Stability analysis and control synthesis for positive semi-Markov jump systems with time-varying delay

Applied Mathematics and Computation 332 (2018) 363–375

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

Stability analysis and control synthesis for positive semi-Markov jump systems with time-varying delay Lei Li a, Wenhai Qi a,∗, Xiaoming Chen b, Yonggui Kao c, Xianwen Gao d, Yunliang Wei e a

School of Engineering, Qufu Normal University, Rizhao 276826, China College of Automation Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China c Department of Mathematics, Harbin Institute of Technology, Weihai 150001, China d College of Information Science and Engineering, Northeastern University, Shenyang 110819, China e School of Mathematical Science, Qufu Normal University, Qufu 273165, China b

a r t i c l e

i n f o

Keywords: Semi-Markov jump systems Semi-Markov process Time-varying delay Linear programming

a b s t r a c t This paper deals with stability analysis and control synthesis for positive semi-Markov jump systems (S-MJSs) with time-varying delay, in which the stochastic semi-Markov process related to nonexponential distribution is considered. The main motivation for this paper is that the positive condition sometimes needs to be considered in S-MJSs and the controller design methods in the existing have some conservation. To deal with these problems, the weak infinitesimal operator is firstly derived from the point of view of probability distribution under the constraint of positive condition. Then, some sufficient conditions for stochastic stability of positive S-MJSs are established by implying the linear Lyapunov– Krasovskii functional depending on the bound of time-varying delay. Furthermore, an improved stabilizing controller is designed via decomposing the controller gain matrix such that the resulting closed-loop system is positive and stochastically stable in standard linear programming. The advantages of the new framework lie in the following facts: (1) the weak infinitesimal operator is derived for S-MJSs with time-varying delay under the constraint of positive condition and (2) the less conservative stabilizing controller is designed to achieve the desired control performance. Finally, three examples, one of which is the virus mutation treatment model, are given to demonstrate the validity of the main results. © 2018 Elsevier Inc. All rights reserved.

1. Introduction Positive systems [1,2] are a special class of dynamic systems, in which the state variables and output signals remain nonnegative whenever both the initial condition and the input signal are nonnegative. Positive systems have extensive applications in communication systems [3], virus mutation systems [4], system ecology [5,6], market economics, chemical industry, and environmental science, and have attracted ever-increasing research interests from different fields, so as to promote the development of positive system theory [7–12]. To mention a few, a necessary and sufficient condition for the existence of common linear copositive Lyapunov function was proposed for switched systems with two constituent linear time-invariant systems [7]. The synthesis of state-feedback controllers for positive linear systems was solved in terms of ∗

Corresponding author Tel.: +8615253875945. E-mail address: [email protected] (W. Qi).

https://doi.org/10.1016/j.amc.2018.02.055 0 096-30 03/© 2018 Elsevier Inc. All rights reserved.

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L. Li et al. / Applied Mathematics and Computation 332 (2018) 363–375

