1 A , i4 di3
A¯ di4 = −A−1 A , i4 di4
F¯i2 = −A−1 F . i4 i2
Lemma 2. [23] Assume that (E, Ai ), ∀i ∈ N, is regular and causal. Then system (1) with (2) is positive if and only if system (3) is positive. Definition 3. [28] System (1) with w(k ) = 0 is said to be (i) regular, causal, if the pair (E, Ai ), ∀i ∈ N, is regular, causal; (ii) uniformly asymptotically stable, if for any ε > 0, there is a δ (ε ) > 0 such that for arbitrary switching signal σ (k), sup−d≤θ ≤0 ϕ (θ ) < δ implies x(k) < ε for k ≥ 0 and there is also a δ > 0 such that for arbitrary switching signal σ (k), sup−d≤θ ≤0 ϕ (θ ) < δ implies x(k) → 0 as k → ∞. Definition 4. Assume that system (1) with w(k ) = 0 is regular, causal, uniformly asymptotically stable. For a given scalar γ > 0 and the initial condition ϕ (k ) = 0, k = −d, −d + 1, . . . , 0, system (1) is said to be uniformly asymptotically stable with an l1 -gain performance if the following condition is satisfied: ∞
z ( k ) 1 < γ
k=0
∞
w(k )1 , ∀w(k ) ∈ l1 ;
k=0
system (1) is said to be uniformly asymptotically stable with an l∞ -gain performance if the following condition is satisfied:
sup z(k )∞ < γ sup w(k )∞ , k≥0
k≥0
∀w(k ) ∈ l∞ .
(4)
In this paper, our aim is to investigate the positivity, asymptotical stability, l1 -gain performance, and l∞ -gain performance for the switched singular delayed system (1). 3. Main results This section will focus on the analysis of positivity, stability, l1 -gain and l∞ -gain performance. 3.1. Positivity analysis Before analyzing the stability and system performance, we firstly present a necessary and sufficient condition for the positivity of system (1). Theorem 1. Assume that (E, Ai ), ∀i ∈ N, is regular and causal. System (1) with (2) is positive if and only if A¯ i1 0, A¯ di1 0, A¯ di2 0, F¯i1 0, A¯ i3 0, A¯ di3 0, A¯ di4 0, F¯i2 0, Ci1 0, Ci2 0, Di 0, ∀i ∈ N, where A¯ i1 , A¯ di1 , A¯ di2 , F¯i1 , A¯ i3 , A¯ di3 , A¯ di4 , F¯i2 , Ci1 , Ci2 are defined in (3). Proof. System (1) with (2) is regular and causal due to that (E, Ai ), ∀i ∈ N, is regular and causal. Sufficiency. We will show that system (1) is positive. We first prove that the solution x(k ) = [xT1 (k ) d]. It follows from Lagrange’s formula and the first equation of (3) that
x1 (k ) = A¯ ki1 x1 (0 ) +
k−1
θ =0
xT2 (k )]T is positive on [0,
θ −1 (A¯ x (θ − d ) + A¯ x (θ − d ) + F¯ w(θ )). A¯ k− i1 di1 1 di2 2 i1
Since A¯ i1 0, it follows that A¯ ki1 0, k ≥ 0. From x1 (0) 0, we have A¯ ki1 x1 (0 )0. Moreover, since A¯ di1 0, A¯ di2 0, F¯i1 0, by the same way we have A¯ k−θ −1 A¯ di1 x1 (θ − d )0, A¯ k−θ −1 A¯ di2 x2 (θ − d )0, A¯ k−θ −1 F¯i1 w(θ )0, for all 0 ≤ θ ≤ k − 1 ≤ d. Hence, we have x1 (k) 0, ∀k ∈ [0, d].
