Stability and stabilization for discrete-time switched systems with asynchronism

Applied Mathematics and Computation 338 (2018) 520–536

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Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc

Stability and stabilization for discrete-time switched systems with asynchronismR Shuang Shi a, Zhongyang Fei a,∗, Zhenpeng Shi a, Shunqing Ren b a b

Research Institute of Intelligent Control and Systems, Harbin Institute of Technology, Harbin 150000 China Space Control and Inertial Technology Research Center, Harbin Institute of Technology, Harbin 150000 China

a r t i c l e

i n f o

Keywords: Discrete-time switched system H∞ control Asynchronous switching Persistent dwell time

a b s t r a c t In this paper, the H∞ control is investigated for a class of discrete-time switched systems. The switching delay between the mode and controller, which leads to the asynchronism, is taken into consideration. The switching signal is considered to be constrained by persistent dwell time (PDT), which is known to be more general than the common used dwell time or average dwell time. Sufficient conditions to guarantee the asymptotic stability and 2 -gain are derived under a PDT scheme. By considering that actual controllers are subjected to norm-bounded gain perturbations, non-fragile controllers are designed based on both state feedback and output feedback stategies. Finally, the effectiveness of the provided methods is illustrated by two examples. © 2018 Elsevier Inc. All rights reserved.

1. Introduction Switched systems have been attracting considerable attention since they provide a unified framework for mathematical modeling dynamic processes displaying switching features [1–3]. Practical applications of switched systems are extensive, including but not limited to manipulator robots, networked control systems, biological systems, flight control systems and power electronics [4–8]. Quantities of fundamental results about switched systems have been broadly probed [9–13]. The switching mechanisms, which orchestrate the switching among subsystems, play significant roles in stability and system performance of switched systems. Generally, the switching rules could be classified into random and deterministic ones, which result from the system and the designers’ intervention [14]. Thereinto, dwell time (DT) and average dwell time switching are two classes of important deterministic switching rules. Primary stability analysis and other issues for switched systems with these two switching rules have been intensively reported. As a matter of fact, there exists another significant switching signal, the persistent dwell time (PDT) switching, which is known to be more general since it can cover both DT and ADT switching as special cases [15,16]. Under the PDT scheme, there exists an infinite number of portions with dwell time no smaller than a fixed value, during which no switching occurs, and the intervals with this property are separated by a period of persistence with upper bound [17]. It indicates that PDT switching could be applied to describe a switched system with both fast and slow switching. Uniformly asymptotically stable criteria are established for the switched systems

R This research is partially supported by the National Science Foundation of China (61503094, 61790564), the Fundamental Research Funds for the Central Universities and the 111 Project (No. B16014). ∗ Corresponding author. E-mail addresses: [email protected] (S. Shi), [email protected] (Z. Fei), [email protected] (Z. Shi), [email protected] (S. Ren).

https://doi.org/10.1016/j.amc.2018.06.049 0 096-30 03/© 2018 Elsevier Inc. All rights reserved.

S. Shi et al. / Applied Mathematics and Computation 338 (2018) 520–536

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with PDT switching in [15]. In [18], the result is extended to adaptive control with PDT switching for switched nonlinear systems. However, the research on switched systems with this meaningful type of switching signals is still limited so far. Generally speaking, state feedback controller is widely adopted to guarantee the stability and performance of control system in the existing literature [19,20]. However, it may be difficult or even impossible to obtain the measurement of a fraction of state variables directly under certain circumstances. Therefore, output feedback control is more practical in such situation [21–23]. Meanwhile, it is noted that the controller design by using H∞ synthesis techniques may be very sensitive, or fragile, with respect to errors in the controller coefficients [24]. Owing to this, it is necessary to design a controller for a given plant such that the controller is insensitive to some amount of error with respect to its gains, i.e., the designed controller is non-fragile. Meanwhile, it takes a certain time to identify the active subsystems before applying the corresponding matched controllers, which leads to the switching delay between system modes and controllers, i.e., the asynchronous phenomenon. Considerable efforts have been devoted to address this issue [20,25–29]. To mention a few, in [20], H∞ control for discretetime systems is investigated under asynchronous switching, and in [29], output feedback control is used to stabilize the switched systems with asynchronism. It is particularly noteworthy that aforementioned results are all based on ADT switching. The asynchronous control for switched systems with PDT is more general and meaningful, but to the best of the authors’ knowledge, there are no results on this topic. In this paper, we concern with non-fragile H∞ control for discrete-time switching systems with asynchronous switching. Under the PDT switching scheme, a non-weighted disturbance attenuation is derived, which is with more flexible physical meaning than the weighted one widely applied under ADT switching. Mode-dependent controllers are designed based on both state feedback and output feedback. The effectiveness is illustrated by two examples. The remainder of this paper is organized as follows. In Section 2, necessary definitions and lemmas are reviewed for further proceeding. In Section 3, criteria to guarantee the stability and 2 -gain are derived for discrete-time switched systems. In Section 4, sufficient conditions for the solvability of the non-fragile H∞ controller design are presented based on both state feedback and output feedback, respectively. In Section 5, two examples are addressed to demonstrate the feasibility and effectiveness of the proposed techniques. Finally, we conclude the paper in Section 6. Notation: The notation used in this paper is fairly standard. Rn represents the n-dimensional Euclidean space, Z+ represents the set of nonnegative integers, and Z≥s represents the set {k ∈ Z+ |k ≥ s}. The notation  ·  refers to the Euclidean vector norm. ∞l2 [0, ∞) is the space of square summable infinite sequence, and for ω = {ω (k )} ∈ l2 [0, ∞ ), its norm is given by  ω2 = k=0 ω (k )ω (k ). A function β : [0, ∞ ) → [0, ∞ ) is said to be of class K∞ if it is continuous, strictly increasing, unbounded and β (0 ) = 0. The superscripts “” and “−1” represent matrix transposition and matrix inverse, respectively. Meanwhile, we use ∗ as an ellipsis for the terms that are introduced by symmetry in symmetric block matrices or long matrix expressions. The notation X > 0(≥ 0) means that X is real symmetric and positive definite (semi-positive definite). 2. Problem formulation Consider a class of discrete-time switched linear systems:

x(k + 1 ) =Aσ (k ) x(k ) + Bσ (k ) u(k ) + Dσ (k ) ω (k ),

(1)

y(k ) =Cyσ (k ) x(k ) + Dyσ (k ) ω (k ),

(2)

z(k ) =Czσ (k ) x(k ) + Bzσ (k ) u(k ) + Dzσ (k ) ω (k ),

(3)

where x(k ) ∈ u (k ) ∈ y (k ) ∈ z (k ) ∈ are the state, control input, measured output, controlled output, respectively, and ω (k ) ∈ Rr is the exogenous disturbance which belongs to 2 [0, ∞). σ (k ) : [0, ∞ ) → L = {1, 2, . . . , M} is the switching signal which is a piecewise constant function depending on time k. Aσ (k) , Bσ (k) , Dσ (k) , Cyσ (k) , Dyσ (k) , Czσ (k) , Bzσ (k) and Dzσ (k) are matrix functions with appropriate dimensions of the jumping process σ (k). In this paper, we aim to study H∞ control for asynchronously switched systems with PDT switching, and design a set of state or output feedback controllers for switched linear systems (1)–(3). The following definitions and lemmas are introduced for later development. Rn ,