linear programming problem, including the requirement of positiveness of the controller and its extension to uncertain plants [9]. Under simplifying assumptions, viral mutation treatment dynamics could be viewed as a positive switched linear system [11]. Using linear co-positive Lyapunov functions, results for the synthesis of stabilizing, guaranteed performance and optimal control laws for switched linear systems were presented. By the use of simple linear performance index, linear Lyapunov function, cone invariant set with linear form and linear computation tool, a new model predictive control framework was proposed for positive systems subject to input/state constraints and interval/polytopic uncertainty [12]. On the other hand, time delay exists widely in practical systems, which has been regard as an important factor to make performance worse and system out of control [13–15]. Time delay increases the difficulty of theoretical analysis and engineering application, and also brings some challenges to analysis and synthesis of dynamics. Over the past decades, considerable attention has been devoted to positive systems with time delay (see e.g., [16–19]). For continuous-time delayed linear systems [16], some necessary and sufficient conditions were presented for the existence of controllers with bounded control satisfying the positivity constraint. By virtue of the positivity and linearity of the system, an explicit expression of the L∞ -gain of positive systems with distributed delays was given in terms of system matrices [17]. By using a copositive Lyapunov–Krasovskii functional and the average impulsive interval method, a sufficient criterion of global exponential stability for delayed impulsive positive systems was established in terms of linear programming problems [18]. For switched nonlinear positive systems with time-varying delay [19], by using the nonlinear Lyapunov–Krasovskii functional, sufficient conditions for absolute exponential L1 stability and a state-feedback control law were established in terms of linear programming. Furthermore, as a special kind of hybrid systems [20–26], Markov jump systems (MJSs) [27–38] have shown some advantages of describing various physical systems, such as economics systems, manufacturing systems, power systems, networkbased control systems, and other aspects. Recently, many results about positive MJSs have been reported; for details, see [39–48]. To mention a few, for positive Markov jump linear systems [39], the equivalent relations between exponential mean stability and 1-moment stability were built and various sufficient conditions for exponential almost-sure stability were worked out with different levels of conservatism. For positive systems with Markovian jump parameters [40], the problems of stochastic stability and mode-dependent state-feedback controller design in both continuous-time and discrete-time contexts were addressed. By constructing a linear stochastic Lyapunov function and introducing an equivalent deterministic positive discrete-time linear system, necessary and sufficient conditions for stochastic stability and 1 -gain performance of the positive discrete-time MJLS were derived and the existence of the desired positive 1 -gain filter was provided by solving a standard linear programming problem [41]. By using a copositive Lyapunov function approach, a computable sufficient condition for positive Markov jump linear systems with switching transition probability was proposed in the framework of dwell time to guarantee the mean stability [42]. Necessary and sufficient conditions of mean stability and stabilization were established for both continuous-time and discrete-time positive Markov jump systems in terms of standard linear programming [43]. For Takagi–Sugeno fuzzy positive Markovian jump systems [48], by constructing a linear copositive Lyapunov function, a positive L1 state-bounding observer was designed in linear programming. As we know, the sojourn time of the transition rate in MJSs follows unique memoryless exponential distribution, which means that the transition rate becomes time-invariant independent of the sojourn time. Compared with MJSs, the ST in S-MJSs obeys a more general nonexponential distribution including Weibull distribution, phase type distribution, Gaussian distribution, and so on, which leads to the time-varying characteristic of the transition rate matrix and brings both challenges and chances to analysis and synthesis of dynamics. Very recently, S-MJSs have many applications including hospital planning, multiple-bus systems, and credit risk systems. To date, it is noted that analysis and synthesis of S-MJSs have attracted a wide range of research interests (see e.g., [49–61]). For semi-Markov jump linear systems with norm-bounded uncertainties [50], the robust stochastic stability condition and the robust control design problem were addressed. For a class of continuous-time semi-Markovian jump systems [51], the possibility to explore more refined probability models on the sojourn time was offered by considering the mode transition-dependent sojourn-time distributions instead of modedependent ones. By using a supplementary variable technique and a plant transformation, a finite phase-type semi-Markov process was transformed into a finite Markov chain and a sliding mode controller was synthesized to guarantee the associated Markovian jump systems satisfying the reaching condition [52]. Robust passivity-based sliding mode controller for uncertain singular systems with semi-Markov switching and actuator failures was designed to ensure the closed-loop system to be stochastically admissible and robustly passive [54]. For a class of stochastic neutral-type neural networks with both semi-Markovian jump parameters and mixed time delays [57], a Luenberger-type observer was designed to guarantee the filter error dynamics mean-square exponentially stable with an expected decay rate and an attenuation level. For a class of semi-Markovian jump systems with external disturbance and sensor fault [59], an observer-based sliding mode control scheme was proposed to stabilize the fault closed-loop system and the reachability of the proposed sliding mode surface could be guaranteed. In this paper, the problems of stability analysis and control synthesis for positive S-MJSs with time-varying delay are considered. It is necessary to point out the differences between the present work and some existing relative works [6,9,39– 61]. First, for controller design method, the iterative optimization method has been introduced in [6] to compute the state feedback controller gain matrices, which increases the calculation complexity especially in the presence of high-order system. The parameter selection method has been presented in [40,44,45,47] with the rank of controller gain matrix confined to one. The dual system design method [9,43] could only be applied to the nominal system and could not be extended to more complex systems with nonlinearity and parameter uncertainties. Second, in the literatures [39–48], the sojourn time

L. Li et al. / Applied Mathematics and Computation 332 (2018) 363–375

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obeying the exponential distribution limits application scope of MJSs that are unable to describe more complex models. Third, the positive condition was not considered in [49–61] while the model in this paper includes the constraint of positive condition. The problems of stability analysis and control synthesis for positive S-MJSs with time-varying delay are still open all the time. We should notice that, compared with positive MJSs, it is very difficult to handle positive S-MJSs. Meantime, the system behaviors are affected by many complicated factors including the semi-Markov process, the time delay, and the positivity. Additionally, since the semi-Markov process exists in dynamics, many existing results for positive MJSs can not be directly applied to positive S-MJSs. Therefore, the following two essential difficulties during the problems of stability analysis and control synthesis for positive S-MJSs with time-varying delay should be solved: Q1: Compared with general S-MJSs [49–61], how to derive the weak infinitesimal operator under the constraint of positive condition? Q2: Compared with controller design method [6,9,40,43–45,47], how to design the less conservative stabilizing controller to achieve the desired control performance? These above aspects motivate our current research work of this paper. The main contributions are given as follows: (i) By implying the linear Lyapunov–Krasovskii functional, sufficient conditions for stochastic stability of positive S-MJSs are proposed; (ii) Based on the stochastic stability criteria, an improved stabilizing controller design method by decomposing the controller gain matrix is proposed to guarantee the resulting closed-loop system not only stochastically stable, but also positive. Notation. A ≥≥ 0 ( ≤ ≤ 0, > > 0, < < 0) means that A is nonnegative (nonpositive, positive, negative); 1-norm of ||x||1  stands for ||x||1 = nk=1 |xk |, where xk is the kth element of x ∈ Rn ; 1n denotes all-ones vector in Rn . 1rn is rth element of the vector 1n is one. E{ · } stands for the mathematical expectation. Fi (h) means the cumulative distribution function of sojourn time when system remains in mode i;  is the weak infinitesimal operator; qij is the probability intensity from mode i to mode j; λi (h) means the transition rate of system jumping from mode i. 2. Problem statement and preliminaries Consider the following S-MJSs with time delay:

x˙ (t ) = A(rt )x(t ) + Ad (rt )x(t − d (t )), x(t0 + θ ) =

ϕ (θ ), ∀θ ∈ [−d, 0],

(1)

where x(t ) ∈ is the state vector; d(t) satisfies 0 < d(t) ≤ d, d˙ (t ) ≤ μ < 1 with known real constants d and μ. {rt , t ≥ 0} represents a semi-Markov process in S = {1, 2, . . . , N } with the probability transitions as Rn



P r{rt+¯ = j|rt = i} =

¯ + o() ¯ , λi j (h ) ¯ + o() ¯ , 1 + λii (h )

i = j, i = j,

¯ /) ¯ = 0 and λij (h) ≥ 0 denotes the transition where h ≥ 0 denotes the sojourn time and has no connection with t, lim (o( ¯ →0  N rate from mode i to mode j for i = j, and j =1, j =i λi j (h ) = −λii (h ). In practice, the transition rate λij (h) is generally bounded ¯ i j are real constant scalars. ¯ i j , where λij and λ as λ ≤ λi j (h ) ≤ λ ij

Remark 1. Compared with the probability distribution function of the sojourn time following exponential distribution in [39–48], the nonexponential distribution is extended in this paper such that the transition rate matrix becomes time-varying matrix depending only on h. In continuous-time jump systems, the sojourn time h is a random variable following continuous probability distribution F. When F is the Weibull distribution, the cumulative distribution function of F is

F (h ) =

⎧ ⎪ ⎨

 β 

1 − exp −

⎪ ⎩

0,

h

γ

,

h ≥ 0, h < 0,

and the probability distribution function is

⎧  β  ⎪ ⎨ β β −1 h h exp − , f (h ) = γ β γ ⎪ ⎩ 0,

h ≥ 0, h < 0,

which means that the transition rate function is λ(h ) =

f (h ) 1−F (h )

= ββ hβ −1 . If we set β = 1 in Weibull distribution, the semiγ

Markovian process reduces to a standard Markov process where the sojourn time at each mode obeys exponential distribution. In such case, system (1) becomes continuous-time MJSs with λi j (h ) = dent of h, which means that MJSs are the special case of S-MJSs. For rt = i ∈ S, A(rt ) and Ad (rt ) are, respectively, denoted as Ai and Adi .

f i j (h ) 1−Fi j (h )

=

λi j e

λi j h

λ h 1−(1−e i j )

= λi j , where λij is indepen-

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Definition 1 [1,2]. System (1) is said to be positive if ϕ (θ ) ≥ ≥ 0, then x(t) ≥ ≥ 0 holds, for t > 0. Lemma 1 [1,2]. Consider the positive system as

x˙ (t ) = Ax(t ) + Ad x(t − d (t )), x(t0 + θ ) =

ϕ (θ ), ∀θ ∈ [−d, 0].

(2)

System (2) is positive if and only if A is Metzler matrix, Ad ≥≥ 0.

∞  Definition 2 [22]. System (1) is said to be stochastically stable if for ϕ (θ ) ≥ ≥ 0 and r0 ∈ S, E 0 ||x(t )||1 dt |(ϕ (θ ), r0 ) < ∞ holds. For controller given by u(t ) = Ki x(t ), ∀i ∈ S, the resulting closed-loop system is denoted as x˙ (t ) = (Ai + Bi Ki )x(t ) + Adi x(t − d (t )), x(t0 + θ ) =

ϕ (θ ), ∀θ ∈ [−d, 0].

(3)

In this paper, our aim is to construct sufficient conditions for stochastic stability and stabilizing controller design in standard linear programming. 3. Stochastic stability Theorem 1. If there exist χ i , β 1i , β 2i , β 1 , β2 ∈ Rn+ , ∀i ∈ S, such that

(Ai + Bi Ki )T χi + β1i + β2i + dβ1 + dβ2 +

N 

λi j (h )χ j << 0,

(4)

j=1

ATdi χi − (1 − μ )β1i << 0, N 

(5)

N 

λi j (h )β1 j ≤≤ β1 ,

j=1

λi j (h )β2 j ≤≤ β2 ,

(6)

j=1

then system (3) is stochastically stable. Proof. Choose the linear Lyapunov–Krasovskii functional

V (x(t ), t , i ) = V1 (x(t ), t , i ) + V2 (x(t ), t , i ) + V3 (x(t ), t , i ),

(7)

where

V1 (x(t ), t , i ) = xT (t )χi ,  t  V2 (x(t ), t , i ) = xT (s )β1i ds + t −d (t )

V3 (x(t ), t , i ) =



0 −d



t

t−d

t

t+θ

xT (s )β2i ds,

xT (s )(β1 + β2 )dsdθ ,

(8)

where χ i , β 1i , β 2i , β 1 , β2 ∈ Rn+ and N 

λi j (h )β1 j ≤≤ β1 ,

j=1

N 

λi j (h )β2 j ≤≤ β2 .