i1
i1
i1
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On the other hand, according to A¯ i3 0, A¯ di3 0, A¯ di4 0, F¯i2 0, x1 (k) 0, x1 (k − d )0, x2 (k − d )0, k ∈ [0, d] and from the second equation of (3) we have x2 (k) 0, ∀k ∈ [0, d]. Thus, the solution x(k) of system (1) is positive on [0, d]. Moreover, from Ci1 0, Ci2 0, Di 0 and the third equation of (3), we can directly get z(k) 0 on [0, d]. Using the step method, we can extend the consideration for the intervals [d, 2d], [2d, 3d], etc. Repeating this procedure iteratively, it follows that x(k) 0, z(k) 0, ∀k ≥ 0. Necessity. Suppose that system (1) is positive. Consider the first equation of (3) on [0, d]. We first prove A¯ i1 0. ad absurdum, A¯ i1 is not nonnegative. That is, suppose there exist some indices p, q ∈ {1, 2, . . . , r} such that [A¯ i1 ] pq < 0. Take the initial condition:
ϕ1 (0 ) = eq , ϕ1 (θ ) = 0, θ ∈ [−d, 0 ); ϕ2 (θ ) = 0, w(θ ) = 0, θ ∈ [−d, 0], where eq , q = 1, 2, . . . , r denotes the qth vector of canonical basis in Rr , and the solution on [0, d] is x1 (1 ) = A¯ i1 eq , which implies that [x1 (1 )] p = [A¯ i1 ] pq < 0, this contradicts the fact that x1 (1) 0. Therefore, we conclude that A¯ i1 0. We now prove A¯ di1 0, ad absurdum, suppose there exist some indices p, q ∈ {1, 2, . . . , r} such that [A¯ di1 ] pq < 0. Take the initial condition:
ϕ1 (θ ) = 0, θ ∈ (−d, 0], ϕ1 (−d ) = eq ; ϕ2 (θ ) = 0, w(θ ) = 0, θ ∈ [−d, 0], where eq , q = 1, 2, . . . , r denotes the qth vector of canonical basis in Rr , and the solution is x1 (1 ) = A¯ di1 eq , which implies that [x1 (1 )] p = [A¯ di1 ] pq < 0, this is a contradiction to the fact that x1 (1) 0. Therefore, A¯ di1 0. To prove A¯ di2 0, we choose the initial condition:
ϕ2 (θ ) = 0, θ ∈ (−d, 0], ϕ2 (−d ) = eq ; ϕ1 (θ ) = 0, w(θ ) = 0, θ ∈ [−d, 0], where eq , q = r + 1, r + 2, . . . , n denotes the qth vector of canonical basis in Rn−r , then by the same argument we obtain A¯ di2 0. To prove F¯i1 0, we choose the initial condition:
ϕ1 (θ ) = 0, ϕ2 (θ ) = 0, w(θ ) = eq , θ ∈ [−d, 0], where eq , q = 1, 2, . . . , n denotes the qth vector of canonical basis in Rn , then by the same argument we obtain F¯i1 0. We now show that A¯ i3 0, A¯ di3 0, A¯ di4 0, F¯i2 0. To this end, consider the second equation of (3) on [0, d], choosing the initial condition:
ϕ1 (0 ) = eq , ϕ1 (θ ) = 0, ϕ2 (θ ) = 0, θ ∈ [−d, 0 ); w(θ ) = 0, θ ∈ [−d, 0], we have x2 (0 ) = A¯ i3 eq 0, q = 1, 2, . . . , r, then it gives A¯ i3 0. Choosing the initial condition:
ϕ1 (−d ) = eq , ϕ1 (θ ) = 0, θ ∈ (−d, 0]; ϕ2 (θ ) = 0, θ ∈ [−d, 0 ); w(θ ) = 0, θ ∈ [−d, 0], the solution x2 (0 ) = A¯ di3 eq 0, q = 1, 2, . . . , r is obtained, then it gives A¯ di3 0. Choosing the initial condition:
ϕ2 (−d ) = eq , ϕ2 (θ ) = 0, θ ∈ (−d, 0 ); ϕ1 (θ ) = 0, w(θ ) = 0, θ ∈ [−d, 0], we have x2 (0 ) = A¯ di4 eq 0, q = r + 1, r + 2, . . . , n, then it gives A¯ di4 0. Choosing the initial condition:
ϕ2 (θ ) = 0, θ ∈ [−d, 0 ); ϕ1 (θ ) = 0, w(θ ) = eq , θ ∈ [−d, 0], we may get x2 (0 ) = F¯i2 eq 0, q = 1, 2, . . . , n, then it gives F¯i2 0. Furthermore, we show that Ci1 0, Ci2 0, Di 0. Considering the third equation of (3) on [0, d], choosing the initial condition:
ϕ1 (0 ) = eq , ϕ1 (θ ) = 0, θ ∈ [−d, 0 ); ϕ2 (θ ) = 0, w(θ ) = 0, θ ∈ [−d, 0], we can obtain z(0 ) = Ci1 eq , q = 1, 2, . . . , r, then it gives Ci1 0. Choosing the initial condition:
