# Output Regulation of Switched Stochastic Systems with Sampled-Data Control

Authors: Xiaoxiao Dong; Hui Lan
Journal: Advances in Mathematical Physics
 (2023)

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## Abstract

This paper studies the output regulation problem for a class of switched stochastic systems with sampled-data control. Solutions to the output regulation problem are given in two situations. On the one hand, the exogenous signal is assumed to be a constant. By designing a sampled-data state feedback controller, we obtain that the closed-loop system is mean-square exponentially stable and the regulation output tends to zero. On the other hand, the exogenous signal is assumed to be time-varying with bounded derivative. By constructing a class of Lyapunov-Krasovskii functional and a switching rule which satisfies the average dwell time, sufficient conditions for the solvability of practical output regulation problem are given for switched stochastic systems. Finally, numerical examples are given to illustrate the effectiveness of the method.


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## Body

## 1. Introduction

Switched system is a typical class of hybrid system, which is composed of a series of subsystems and a switching rule which determines the active subsystem [1]. These systems have been applied to many practical engineering problems, such as flight control [2], communication engineering [3, 4], and network control [5, 6]. Therefore, switched systems and switching control are paid more and more attention [7–10]. As is well known, the random factors may result in the performance degeneration of practical systems. Consequently, it is very important to discuss the influence of random factors on the systems and there are many achievements about switched systems with random factors [11]. The related research results of switched stochastic systems have also been widely used in water quality engineering [12], industrial process control, biomedical engineering, and so on [13–18], although the switched system with random factors is more complex than the general switched system.Output regulation problem (ORP) is aimed at designing the controllers to achieve the asymptotic tracking and disturbance rejection for reference signals. A majority of existing literature on output regulation focus on non-switched systems [19–21]. Nevertheless, the results on the ORP of switched systems are still few. Wang et al. have constructed a class of multiple linear Lyapunov functions and solved the ORP of positive switched systems by designing two kinds of controllers [22]. Reference [23] discusses the solvability of the ORP for discrete switched systems, and in order to utilize the communication network effectively, the designed feedback controller adopts the quantized output, and the solution to the problem is given by different coordinate transformations. There are still many problems of output regulation of switched systems to be studied.With the development of digital control technology, many practical control systems adopt sampled-data control. The controller designed based on the time sampled-data is usually composed of time-triggered continuous step signal samplers and zero-order hold devices; thus, it can be directly implemented on the digital platform. The application of sampled-data control can improve the efficiency of system control and the utilization of the controller. In addition, the time-based sampling control will not trigger infinite times in a limited time interval. Therefore, Zeno phenomenon does not occur, and the controlled system under this kind of control strategy has the advantages of efficient execution and predictability, which meets the requirements of the practical engineering field [24]. In [25], the stability of the closed-loop system is analyzed directly, and the ORP of the linear system under aperiodic sampled-data control is solved. By extending the Halanay inequality to time-varying sampled-data systems, Zhang et al. obtained sufficient conditions for the stability of the closed-loop system [26]. Due to the wide application of switched systems, many scholars extend the relevant results of nonswitched systems under sampled-data control to switched systems. By constructing a new class of nonpositive definite functionals and giving a sampled-data feedback controller, Lian et al. obtained exponential stability of linear switched systems under asynchronous switching [27]. In [28], Fu et al. designed a sampled-data controller by using a fixed sampling period to ensure the stability of a class of switched neutral systems under asynchronous switching. The research results of ORP based on sampled-data have also been widely used in intelligent control fields such as mobile robots [29]. By virtue of the above analysis, the research on the ORP of switched stochastic systems under sampled-data control is essential. However, up to now, the results about this problem are not available.This paper studies the ORP of a class of switched stochastic systems under a periodic sampling mechanism. For the exogenous signal generated by the exosystem, the constant signal and the time-varying signal with derivative bounded are considered, respectively. The main contributions are listed as follows: (1) the output regulation problem of stochastic switched systems is considered. In fact, even if the output regulation problem of each subsystem of the switched system is solvable, the output regulation problem of the switched system may not be solvable under any switching signal. Therefore, it is necessary to give the relationship between the average dwell time (i.e., the conditions to be satisfied by the switching signal) and the sampling period to obtain the solvable conditions of the output regulation problem of switched stochastic systems. This paper gives the relationship between the average dwell time and the sampling period. Meanwhile, this paper considers the asynchronous switching between the subsystem and the controller of the switched stochastic systems. (2) The sampled-data controller designed by time trigger is used to solve the output regulation problem of switched stochastic systems for the first time. The sampling mechanism applied in this paper will not trigger infinitely in a finite interval; that is, Zeno phenomenon will not occur. In addition, this paper also considers both output regulation and practical output regulation and gives sufficient conditions for the solvability of the problem.