(9)

j=1

Along with system (3), one has the weak infinitesimal operator [47] as

 V1 (x(t ), t , i ) = lim

¯ →0 

1 ¯ ,t +  ¯ , r ¯ )|rt = i} − V1 (x(t ), t , i )] [E {V1 (x(t + ) t+ ¯ 

1 = lim ¯ ¯ →0   1 = lim ¯ ¯ → 0  1 = lim ¯ ¯ →0  

  

N  j=1, j=i

N  P r{rt+¯ = j, rt = i} j=1, j=i



¯ χ j + P r{g ¯ = i|gt = i} x (t + ) ¯ χi − x (t )χi P r{rt+¯ = j|rt = i}x (t + ) t+ T

P r{rt = i}

¯ χj + x (t + ) T

T

P r{rt+¯ = i, rt = i} P r{rt = i}

T

 ¯ χi − xT (t )χi x (t + ) T



N ¯ − Fi (h ))  ¯ qi j (Fi (h + ) ¯ χ j + 1 − Fi (h + ) xT (t + ) ¯ χi − xT (t )χi . xT (t + ) 1 − Fi (h ) 1 − Fi (h )

j=1, j=i

(10)

L. Li et al. / Applied Mathematics and Computation 332 (2018) 363–375

367

¯ is According to the relevant knowledge of calculus, the first order approximation of x(t + )

¯ = x(t ) + x˙ (t ) ¯ + o() ¯ = x(t ) + [(Ai + Bi Ki )x(t ) + Adi x(t − d (t ))] ¯ + o() ¯ , x(t + ) ¯ → 0. when  Then, we have



1  V1 (x(t ), t , i ) = lim ¯ ¯ →0  

(11)

N ¯ − Fi (h ))  qi j (Fi (h + ) ¯ + o()) ¯ Tj χ (x(t ) + ((Ai + Bi Ki )x(t ) + Adi x(t − d (t ))) 1 − Fi (h )

j=1, j=i

¯ 1 − Fi (h + ) ¯ + o()) ¯ Ti χ − xT (t )χ (x(t ) + ((A + Bi Ki )x(t ) + Adi x(t − d (t ))) i 1 − Fi (h )

+



1 = lim ¯ ¯ →0 



N ¯ − Fi (h ))  qi j (Fi (h + ) ¯ + o()) ¯ T χj (x(t ) + ((Ai + Bi Ki )x(t ) + Adi x(t − d (t ))) 1 − Fi (h )

j=1, j=i



¯ ¯ 1 − Fi (h + ) ¯ + o()) ¯ T χi − Fi (h + ) − Fi (h ) xT (t )χi . + ((Ai + Bi Ki )x(t ) + Adi x(t − d (t ))) 1 − Fi (h ) 1 − Fi (h ) (12) According to lim

¯ →0 

¯ −F (h ) Fi (h+) i 1−Fi (h )

= 0, lim

¯ →0 



¯ 1−Fi (h+) 1−Fi (h )

= 1, lim

¯ →0 

¯ −F (h ) Fi (h+) i ¯ 1−F (h )) ( i

 V1 (x(t ), t , i ) = xT (t )(Ai + Bi Ki )T χi + xT (t − d (t ))ATdi χi +

N 

= λi (h ), and λi j (h ) = qi j λi (h ), i = j, we have



λi j (h )χ j .

(13)

j=1

Similarly, we obtain



 V2 (x(t ), t , i ) = xT (t )(β1i + β2i ) − (1 − d˙ (t ))xT (t − d (t ))β1i  +

t

x (s ) T

t −d (t )

N 

λi j (h )β1 j ds − x (t − d )β2i +



t

T

j=1

x (s ) T

t−d

N 

   t  V3 (x(t ), t , i ) = xT (t )(dβ1 + dβ2 ) − xT (s )(β1 + β2 )ds t−d    t  t ≤ xT (t )(dβ1 + dβ2 ) − xT (s )β1 ds − xT (s )β2 ds . t −d (t )

 λi j (h )β2 j ds ,

j=1

(14)

t−d

From the condition d˙ (t ) ≤ μ < 1, one has

−(1 − d˙ (t ))xT (t − d (t ))β1i ≤ −(1 − μ )xT (t − d (t ))β1i . Then, we have



(15)



 V (x(t ), t , i ) ≤ x (t ) (Ai + Bi Ki ) χi + β1i + β2i + dβ1 + dβ2 + T

T

N 

 λi j (h )χ j

j=1



+ xT (t − d (t ))(ATdi χi − (1 − μ )β1i ) − xT (t − d )β2i . According to (4)–(6), we get



 V (x(t ), t , i ) ≤ xT (t ) (Ai + Bi Ki )T χi + β1i + β2i + dβ1 + dβ2 +

N 

(16)

 λi j (h )χ j

j=1

< 0, which means that system (3) is stochastically stable.