ϕ2 (0 ) = eq , ϕ2 (θ ) = 0, θ ∈ [−d, 0 ); ϕ1 (θ ) = 0, w(θ ) = 0, θ ∈ [−d, 0], we can get z(0 ) = Ci2 eq 0, q = r + 1, r + 2, . . . , n, then it gives Ci2 0. Choosing the initial condition:
ϕ1 (θ ) = 0, ϕ2 (θ ) = 0, w(θ ) = eq , θ ∈ [−d, 0], the solution z(0 ) = Di eq 0, q = 1, 2, . . . , n is obtained, then it gives Di 0. This completes the proof. In the sequence, 3.2. Stability analysis In this section, based on Theorem 1, some sufficient conditions will be provided to ensure the asymptotical stability of the positive system (1) (w(k ) = 0), namely, the following system (5):
Ex(k + 1 ) = Aσ (k) x(k ) + Adσ (k) x(k − d ), x(θ ) = ϕ (θ ), θ = −d, −d + 1, . . . , 0,
(5)
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where E, Ai , and Adi , i ∈ N, satisfy (2). It is easy to see that system (5) can be written as:
x1 ( k + 1 ) = Aσ ( k ) 1 x1 ( k ) + A σ ( k ) 2 x2 ( k ) + A d σ ( k ) 1 x1 ( k − d ) + A d σ ( k ) 2 x2 ( k − d ) , 0 = Aσ ( k ) 3 x 1 ( k ) + A σ ( k ) 4 x 2 ( k ) + A d σ ( k ) 3 x 1 ( k − d ) + A d σ ( k ) 4 x 2 ( k − d ) .
(6)
By introducing the following vector
xˆ(k ) = [xT1 (k ), xT2 (k − 1 ), xT1 (k − 1 ), xT2 (k − 2 )]T ,
(7)
system (6) is rewritten as
Eˆσ (k) xˆ(k + 1 ) = Aˆ σ (k) xˆ(k ) + Aˆ dσ (k) xˆ(k − d + 1 ), where for each i ∈ N,
⎡
Ir 0 ⎢ Eˆi = ⎣ 0 0
−Ai2 −Ai4 0 0
0 0 Ir 0
0 0 0
⎤ ⎥ ⎦,
In−r
⎡
Ai1 A ⎢ Aˆ i = ⎣ i3 Ir 0
0 0 0 In−r
(8)
0 0 0 0
⎤
0 0⎥ , 0⎦ 0
⎡
0 0 ⎢ Aˆ di = ⎣ 0 0
Adi2 Adi4 0 0
Adi1 Adi3 0 0
⎤
0 0⎥ . 0⎦ 0
(9)
Remark 2. By Definitions 1 and 3, one can see that the regularity, causality and uniform asymptotic stability of system (5) are equivalent to those of system (6). Lemma 3. System (5) is regular, causal and uniformly asymptotically stable, if and only if system (8) is a switched standard linear system (i.e. the matrix Eˆi is nonsingular for each i ∈ N ) and uniformly asymptotically stable. Proof. First, from Remark 2, that system (5) is regular, causal and uniformly asymptotically stable is equivalent to that system (6) is regular, causal and uniformly asymptotically stable. From Definition 1 and Lemma 1, that system (6) is regular, causal is equivalent to that matrix Ai4 is nonsingular for each i ∈ N; then from (9), it follows that the matrix Ai4 is nonsingular is equivalent to that the matrix Eˆi in system (8) is nonsingular. From Definition 3, that system (6) is uniformly asymptotically stable is equivalent to that x1 (k) → 0, x2 (k) → 0 as k → ∞; then from (7), x1 (k) → 0, x2 (k) → 0 as k → ∞, it is shown that system (8) is uniformly asymptotically stable. Thus, the conclusion is obtained. The proof is completed. Remark 3. Based on (2) and (7), system (5) is transformed into system (8) with delay d − 1. Based on Lemma 3, the regularity, causality and uniform asymptotical stability for system (5) can be analyzed by investigating the uniform asymptotic stability for the standard linear switched system (8). Theorem 2. Assume that (E, Ai ), ∀i ∈ N, is regular and causal. System (5) with (2) is positive and uniformly asymptotically stable, if 2n conditions in Theorem 1 are satisfied, and for each i ∈ N, there exist vectors νi ∈ R2n + and υ ∈ R+ satisfying the following LPs:
Aˆ Ti Eˆi−T ν j − νi + υ ≺ 0,
(10)
Aˆ Tdi Eˆi−T ν j − υ ≺ 0.