Rm ,

R p,

Rq

Definition 1. [15] Consider switching instants k0 , k1 , …, ks , … with k0 = 0. A positive constant τ D is said to be the persistent dwell time if there exists an infinite number of disjoint intervals of length no smaller than τ D on which σ (k) is constant, and consecutive intervals with this property are separated by no more than T, where T is called the period of persistence. As a matter of fact, the interval is divided into a number of stages in PDT switching, while each stage consists of the running time of a certain subsystem (termed as τ -portion) and the period of persistence (termed as T-portion). When considering the system is active at the p-th stage, p ∈ Z≥1 , the switching occurs at ks p , ks p +1 , …, ks p+1 . It is worth mentioning that ks p +1 stands for the next switching instant after ks p at the p-th stage, and ks p+1 represents the instant switching into ( p + 1 )-th stage. In the τ -portion, i.e., k ∈ [ks p , ks p +1 ), a certain subsystem is activated with the running time τ p ≥ τ D . In the T-portion, i.e., k ∈ [ks p +1 , ks p+1 ), the actual running time Tp satisfies

522

S. Shi et al. / Applied Mathematics and Computation 338 (2018) 520–536 N (ks p +1 ,ks p+1 )

Tp =



T (ks p +r , ks p +r+1 ) ≤ T ,

r=1

where T (ku , kw ) = kw − kv , and N (ku , kw ) denotes the total switching times during the interval [ku , kw ) and it holds that N (ku , kw ) = N (ku , kv ) + N (kv , kw ) for any 0 ≤ ku ≤ kv ≤ kw . Remark 1. Definition 1 indicates that during the T-portion, arbitrary switching could be occur among the subsystems as long as the actual running time of any activated mode is always less than τ D . As pointed in [15,17], The PDT switching is more general than DT and ADT switching. Here, let SDT (τD ), SADT (τD , N0 ) and SPDT (τD , T ) denote the sets of switching signals with DT property normalized by τ D > 0, ADT property normalized by τ D and N0 > 0, and PDT property normalized by τ D and T > 0, respectively. It is straightforward to arrive at

SDT (τD ) = SADT (τD , 1 ) = SPDT (τD , 0 ) ⊂ SADT (τD , N0 ) ⊂ SPDT (δτD , T ),

with ∀τ D > 0, ∀N0 ≥ 1, δ ∈ (0, 1), and T = δτD (N0 − δ )/(1 − δ ) [15]. Consequently, one can conclude that the DT switching can be termed as a special case of ADT or PDT switching, furthermore, the PDT switching is more general than ADT switching. Definition 2 [1]. The switched system

x(k + 1 ) = fσ (k ) (x(k ))

(4)

is globally uniformly asymptotically stable (GUAS) under certain switching signal σ if for initial condition x(k0 ), there exists a class of K∞ function κ such that the solution of the system (4) satisfies the inequality x(k) ≤ κ (x(k0 )), ∀k ≥ k0 , and x(k) → 0 as k → ∞. Definition 3 [10]. For γ > 0, the switched system

x(k + 1 ) = fσ (k ) (x(k ), ω (k )),

(5)

z(k ) =gσ (k ) (x(k ), ω (k )) is said to be GUAS with an 2 -gain, if under zero initial condition, the systems (1)–(3) are GUAS and ∞  s=k ω (s )ω (s ) holds for any non-zero ω (s) ∈ 2 [0, ∞).

(6) ∞

s=k0

z  ( s )z ( s )

≤

0

Lemma 1 [28]. For any real matrices R, S of compatible dimensions and any real scalar ρ > 0, the following inequality holds:

R S + S  R ≤ ρ R R +

1

ρ

S S.

Lemma 2 [21]. For real matrices Q, R, S, W of compatible dimensions, the following conditions are equivalent: (i) there exist real matrices Q, R, S, W such that



Q S − WR



∗ < 0; W + W

(ii) there exist real matrices Q, R, S such that

Q < 0, Q + R S + S  R < 0. 3. Stability and 2 -gain analysis To facilitate further development, we address the stability and 2 -gain analysis of asynchronously switched systems under PDT switching in this section. Since the asynchronous switching is taken into consideration in this paper, the mismatched controllers will be applied to the system for a certain time. When a switching happens, the mismatched time ϕ (k) satisfies 0 ≤ ϕ (k) ≤ ϕ max . Without loss of generality, the maximum mismatched time ϕ max is assumed to be known a priori. It is noted that the Lyapunov function could increase when the mismatched controllers are activated. Within the interval [ku , kw ), the separation is given as [ku , kw ) = T↑ (ku , kw ) ∪ T↓ (ku , kw ), while T↑ (ku , kw ) and T↓ (ku , kw ) stand for the unions of dispersed intervals during which the Lyapunov function is increasing and decreasing, respectively. Meanwhile, the length of T↑ (ku , kw ) and T↓ (ku , kw ) are denoted as T↑ (ku , kw ) and T↓ (ku , kw ), respectively, and T (ku , kw ) = T↑ (ku , kw ) + T↓ (ku , kw ). When considering the worst case, i.e., the Lyapunov function always increases during the mismatched period, it is obvious that ϕmax = max T↑ (ks p , ks p +1 ).

S. Shi et al. / Applied Mathematics and Computation 338 (2018) 520–536

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3.1. Stability analysis A stability criterion for the switched system (4) is established firstly in this subsection. Lemma 3. Consider the switched system (4), and let 0 < α < 1, β > 0 and μ > 1 be given constants. For a prescribed period of persistence T, suppose that there exist a family of functions Vσ (k ) : Rn → R, σ (k ) ∈ L, and two class K∞ functions κ 1 and κ 2 such that ∀σ (k ) = i ∈ L,

κ1 (x(k )) ≤ Vi (x(k )) ≤ κ2 (x(k )),

(7)



−αVi (x(k )), ∀k ∈ T↓ (kl , kl+1 ), βVi (x(k )), ∀k ∈ T↑ (kl , kl+1 ),

Vi (x(k )) ≤

(8)

and ∀(σ (ks ) = i, σ (ks−1 ) = j ) ∈ L × L , i = j,

Vi (x(kl )) ≤ μV j (x(kl )).

(9)

Then, the switched system is GUAS for any PDT switching signals satisfying

(T + 1 ) ln μ + T ln(1 + β ) + ϕmax ln θ , ln(1 − α ) where θ = (1 + β )/(1 − α ).