(17) 

Remark 2. For the first question (Q1), there are no relevant literatures for reference. Different from general S-MJSs [49–61], we need to derive the weak infinitesimal operator from the point of view of probability distribution under the constraint of positive condition (see formulas (10)–(13)).

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L. Li et al. / Applied Mathematics and Computation 332 (2018) 363–375

4. Controller design Here, we shall answer the second question (Q2). During the controller design of positive systems, three cases are discussed as follows: (i) All the elements of Bi are greater than or equal to 0; (ii) The elements of Bi appear greater than or equal to 0 and less than or equal to 0 simultaneously; (iii) All the elements of Bi are less than or equal to 0. When all the elements of Bi are greater than or equal to 0, we derive the Theorem 1. Theorem 2. If there exist χ i , β 1i , β 2i , β 1 , β2 ∈ Rn+ , κ pi , κi ∈ Rn , ∀i ∈ S, such that (5) and

ATi χi + κi + β1i + β2i + dβ1 + dβ2 +

N 

λi j χ j << 0,

(18)

λ¯ i j χ j << 0,

(19)

j=1

ATi χi + κi + β1i + β2i + dβ1 + dβ2 +

N  j=1

N 

N 

λi j β1 j ≤≤ β1 ,

j=1

N 

λ¯ i j β1 j ≤≤ β1 ,

j=1

N 

λi j β2 j ≤≤ β2 ,

j=1

λ¯ i j β2 j ≤≤ β2 ,

(20)

j=1

p T m Ai 1Tm BiT χi + Bi p=1 1m κ pi + εi I ≥≥ 0,

(21)

κ pi ≤≤ κi ,

(22)

then system (3) is positive and stochastically stable. Moreover, the controller can be given as

u(t ) = Ki x(t ) =

p T m p=1 1m κ pi x(t ). T T 1m Bi χi

(23)

Proof. Since 1Tm > 0, Bi >> 0, and χ i > 0, we have 1Tm BiT χi > 0. This together with (21) gives that Ai + Bi εi

1Tm BiT χi

I ≥≥ 0. Following Eq. (23), it is clear that Ai + Bi Ki +

εi

1Tm BiT χi

p m 1 κT p=1 m pi

1Tm BiT χi

+

I ≥≥ 0, which means that system (3) is positive from

Lemma 1. p T p m 1 p κ T ≤≤ m 1 p κ T = 1 κ T . From (22), we can get 1m κ pi ≤≤ 1m κiT and p=1 m i m pi p=1 m i Then, one has

KiT BiT χi ≤≤ κi .

(24)

¯ i j , where θ1 + θ2 = 1 and θ 1 > 0, θ 2 > 0. Multiplying (18) by For a specific h, λij (h) can be written as λi j (h ) = θ1 λi j + θ2 λ θ 1 and (19) by θ 2 leads to

ATi χi + κi + β1i + β2i + dβ1 + dβ2 +

N 

(θ1 λi j + θ2 λ¯ i j )χ j + CiT 1s << 0.

(25)

j=1

¯ i j ] can be achieved, which means that (4) holds. By tuning θ 1 and θ 2 , all possible λi j (h ) ∈ [λi j , λ Similar method to (20), we can get (6).  Remark 3. During the proof of Theorem 1, the matrix decomposition method is proposed to transform present condition into linear programming, that is



Ki = [Km1 n1 i ]m×n = 11m × K11i



+ 12m × K21i m1 = 1, . . . , m,

K22i

···

K12i



···

K1ni





K2ni + · · · + 1m m × K m1i

K m2i

···



Kmni ,

n1 = 1, . . . , n.

(26)

According to (26), the conditions (21) and (22) can be obtained. When the parameters of Bi appear greater than or equal to 0 and less than or equal to 0 simultaneously, we derive the Corollary 1. Corollary 1. If there exist χ i , β 1i , β 2i , β 1 , β2 ∈ Rn+ , κ pi , κ¯ i , κ i ∈ Rn , ∀i ∈ S, such that (5), (20), (21) and

ATi χi + κ¯ i + β1i + β2i + dβ1 + dβ2 +

N  j=1

λi j χ j + CiT 1s << 0,

(27)

L. Li et al. / Applied Mathematics and Computation 332 (2018) 363–375

ATi χi + κ¯ i + β1i + β2i + dβ1 + dβ2 +

N 

369

λ¯ i j χ j + CiT 1s << 0,

(28)

j=1

1Tm BiT χi > 0,

(29)

pT T 1m Bi χi < 0,

p = 1, 2, · · · , p1 < m,

(30)

pT T 1m Bi χi > 0,

p = p1 + 1, p1 + 2, · · · , m,

(31)