(11)
Proof. By Theorem 1, system (5) with (2) is regular, causal and positive. Based on Lemma 3, to prove that system (5) is uniformly asymptotically stable, it only required to prove that system (8) is uniformly asymptotically stable. We construct the following Lyapunov functional candidate
V (k ) = xˆT (k )νσ (k) +
k−1
xˆT (l )υ ,
(12)
l=k−d+1
where vectors ν σ (k) 0, υ 0. Let the mode at time k and k + 1 be i and j, respectively, that is σ (k ) = i and σ (k + 1 ) = j for any i, j ∈ N. One can obtain that
V (k ) = V (k + 1 ) − V (k ) = xˆT (k + 1 )ν j +
k
xˆT (l )υ − xˆT (k )νi −
l=k−d+2
k−1
xˆT (l )υ
l=k−d+1
= (xˆT (k )Aˆ Ti Eˆi−T + xˆT (k − d + 1 )Aˆ Tdi Eˆi−T )ν j − xˆT (k )νi + xˆT (k )υ − xˆT (k − d + 1 )υ = xˆT (k )[Aˆ Ti Eˆi−T ν j − νi + υ ] + xˆT (k − d + 1 )[Aˆ Tdi Eˆi−T ν j − υ ]. From (10) and (11), it is obtained that V(k) ≤ 0. Then system (8) is uniformly asymptotically stable. The proof is completed.
(13)
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3.3. l1 -gain and l∞ -gain analysis In consideration of the exogenous disturbance, to analyze the disturbance attenuation performance, we consider the l1 -gain performance and l∞ -gain performance for positive system (1) in this section, respectively. From (2) and (7), positive system (1) is rewritten as
Eˆσ (k) xˆ(k + 1 ) = Aˆ σ (k) xˆ(k ) + Aˆ dσ (k) xˆ(k − d + 1 ) + Fˆσ (k) w(k ), z(k ) = Cˆ1σ (k) xˆ(k ) + Cˆ2σ (k) xˆ(k + 1 ) + Dσ (k) w(k ),
where for each i ∈ N,
Fˆi = Fi1T
Fi2T
T
0 0 ,
Cˆ1i = [Ci1 0 0 0],
(14)
Cˆ2i = [0 Ci2 0 0].
(15)
A sufficient condition for the uniformly asymptotical stability with an l1 -gain performance of positive system (1) is proposed in the following theorem. Theorem 3. Assume that (E, Ai ), ∀i ∈ N, is regular and causal. For a given scalar γ > 0, system (1) with (2) is positive and uniformly asymptotically stable with an l1 -gain performance, if conditions in Theorem 1 are satisfied, and for each i ∈ N, there exist vectors 2n νi ∈ R2n + and υ ∈ R+ satisfying the following LPs: T T Aˆ Ti Eˆi−T ν j − νi + υ + Cˆ1i 1m + Aˆ Ti Eˆi−T Cˆ2i 1m ≺ 0,
(16)
T Aˆ Tdi Eˆi−T ν j − υ + Aˆ Tdi Eˆi−T Cˆ2i 1m ≺ 0,
(17)
T 1m + DTi 1m − γ 1q ≺ 0. FˆiT Eˆi−T ν j + FˆiT Eˆi−T Cˆ2i
(18)
Proof. First, note that (16) and (17) implies (10) and (11). When w(k ) = 0, by Theorem 2, system (1) with (2) is regular, causal, positive and uniformly asymptotically stable. Next, to prove that positive system (1) satisfies the l1 -gain performance for any w(k ) ∈ l1 , it is only required to prove system (14) satisfies the l1 -gain performance. Construct Lyapunov functional (12), and let the mode at time k and k + 1 be i and j, that is σ (k ) = i and σ (k + 1 ) = j. Similar to the proof in Theorem 2, it follows that
V (k ) = V (k + 1 ) − V (k ) = (xˆT (k )Aˆ Ti + xˆT (k − d + 1 )Aˆ Tdi + wT (k )FˆiT )Eˆi−T ν j − xˆT (k )νi + xˆT (k )υ − xˆT (k − d + 1 )υ = xˆT (k )[Aˆ Ti Eˆi−T ν j − νi + υ ] + xˆT (k − d + 1 )[Aˆ Tdi Eˆi−T ν j − υ ] + wT (k )[FˆiT Eˆi−T ν j ].