τD > τD∗ = −

(10)

Proof. Consider σ (ks p ) = m, σ (ks p+1 ) = n, and an arbitrary switching occurs within Tp . Since Vi (x(k )) = Vi (x(k + 1 )) − Vi (x(k )), combining (8) with (9), we obtain

Vn (x(ks p+1 )) ≤μ(1 + β )

T↑ (ks p+1 ,ks p+1 −1 )

(1 − α )T↓ (ks p+1 ,ks p+1 −1 )Vσ (ks p+1 −1 ) (x(ks p+1 −1 ))

≤··· ≤μ

N (ks p +1 ,ks p+1 )

(1 + β )T↑ (ks p+1 ,ks p +1 ) (1 − α )T↓ (ks p+1 ,ks p +1 )

× μ(1 − α )ks p +1 −ks p −ϕ (ks p ) (1 + β )ϕ (ks p )Vm (x(ks p )) ≤ μT +1 (1 + β )T (1 − α )τD θ ϕmax Vm (x(ks p )). Denote ζ = μT +1 (1 + β )T (1 − α )τD θ ϕmax . If the PDT switching signal satisfies (10), we have 0 < ζ < 1. Thus, we conclude that

Vσ (ks p ) (x(ks p )) ≤ ζ p−1Vσ (ks ) (x(ks1 )). 1





It is noted that ks1 = k0 , from (7), we have x(ks p ) ≤ κ1−1 (ζ p−1 κ2 (x(k0 ) )). Furthermore, from (8) and (9), ∀k ∈

[ks p , ks p+1 ), x(k) ≤ κ 3 x(k0 ) holds, where κ3 (· ) = κ1−1 (μT (1 + β )T +ϕmax κ2 (κ1−1 (ζ p−1 κ2 (· ))). According to Definition 2, one can conclude that the switched system (4) under asynchronous switching is GUAS, which completes the proof.  3.2. 2 -Gain analysis Based on Lemma 3, we present the criterion on 2 -gain analysis of switched nonlinear systems (5) and (6) under asynchronous switching. Lemma 4. Consider the switched systems (5) and (6), and let 0 < α < 1, β > 0 and μ > 1 be given constants. For a prescribed period of persistence T, suppose that there exist a family of functions Vσ (k ) : Rn → R, σ (k ) ∈ L, two class K∞ functions κ 1 and κ 2 , and a scalar γ such that (7) and (9) hold, and ∀σ (k ) = i ∈ L,



Vi (x(k )) ≤

−αVi (x(k )) − (k ), ∀k ∈ T↓ (kl , kl+1 ), βVi (x(k )) − (k ), ∀k ∈ T↑ (kl , kl+1 ),

(11)

where (k ) = z (k )z(k ) − γ 2 ω (k )ω (k ). Then, for any PDT switching signal satisfying (10), the switched systems (5) and (6) are GUAS with an 2 -gain no greater than



γl =

μ1 +T +1 θ 2 +T +ϕmax

1 − α¯ γ, 1−ϑ

(12)

where α¯ = 1 − α , β¯ = 1 + β , θ = β¯ /α¯ , 1 = τT +1 , 2 = T τ+ϕ+max , and ϑ = μ1 αθ ¯ 2 . D +T D T Proof. First of all, consider ω(k) ≡ 0, (8) can be ensured by (11). Therefore, one can conclude that the switched system is GUAS from Lemma 3.

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S. Shi et al. / Applied Mathematics and Computation 338 (2018) 520–536

Then, consider ω(k) = 0, from (9) and (11), we have

Vσ (ks

p+1

¯ ks p+1 −ks p+1 −1 Vσ (k ) (x (ks p+1 )) ≤μβ s

ks p+1 −1 p+1

−1 )



(x(ks p+1 −1 )) −

β¯ ks p+1 −1−l (l )

l=ks p+1 −1

≤... ≤μ

N (ks p +1 ,ks p+1 )

≤μ

N (ks p ,ks p+1 )

β¯ ks p+1 −ks p +1 Vσ (ks p +1 ) (x(ks p +1 )) −

ks p+1 −1



μN (l,ks p+1 ) β¯ ks p+1 −1−l (l )

l=ks p +1

β¯ ks p+1 −ks p +1 α¯ ks p +1 −ks p θ ϕ (ks p )Vσ (ks p ) (x(ks p ))

ks p +1 −1



−

μN (l,ks p+1 ) β¯ ks p+1 −ks p +1 α¯ ks p +1 −1−l (l )

l=ks p +ϕ (ks p ) ks p +ϕ (ks p )−1



−

μN (l,ks p+1 ) β¯ ks p+1 −ks p +1 α¯ ks p +1 −1−l θ ks p +ϕ (ks p )−1−l (l )

l=ks p ks p+1 −1



− ... −

μN (l,ks p+1 ) β¯ ks p+1 −1−l (l )

l=ks p +1

≤μ

N (ks1 +1 ,ks p+1 )

β¯ ks p+1 −ks p +1 +ϕ (ks p ) α¯ ks p +1 −ks p θ ϕ (ks p )

. . . β¯ ks2 −ks1 +1 α¯ ks1 +1 −ks1 −ϕ (ks1 ) θ ϕ (ks1 )Vσ (ks ) (x(ks1 )) 1 −μ

N (ks1 +1 ,ks p+1 )

β¯ ks p+1 −ks p +1 +ϕ (ks p ) α¯ ks p +1 −ks p θ ϕ (ks p )

ks1 +1 −1



. . . β¯ ks2 −ks1 +1

α¯ ks1 +1 −1−l (l )

l=ks1 +ϕ (ks1 )

−μ

N (ks1 +1 ,ks p+1 )

. . . β¯ ks2 −ks1 +1

β¯ ks p+1 −ks p +1 +ϕ (ks p ) α¯ ks p +1 −ks p θ ϕ (ks p )

ks1 +ϕ (ks1 )−1



α¯ ks1 +1 −1−l θ ks1 +ϕ (ks1 )−1−l (l )

l=ks1 ks p+1 −1



− ...−

μN (l,ks p+1 ) β¯ ks p+1 −1−l (l ),

l=ks p +1

where α¯ = 1 − α , β¯ = 1 + β , and θ = β¯ /α¯ . Note that ks1 = k0 , we have Vσx(k ) (x(ks1 )) = 0 under zero initial condition. Therefore, for ∀k ∈ [ks p , ks p+1 ), one can arrive s1

at k−1 

μN (l,k) α¯ T↓ (l,k) β¯ T↑ (l,k) (l ) ≤ 0.

(13)

l=k0

Owing to the fact that



0 ≤N (l, k ) ≤

0 ≤T↑ (l, k ) ≤



k−l + 1 ( T + 1 ), τD + T



k−l + 1 (T + ϕmax ), τD + T

From (13), we obtain that k−1  l=k0

α¯ k−1−l z (l )z(l ) ≤

k−1 



μ

k−l τD +T +1

(T +1 ) k−1−l

α¯



θ

k−l τD +T +1

(T +ϕmax )

γ 2 ω  ( l )ω ( l )

l=k0

= μ1 +T +1 θ 2 +T +ϕmax γ 2

k−1  l=k0

ϑ k−1−l ω (l )ω (l ).

S. Shi et al. / Applied Mathematics and Computation 338 (2018) 520–536

525

where 1 = τT +1 , 2 = T τ+ϕ+max , and ϑ = μ1 αθ ¯ 2 . It is noted that 0 < ϑ < 1 can be ensured by (10). Therefore, we can get D +T D T ∞ k−1  

α¯ k−1−l z (l )z(l ) ≤ μ1 +T +1 θ 2 +T +ϕmax γ 2

k=k0 +1 l=k0

∞ k−1  

ϑ k−1−l ω (l )ω (l ),

k=k0 +1 l=k0

which indicates that ∞  ∞ 

α¯ k−1−l z (l )z(l ) ≤ μ1 +T +1 θ 2 +T +ϕmax γ 2

l=k0 k=l+1

∞  ∞ 

ϑ k−1−l ω (l )ω (l ).

l=k0 k=l+1

Then, we get that ∞ 

z (l )z(l ) ≤ γl2 ω (l )ω (l ),

l=k0

where γ l is defined in (12). Consequently, the switched systems (5) and (6) are GUAS with an 2 -gain no greater than γ l . The proof is completed.