## 2. Problem Formulation and Preliminaries

Consider the following switched stochastic system described by(1)dxt=Aσtxt+Bσtut+Hσtωtdt+fσtt,xtdϖt,et=Cσtxt+Dσtωt,and the exosystem is described by
(2)dωt=Sωtdt,where xt∈Rm is the state vector of the system, ωt∈Rp is called the exogenous signal, et∈Rq represents the regulated output, σ:0,∞⟶IN=1,⋯,n is the switching signal, and it is a class of the piecewise right continuous constant value functions, and ut∈Rn is control input. Brownian motion ϖt with n-dimensional satisfies Edϖt=0 and Edϖ2t=dt. σ=i, i∈IN denotes that the ith subsystem is active. Ai, Bi, Hi, Ci, Di, and S are constant matrices; the nonlinear coefficient fi∈Rr satisfies the local Lipschitzness and the linear growth condition. Suppose that the matrix S has nonnegative real parts and the pair Ai,Bi:i∈IN is stable.The sampled-data state feedback controllers are designed as follows:(3)ut=Kixtk+Liωtk,∀t∈tk,tk+1,where Ki and Li, i∈IN, are constant matrices with suitable dimensions.Then, the closed-loop system composed of the system (1) and the controller (3) is represented as
(4)dxt=Aσtxt+BσtKσtxtk+BσtLσtωtk+Hσtωtdt+fσtt,xtdϖt,et=Cσtxt+Dσtωt.Letx~t=xt−xtk and ω~t=ωt−ωtk; the formula (4) can be rewritten as
(5)dxt=A¯σtxt+B¯σtx~t+M¯σtωt+L¯σtω~tdt+fσtt,xtdϖt,et=Cσtxt+Dσtωt,where A¯i=Ai+BiKi, B¯i=−BiKi, M¯i=Hi+BiLi, and L¯i=−BiLi.The main results are based on the following assumptions:Assumption 1.
(see [25]).
There exists a matrixΠ with appropriate dimension, satisfying the matrix equations
(6)ΠS=A¯iΠ+M¯i,0=CiΠ+Di,i∈IN.
According to equation (6), we have the coordinate transformation x¯=x−Πω; then, the closed-loop system (5) is transformed into the following form:
(7)dx¯t=A¯σtx¯t+B¯σtx¯~t+B¯σtΠ+L¯σtω~tdt+fσtt,x¯t+Πωtdϖt,e=Cσtx¯t.Assumption 2.
(see [11]).
For nonlinear functionsfit,x¯t+Πωt, there exist the constant matrices Ji, i∈IN such that
(8)trfiTt,x¯t+Πωtfit,x¯t+Πωt≤Jix¯t+Πωt2.Definition 3.
(see [30]).
Switched stochastic system (1) is said to be exponentially stable in mean square, if there exist β1>0 and β2>0, such that
(9)Ext2≤β1e−β2t−t0Ext02.Definition 4.
(see [27]).
If there existτa>τd and N0≥1, such that
(10)Nσt1,t2≤N0+t2−t1τa,where the notation Nσt1,t2 denotes the number of switches in the interval t1,t2, 0≤t1≤t2; then, τa>0 is called the average dwell time (ADT). The interval between any two switches is at least dwell time τd>0.
Next, the ORP of the switched stochastic system (1) based on sampled data is described as follows.
By designing an appropriate switching signal and sampled-data feedback controllers, such that(i)
whenω̇t=0, the regulated output of the closed-loop system (7) satisfies
(11)limt⟶∞Eet=0,i.e., the ORP of the closed-loop system is solvable(ii)
if there exists a constanta1>0 and for any ε~>0, when ω̇t≤a1, the regulated output of the closed-loop system (7) satisfies
(12)limt⟶∞Esupet<ε~,i.e., the PORP of the closed-loop system is solvableWe considered the asynchronous switching phenomenon caused by the mismatch between the sampling instants and the switching instants; this phenomenon will last until the next sampling instant.tk,k=0,1,⋯t0<t1<⋯ denote the sequence of sampling instants; t^s,s=1,2,⋯ denote the sequence of switching instants. Assume that the sampling period τs is fixed. To ensure that there is at most one switch in each sampling interval, the period τs satisfies τs≤τd.In the following sections, we will give the sufficient conditions for the solvability of the output regulation problems of the closed-loop system (7).