(19)
When the initial condition xˆ(k ) = 0, k = −d + 1, . . . , −1, 0, which yields V (0 ) = 0, then from (19), it follows that T
(z(k )1 − γ w(k )1 ) =
k=0
T
(z(k )1 − γ w(k )1 + V (k )) −
k=0
=
T
V (k )
k=0
T
(z(k )1 − γ w(k )1 + V (k )) − V (T + 1 ) + V (0 )
k=0
≤
T [z(k )1 − γ w(k )1 + V (k )] k=0
≤
T T T [xˆT (k )Cˆ1i 1m + (xˆT (k )Aˆ Ti + xˆT (k − d + 1 )Aˆ Tdi + wT (k )FˆiT )Eˆi−T Cˆ2i 1m + wT (k )DTi 1m k=0
− γ wT (k )1q + V (k )] ≤
T
{xˆT (k )[Aˆ Ti Eˆi−T ν j − νi + υ + Cˆ1iT 1m + Aˆ Ti Eˆi−T Cˆ2iT 1m ]
k=0 T + xˆT (k − d + 1 )[Aˆ Tdi Eˆi−T ν j − υ + Aˆ Tdi Eˆi−T Cˆ2i 1m ] T 1m + DTi 1m − γ 1q ]}. + wT (k )[FˆiT Eˆi−T ν j + FˆiT Eˆi−T Cˆ2i
Therefore, it is obtained from (16)–(18) that for any w(k ) ∈ l1 , The proof is completed.
∞
k=0
z(k )1 < γ
∞
k=0
(20)
w(k )1 .
In the following, a sufficient condition for the uniformly asymptotical stability with an l∞ -gain performance of positive system (1) is proposed.
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Theorem 4. Assume that (E, Ai ), ∀i ∈ N, is regular and causal. For a given scalar γ > 0, system (1) with (2) is positive and uniformly asymptotically stable with an l∞ -gain performance, if conditions in Theorem 1 are satisfied, and for each i ∈ N, there exist vectors 2n νi ∈ R2n + and υ ∈ R+ such that (16) and (17) and the following LP holds: T 1Tq FˆiT Eˆi−T ν j + 1Tq FˆiT Eˆi−T Cˆ2i 1m + 1Tq DTi 1m − γ ≺ 0.
(21)
Proof. First, note that (16) and (17) implies (10) and (11). When w(k ) = 0, by Theorem 2, system (1) with (2) is regular, causal, positive and uniformly asymptotically stable. Next, to prove that positive system (1) satisfies an l∞ -gain performance for any w(k ) ∈ l∞ , it is only need to prove system (14) satisfies an l∞ -gain performance. From (14), we have
z(k )∞ ≤ Cˆ1i xˆ(k )∞ + Cˆ2i Eˆi−1 Aˆ i xˆ(k )∞ + Cˆ2i Eˆi−1 Aˆ di xˆ(k − d + 1 )∞ + Cˆ2i Eˆi−1 Fˆi w(k )∞ + Di w(k )∞ ≤ Cˆ1i xˆ(k ) + Cˆ2i Eˆ −1 Aˆ i xˆ(k ) + Cˆ2i Eˆ −1 Aˆ di xˆ(k − d + 1 ) + Cˆ2i Eˆ −1 Fˆi w(k ) + Di w(k )1 1
i
1
i
1
i
1
T T T ≤ xˆT (k )Cˆ1i 1m + xˆT (k )Aˆ Ti Eˆi−T Cˆ2i 1m + xˆT (k − d + 1 )Aˆ Tdi Eˆi−T Cˆ2i 1m T + w(k )∞ 1Tq FˆiT Eˆi−T Cˆ2i 1m + w(k )∞ 1Tq DTi 1m .
(22)
Construct Lyapunov functional (12), and let σ (k ) = i and σ (k + 1 ) = j. Similar to the proof in Theorem 3, it follows that,
V (k ) + z(k )∞ − γ w(k )∞ ≤ xˆT (k )[Aˆ Ti Eˆi−T ν j − νi + υ + Cˆ1iT 1m + Aˆ Ti Eˆi−T Cˆ2iT 1m ] T + xˆT (k − d + 1 )[Aˆ Tdi Eˆi−T ν j − υ + Aˆ Tdi Eˆi−T Cˆ2i 1m ] T 1m + 1Tq DTi 1m − γ ] + w(k )∞ [1Tq FˆiT Eˆi−T ν j + 1Tq FˆiT Eˆi−T Cˆ2i
≤ 0.