Remark 2. Compared with the weighted 2 -gain for switched systems with ADT switching, the guaranteed 2 -gain in Lemma 4 is non-weighted, which is more explicit in practice. Remark 3. It is well to be reminded that the worst cases are considered in the proof of Lemmas 3 and 4. We assume that there exists T times of switching during each T-portion, and the Lyapunov function always increase in the T-portion, since the actual switching in the T-portion is arbitrary. 4. H∞ Control In this section, we aim to design a set of non-fragile controllers for discrete-time switched linear systems (1)–(3) such that the corresponding closed-loop system is GUAS with an non-weighted 2 -gain. 4.1. State feedback case Consider the state feedback controllers with the following form:

u(k ) = (Ksσ (k−ϕ (k )) + Ksσ (k−ϕ (k )) (k ))x(k ),

(14)

where Ksσ (k) is the controller gain to be designed, and Ksσ (k) is unknown matrix which stands for the time-varying gain perturbation, and is assumed to be of the norm-bounded type,

Ksσ (k ) = Jsσ (k ) Lsσ (k ) (k )Msσ (k ) ,

(15)

where Jsσ (k) , Msσ (k) are known real constant matrices with appropriate dimensions, and Lsσ (k ) : R → Rm×n , σ (k ) ∈ L is unknown time-varying function with Lebesgue-measurable elements satisfying L (k )Lsσ (k) (k ) ≤ I, ∀k ≥ 0. Hence, assuming sσ ( k ) that kl is the switching instant and σ (kl ) = i ∈ L, σ (kl − 1 ) = j ∈ L, i = j, then we obtain the resulting closed-loop systems by combining (1)–(3) with (14),

∀k ∈ T↓ (kl , kl+1 ),  x(k + 1 ) = A¯ si (k )x(k ) + Bsi ω (k ) , z(k ) = C¯si (k )x(k ) + Dsi ω (k ) ∀k ∈ T↑ (kl , kl+1 ),  x(k + 1 ) = A˜si j (k )x(k ) + Bsi ω (k ) , z(k ) = C˜si j (k )x(k ) + Dsi ω (k )

(16)

(17)

where

A¯ si (k ) =Ai + Bi (Ksi + Jsi Lsi (k )Msi ), A˜si j (k ) = Ai + Bi (Ks j + Js j Ls j (k )Ms j ), Bsi = Di , C¯si (k ) =Ci + Bzi (Ksi + Jsi Lsi (k )Msi ), C˜si j (k ) = Ci + Bzi (Ks j + Js j Ls j (k )Ms j ), Dsi = Dzi . For further controller design, we firstly address a sufficient criterion which guarantees the stability and 2 -gain of the closed-loop systems (16) and (17) based on Lemma 4.

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S. Shi et al. / Applied Mathematics and Computation 338 (2018) 520–536

Lemma 5. Consider the switched systems (16) and (17), let 0 < α < 1, β > 0, μ > 1 and ν > 0 be given constants. For a prescribed period of persistence T, suppose that there exist matrices P¯i > 0, Xi , ∀i ∈ L, and a scalar γ such that ∀(i, j ) ∈ L × L, i = j,

⎡ ¯ i11  ⎢ ∗ ⎣ ∗

0

−γ 2 I ∗ ∗

∗

⎡˜ i j11 ⎢ ∗ ⎣

0 −γ 2 I ∗ ∗

∗ ∗

⎤

Xi A¯  (k ) si Bsi −P¯i ∗

Xi C¯si (k ) Dsi ⎥ ⎦ <0, 0 −I

X j A˜ (k ) si j Bsi −P¯i ∗

(18)

⎤

X j C˜sij (k ) Dsi ⎥ ⎦ < 0, 0 −I

(19)

P¯j − μP¯i ≤0,

α¯ (ν 2 P¯i

(20) − ν Xi

β¯ (ν 2 P¯i

− ν X j

¯ i11 = ˜ i j11 = where  − ν Xi ),  − ν X j ), α¯ = (1 − α ) and β¯ = 1 + β . Then, for any PDT switching signal satisfying (10), the switched systems (16) and (17) are GUAS with an 2 -gain no greater than (12). Proof. Consider the Lyapunov function given by the following quadratic form,

Vi (x(k )) = x (k )Pi x(k ), ∀σ (k ) = i ∈ L, where Pi =

P¯i−1 .

(21)

From (16), (17) and (21), we obtain ∀(i, j ) ∈ L × L, i = j,

¯ ai η (k ), Vi (x(k )) + αVi (x(k )) + (k ) =η (k )

(22)

˜ bi j η (k ), Vi (x(k )) − β Vi (x(k )) + (k ) =η (k )

(23)

Vi (x(kl )) − μV j (x(kl )) =x (kl )(Pi − μPj )x(kl ),

(24)

and



where η (k ) = x (k )



¯ ¯ ai = ai11  ∗

 ω (k ) , and





˜ bi j11  (k )Pi B¯si − C¯si (k )D¯ si ˜ A¯  si , bi j = B¯si Pi B¯si − D¯ si D¯ si − γ 2 I ∗



A˜ (k )Pi B˜si − C˜sij (k )D¯ si si j , B¯si Pi B¯si − D¯ si D¯ si − γ 2 I

¯ ai11 = A¯  (k )Pi A¯ si (k ) − C¯ (k )C¯si (k ) − α¯ Pi , and  ˜ bi j11 = A¯  (k )Pi A¯ si j (k ) − C¯ (k )C¯si j (k ) − β¯ Pi . with  si si si j si j  On the other hand, since (Xi − ν P¯i ) Pi (Xi − ν P¯i ) ≥ 0 holds when Pi > 0, one can get −Xi Pi Xi ≤ ν 2 P¯i − ν Xi − ν Xi . By us¯ ai < 0. Similarly, (19) can guarantee  ˜ bi j < 0. Thus, according to (22) and (23), ing Schur complement, (18) implies that  (11) is satisfied. Furthermore, noticing that Pi = P¯i−1 , (20) indicates that Pi − μPj ≤ 0. Then from (24) we know that (9) holds. Consequently, the switched systems (16)and (17) are GUAS with an 2 -gain according to Lemma 4, which ends the proof.  Based on Lemma 5, we present the existence conditions of a set of asynchronous stabilizing controllers for the switched systems (1)–(3). Theorem 1. Consider the switched systems (1)–(3), let 0 < α < 1, β > 0, μ > 1 ,ν > 0, and  > 0 be given constants. For a prescribed period of persistence T, suppose that there exist matrices P¯i > 0, Xi , Yi , ∀i ∈ L, and a scalar γ such that ∀(i, j ) ∈ L × L, i = j, (20) holds, and

⎡ ¯ i11  ⎢ ∗ ⎢ ⎢ ∗ ⎣ ∗ ∗

⎡ ˜ i j11  ⎢ ∗ ⎢ ⎢ ∗ ⎣ ∗ ∗

0

−γ 2 I ∗ ∗ ∗ 0 −γ 2 I ∗ ∗ ∗

¯ i13 

¯ i14 

¯ i33 

¯ i34  ¯ i44 

D i ∗ ∗

˜ i j13 

D i ˜ i j33 ∗ ∗

D zi ∗

˜ i j14 

D zi ˜ i j34 ˜ i j44  ∗

⎤

Xi Msi 0 ⎥ ⎥ 0 ⎥ < 0, ⎦ 0 − I

(25)