## 3. The Situation ofω̇t=0

This section gives a solution to the ORP of the closed-loop system when the exogenous signal is constant.Theorem 5.
Whenω̇t=0, the system (1) which satisfies Assumptions 1 and 2 is considered. For given positive constants λ1, λ2, τs, ι<1, μ≥1, and εκ, if there exist positive definite matrices P^κ, E^κ, F^κ, W^1κ, and W^2κ and appropriate dimension matrices K^i, Y≥0, Y¯≥0, and Π, κ=i,j∈IN, i≠j, such that
(13)Ω^i=Δ¯1100Δ¯140P^iJiT∗Δ¯220000∗∗Δ¯33000∗∗∗Δ¯4400∗∗∗∗τsY55iP^iΠTJiT∗∗∗∗∗−εi−1I<0,Φ^i=Y11iY12iY13iY14iY15iW^1i∗Y22iY23iY24iY25i0∗∗Y33iY34iY35i0∗∗∗Y44iY45iW^2i∗∗∗∗Y55i0∗∗∗∗∗0≥0,(14)Σ^ij=Ω^ijχ^1∗χ^2<0,Φ¯^j=Y¯χ^3∗0≥0,(15)P^i≤μP^j,E^i≤μE^j,F^i≤μF^j,I≤εκP^κ,∀i,j∈IN,where
(16)Ω^ij=∇¯1100∇¯140∗∇¯22000∗∗∇¯3300∗∗∗∇¯440∗∗∗∗τsY¯55j,χ^1=00P^jJjT00000000000P^jΠTJjT,χ^2=∇¯6600∗∇¯770∗∗−εj−1I,χ^3=W∧1jT00W∧2jT000T,Δ¯11=λ1P^i+E^i+F^i+AiP^i+PiAiT+τsY11i+W^1i+W^1iT,Δ¯14=BiK^i+W^2iT−W^1i+τsY14i,Δ¯22=−e−λ1τsE^i+τsY22i,Δ¯33=−1−ιe−ιλ1τsF^i+τsY33i,Δ¯44=−W^2i−W^2iT+τsY44i,∇¯^11=−λ2P^j+E^j+F^j+AjP^j+P^jAjT+τsY¯11j+W^1j+W^1jT,∇¯14=BjK^i+W^2jT−W^1j+τsY¯14j,∇¯22=−e−λ1τsE^j+τsY¯22j,∇¯33=−1−ιe−ιλ1τsF^j+τsY¯33j,∇¯44=−W^2j−W^2jT+τsY¯44j,∇¯66=−λ1+λ2e−λ1τsτsE^j+τsY¯66j,∇¯77=−λ1+λ2e−ιλ1τsτsF^j+τsY¯77j,then the ORP of the closed-loop system (4) is solvable for any switching signal which satisfies ADT defined by
(17)τa>τa∗=lnμ+λ1+λ2τsλ1,where Ki=K^iP^i−1, Li=−KiΠ, the matrix Y is composed of Yi0j0i, i0,j0=1,2,⋯,5, and the matrix Y¯ is composed of Y¯i0j0j, i0,j0=1,2,⋯,7.Proof.
Sinceω̇t=0, thus, we get ωt=ω0, ω~t=0, t≥0. And according to the previous coordinate transformation x~t=xt−xtk and ω~t=ωt−ωtk, system (7) becomes
(18)dx¯t=Aσtx¯t+BσtKσtx¯t−ςtdt+fσtt,x¯t+Πωtdϖte=Cσtx¯t,where ςt=t−tk<tk+1−tk=τs and ς̇t=1.
Letgit=Aix¯t+BiKix¯t−ςt and fit=fit,x¯t+Πωt. When the ith subsystem is active, the closed-loop system (18) can be rewritten as
(19)dx¯t=gσttdt+fσttdϖ,e=Cσtx¯t.(i)
Synchronous switching interval
Assume that the subsystemi is active on the interval tk,tk+1, and it is synchronous with the controller on the whole interval, i.e., σtk+1=σtk=i∈IN. For system (19), construct the L-K functional candidates as follows:
(20)Vit=x¯TtPix¯t+∫t−τstx¯TsEieλ1s−tx¯sds+∫t−ιςttx¯TsFieλ1s−tx¯sds,where Pi, Ei, and Fi are positive definite symmetric matrices. From Itô’s differential formula, we get the stochastic differential
(21)dVit=LVitdt+2x¯TtPifitdϖt,with
(22)LVit=x¯TtEix¯t−e−λ1τsx¯Tt−τsEix¯t−τs−λ1∫t−τstx¯TsEieλ1s−tx¯sds+x¯TtFix¯t−1−ιe−ιλ1ςtx¯Tt−ιςtFix¯t−ιςt−λ1∫t−ιςtteλ1s−tx¯TsFix¯sds+2x¯TtPigit+fiTtPifit.
According to the Newton-Leibniz formula, the following equation is true:(23)2x¯TtW1i+x¯Tt−ςtW2ix¯t−x¯t−ςt−∫t−ςttx¯̇sds=0,where W1i and W2i are positive definite matrices with suitable dimension. According to Assumption 2 and condition (15), there exist εi>0 and constant matrices Ji, such that
(24)trfiTtPifit≤εix¯t+ΠωtTJiTJix¯t+Πωt.
By combining (22)–(24), we have
(25)LVit+λ1Vit≤x¯TtEix¯t−e−λ1τsx¯Tt−τsEix¯t−τs+x¯TtFix¯t−1−ιe−ιλ1τsx¯Tt−ιςtFix¯t−ιςt+2x¯TtPiAix¯t+2x¯TtPiBiKix¯t−ςt+εix¯TtJiTJix¯t+εiωTtΠTJiTJix¯t+εix¯TtJiTJiΠωt+εiωTtΠTJiTJiΠωt+λ1x¯TtPix¯t+2x¯Tt−ςtW2ix¯t−2x¯Tt−ςtW2ix¯t−ςt+2x¯TtW1ix¯t−2x¯TtW1ix¯t−ςt−2∫t−ςttx¯TtW1ix¯̇sds−2∫t−ςttx¯Tt−ςtW2ix¯̇sds.
Consider the arbitrary matrices(26)Xi=X11i⋯X15i⋮⋱⋮X51i⋯X55i,which satisfy the following inequality:
(27)τsη1TtXiη1t−∫t−ςttη1TtXiη1tds≥0.By combining(23)–(27) and Schur complement lemma, we obtain
(28)LVit+λ1Vit≤η1TtΩiη1t−∫t−ςttη2Tt,sΦiη2t,sds,where
(29)η1Tt=x¯Ttx¯Tt−τsx¯Tt−ιςtx¯Tt−ςtωTt,η2Tt,s=η1Ttx¯̇Ts,Ωi=Δ1100Δ14τsX15iJiT∗Δ220000∗∗Δ33000∗∗∗Δ4400∗∗∗∗τsX55iΠTJiT∗∗∗∗∗−εi−1I,Φi=X11iX12iX13iX14iX15iW1i∗X22iX23iX24iX25i0∗∗X33iX34iX35i0∗∗∗X44iX45iW2i∗∗∗∗X55i0∗∗∗∗∗0,Δ11=λ1Pi+Ei+Fi+PiAi+AiTPi+τsX11i+W1i+W1iT,Δ14=PiBiKi+W2iT−W1i+τsX14i,Δ22=−e−λ1τsEi+τsX22i,Δ33=−1−ιe−ιλ1τsFi+τsX33i,Δ44=−W2i−W2iT+τsX44i.For the inequality (13), we let P^i=Pi−1, K^i=KiPi−1, W^1i=Pi−1W1iPi−1, W^2i=Pi−1W2iPi−1, E^i=Pi−1EiPi−1, F^i=Pi−1FiPi−1, and Yi0j0i=Pi−1Xi0j0iPi−1, i0,j0=1,2,⋯,5. Multiply diagPi,Pi,Pi,Pi,Pi,I on both sides of inequality Ω^i<0, and multiply both sides of inequality Φ^i≥0 by diagPi,Pi,Pi,Pi,Pi,Pi. Then, formula (28) becomes