(23)
Then it is derived that
V (k ) ≤ V (0 ) −
k−1
(z(s )∞ − γ w(s )∞ ).
(24)
s=0
Notice that
w(k )∞ ≤ sup w(k )∞ . k≥0
Under zero initial conditions, from (24), we have
z(k )∞ ≤ γ k sup w(k )∞ , k≥0
which implies that,
sup z(k )∞ ≤ γ k sup w(k )∞ . k≥0
k≥0
Then according to Definition 4, positive switched singular system (1) is uniformly asymptotically stable with an l∞ -gain performance. The proof is completed. Remark 4. In LPs (16)–(18), it is easy to design the state feedback controller for the switched singular system (1) by using LP technique, this is the advantage of transforming system (1) into system (14) equivalently in this paper. Remark 5. The stability for system (8) and performance analysis for system (14) can also be solved by transforming systems (8) and (14) into delay-free discrete-time standard positive switched systems by state-augmentation approach, respectively. But if the time delay is large or the order of system is high, the order of the transformed system will be very high, which will lead to lots of computations. The method proposed in this paper avoids the deficiency caused by the method of state-augmentation, and the orders of systems (8) and (14) are independent of the time delay. Remark 6. As stated in [28], the derivative matrix E is assumed to be switch-mode-independent in this paper. If E is also switchmode-dependent, then E is changed to Ei , ∀i ∈ N, and the transformation matrices P and Q should become Pi and Qi so that I 0 ]. 0 0
Pi Ei Qi = [ ri
In this case, the state of the transformed system becomes x(k ) = [xTi1 (k )
xTi2 (k )]T with xTi1 (k ) ∈ Rri and xTi2 (k ) ∈
Rn−ri , which means that there does not exist one common state space coordinate basis for different subsystems; thus it is rather complicated to discuss the transformed system. Hence, some assumptions for Ei (for example, Ei , i ∈ N, have the same right zero subspace) should be given so that Qi remains the same; in this case, the method presented in this paper is also valid. Nonetheless, the general case with E being switch-mode-dependent is an interesting problem for future investigation via other methods.
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Fig. 1. Switching signal.
4. Numerical examples In this section, three examples are provided to illustrate the results. Example 1. First, we consider the asymptotical stability with an l1 -gain performance for system (1). We consider system (1) with two modes, i.e., N = {1, 2}, and E = [10 00],
0.2 0.1
−6 , −6
0.3 A2 = 0.1
−3 , −2
A1 =
0.2 , 0.1
F1 =
0.1 , 0.1
C1 = [0.1 0.2],
D1 = 0.1,
0.2 = 0.0
0.3 , 0.2
0.2 F2 = , 0.1
C2 = [0.2 0.1],
D2 = 0.1.
Ad2
0.1 0.0
Ad1 =
Direct calculation shows that
det(sE − A1 ) = 6s − 0.6 = 0,
for
s = 0,
deg(det(sE − A1 )) = deg(6s − 0.6 ) = rankE = 1, det(sE − A2 ) = 2s − 0.3 = 0,
for
s = 0,
deg(det(sE − A2 )) = deg(2s − 0.3 ) = rankE = 1. Thus the switched singular system (1) is regular and causal. Furthermore, by simple computation, we obtain
A¯ 11 = 0.1000, A¯ 13 = 0.0167, A¯ 21 = 0.1500, A¯ 23 = 0.0500,
A¯ d11 = 0.1000, A¯ d13 = 0.0000,
A¯ d12 = 0.1000, F¯11 = 0.0000, A¯ d14 = 0.0167, F¯12 = 0.0167, A¯ d21 = 0.2000, A¯ d22 = 0.0000, F¯21 = 0.0500, A¯ d23 = 0.0000, A¯ d24 = 0.1000, F¯22 = 0.0500.