κ i ≤≤ κ pi ≤≤ κ¯ i ,

(32)

then system (3) with controller (23) is positive and stochastically stable. Proof. By (27)–(32), we get p1 pT T m m κ pi 1mpT BiT χi p= p=1 κ pi 1mpT BiT χi p=1 p1 +1 κ pi 1m Bi χi = + 1Tm BiT χi 1Tm BiT χi 1Tm BiT χi   p1 pT T pT T pT T m m κ i p=1 1m Bi χi κ¯ i p= p= p1 +1 1m Bi χi p1 +1 1m Bi χi << + = κi 1 − 1Tm BiT χi 1Tm BiT χi 1Tm BiT χi

KiT BiT χi =

pT T pT T m m p= p= p1 +1 1m Bi χi p1 +1 1m Bi χi = κ + ( κ ¯ − κ ) i i i 1Tm BiT χi 1Tm BiT χi << κ i + (κ¯ i − κ i ) = κ¯ i .

+ κ¯ i

(33)



The rest of the proof is similar to Theorem 2 and is omitted.

When all the parameters of Bi are less than or equal to 0, we derive the Corollary 2. Corollary 2. If there exist χ i , β 1i , β 2i , β 1 , β2 ∈ Rn+ , κ pi , κi ∈ Rn , ∀i ∈ S, such that (5), (18)–(20), (22) and p T m Ai 1Tm BiT χi + Bi p=1 1m κ pi + εi I ≤≤ 0,

(34)

then system (3) with controller (23) is positive and stochastically stable. Proof. Since 1Tm > 0, Bi << 0, and χ i > 0, we have 1Tm BiT χi < 0. This together with (34) gives that Ai + Bi εi

1Tm BiT χi

I ≥≥ 0. Following Eq. (23), it is clear that Ai + Bi Ki +

εi

1Tm BiT χi

Lemma 1. The rest of the proof is similar to Theorem 2 and is omitted.

p m 1 κT p=1 m pi

1Tm BiT χi

+

I ≥≥ 0, which means that system (3) is positive from



Remark 4. Compared with some existing relative works [6,9,40,44,45,47], the advantages of the matrix decomposition method are given as follows: (i) It proposes a simple controller form containing three cases of the parameter Bi ; (ii) The conditions can be easily computed in standard linear programming; (iii) It is easily extended to other issues of positive systems, such as finite-time control and filter design. Compared with the iterative optimization method [6], the state feedback controller can be solved directly, which reduces the calculation complexity. 5. Numerical example Example 1. Consider the virus mutation treatment model, taken from [11], described by

x˙ (t ) = (R(rt ) − δ I + ε M )x(t ) + B (rt )u(t ),

(35)

where x(t ) ∈ represents two different viral genotypes; u(t ) ∈ is the control input; {rt , t ≥ 0} represents a semi-Markov process in S = {1, 2}; ε , δ , and M = [Mmn ] are the mutation rate, the death or decay rate, and the system matrices; Mmn ∈ {0, 1} means the genetic connections between genotypes. The parameter values are given as: R2



R1 =

 B2 =

0.05 0.7 0.1 0

R2



0.8 , 0.25



0.1 , 0.5



0.06 R2 = 0.6



0.9 , 0.26

δ = 0.15, ε = 0.001.



0 M= 1



1 , 0



0.1 B1 = 0



0.2 , 0.3

370

L. Li et al. / Applied Mathematics and Computation 332 (2018) 363–375

Switching signal 3

2.5

System Mode

2

1.5

1

0.5

0

0

5

10

15

20

Time Fig. 1. System mode of Example 1.

that there exists the time-varying delay in the virus mutation treatment model (35) with Ad1 =  In addition,  we assume  0.1 0 0.1 0 , Ad2 = , d (t ) = 0.2(1 − sin(0.5t )), d = 0.4, μ = 0.1. 0 0.1 0 0.1 The transition rate matrices are supposed as



λ=

−0.5 0.8



0.5 , −0.8



λ¯ =

By Theorem 3 in [47], we get



−2.1824 K1 = −2.1824

−1.0 1.4



−2.5948 , −2.5948



1.0 . −1.4



−1.1628 K2 = −1.1628



−3.8944 . −3.8944

From the controller gain matrix K1 and K2 , we can see that the rank of controller gain matrix is confined to one, which may have some conservativeness. Now, we apply Theorem 2 in this paper to get



−2.6394 K1 = −2.2718



−2.5126 , −2.4314



−1.5051 K2 = −1.1151



−3.5628 . −3.4011

From the controller gain matrix K1 and K2 , we can see that the rank of controller gain matrix is not confined to one, which may reduce some conservativeness. Then, one has