(30)LVit+λ1Vit≤ηTtΩiηt<0.Thus, we have(31)LVit≤−λ1Vit.Combining (21) and (31), we can derive that
(32)deλ1tVit=λ1eλ1tVitdt+eλ1tdVit≤eλ1tλ1Vitdt+−λ1Vitdt+2x¯TtPifitdϖt=2eλ1tx¯TtPifitdϖt.Integrate both sides of (32) from t to tk, and take expectation
(33)EVit≤e−λ1t−tkEVitk.(ii)
Asynchronous switching intervalFor∀t∈tk,tk+1, suppose that a switch has occurred at instant t^s, t^s∈tk,tk+1, and it satisfies t−t^s<τs. Then, the subsystem i is active on the interval tk,t^s; the subsystem j is active on t^s,tk+1. When t∈tk,t^s, we obtain the result from (i).
(34)EVit≤e−λ1t−tkEVitk.Whent∈t^s,tk+1, the closed-loop system has the following forms:
(35)dx¯t=gijtdt+fjtdϖ,e=Cjx¯t,where gijt=Ajx¯t+BjKix¯t−ςt. For system (35), construct the L-K functional candidates as follows:
(36)Vjt=x¯TtPjx¯t+∫t−τstx¯TsEjeλ1s−tx¯sds+∫t−ιςttx¯TsFjeλ1s−tx¯sds,where Pj, Ej, and Fj are positive definite symmetric matrices. From Itô’s differential formula, we get
(37)dVjt=LVjtdt+2x¯TtPjfjtdϖt,with
(38)LVjt=x¯TtEjx¯t−e−λ1τsx¯Tt−τsEjx¯t−τs−λ1∫t−τstx¯TsEjeλ1s−tx¯sds+x¯TtFjx¯t−1−ιe−ιλ1ςtx¯Tt−ιςtFjx¯t−ιςt−λ1∫t−ιςtteλ1s−tx¯TsFjx¯sds+2x¯TtPjgijt+fjTtPjfjt.According to the Newton-Leibniz formula, we have(39)2x¯TtW1j+x¯Tt−ςtW2jx¯t−x¯t−ςt−∫t−ςttx¯̇sds=0,where W1j and W2j are positive definite matrices with suitable dimension. And there exist εj and constant matrices Jj, such that
(40)trfjTtPjfjt≤εjx¯t+ΠωtTJjTJjx¯t+Πωt.From (38) and (40), we get
(41)LVjt−λ2Vjt≤x¯TtEjx¯t−e−λ1τsx¯Tt−τsEjx¯t−τs+x¯TtFjx¯t−1−ιe−ιλ1τsx¯Tt−ιςtFjx¯t−ιςt+2x¯TtPjAjx¯t+2x¯TtPjBjKix¯t−ςt+εjx¯TtJjTJjx¯t+εjωTtΠTJjTJjx¯t+εjx¯TtJjTJjΠωt+εjωTtΠTJjTJjΠωt−λ2x¯TtPjx¯t+2x¯Tt−ςtW2jx¯t−2x¯Tt−ςtW2jx¯t−ςt+2x¯TtW1jx¯t−2x¯TtW1jx¯t−ςt−2∫t−ςttx¯TtW1jx¯̇sds−2∫t−ςttx¯Tt−ςtW2jx¯̇sds−λ1+λ2e−λ1τsτs∫t−τstx¯TsdsEj∫t−τstx¯sds−λ1+λ2e−ιλ1τsτs∫t−ιςttx¯TsdsFj∫t−ιςttx¯sds.Consider the arbitrary matrices(42)X¯j=X¯11j⋯X¯17j⋮⋱⋮X¯71j⋯X¯77j,which satisfies
(43)τsη~1TtX¯jη~1t−∫t−ςttη~1TtX¯jη~1tds≥0.By combining(41)–(43) and the Schur complement lemma, we have
(44)LVjt−λ2Vjt≤η~1TtΣijη~1t−∫t−ςttη2Tt,sΦ¯ijη2t,sds,where
(45)η~1Tt=x¯Ttx¯Tt−τsx¯Tt−ιςtx¯Tt−ςtωTt∫t−τstx¯Tsds∫t−ιςttx¯Tsds,η~2Tt,s=η~1Ttx¯̇Ts,Σij=Ωijχ1∗χ2,Φ¯j=X¯jχ3∗0,W1jT00W2jT000T,Ωij=∇1100∇14τsX¯15j∗∇22000∗∗∇3300∗∗∗∇440∗∗∗∗τsX¯55j,χ1=00JjT00000000000ΠTJjT,χ2=∇6600∗∇770∗∗−εj−1I,∇11=−λ2Pj+Ej+Fj+PjAj+AjTPj+τsX¯11j+W1j+W1jT,∇14=PjBjKi+W2jT−W1j+τsX¯14j,∇22=−e−λ1τsEj+τsX¯22j,∇33=−1−ιe−ιλ1τsFj+τsX¯33j,∇44=−W2j−W2jT+τsX¯44j,∇55=τsX¯55j,∇66=−λ1+λ2e−λ1τsτsEj+τsX¯66j,∇77=−λ1+λ2e−ιλ1τsτsFj+τsX¯77j.For the inequality (14), we let P^j=Pj−1, W^1j=Pj−1W1jPj−1, W^2j=Pj−1W2jPj−1, E^j=Pj−1EjPj−1, F^j=Pj−1FjPj−1, and Y¯i0j0j=Pj−1X¯i0j0jPj−1, i0,j0=1,2,⋯,7. Multiply diagPj,Pj,Pj,Pj,Pj,Pj,Pj,I on both sides of inequality Σ^ij<0, and multiply both sides of inequality Φ¯^j≥0 by diagPj,Pj,Pj,Pj,Pj,Pj,Pj,Pj. Then, formula (44) becomes
(46)LVjt−λ2Vjt≤η¯TtΣijη¯t<0.Thus, we have(47)LVjt≤λ2Vjt.Combining (21) and (47), we can derive that
(48)de−λ2tVjt≤2e−λ2tx¯TtPjfjtdϖt.Integrate both sides of (48) from t^s to t, and take expectation
(49)EVjt≤eλ2t−t∧sEVjt^s.For the entire intervaltk,tk+1, combining (15), (34), and (49), we have
(50)EVt<μe−λ1t−tk+λ1+λ2τsEVtk.According to the inequality (15), (50), and Definition 4, we can obtain
(51)EVt≤celnμ+λ1+λ2τs/τa−λ1t−t0EVt0,where c=elnμ+λ1+λ2τsN0. According to the constructed L-K functional, we get
(52)EVt≥Ex¯TtPix¯t≥λmPiEx¯t2≥γ1Ex¯t2,EVt0≤γ2Ex¯t02,where γ1=mini∈INλmPi and γ2=maxi∈INλMPi+τsλMEi+ιτsλMFi. From (51) and (52), we have
(53)Ex¯t2≤γ2γ1celnμ+λ1+λ2τs/τa−λ1t−t0Ex¯t02.Under condition (17), the above formula means that
(54)limt⟶∞Eet=0.Remark 6.
Whenωt≡0, the switched stochastic system (1) is simplified as
(55)dxt=Aσtxt+Bσtutdt+fσtt,xtdϖt.
Based on the result (53), we obtain
(56)Ex¯t2≤γ2γ1celnμ+λ1+λ2τs/τa−λ1t−t0Ex¯t02,i.e., the system (55) is mean-square exponentially stable.