Therefore, by Theorem 1 system (1) is positive. Let d = 2, γ = 0.5, then solving the LPs (16)–(18) in Theorem 3, we can obtain the following feasible solutions
⎡
ν1
⎤
2.7231 ⎢2.8391⎥ =⎣ , 1.4550⎦ 1.5281
⎡
ν2
⎤
2.9665 ⎢2.9542⎥ =⎣ , 1.4759 ⎦ 1.4898
⎡
υ
⎤
0.6529 ⎢1.0258⎥ =⎣ . 1.0965⎦ 0.9456
According to Theorem 3, we can conclude that positive system (1) is asymptotically stable with an l1 -gain performance. The simulation results are shown in Figs. 1 and 2, where the initial condition is x(θ ) = [2 3]T , θ = −2, −1, 0, and the external disturbance is w(k ) = 0.5e−0.5k . Fig. 1. shows the switching signal σ (k) and Fig. 2 shows the state trajectory over 0–40. It is easy to see that the singular system is positive and asymptotically stable.
S. Li, Z. Xiang / Applied Mathematics and Computation 275 (2016) 95–106
103
Fig. 2. State trajectory of the given system.
Example 2. To illustrate better the effectiveness of the method in Example 1, we provide a numerical example, where the order and number of A are equal to three. We consider the asymptotical stability with an l1 -gain performance for system (1) with three 1 0 0 1 0], 0 0 0
modes, i.e., N = {1, 2, 3}, and E = [0
A1 =
A2 =
A3 =
0.2 0.16 0.1
0.16 0.2 0.1
−4.7 −4.7 , −3
0.3 0.15 0.1
0.15 0.3 0.1
−1 −1 , −2
0.3 0.15 0.1
0.15 0.3 0.1
−2.8 −2.8 , −2
Ad1 =
Ad2 =
C1 = [0.1 0.1 0.2],
0.1 0 0
0.1 0 0
0 0.1 0
0.1 0 0
Ad3 =
C2 = [0.2 0.2 0.1],
0 0.1 0
0.2 0.2 , 0.1
F1 =
0.3 0.3 , 0.2
F2 =
0 0.1 0
0.3 0.3 , 0.2
D1 = 0.1,
0.2 0.2 , 0.1
F3 =
0.2 0.1 , 0.1
D2 = 0.1,
0.2 0.2 , 0.1
D3 = 0.1,
C3 = [0.1 0.2 0.1].
Direct calculation shows that
det(sE − A1 ) = 3s2 − 0.26s − 1.6864 = 0,
for s = 0,
deg(det(sE − A1 )) = deg(3s − 0.26s − 1.6864 ) = rankE = 2, 2
det(sE − A2 ) = 2s2 − s + 0.105 = 0,
for s = 0,
deg(det(sE − A2 )) = deg(2s − s + 0.105 ) = rankE = 2, 2
det(sE − A3 ) = 2s2 − 0.64s + 0.051 = 0,
s = 0,
for
deg(det(sE − A3 )) = deg(2s2 − 0.64s + 0.051 ) = rankE = 2, Thus the switched singular system (1) is regular and causal. Furthermore, by simple computation, we obtain
0.0433 A¯ 11 = 0.0033
0.0033 , 0.0433
A¯ 13 = [0.0333 0.0333],
0.25 A¯ 21 = 0.1
0.1 , 0.25
A¯ 23 = [0.05 0.05],
0.1 A¯ d11 = 0
A¯ d13 = [0 0],
0.1 A¯ d21 = 0
A¯ d23 = [0 0],
0 , 0.1
A¯ d14 = 0.0333,
0 , 0.1
F¯12 = 0.0333,
0.2 A¯ d22 = , 0.2
A¯ d24 = 0.1,
0.0433 A¯ d12 = , 0.0433
F¯22 = 0.05,
0.0433 F¯11 = , 0.0433
0.15 F¯21 = , 0.15
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Fig. 3. Switching signal.
0.16 A¯ 31 = 0.01
0.01 , 0.16
0.1 A¯ d31 = 0
A¯ 33 = [0.05 0.05, A¯ d33 = [0 0],
0 , 0.1
A¯ d34 = 0.1,
0.02 A¯ d32 = , 0.02
0.06 F¯31 = , 0.06
F¯32 = 0.05.