⎤

X j Msj 0 ⎥ ⎥ 0 ⎥ < 0, ⎦ 0 − I

(26)

S. Shi et al. / Applied Mathematics and Computation 338 (2018) 520–536

527

where

¯ i11 =α¯ (ν 2 P¯i − ν X  − ν Xi ),  ¯ i13 = X  A + Y  B ,  ¯ i14 = X C  + Y  B ,  i i i i i i zi i zi ¯ i33 = − P¯i + i Bi Jsi J  B ,  ¯ i34 = i Bi Jsi J  B ,  ¯ i44 = −I + i Bzi Jsi J  B ,  si i si zi si zi ˜ i j11 =β¯ (ν 2 P¯i − ν X  − ν X j ),  ˜ i j13 = X  A + Y  B ,  ˜ i j14 = X C  + Y  B ,  j j i j i j zi j zi ˜ i j33 = − P¯i + i Bi Js j J  B ,  ˜ i j34 = i Bi Js j J  B ,  ˜ i j44 = −I + i Bzi Js j J  B .  sj i s j zi s j zi Then, for any PDT switching signal satisfying (10), the switched systems (1)–(3) is GUAS with an 2 -gain no greater than (12). Moreover, the admissible controller in the form of (14) can be given by Ksi = Yi Xi−1 . Proof. According to Schur complement, (25) is equivalent to

⎡ ¯ i11 +  ˆ i11  ∗ ⎢ ⎣ ∗ ∗

0

−γ I ∗ ∗ 2

¯ i13 

¯ i14 

⎤

⎥ ¯ i34 ⎦ < 0,  ¯ i44 

D i

D zi

¯ i33  ∗

(27)

ˆ i11 = 1 X  M Msi Xi . where   i si It is noted that L (k )Lsσ (k) (k ) ≤ I, ∀k ≥ 0, from which we obtain that sσ ( k )

1



ˆ Xi Msi L si (k )Lsi (k )Msi Xi ≤ i11

holds for ∀k ≥ 0. Hence, from (27) we have

⎡ ¯ i11  ⎢ ∗ ⎣ ∗

0

−γ 2 I ∗ ∗

∗

where



Rsi = 0

¯ i13  D i −P¯i ∗

Jsi B i

0

¯ i14  D zi 0 −I

⎤

1 ⎥ ⎦ +  Rsi Rsi +  Ssi (k )Ssi (k ) < 0,





Jsi B , Ssi (k ) = Lsi (k )Msi Xi zi

0

0

(28)



0 .

According to Lemma 1, we know that for ∀ > 0,   R si Ssi (k ) + Ssi (k )Rsi ≤  Rsi Rsi +

Hence, from (28), we have

⎡ ¯ i11  ⎢ ∗ ⎣ ∗ ∗

0

−γ I ∗ ∗ 2

¯ i13 

¯ i14 

−P¯i ∗

0 −I

D i

D zi

1



Ssi (k )Ssi (k ).

⎤ ⎥ ⎦ + Rsi Ssi (k ) + Ssi (k )Rsi < 0.

(29)

It is noted that Yi = Ksi Xi . Combining (16) and (17), (29) can be rewritten as (18), which indicates that (25) can ensure (18). By the similar manipulation, it also holds that (19) can be guaranteed by (26). According to Lemma 5, the stability and 2 -gain of the switched systems (1)–(3) is guaranteed with the controller Ksi = Yi Xi−1 . The proof is completed.  In the absence of asynchronous switching, i.e., the mismatched time ϕ (k ) = 0 for ∀k ≥ 0. Then, we can obtain the following corollary. Corollary 1. Consider the switched systems (1)–(3), let 0 < α < 1, μ > 1 ,ν > 0, and  > 0 be given constants. For a prescribed period of persistence T, suppose that there exist matrices P¯i > 0, Xi , Yi , ∀i ∈ L, and a scalar γ , ∀i ∈ L, such that ∀(i, j ) ∈ L × L, i = j, (20) and (25) hold. Then, for any PDT switching signal satisfying

τD > τD∗ = −

(T + 1 ) ln μ − T, ln(1 − α )

(30)

the switched systems (16) and (17) are GUAS with an 2 -gain no greater than



γl =

μ1 +T +1

1 − α¯ γ, 1 − μ1 α¯

where α¯ = 1 − α , and 1 = τT +1 . Moreover, the admissible controller can be given by Ksi = Yi Xi−1 . D +T

(31)

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S. Shi et al. / Applied Mathematics and Computation 338 (2018) 520–536

Fig. 1. The PDT switching signal σ (k) with τD = 5, T = 12.

4.2. Output feedback case In this subsection, our objective is to design a set of dynamic output feedback controllers with the following structure:

xc (k + 1 ) =(Aoσ (k−ϕ (k )) + Aoσ (k−ϕ (k )) (k ))xc (k ) + (Boσ (k−ϕ (k )) + Boσ (k−ϕ (k )) (k ))y(k ),

(32)

u(k ) =(Coσ (k−ϕ (k )) + Coσ (k−ϕ (k )) (k ))xc (k ) + (Doσ (k−ϕ (k )) + Doσ (k−ϕ (k )) (k ))y(k ),

(33)

where xc (k ) ∈ is the controller state satisfying nc ≤ n, real constant matrices Aoσ (k−ϕ (k )) , Boσ (k−ϕ (k )) , Coσ (k−ϕ (k )) , Doσ (k−ϕ (k )) are the desired controller gains to be designed, and Aoσ (k) , Boσ (k) , Coσ (k) , Doσ (k) are unknown matrices which stand for the time-varying gain perturbation, and are assumed to be of the norm-bounded type, R nc



Aoσ (k ) Coσ (k )



Boσ (k ) = Joσ (k ) Loσ (k ) (k )Moσ (k ) , Doσ (k )

(34)

where Joσ (k) , Moσ (k) are known real constant matrices with appropriate dimensions, and Loσ (k ) : R → R(nc +m )×(nc + p) , σ (k ) ∈ L is unknown time-varying function satisfying L (k )Loσ (k) (k ) ≤ I, ∀k ≥ 0. Thus, considering kl is the switching instant and oσ (k ) σ (kl ) = i ∈ L, σ (kl − 1 ) = j ∈ L, i = j, then we get the augmented systems by combining (1)–(3) with (32) and (33),

∀k ∈ T↓ (kl , kl+1 ),  xˆ(k + 1 ) = A¯ oi (k )xˆ(k ) + B¯oi (k )ω (k ) , z(k ) = C¯oi (k )xˆ(k ) + D¯ oi (k )ω (k )

(35)

∀k ∈ T↑ (kl , kl+1 ),  xˆ(k + 1 ) = A˜oi j (k )xˆ(k ) + B˜oi j (k )ω (k ) , z(k ) = C˜oi j (k )xˆ(k ) + D˜ oi j (k )ω (k ) 

where xˆ(k ) = x (k )



x c (k )

(36)