## 4. The Situation ofω̇t≤a1

For the case that the exogenous signal is time-varying with bounded derivative, this section gives the solution to the PORP of the closed-loop system.(i)
Synchronous switching intervalTheorem 7.
If there exists a constanta1>0, such that ω̇t≤a1, t≥0, consider the closed-loop system (1) which satisfies Assumptions 1 and 2 and the following conditions:
(1)
For given positive constantsλ⌢1, λ⌢2, λ¯1, λ¯2, τs, ε⌢κ, μ≥1, ι⌢<1, and any ε~>0, if there exist positive definite matrices Z≥0, Z¯≥0, K⌢i, and L⌢i, κ=i,j∈IN and i≠j, such that
(57)Λ^i=Ω¯^iΓ^1∗−ε⌢i−1I<0,Ξ^i=ZΓ^3∗0≥0,(58)Λ¯^ij=Ω¯^ijχ¯^1∗−ε⌢j−1I<0,Ξ¯^ij=Z¯χ¯^2∗0≥0,(59)P⌢^i≤μP⌢^j,E⌢^i≤μE⌢^j,F⌢^i≤μF⌢^j,I≤ε⌢κP⌢^κ,∀i,j∈IN,where
(60)Ω¯^i=ζ¯1100ζ¯14τsZ15iζ¯16∗ζ¯220000∗∗ζ¯33000∗∗∗ζ¯4400∗∗∗∗τsZ55i0∗∗∗∗∗ζ¯66,Γ^1=P⌢^iJiT000P⌢^iΠTJiT0,Γ^3=W⌢^1i00000W⌢^2i0000R⌢i,Γ^3=W⌢^1i00000W⌢^2i0000R⌢i,Ω¯^ij=ΔΩ¯^ij00∗ρ¯770∗∗ρ¯88,ΔΩ¯^ij=ρ¯1100ρ¯14τsZ¯15jρ¯16∗ρ¯220000∗∗ρ¯33000∗∗∗ρ¯4400∗∗∗∗τsZ¯55j0∗∗∗∗∗ρ¯66,χ¯^1=JjP⌢∧j000JjΠP⌢∧j000T,χ¯^2=W⌢∧1jT00W⌢∧2jT000000000R⌢jT00T,ζ¯11=λ⌢1P⌢^i+E⌢^i+F⌢^i+P⌢^iAi+AiTP⌢^i+τsZ11i+W⌢^1i+W⌢^1iT,ζ¯14=BiK⌢i+W⌢^2iT−W⌢^1i+τsZ14i,ζ¯16=−BiΠK⌢i−BiL⌢i+τsZ16i,ζ¯33=−1−ιe−ιλ⌢1τsF⌢^i+τsZ33i,ζ¯22=−e−λ⌢1τsE⌢^i+τsZ22i,ζ¯44=−W⌢^2iT−W⌢^2i+τsZ44i,ζ¯66=R⌢i+R⌢iT−2λ⌢1N⌢i+τsZ66i,ρ¯11=−λ⌢2P⌢^j+E⌢^j+F⌢^j+P⌢^jAj+AjTP⌢^j+τsZ¯11i+W⌢^1j+W⌢^1jT,ρ¯14=BjK⌢i+W⌢^2jT−W⌢^1j+τsZ14j,ρ¯22=−e−λ⌢1τsE⌢^j+τsZ¯22j,ρ¯33=−1−ι⌢e−ι⌢λ⌢1τsF⌢^j+τsZ¯33j,ρ¯16=−BjΠK⌢i−BjL⌢i+τsZ¯16i,ρ¯44=−W⌢^2jT−W⌢^2j+τsZ¯44j,ρ¯66=R⌢j+R⌢jT−2λ¯2N⌢j+τsZ¯66j,ρ¯77=−λ⌢1+λ⌢2e−λ⌢1τsτsE⌢^j+τsZ¯77j,ρ¯88−−λ⌢1+λ⌢2e−ι⌢λ⌢1τsτsF⌢^j+τsZ¯88j(2)
If there exists the matrixΠ satisfying the matrix equations (6),then the PORP of the closed-loop system (4) is solvable for any switching signal which satisfies ADT defined by
(61)τa>τa∗=λ⌢1+λ⌢2τs+lnμλ⌢1,where Ki=K⌢iP⌢i−1 and Li=L⌢iP⌢i−1. And the matrix Z is composed of Zi0j0i, i0,j0=1,2,⋯,6; the matrix Z¯ is composed of Z¯i0j0j, i0,j0=1,2,⋯8Proof.
Whenω̇t≤a1, the closed-loop system has the following form:
(62)dx¯t=Aσtx¯t+BσtKσtx¯t−ςt+Gσtω~tdt+fσtt,x¯t+Πωtdϖt,e=Cσtx¯t,where Gi=B¯iΠ+L¯i. Let g¯it=Aix¯t+BiKix¯t−ςt+Giω~t, fit=fit,x¯t+Πωt. When the ith subsystem is activated, (62) is equivalent to the following form:
(63)dx¯t=g¯σttdt+fσttdϖ,e=Cσtx¯t.Also, assume that the subsystemi is activated on the interval tk,tk+1. For the system (63), construct the L-K functional candidate as follows:
(64)Vit=x¯TtP⌢ix¯t+∫t−τstx¯TsE⌢ieλ⌢1s−tx¯sds+∫t−ι⌢ςttx¯TsF⌢ieλ⌢1s−tx¯sds,where P⌢i, E⌢i, and F⌢i are positive definite symmetric matrices. Besed on Itô’s differential formula and (64), we get the stochastic differential
(65)dVit=LVitdt+2x¯TtP⌢ifitdϖt.According to the Newton-Leibniz formula, we have the following equations:(66)2x¯TtW⌢1i+x¯Tt−ςtW⌢2ix¯t−x¯t−ςt−∫t−ςttx¯̇sds=0,2ω~TtRiω~t−∫t−ςttω̇sds=0,where W⌢1i, W⌢2i, and Ri are positive definite matrices with suitable dimension. By virtue of Assumption 2 and condition (59), there exist ε⌢i>0 and constant matrices J⌢i, such that
(67)trfiTtP⌢ifit≤ε⌢ix¯t+ΠωtTJ⌢iTJ⌢ix¯t+Πωt.By combining (64)–(67), we have