A1 + B1 K1 =

−0.8183 0.0195



0.0635 , −0.6294



A2 + B2 K2 =



−0.3520 0.0434

0.2046 , −1.5905

which means that the closed-loop system  is positive.  For given initial conditions x(0 ) = 0.3 0.25 and r0 = 1, Figs. 1 and 2 plot the stochastic switching rule following a semi-Markov process rt and the state response x(t) of the closed-loop system, which indicates the proposed state feedback controller works well. These figures demonstrate the applicability and correctness of our results. Example 2. Consider the system (3) with the parameters given as



A1 =

1.2 0.3



0.6 , −0.9



0.9 A2 = 0.2



0.6 , −1.6



Ad1

0.2 = 0



0 , 0.1



Ad2

0.1 = 0



0 , 0.2

L. Li et al. / Applied Mathematics and Computation 332 (2018) 363–375

371

0.35 x1 x2 0.3

State resopnse

0.25

0.2

0.15

0.1

0.05

0

0

5

10

15

20

Time Fig. 2. State trajectory of the closed-loop system of Example 1.

 B1 =



−0.02 −0.01

0.15 , −0.1

 B2 =



−0.05 −0.01

0.2 . −0.1

The lower and upper bounds of the transition rate matrix are supposed as



−0.6 0.8

λ=





0.6 , −0.8

λ¯ =



−1.0 1.4

1.0 . −1.4

Let d (t ) = 0.3(1 − sin(0.5t )), then d = 0.6, μ = 0.15. By using the parameter selection method [40,44,45,47], we can not get the controller gain matrices since the parameter Bi needs to satisfy Bi ≥≥ 0. However, the elements of the parameter Bi are partly negative in this example. Now, we apply Corollary 1 to get





−83.8524 K1 = −90.7358 Then, we have



−49.1028 , −33.8708



−4.0251 A1 + B1 K1 = 0.9883



−33.3482 K2 = −36.1467



0.4297 , −2.4232

−5.0282 . −1.9797



−2.9945 A2 + B2 K2 = 0.4798



0.7069 , −1.9049

which means that the closed-loop system  is positive.  For given initial conditions x(0 ) = 0.5 0.8 and r0 = 2, Figs. 3 and 4 describe the stochastic switching rule following a semi-Markov process rt and the state response x(t) of the closed-loop system. These figures demonstrate the applicability and correctness of our results. Example 3. Consider the system (3) with the parameters given as



A1 =

 B1 =

0.4 0.5 −0.1 −0.1



0.6 , −0.8







−1.0 A2 = 0.2

−0.15 , −0.01



B2 =

0.5 , 0.3

−0.1 −0.2



Ad1



0.1 = 0



0.1 , 0.1



Ad2

−0.12 . −0.01

The lower and upper bounds of the transition rate matrix are supposed as



−0.4 λ= 0.6



0.4 , −0.6



−0.8 λ¯ = 1.0



0.8 . −1.0

0.2 = 0



0 , 0.2

372

L. Li et al. / Applied Mathematics and Computation 332 (2018) 363–375

Switching signal 3

System Mode

2.5

2

1.5

1

0.5

0

0

5

10

15

20

Time Fig. 3. System mode of Example 2.

0.8 x1 x2

0.7

State resopnse

0.6 0.5 0.4 0.3 0.2 0.1 0

0

5

10

15

Time Fig. 4. State trajectory of the closed-loop system of Example 2.

20

L. Li et al. / Applied Mathematics and Computation 332 (2018) 363–375

373

Switching signal 3

System Mode

2.5

2

1.5

1

0.5

0

0

5

10

15

20

Time Fig. 5. System mode of Example 3.

0.7 x1 x2 0.6

State resopnse

0.5

0.4

0.3

0.2

0.1

0

0

5

10

15

Time Fig. 6. State trajectory of the closed-loop system of Example 3.

20

374

L. Li et al. / Applied Mathematics and Computation 332 (2018) 363–375

Let d (t ) = 0.3(1 − sin(t )), then d = 0.6, μ = 0.3. By using the parameter selection method [40,44,45,47], we can not get the controller gain matrices since the parameter Bi needs to satisfy Bi ≥≥ 0. However, all the elements of the parameter Bi are negative in this example. Now, we apply Corollary 2 to get