, and

A¯ oi (k ) =Ai + Bi (Koi + Joi Loi (k )Moi )Cyi , B¯oi (k ) = Di + Bi (Koi + Joi Loi (k )Moi )Dyi , C¯oi (k ) =Czi + Bzi (Koi + Joi Loi (k )Moi )Cyi , D¯ oi (k ) = Dzi + Bzi (Koi + Joi Loi (k )Moi )Dyi , A˜oi j (k ) =Ai + Bi (Ko j + Jo j Lo j (k )Moi )Cyi , B˜oi j (k ) = Di + Bi (Ko j + Jo j Lo j (k )Moi )Dyi , C˜oi j (k ) =Czi + Bzi (Ko j + Jo j Lo j (k )Moi )Cyi , D˜ oi j (k ) = Dzi + Bzi (Ko j + Jo j Lo j (k )Moi )Dyi , with



A Ai = i 0





0 0 , Bi = 0 I



 



Bi Di 0 , Di = , Cyi = 0 0 Cyi







I 0 , Dyi = , 0 Dyi

S. Shi et al. / Applied Mathematics and Computation 338 (2018) 520–536

529

Fig. 2. The state response of the closed-loop system x(k) under the PDT switching signal σ (k) in Fig. 1.





and

 Koi =



Bzi , C¯zi = Czi

Bzi = 0

Aoi Coi



0 , Dzi = Dzi ,



Boi . Doi

Remark 4. Actually, the dynamic output feedback controllers presented in (32) and (33) are equal to full-order ones when nc = n. When nc < n, they are reduced-order. Particularly, the controllers degrade to static ones when nc = 0. In this situation, matrices Aoi , Boi ,Coi , ∀i ∈ L are non-existent. Before further proceeding, we will arrive at a modified bounded real lemma for the discrete-time switched systems (35) and (36) in the following lemma. Lemma 6. Consider the switched systems (35) and (36), let 0 < α < 1, β > 0, μ > 1, ν > 0 be given constants. For a prescribed period of persistence T, suppose that there exist matrices Qi > 0, Zi , ∀i ∈ L, and a scalar γ such that ∀(i, j ) ∈ L × L, i = j,

⎡

−α¯ Qi ⎢ ∗ ⎣ ∗ ∗

⎡ ¯ −β Qi ⎢ ∗ ⎣ ∗ ∗

0

−γ 2 I ∗ ∗ 0 −γ 2 I ∗ ∗

(k )Zi A¯  oi  ¯ Boi (k )Zi ν 2 Qi − ν Zi − ν Zi ∗ A˜ (k )Zi oi j  B˜oi j (k )Zi

ν 2 Qi − ν Zi − ν Zi

⎤

C¯oi (k )  (k )⎥ D¯ oi < 0, 0 ⎦ −I

⎤

C˜oi j (k )  D˜ oi (k )⎥ j

∗

(37)

0 −I

⎦ < 0,

Q i − μQ j ≤ 0 ,

(38)

(39)

where α¯ = (1 − α ) and β¯ = 1 + β . Then, for any PDT switching signal satisfying (10), the switched systems (35) and (36) are GUAS with an 2 -gain no greater than (12). Proof. Consider the Lyapunov function given as follow:

Vi (xˆ(k )) = xˆ (k )Qi xˆ(k ), ∀σ (k ) = i ∈ L,

(40)

According to Schur complement, (37) is equivalent to



ϒ¯ i11 ∗



(k )Qi B¯oi − C¯oi (k )D¯ oi A¯  oi < 0,   ¯ B¯oi Pi B¯oi − D¯ oi Doi − γ 2 I

(41)

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S. Shi et al. / Applied Mathematics and Computation 338 (2018) 520–536

Fig. 3. Evolution of γ s (k) of the closed-loop system under the PDT switching signal σ (k) in Fig. 1.

Fig. 4. The PDT switching signal σ (k) with τD = 6, T = 10.

¯ i11 = A¯  (k )Qi A¯ oi (k ) − C¯ (k )C¯oi (k ) − α¯ Qi . From (35), (36), (40) and (41), we obtain that ∀k ∈ T↓ (kl , kl+1 ), Vi (x(k )) + with ϒ oi oi αVi (x(k )) + (k ) < 0. At the same time, for ∀k ∈ T↑ (kl , kl+1 ), Vi (x(k )) − βVi (x(k )) + (k ) < 0 holds if (38) holds. Hence, we conclude that (11) can be ensured by (37) and (38). Furthermore, for ∀(i, j ) ∈ L × L, i = j, (39) can guarantee (9). Therefore, the stability and 2 gain of the switched systems (35) and (36) can be deduced according to Lemma 4. The proof is completed. 

Applying Lemma 6, a unified framework for non-weighted H∞ output feedback control is provided for the discrete-time switched system (1)-(3) in the presence of asynchronous switching.

Theorem 2. Consider the switched systems (1)–(3), let 0 < α < 1, β > 0, ν > 0, μ > 1,  > 0, and ε > 0 be given constants. For a prescribed period of persistence T, suppose that there exist matrices Qi > 0, Ui , Vi , Zi , ∀i ∈ L, and a scalar γ such that ∀(i, j ) ∈

S. Shi et al. / Applied Mathematics and Computation 338 (2018) 520–536

531

Fig. 5. The state response of the closed-loop system x(k) under switching signal σ (k) in Fig. 4.

L × L, i = j, (39) holds, and

⎡¯ i11 ⎢ ∗ ⎢ ⎢ ∗ ⎢ ⎢ ∗ ⎣ ∗ ∗

⎡˜ i j11 ⎢ ∗ ⎢ ⎢ ∗ ⎢ ⎢ ∗ ⎣ ∗ ∗

¯ i12  ¯ i22 ∗ ∗ ∗ ∗

˜ i j12  ˜ i j22  ∗ ∗ ∗ ∗

¯ i13  ¯ i23 ¯ i33  ∗ ∗ ∗

¯ i14  ¯ i24 0 −I ∗ ∗

˜ i j13  ˜ i j23  ˜ i j33  ∗ ∗ ∗

ε CyiVi ε DyiVi ¯ i35  ¯ i45  ¯ i55  ∗

˜ i j14  ˜ i j24  0 −I ∗ ∗

ε CyiV j ε DyiV j ˜ i j35  ˜ i j45  ˜ i j55  ∗

⎤

0 0 ⎥ ⎥ Zi Bi Joi ⎥ < 0, ⎥ Bzi Joi ⎥ ⎦ 0 − I

(42)

⎤

0 0 ⎥ ⎥ Zi Bi Jo j ⎥ < 0, ⎥ Bzi Jo j ⎥ ⎦ 0 − I

(43)

where α¯ = 1 − α , β¯ = 1 + β , and

¯ i11 = − α¯ Qi +  C  M Moi Cyi ,  ¯ i12 =  C  M Moi Dyi ,  yi oi yi oi ¯ i13 =A Z  + C V  B  ,  ¯ i14 = C  + C V  B  ,  i i yi i i zi yi i zi ¯ i22 = − γ 2 I +  D  M Moi Dyi ,  ¯ i23 = D  Z  + D V  B  ,  ¯ i24 = D  + D V  B  ,  yi oi i i yi i i zi yi i zi ¯ i33 =ν 2 Qi − ν Z  − ν Zi ,  ¯ i35 = Zi Bi − BiUi ,  ¯ i45 = Bzi − BziUi ,  ¯ i55 = −εU  − εUi ,  i i ˜ i j11 = − β¯ Qi +  C  M Mo j Cyi ,  ˜ i j12 =  C  M Mo j Dyi ,  yi o j yi o j ˜ i j13 =A Z  + C V  B  ,  ˜ i j14 = C  + C V  B  ,  i i yi j i zi yi j zi ˜ i j22 = − γ 2 I +  D  M Mo j Dyi ,  ˜ i j23 = D  Z  + D V  B  ,  ˜ i j24 = D  + D V  B  ,  yi o j i i yi j i zi yi j zi ˜ i j33 =ν 2 Qi − ν Z  − ν Zi ,  ˜ i j35 = Zi Bi − BiU j ,  ˜ i j45 = Bzi − BziU j ,  ˜ i j55 = −εU  − εU j ,  i j Then, for any PDT switching signal satisfying (10), the switched systems (1)–(3) is GUAS with an 2 -gain no greater than (12). Moreover, the admissible controller in the form of (32) and (33) can be given by Koi = Ui−1Vi .