(68)LVit+λ⌢1Vit≤x¯TtE⌢ix¯t−e−λ⌢1τsx¯Tt−τsE⌢ix¯t−τs+x¯TtF⌢ix¯t−1−ι⌢e−ι⌢λ⌢1τsx¯Tt−ι⌢ςtF⌢ix¯t−ι⌢ςt+2x¯TtP⌢iAix¯t+2x¯TtP⌢iBiKix¯t−ςt−2x¯TtP⌢iBiΠKiω~t−2x¯TtP⌢iBiLiω~t+ε⌢ix¯TtJ⌢iTJ⌢ix¯t+ε⌢iωTtΠTJ⌢iTJ⌢ix¯t+ε⌢ix¯TtJ⌢iTJ⌢iΠωt+ε⌢iωTtΠTJ⌢iTJ⌢iΠωt+λ⌢1x¯TtP⌢ix¯t+2x¯TtW⌢1ix¯t−2x¯TtW⌢1ix¯t−ςt+2x¯Tt−ςtW⌢2ix¯t−2x¯Tt−ςtW⌢2ix¯t−ςt+2ω~TtRi−λ¯iNiω~t+2λ¯iω~TtNiω~t−2∫t−ςttx¯TtW⌢1ix¯̇sds−2∫t−ςttx¯Tt−ςtW⌢2ix¯̇sds−2∫t−ςttω~TtRiω̇sds.Consider the arbitrary matrices(69)X~i=X~11i⋯X~16i⋮⋱⋮X~61i⋯X~66i,which satisfies
(70)τsφ1TtX~iφ1t−∫t−ςttφ1TtX~iφ1tds≥0.By combining (68)–(70)) and the Schur complement lemma, we obtain
(71)LVit+λ⌢1Vit≤φ1TtΛiφ1t+2λ¯1ω~TtNiω~t−∫t−ςttφ2Tt,sΞiφ2t,sds,where
(72)φ1Tt=x¯Ttx¯Tt−τsx¯Tt−ιςtx¯Tt−ςtωTtω~Tt,φ2Tt,s=φ1Ttx¯̇Tsω̇Ts,ΛiΩ¯iΓ1∗−ε⌢i−1I,Ξi=X~iΓ3∗0,Ω¯^i=ζ1100ζ14τsX~15iζ16∗ζ220000∗∗ζ33000∗∗∗ζ4400∗∗∗∗τsX~55i0∗∗∗∗∗ζ66,Γ1=JiT000ΠTJiT0,Γ3=W⌢1i00000W⌢2i0000Ri,ζ11=λ⌢1P⌢i+E⌢i+F⌢i+P⌢iAi+AiTP⌢i+τsX~11i+W⌢1i+W⌢1iT,ζ14=P⌢iBiKi+W⌢2iT−W⌢1i+τsX~14i,ζ22=−e−λ⌢iτsE⌢i+τsX~22i,ζ33=−1−ιe−ιλ⌢1τsF⌢i+τsX~33i,ζ44=−W⌢2i−W⌢2iT+τsX~44i,ζ16=−P⌢iBiΠKi−P⌢iBiLi+τsX¯16i,ζ66=Ri+RiT−2λ¯1Ni+τsX~66i,λ¯1=maxi∈INλ¯i.For the inequality (57), we let P⌢^i=P⌢i−1, K⌢i=KiP⌢i−1, L⌢i=LiP⌢i−1, E⌢^i=P⌢i−1E⌢iP⌢i−1, F⌢^i=P⌢i−1F⌢iP⌢i−1, R⌢i=P⌢i−1RiP⌢i−1, N⌢i=P⌢i−1NiP⌢i−1, W⌢^1i=P⌢i−1W⌢1iP⌢i−1, W⌢^2i=P⌢i−1W⌢2iP⌢i−1, and Zi0j0i=P⌢i−1X~i0j0iP⌢i−1,i0,j0=1,2,⋯,6. Multiply diagP⌢i,P⌢i,P⌢i,P⌢i,P⌢i,P⌢i,I on both sides of inequality Λ^i<0, and multiply both sides of inequality Ξ^i≥0 by diagP⌢i,P⌢i,P⌢i,P⌢i,P⌢i,P⌢i,P⌢i,P⌢i. Then, formula (71) becomes
(73)LVit≤−λ⌢1Vit+2λ¯1ω~TtNiω~t.Sinceω̇t≤a1, thus, we have
(74)ω~t≤∫tktω̇τdτ≤a1t−tk≤a1τs.Combining the above formula with (73), we have
(75)LVit≤−λ⌢1Vit+λ~1ω~t2≤−λ⌢1Vit+λ~1a1τs2,where λ~1=2maxi∈INλ¯1Ni. From (65) and (75), we can derive that
(76)deλ⌢1tVit=λ⌢1eλ⌢1tVitdt+eλ⌢1tdVit≤eλ⌢1tλ⌢1Vitdt+−λ⌢1Vit+λ~1a1τs2dt+2x¯TtP⌢ifitdϖt=eλ⌢1tλ~1a1τs2dt+2x¯TtP⌢ifitdϖt.Integrate both sides of (76) from t to tk, and take expectation
(77)EVit≤e−λ⌢1t−tkEVitk+E∫tkte−λ⌢1t−τλ~1a1τs2dτ.(ii)
Asynchronous switching intervalFor∀t∈tk,tk+1, we also suppose that one switch has occurred at the instant t^s and asynchronous switching occurs on the interval tk,tk+1. When t∈tk,t^s, we obtain the result from (i).
(78)EVit≤e−λ⌢1t−tkEVitk+E∫tkte−λ⌢1t−τλ~1a1τs2dτ.Whent∈t^s,tk+1, the closed-loop system has the following form:
(79)dx¯t=g¯ijtdt+fjtdϖ,(80)e=Cjx¯t,where g¯ijt=Ajx¯t+BjKix¯t−ςt+Gjω~t, Gj=−BjKiΠ−BjLi, and fjt=fjt,x¯t+Πωt. For the system (79), construct the L-K functional candidate as follows:
(81)Vjt=x¯TtP⌢jx¯t+∫t−τstx¯TsE⌢jeλ⌢1s−tx¯sds+∫t−ι⌢ςttx¯TsF⌢jeλ⌢1s−tx¯sds,where P⌢j, E⌢j, and F⌢j are positive definite symmetric matrices. Besed on Itô’s differential formula and (81), we get the stochastic differential
(82)dVjt=LVjtdt+2x¯TtP⌢jfjtdϖt.According to the Newton-Leibniz formula, we have(83)2x¯TtW⌢1j+x¯Tt−ςtW⌢2jx¯t−x¯t−ςt−∫t−ςttx¯̇sds=0,2ω~TtRjω~t−∫t−ςttω̇sds=0,where W⌢1j, W⌢2j, and Rj are positive definite matrices with suitable dimension. And there exist ε⌢j and constant matrices J⌢j, such that
(84)trfjTtP⌢jfjt≤ε⌢jx¯t+∏ωtTJ⌢jTJ⌢jx¯t+∏ωt.From (81)–(84), we get
(85)LVjt−λ⌢2Vjt≤x¯TtE⌢jx¯t−e−λ⌢1τsx¯Tt−τsE⌢jx¯t−τs+x¯TtF⌢jx¯t−1−ι⌢e−ι⌢λ⌢1τsx¯Tt−ι⌢ςtF⌢jx¯t−ι⌢ςt+2x¯TtP⌢jAjx¯t+2x¯TtP⌢jBjKix¯t−ςt−2x¯TtP⌢jBj∏Kiω~t−2x¯TtP⌢jBjLiω~t+ε⌢jx¯TtJ⌢jTJ⌢jx¯t+ε⌢jωTt∏TJ⌢jTJ⌢jx¯t+ε⌢jx¯TtJ⌢jTJ⌢j∏ωt+ε⌢jωTt∏TJ⌢jTJ⌢j∏ωt−λ⌢2x¯TtP⌢jx¯t+2x¯TtW⌢1jx¯t−2x¯TtW⌢1jx¯t−ςt+2x¯Tt−ςtW⌢2jx¯t−2x¯Tt−ςtW⌢2jx¯t−ςt+2ω~TtRj−λ¯jNjω~t+2λ¯jω~TtNjω~t−λ⌢1+λ⌢2e−λ⌢1τsτs∫t−τstx¯TsdsE⌢j∫t−τstx¯s−λ⌢1+λ⌢2e−ι⌢λ⌢1τsτs∫t−ι⌢ςttx¯TsdsF⌢j∫t−ι⌢ςttx¯sds−2∫t−ςttx¯TtW⌢1jx¯̇sds−2∫t−ςttx¯Tt−ςtW⌢2jx¯̇s−2∫t−ςttω~TtRjω̇sds.Consider the arbitrary matrices(86)X⌢j=X⌢11j⋯X⌢18j⋮⋱⋮X⌢81j⋯X⌢88j,which satisfies