4.2414 K1 = 4.5387 Then, we have

A1 + B1 K1 =





2.3124 , 2.2672

−0.7050 0.0305



0.6332 K2 = 0.7786



2.2483 . 2.2397



0.0287 , −1.0539

 A2 + B2 K2 =

−1.1568 0.0656



0.0064 , −0.1721

which means that the closed-loop system  is positive.  For given initial conditions x(0 ) = 0.4 0.6 and r0 = 2, Figs. 5 and 6 describe the stochastic switching rule following a semi-Markov process rt and the state response x(t) of the closed-loop system. These figures demonstrate the applicability and correctness of our results. 6. Conclusions This study has developed stability analysis and control synthesis for positive S-MJSs with time-varying delay. Sufficient conditions for positivity and stochastic stability criteria in terms of linear programming have been proposed by using the weak infinitesimal operator from the viewpoints of probability distribution firstly. Based on these criteria, an improved stabilizing controller design method by decomposing the controller gain matrix has been proposed to guarantee the closedloop system positive and stochastically stable, which is simple and easily computed. In future work, the results can be easily extended to other issues of positive S-MJSs, such as finite time control and filter design. Acknowledgments This work is supported by Natural Science Foundation of Shandong under Grant Nos. BS2015NJ014, ZR2017QF001, ZR2017MF063 and ZR2016FQ09, National Natural Science Foundation of China under Grant Nos. 61703231 and 61503184, and Postdoctoral Science Foundation of China under Grant Nos. 2017M612235 and 2016M602112. References [1] L. Farina, S. Rinaldi, Positive Linear Systems: Theory and Applications, Wiley, New York, 20 0 0. [2] T. Kaczorek, Positive 1D and 2D Systems, Springer, London, 2002. [3] R. Shorten, D. Leith, J. Foy, R. Kilduff, Towards an analysis and design frame work for congestion control in communication networks, in: Proceedings of the 12th Yale Workshop on Adaptive and Learning Systems, New Haven, CT, USA, 2003. [4] J. Ferreira, R.H. Middleton, A preliminary analysis of HIV infection dynamics, Proceedings of Irish Signals and Systems Conference, Galway, Ireland, 2008. [5] P. Li, J. Lam, Z. Shu, H∞ positive filtering for positive linear discrete-time systems: an augmentation approach, IEEE Trans. Autom. Control 55 (10) (2010) 2337–2342. [6] X.M. Chen, J. Lam, P. Li, Z. Shu, 1 -induced norm and controller synthesis of positive systems, Automatica 49 (5) (2013) 1377–1385. [7] O. Mason, R. Shorten, On linear copositive Lyapunov functions and the stability of switched positive linear systems, IEEE Trans. Autom. Control 52 (7) (2007) 1346–1349. [8] E. Fornasini, M. Valcher, Linear copositive lyapunov functions for continuous-time positive switched systems, IEEE Trans. Autom. Control 55 (8) (2010) 1933–1937. [9] M.A. Rami, F. Tadeo, Controller synthesis for positive linear systems with bounded controls, IEEE Trans. Circ. Syst. II Express Briefs 54 (2) (2007) 151–155. [10] X.M. Chen, M. Chen, W.H. Qi, J. Shen, Dynamic output-feedback control for continuous-time interval positive systems under L1 performance, Appl. Math. Comput. 289 (2016) 48–59. [11] E. Hernandez-Varga, P. Colaneri, R. Middleton, F. Blanchini, Discrete-time control for switched positive systems with application to mitigating viral escape, Int. J. Robust Nonlinear Control 21 (10) (2011) 1093–1111. [12] J.F. Zhang, X.L. Jia, R.D. Zhang, Y. Zuo, A model predictive control framework for constrained uncertain positive systems, Int. J. Syst. Sci. 49 (4) (2018) 884–896. [13] W.H. Qi, Y.G. Kao, X.W. Gao, Y.L. Wei, Controller design for time-delay system with stochastic disturbance and actuator saturation via a new criterion, Appl. Math. Comput. 320 (1) (2018) 535–546. [14] G.D. Zong, R.H. Wang, W.X. Zheng, L.L. Hou, Finite-time H∞ control for discrete-time switched nonlinear systems with time delay, Int. J. Robust Nonlinear Control 25 (6) (2015) 914–936. 2015 [15] W.H. Qi, J.H. Park, J. Cheng, Y.G. Kao, X.W. Gao, Anti-windup design for stochastic markovian switching systems with mode-dependent time-varying delays and saturation nonlinearity, Nonlinear Anal. Hybrid Syst. 26 (2017) 201–211. [16] X.W. Liu, Constrained control of positive systems with delays, IEEE Trans. Autom. Control 54 (7) (2009) 1596–1600. [17] J. Shen, J. Lam, L∞ -gain analysis for positive systems with distributed delays, Automatica 50 (1) (2014) 175–179. [18] Y.W. Wang, J.S. Zhang, M. Liu, Exponential stability of impulsive positive systems with mixed time-varying delays, IET Control Theory Appl. 8 (15) (2014) 1537–1542. [19] J.F. Zhang, X.D. Zhao, X.S. Cai, Absolute exponential L1 -gain analysis and synthesis of switched nonlinear positive systems with time-varying delay, Appl. Math. Comput. 284 (2016) 24–36. [20] X.H. Chang, J. Xiong, J.H. Park, Fuzzy robust dynamic output feedback control of nonlinear systems with linear fractional parametric uncertainties, Appl. Math. Comput. 291 (2016) 213–225. [21] H.L. Ren, G.D. Zong, T.S. Li, Event-triggered finite-time control for networked switched linear systems with asynchronous switching, IEEE Trans. Syst. Man Cybern. Syst. http://dx.doi.org/10.1109/TSMC.2017.2789186.

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