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S. Shi et al. / Applied Mathematics and Computation 338 (2018) 520–536

Proof. By applying Schur complement, (42) is equivalent to

⎡¯ i11 ⎢ ∗ ⎢ ⎢ ∗ ⎣

¯ i12  ¯ i22

¯ i13  ¯ i23 ¯ i33 

∗ ∗ ∗

∗ ∗

⎤ ε CyiVi ε DyiVi ⎥ ⎥ ¯ i35 ⎥ +  ¯ i45 ⎦  ¯ i55 

¯ i14  ¯ i24 0 −I ∗

∗ ∗

⎡0

0 0 ∗ ∗ ∗

∗ ∗  ⎣∗ ∗ 1⎢ ⎢

0 0 Zi Bi Joi Joi Bi Zi ∗ ∗

0 0 Zi Bi Joi Joi Bzi Bzi Joi Joi Bzi ∗

⎤

0 0⎥ 0⎥ ≤ 0 . ⎦ 0 0

(44)

On the other hand, noticing that L (k )Loσ (k) (k ) ≤ I, for ∀k ≥ 0, then we have ∀σ (k ) = i ∈ L, oσ (k )







   Cyi Moi Lsi (k )    Dyi Moi Lsi (k )







Lsi (k )Moi Cyi

Lsi (k )Moi Dyi ≤

  Cyi Moi Moi Dyi .  Dyi Moi Moi Dyi

 Cyi Moi Moi Cyi ∗

Combining (44), by Lemma 1, we obtain that ∀ > 0,

⎡

−α¯ Qi ⎢ ∗ ⎢ ⎢ ∗ ⎣ ∗ ∗ where

0 −γ I ∗ ∗ ∗ 2

¯ i13  ¯ i23  ¯ i33 

¯ i14  ¯ i24  0 −I ∗

∗ ∗



Roi (k ) = Lsi (k )Moi Cyi

⎤ ε CyiVi  ⎥ ε DyiVi ⎥  ¯ i35 ⎥ + R  oi (k )Soi + Soi Roi (k ) ≤ 0, ⎦ ¯ i45  ¯ i55 

Lsi (k )Moi Dyi

Here, we denote

⎡

−α¯ Qi ⎢ ∗ Qi (k ) =⎣ ∗ ∗



Si = 0

0

0

−γ 2 I ∗ ∗

¯ i13 (k ) ¯ i23 (k ) ¯ i33 

0



0

⎤ ¯ i14 (k ) ¯ i24 (k )⎥, R = U −1 V C ⎦ i i yi i 0 −I

∗

Bi Zi − Ui Bi



0 , Soi = 0

0

(45)

Joi Bi Zi

Vi Dyi

0

Joi Bzi



0 .



0 ,



Bzi − Ui Bzi , Wi = εUi ,

where

¯ i13 + C  M L (k )J  B  Z  ,  ¯ i14 (k ) =  ¯ i14 + C  M L (k )J  B  , ¯ i13 (k ) = yi oi oi oi i i yi oi oi oi zi ¯ i23 + D  M L (k )J  B  Z  ,  ¯ i24 (k ) =  ¯ i24 + D  M L (k )J  B  , ¯ i23 (k ) = yi oi oi oi i i yi oi oi oi zi and ε is a positive scalar. Then, (45) can be rewritten as



Qi ( k ) ∗



Si − R Wi i < 0,  Wi + Wi

which is equivalent to  Qi ( k ) + R i Si + Si Ri ≤ 0,

(46) Ui−1Vi .

according to Lemma 2. Consider Ksi = Then we obtain that (46) can be rewritten as (37) by combining (35), which indicates that (42) can ensure (37). By applying the similar manipulation, one can obtain that (37) holds when (43) is satisfied. Consequently, the stability and non-weighted H∞ performance of the switched systems (1)–(3) is deduced with the dynamic output feedback controller Ksi = Ui−1Vi . The proof is completed.  When ignoring the switching delay between the switching of subsystems and controllers, the synchronous results is obtained. Corollary 2. Consider the switched systems (1)–(3), let 0 < α < 1, μ > 1, ν > 0,  > 0, and ε > 0 be given constants. For a prescribed period of persistence T, suppose that there exist matrices Qi > 0, Ui , Vi , Zi , ∀i ∈ L, and a scalar γ such that ∀(i, j ) ∈ L × L, i = j, (39) and (42) hold. Then, for any PDT switching signal satisfying (30), the switched systems (1)–(3) is GUAS with an 2 -gain no greater than (31). Moreover, the admissible controller can be given by Koi = Ui−1Vi . Remark 5. In [20], state feedback controllers are designed to solve the asynchronous H∞ control for discrete-time switched systems under ADT scheme. Here, we adopt PDT switching, which is more flexible. Meanwhile, we provide both state feedback (in Theorem 1) and output feedback (in Theorem 2) control schemes. Remark 6. The constraint conditions in Theorems 1 and 2 are standard linear matrix inequalities, which can be solved by standard toolboxes. In Theorem 1, there are Mn(3n + 2m + 1 )/2 independent decision variables, while in Theorem 2, this number is M[2(nc + m )(nc + p) + (nc + n )(3nc + 3n + 1 )/2]. From the above formula, the constraints will be more complex with the increasing of subsystems and state dimensions.

S. Shi et al. / Applied Mathematics and Computation 338 (2018) 520–536

533

Fig. 6. The output response of the closed-loop system z(k) under switching signal σ (k) in Fig. 4.