(87)τsφ~1TtX⌢jφ~1t−∫t−ςttφ~1TtX⌢jφ~1tds≥0.By combining(85)–(87) yields and the Schur complement lemma, we have
(88)LVjt−λ⌢2Vjt≤φ~1TtΛ¯ijφ~1t+2λ¯jω~TtNjω~t−∫t−ςttφ~2Tt,sΞjφ~2t,sds,where
(89)φ1Tt=φ1Tt∫t−τstx¯Tsds∫t−ι⌢ςttx¯Tsds,φ~2Tt,s=φ~1Ttx¯̇Tsω̇Ts,Λ¯ij=Ω¯ijχ¯1∗−ε⌢j−1I,Ξ¯j=X⌢jχ¯2∗0,Ω¯ij=ΔΩ¯ij00∗ρ770∗∗ρ88,ΔΩ¯ij=ρ1100ρ14τsX⌢15jρ16∗ρ220000∗∗ρ33000∗∗∗ρ4400∗∗∗∗τsX⌢15j0∗∗∗∗∗ρ66,χ¯1=Jj000JjΠ000T,χ¯2=W⌢1jT000W⌢2jT00000000RjT00T,ρ11=−λ⌢2P⌢j+E⌢j+F⌢j+P⌢jAj+AjTP⌢j+τsX⌢11j+W⌢1j+W⌢1jT,ρ14=P⌢jBjKi+W⌢2jT−W⌢1j+τsX⌢14j,ρ22=−e−λ⌢1τsE⌢j+τsX⌢22j,ρ33=−1−ι⌢e−ι⌢λ1τsF⌢j+τsX⌢33j,ρ16=−P⌢jBjΠKi−P⌢jBjLi+τsX⌢16j,ρ44=−W⌢2j−W⌢2jT+τsX⌢44j,ρ66=Rj+RjT−2λ¯2Nj+τsX⌢66j,λ¯2=maxj∈INλ¯j,ρ77=−λ⌢1+λ⌢2e−λ⌢1τsτsE⌢j+τsX⌢77j,ρ88=−λ⌢1+λ⌢2e−ι⌢λ⌢1τsτsF⌢j+τsX⌢88j.For the inequality (58), we let P⌢^j=P⌢j−1,E⌢^j=P⌢j−1E⌢jP⌢j−1,F⌢^j=P⌢j−1F⌢jP⌢j−1,R⌢j=P⌢j−1RjP⌢j−1,N⌢j=P⌢j−1NjP⌢j−1,W⌢^1j=P⌢j−1W⌢1jP⌢j−1,W⌢^2j=P⌢j−1, and Zi0j0j=P⌢j−1X⌢i0j0iP⌢j−1,i0,j0=1,2⋯,8. Multiply diagP⌢j,P⌢j,P⌢j,P⌢j,P⌢j,P⌢j,P⌢j,P⌢j,I on both sides of inequality Λ¯^ij<0, and multiply both sides of inequality Ξ¯^j≥0 by diagP⌢j,P⌢j,P⌢j,P⌢j,P⌢j,P⌢j,P⌢j,P⌢j,P⌢j,P⌢j. Then, formula (88) becomes
(90)LVjt≤λ⌢2Vjt+2λ¯2ω~TtNjω~t.Sinceω̇t≤a1, formula (90) becomes
(91)LVjt≤λ⌢2Vjt+λ~2ω~t2≤λ⌢2Vjt+λ~2a1τs2,where λ~2=2maxi∈INλ¯2Nj. Combining (82) and (91), we can derive that
(92)de−λ⌢2tVjt≤e−λ⌢2tλ~2a1τs2dt+2e−λ⌢2tx¯TtP⌢jfjtdϖt.Integrate both sides of (93) from t^s to t, and take expectation
(93)EVjt≤eλ⌢2t−t∧sEVjt^s+E∫tkteλ⌢2t−τλ~2a1τs2dτ.For the entire intervaltk,tk+1 and λ~=maxλ~1,λ~2, combining (59), (78), and (93), we have
(94)EVt≤μeλ⌢2t−t∧se−λ⌢1t∧s−tkEVtk+μE∫tkt⌢se−λ⌢1t∧s−τ+λ⌢2t−t∧sλ~a1τs2dτ+E∫t⌢steλ⌢2t−τλ~a1τs2dτ,where the notation Tt,τ represents the total time for asynchronous occurrence on the interval τ,t. And from (59) and (94), we get
(95)EVt≤μe−λ⌢1t−tk+λ⌢1+λ⌢2t−t∧sEVtk+μE∫tkt^se−λ⌢1t−τ+λ⌢1+λ⌢2t−t∧λ~a1τs2dτ+E∫t^ste−λ⌢1t−τ+λ⌢1+λ⌢2t−τλ~a1τs2dτ≤μ2e−λ⌢1t−tk−1+λ⌢1+λ⌢2t−t∧s+tk−t∧s−1EVtk−1+μ2E∫tk−1t^s−1e−λ⌢1t−τ+λ⌢1+λ⌢2t−t∧s+tk−t∧s−1λ~a1τs2dτ+μE∫t^s−1tke−λ⌢1t−τ+λ⌢1+λ⌢2tk−τ+λ⌢1+λ⌢2t−t∧sλ~a1τs2dτ+μE∫tkt^seλ⌢1t−τ+λ⌢1+λ⌢2t−t∧sλ~a1τs2dτ+E∫t^ste−λ⌢1t−τ+λ⌢1+λ⌢2t−τλ~a1τs2dτ≤⋯≤μNσt,t0e−λ⌢1t−t0+λ⌢1+λ⌢2Tt,t0EVt0+E∫t0te−λ⌢1t−τ+λ⌢1+λ⌢2Tt,τ+Nσt,τInμλ~a1τs2dτ.Based on Definition4, the following results hold:
(96)EVt≤c~e−αt−t0EVt0+E∫t0tc~e−αt−τλ~a1τs2dτ,where c~=eN0lnμ+λ⌢1+λ~2N0+1 and α=λ⌢1−τa−1λ⌢1+λ⌢2τs−τa−1lnμ.According to the constructed L-K functional, we obtain that(97)EVt≥Ex¯TtP⌢ix¯t≥λmP⌢iEx¯t2≥γ⌢1Ex¯t2,EVt0≤γ⌢2Ex¯t02,where λm=λminP^i, λM=λmaxP^i, γ⌢1=mini∈INλmP⌢i, and γ⌢2=maxi∈INλMP⌢i+τsλME⌢i+ι⌢τsλMF⌢i.According to the ADT (61), we get α>0. Thus, we have
(98)limt⟶∞supEVt≤limt⟶∞supc~e−αt−t0EVt0+E∫t0tc~e−αt−τλ~a1τs2dτ=limt⟶∞supc~e−αt−t0EVt0+Ec~λ~a1τs2α1−e−αt−t0=limt⟶∞supc~e−αt−t0EVt0−λ~a1τs2α+c~λ~a1τs2α=c~λ~a1τs2α.Then, based on condition (97), for given any ε0>0, letting
(99)0<a1≤ε0τsα⌢γ1c~λ~,from (97)–(99), we have
(100)limt⟶∞supEx¯t≤limt⟶∞supEVt⌢γ1≤c~λ~a1τs2α⌢γ1≤ε0.Then, the regulated output satisfies(101)limt⟶∞Esupet≤limt⟶∞EsupCix¯t≤maxi∈INCiε0=ε~.Therefore, the PORP of the switched stochastic system can be solved.