Therefore, by Theorem 1 system (1) is positive. Let d = 2, γ = 0.5, then solving the LPs (16)–(18) in Theorem 3, we can obtain the following feasible solutions
⎡
ν1
⎤
⎡
0.8736 ⎢ 0.8751 ⎥ ⎢1.2044⎥ ⎥ =⎢ ⎢0.3869⎥, ⎣ ⎦ 0.3890 0.3819
ν2
⎤
1.3099 ⎢ 1.3183 ⎥ ⎢1.2247⎥ ⎥ =⎢ ⎢ 0.3518 ⎥, ⎣ ⎦ 0.3532 0.3795
⎡
ν3
⎤
0.9605 ⎢1.0594⎥ ⎢ 1.1421 ⎥ ⎥ =⎢ ⎢0.3940⎥, ⎣ ⎦ 0.3938 0.3994
⎡
υ
⎤
0.1445 ⎢ 0.1434 ⎥ ⎢0.6077⎥ ⎥ =⎢ ⎢0.2424⎥. ⎣ ⎦ 0.2445 0.1994
According to Theorem 3, we can conclude that positive system (1) is asymptotically stable with an l1 -gain performance. The simulation results are shown in Figs. 3 and 4, where the initial condition is x(θ ) = [0.5 3 4]T , θ = −2, −1, 0, and the external disturbance is w(k ) = 0.5e−0.5k . Fig. 3. shows the switching signal σ (k) and Fig. 4 shows the state trajectory over 0–40. It is easy to see that the singular system is positive and asymptotically stable. Example 3. We consider the asymptotical stability with an l∞ -gain performance for system (1). Consider system (1) with two modes, i.e., N = {1, 2}, and E = [10 00],
0.2 A1 = 0.1
−6 , −7
Ad1
0.3 0.1
−5 , −4
Ad2 =
A2 =
0.1 = 0.0
0.2 , 0.1
0.1 F1 = , 0.1
C1 = [0.1 0.2],
D1 = 0.1,
0.2 0.0
0.3 , 0.2
F2 =
0.2 , 0.1
C2 = [0.2 0.1],
D2 = 0.1.
Direct calculation shows that
det(sE − A1 ) = 7s − 0.8 = 0,
for
s = 0,
deg(det(sE − A1 )) = deg(7s − 0.8 ) = rankE = 1, det(sE − A2 ) = 4s − 0.7 = 0,
for
s = 0,
deg(det(sE − A2 )) = deg(4s − 0.7 ) = rankE = 1. Thus the switched singular system (1) is regular and causal. Furthermore, by simple computation, we obtain
A¯ 11 = 0.1143, A¯ 13 = 0.0143,
A¯ d11 = 0.1000, A¯ d12 = 0.1143, F¯11 = 0.0143, A¯ d13 = 0.0000, A¯ d14 = 0.0143, F¯12 = 0.0143,
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105
Fig. 4. State trajectory of the given system.
A¯ 21 = 0.1750, A¯ 23 = 0.0250,
A¯ d21 = 0.2000, A¯ d22 = 0.0500, F¯21 = 0.0750, A¯ d23 = 0.0000, A¯ d24 = 0.0500, F¯22 = 0.0250.
Therefore, by Theorem 1 system (1) is positive. Let d = 2, γ = 0.5, then solving the LPs (16) and (17), and (21) in Theorem 4, we can obtain the following feasible solutions
⎡
ν1
⎤
1.7497 ⎢ 1.9216 ⎥ =⎣ , 0.9340⎦ 1.0021
⎡
ν2
⎤
1.9330 ⎢1.9484⎥ =⎣ , 0.9412 ⎦ 0.9932
⎡
υ
⎤
0.4042 ⎢0.6874⎥ =⎣ . 0.7081⎦ 0.6234
According to Theorem 4, we can conclude that positive system (1) is asymptotically stable with an l∞ -gain performance. 5. Conclusions In this paper, we have investigated the problems of asymptotical stability, l1 -gain and l∞ -gain analysis for positive switched singular systems with time delay. Firstly, based on the singular value decomposition method, a necessary and sufficient condition for the positivity of the system has been derived. Then by introducing new state vectors, the positive switched singular system is equivalently transformed into a standard positive switched system, and sufficient conditions of asymptotical stability with an l1 -gain performance and an l∞ -gain performance for the time-delay positive switched singular system is obtained, respectively. Finally, three numerical examples are presented to demonstrate the effectiveness of the proposed results. Our future work will be focused on generalizing the obtained results to more general positive switched singular systems where the matrix E is switchmode-dependent. Acknowledgments This work was supported by the National Natural Science Foundation of China under Grant no. 61273120 and the Postgraduate Innovation Project of Jiangsu Province under Grant no. KYLX-0371. References [1] [2] [3] [4]
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