5. Illustrative examples In this section, we present two examples to illustrate the effectiveness of our proposed control scheme. 5.1. Example 1 We firstly introduce a modified population ecological system model borrowed from [31] to demonstrate the validity of proposed control scheme in Theorem 1. Consider an ecological system consisting of two types of population Z1 and Z2 , and suppose the situated environment be subject to switching between two scenarios Eσ , σ = 1, 2. Denote the number of individuals in population Zl as xl , l = 1, 2. Then, the basic mathematical principle of growth of xl can be described as:

  

x˙ 1 a ( 1 − ρ1 σ ) = 1σ x˙ 2 b21σ

 

b12σ a 2 σ ( 1 − ρ2 σ )





x1 c + 1σ u x2 c2σ

where alσ , l = 1, 2, is the maximum per-capita rate of change of Zl ; blmσ , m = 1, 2, l = m, is a transfer coefficient modeling the mutual influence of xl ; and clσ denotes the effect of immigration and emigration on xl . Let vl σ denote the carrying capacity of the population, then ρ lσ stands for the approximate proportion of vl σ and xl . Here, we select a11 = 0.2, a12 = 0.4, a21 = 0.5, a22 = 0.6, b121 = b122 = 0.1, b211 = b212 = 0.2, c11 = 0.8, c21 = 1.2, c12 = 1.1, c22 = 1, ρ11 = ρ12 = 0.5, ρ21 = ρ22 = 2. Then, the discretization model can be achieved as



A1 =



1.1 0.2







0.1 0.8 1.2 , B1 = , A2 = 0.5 1.2 0.2

Some other system parameters are given as:



D1 = D2 = 0 .1



0.1



, Cz1 = Cz2 = 1







0.1 1.1 , B2 = . 0.6 1



1 , Bz1 = 0.1, Bz2 = −0.1, Dz1 = Dz2 = 0.1.

Here, we consider α = 0.45, β = 0.1, μ = 1.03, γ = 0.7746, ϕmax = 2, T = 10, and ν = 1,  = 1. Assume that the actual controllers contain additive   perturbations with the structure (15), where Ls1 (k ) = Ls2 (k ) = sin(k ), ∀k ≥ 0, Js1 = Js2 = 0.1, Ms1 = Ms1 = 0.15 0.15 . Then, the conditions in Theorem 1 can be applied for the controller design for systems (1) and (3). Firstly, we can obtain τD∗ = 4.8747 with the above given parameters from (10). The conditions in Theorem 1 are feasible and a set of state feedback controllers is deduced with



Ks1 = −4.4308





−0.7628 , Ks2 = −4.1524



−0.0963 .

Consider the switching signal constrained by PDT with τ D ≥ 5 and T = 12, a randomly generated switching signal satisfying these constraints is given in Fig. 1. Under such a switching signal, the first stage is [0,16), while [0,6) is the τ -portion, during which the length is greater than the given τ D , and [6,16) is the T-portion, during which 6 switchings occur, and the dwell time of each mode is less than τ D . Thereafter, the system switches to the second stage. It is noted that the switching signal in Fig. 1 only governs the switching of subsystems. There exists a delay between the modes and controllers switching with the maximum switching delay ϕmax = 2. Assume that the disturbance input is in the form of ω (k ) = sin(k )e−0.1k . Under zero initial condition and the switching signal in Fig. 1, we obtain the state responses of the closed-loop system displayed

534

S. Shi et al. / Applied Mathematics and Computation 338 (2018) 520–536

Fig. 7. Evolution of γ s (k) of the closed-loop system under the PDT switching signal σ (k) in Fig. 4.

in Fig. 2, from which we know that the switched system converges by applying designed controllers since the states of the system approach zero with time going by. Here, we introduce



γs (k ) =

k 

z  ( s )z ( s )/

s=0

k 

ω (s )ω (s ),

s=0

which can reflect the impact of disturbance ω(k) to the controlled output z(k). The evolution of γ s (k) is displayed in Fig. 3. It can be seen that the value of γ s (k) is always less than 0.0678, which is far less than the prescribed 2 -gain index. The simulation results illustrate the effectiveness of Theorem 1. 5.2. Example 2 Consider the discrete-time switched linear systems (1)–(3) with three subsystems, the system parameters borrowed from [32].

⎡

−0.7705 ⎢ −0.1812 A1 =⎣ −0.2134 −0.5990



Cy1

1.2946 = −0.0975



Bz1 = 0.4792

⎡

0.5418 ⎢0.2434 A2 =⎣ 0.0403 0.3069

5.1328 1.8524 4.5333 7.8327 −0.1459 −1.1258







⎡

−0.7125 ⎢ −0.1650 A3 =⎣ −0.3273 −0.5494

1.1247 0.8063 2.1735 3.5743





0

 

⎡

⎤

⎤

⎡

⎤



0 , Dz1 = 0.0430,

0

⎤

⎡

1.6261 1.5880 0.1414 ⎥ ⎢0.3930 ,B = 0.7457 ⎦ 2 ⎣ 0.5519 1.7312 1.4598





0.4175 , Cz2 = 0

−0.0980 0.1070 0.0307 0.4024

Cy3 =Cy1 , Dy3 = Dy1 , Bz3 = 0.7209

⎤

1.0224 0.1313 0.2463⎥ ⎢ 0 ⎥ , D = , 0.8199⎦ 1 ⎣0.8995⎦ 1.2308 0.1707

−0.8153 0 , Dy1 = , 1.0753 0

0.3050 0.2852 −0.4102 −0.1759

Cy2 =Cy1 , Dy2 = Dy1 , Bz2 = 0.4939

⎡

−0.1174 0.2942 −0.2259⎥ ⎢0.0802 , B = −0.2665⎦ 1 ⎣ 0.1680 −0.8468 0.3274

0.5738 0.0579

0.0939 , Cz1 = 1 −1.8711 −0.0117 1.8748 0.1802

⎤

1.0885 0.4378 0.1267 1.2444

⎤

1

⎡

0

2.2917 1.7394 0.5089⎥ ⎢0.5328 ,B = 1.3381⎦ 3 ⎣1.3478 1.8611 2.1196





0.7649 , Cz2 = 0

0

1

1.5035 0.6500 0.3795⎥ ⎢ 0 ⎥ ,D = , 1.2047⎦ 2 ⎣ 0.4577 ⎦ 1.6609 0.5369



0 , Dz1 = 0.0665,

⎤

⎡

⎤

1.6432 0 0.4802⎥ ⎢0.2897⎥ ,D = , 0.7228⎦ 2 ⎣0.7538⎦ 1.5727 00968



0 , Dz1 = 0.0769.

Consider α = 0.29, β = 0.1, μ = 1.01, γ = 3.8730, ϕmax = 2, T = 10, and ν = 1,  = 1, ε = 1. Meanwhile, assume that nc = 1 and the actual output feedback controllers contain additive perturbations with the structure (34), where Lo1 (k ) =

S. Shi et al. / Applied Mathematics and Computation 338 (2018) 520–536

Lo2 (k ) = Lo3 (k ) = sin(k ), ∀k ≥ 0,



Jo1 =Jo2 = Jo3 = 0.065

0.065



0.065



, Mo1 = Mo2 = Mo3 = 0.091

0.091

535



0.091 .

According to Theorem 2, we can obtain τD∗ = 5.6590 and a set of output feedback controllers. The switching signal satisfying these constraints is given in Fig. 4. Assume that the disturbance input is in the form of ω (k ) = sin(k )e−0.1k . Under zero initial condition, we obtain the state and output responses of the closed-loop system, displayed in Figs. 5 and 6, respectively, which are convergent. At the same time, the evolution of γ s (k) is displayed in Fig. 7, which is always less than 0.4026. It indicates that the designed PDT switching signals and output feedback controllers by Theorem 2 are effectiveness. 6. Conclusions This work concerns the asynchronous H∞ control for a class of discrete-time switched systems with PDT switching. The criteria to guarantee the stability and 2 -gain for switched systems are established firstly, upon which the control schemes are proposed. Both state feedback and output feedback controllers are designed for switched linear systems. 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