## 5. Simulation

In this section, we will give two numerical examples corresponding to Theorem5 and Theorem 7, which verify the effectiveness of the method.Example 1.
Consider the switched stochastic system (1) with two subsystems, where ω̇t=0 and
(102)A1=−2.100−1,A2=−1.5000.2,B1=0.6−1,B2=1−1.4,C1=C2=−0.50.5,D1=D2=0.3−0.6,H1=1.200.2−0.8,H2=00.80−1.5,S=1001,f1t,xt=f2t,xt=12sintJjxt,j∈1,2,J1=J2=−200−2,and the matrix which satisfies Assumption 1 as follows:
(103)Π=10.50.51.Letλ1=0.8, λ2=1.2, τs=0.5, ι=0.8, and μ=1.6. From (17), we know that the minimum average dwell time is τa∗=1.8375; thus, we construct a switching strategy which satisfies τa=1.9>τa∗. By solving the inequalities (13)–(15) in Theorem 5, we have
(104)K1=−0.53760.5779,K2=−1.32292.1464,L1=0.2416−0.3126,L2=0.2497−1.4850.Under the initial statex¯0=−1.52.5T and ω0=01T, we obtain the state responses of the closed-loop system (18) shown in Figure 1, and the regulated output tends to zero which is shown in Figure 2. Figure 3 describes the time-triggered instants and shows that it only triggered 20 times in 10 seconds. Compared with continuous sampling, fixed-time sampling greatly reduces the waste of resources and greatly improves the efficiency. Based on the above simulation, we can draw the conclusion that the method to solve the ORP in Theorem 5 is effective.Example 2.
Consider the switched stochastic system (1) with two subsystems, where ω̇t≤a1 and
(105)A1=−10.20−1.5,A2=−0.600.1−1.8,B1=−0.20.1,B2=0.3−0.2,C1=C2=−0.40,D1=D2=−10.2,S=1001,f1t,xt=f2t,xt=12sintJjxt,j∈1,2,J1=J2=−200−2,and the matrices which satisfy Assumption 1 as follows:
(106)Π=1001,H1=1.54901.60270.22551.5986,H2=1.81253.8413−0.24170.2391.Figure 1
State responses.Figure 2
Regulated output.Figure 3
Time-triggered instants.Letλ⌢1=0.6,λ⌢2=0.8, τs=0.4, ι⌢=0.5, μ=1.9, λ¯1=λ¯2=0.6, and a1=0.4. By (57), we can obtain the minimum average dwell time τa∗=2.0031; then, we construct a switching strategy with τa=2.1>τa∗. By solving the inequalities (52)–(55) in Theorem 7, we can obtain that
(107)K1=−0.15470.2483,K2=0.01360.3861,L1=−2.10030.7653,L2=−0.7219−1.1905.We choose the initial statex¯0=−2.51T and ω0=01T; then, the PORP with sampled-data control is solved according to Theorem 7. The state responses of the closed-loop system (58) is shown in Figure 4, Figure 5 means that the regulated output is bounded. Figure 6 describes the time-triggered instants, which shows that it only triggered 25 times in 10 seconds, proving that the scheme based on time sampling design can effectively save communication resources. Therefore, the simulation results show that the proposed method is effective.Figure 4
States responses.Figure 5
Regulated output.Figure 6
Time-triggered instants.


## 6. Conclusion

We have solved the ORP of a class of switched stochastic systems via sampled-data control. We have discussed two cases of the controlled system under the constant exogenous signal and the time-varying exogenous signal with bounded derivative, respectively. For these two cases, the appropriate sampled-data state feedback controllers are designed and the stochastic signal is handled by utilizing Itô’s formula. Furthermore, by combining the ADT method and the free-weighting matrix method, the ORP of the closed-loop system can be solved. This problem will be further studied in the future. For example, the sampled-data control problem based on the design switching signal and the tracking control problem of switched stochastic systems based on the results of this paper will be studied.



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