# Finite-Time Asynchronous Stabilization for Nonlinear Hidden Markov Jump Systems with Parameter Varying in Continuous-Time Case

Authors: Lianjun Xiao; Xiaofeng Wang; Lingling Gao
Journal: Complexity
 (2020)

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## Abstract

The finite-time asynchronous stabilization problem has received great attention because of the wide application of actual engineering. In this paper, we consider the problem of finite-time asynchronous stabilization for nonlinear hidden Markov jump systems (HMJSs) with linear parameter varying. Compared with the existing research results on Markov jump systems, this paper considers the HMJSs which contain both the hidden state and the observed state in continuous-time case. Moreover, we consider the parameters of the systems are time varying. The aim of the paper is to design a proper observation-mode-based asynchronous controller such that the closed-loop HMJSs with linear parameter varying be stochastically finite-time bounded withH∞ performance (SFTB-H∞). Then, we give some sufficient conditions to solve the SFTB-H∞ asynchronous controller by considering the stochastic Lyapunov–Krasovskii functional (SLKF) methods. Finally, a numerical example is used to show the validity of the main results.


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## Body

## 1. Introduction

In recent years, hybrid systems have attracted the great attention and research due to their wide application in the industrial control field. Moreover, with the continuous upgrading and changes in the industrial environment, how to model and control hybrid systems in complex environments has become a major research hotspot in the control field. As a kind of special hybrid systems, Markov jump systems (MJSs) have received researchers’ great attention and many constructive results have been made. It should be noted that the synchronous controllers are always considered in many existing results when it comes to the stabilization problem of Markov jump systems. That means the controller of the systems can always track the information of the model of the system in real time. However, such a situation is always impossible to achieve in a Markov jump system with a complex environment. Therefore, the asynchronous characteristic between the system modal and the controller has attracted the attention of many researchers and they began to focus on the various characteristics of the HMJSs. The static output constrained control [1], the output feedback control [2], and the observer-based asynchronous fault detection [3] problems of the HMJSs with the discrete-time state are studied by the authors, respectively. In [4], the resilient asynchronous H∞ control problem of the discrete-time HMJSs with singularly perturbed was considered. For the nonlinear HMJSs, the quantized control [5] and the finite-time L2-gain asynchronous control [6] problems by using T-S fuzzy model approach are studied, respectively. Moreover, many researchers also have carried out some work for the continuous-time HMJSs. The asynchronous H2-controller design and the asynchronous filter design problems are studied for the continuous-time HMJSs in [7, 8], respectively. For more particulars of HMJSs, the interested readers can read references [9, 10].It should be pointed out that many results of the MJSs assume that the parameter matrices are constant matrices or uncertain matrices that satisfy some given known conditions. In actual control systems, the system parameters may have the characteristics of convex polygonal linear parameters varying due to the sensor and actuator failures. For the control systems with such a special situation, it is necessary to design the corresponding nonlinear parameter time-varying controller to stabilization of the systems. Thus, many researchers proposed the linear parameter-varying control methods which laid a solid theoretical foundation for the design of nonlinear parameter time-varying controller. For a kind of parameter-varying system [11], an adaptive algorithm was proposed to achieve the active vibration control by using a new online secondary path estimation method. For bounded parameter variation linear parameter-varying systems, the LMI-based filter design problem is studied in [12]. The robust fault estimation [13] and set-membership fault estimation [14] problem is studied for parameter time-varying systems. For more particulars of parameter-varying systems, the interested readers can read references [15–18].In the abovementioned references, the system analysis and/or some control method were concerned only over an infinite-time interval, which portrayed the asymptotic properties of the HMJSs and parameter-varying systems. However, the transient characteristics in a given finite-time interval is significant in many control systems [19–21] and should be considered simultaneously. For a class of nonlinear systems, the authors studied the finite-time adaptive fuzzy control problem in [22, 23], respectively. In [24], a finite-time control approach was used to solve the problem of accurate trajectory tracking for disturbed surface vehicles. Moreover, many researchers also combine the finite-time control scheme with the networked switched systems [25], nonlinear systems [26, 27], and quadrotors control [28] and have carried out lots of excellent works. The authors studied the design and implementation of bounded finite-time control algorithm problem for the speed regulation of permanent magnet synchronous motor in [29]. By using the state-dependent switching method, the adaptive fuzzy finite-time control of switched nonlinear systems is studied in [30]. The problem of fuzzy finite-time control for switched systems via adding a barrier power integrator is considered in [31]. For more particulars of HMJSs, the interested readers can read references [32–34].However, few results have been reported regarding research on finite-time asynchronous control of HMJSs based on the parameter varying, which is the motivation of this work. Different from some existing results on the finite-time control problem [35, 36], the problem of SFTB-H∞ asynchronous control is studied for continuous-time HMJSs via parameter varying in this paper. Compared with the existing results of asynchronous control for discrete-time HMJSs [37–40], this paper mainly consists of the following threefold contributions:(1)
The asynchronous characteristic between the controller modes and the system modes is characterized by the hidden Markov dynamics. Moreover, we firstly consider the asynchronous stabilization problem for the continuous-time HMJSs with parameter varying models.(2)
Some sufficient conditions will be given to solve the SFTB-H∞ observation-mode-based asynchronous controller by considering the SLKF methods and introducing some auxiliary variables.(3)
Considering the parameter time-varying, the methods of gridding technique and approximate basis function will be used in order to change the infinite LMIs into finite LMIs which can be solved by MATLAB LMI toolbox to get the finite-time asynchronous controller gain directly.The organization of this paper is made of five parts. In Section1, the background of the HMJSs, the parameter time-varying systems, finite-time control scheme, and the notation meaning of this paper are introduced and given. Section 2 introduces the system description of the parameter varying hidden Markov jump systems (PV-HMJSs) and designs a suitable asynchronous controller for the studied systems. Moreover, the main definitions and lemmas are also given in this part. Section 3 gives some sufficient conditions to solve the SFTB-H∞ observation-mode-based asynchronous controller. In Section 4, the simulation experiment of HMJSs via parameter varying with two subsystems is carried out, only to find that the closed-loop HMJSs via parameter varying fulfill the condition of SFTB-H∞ under the action of the designed asynchronous controller. The conclusion and future research work follow in Section 5.Notation: the notation meaning throughout this paper is shown in Table1. Furthermore, we assume that the matrices and notations in this paper are standard with compossible dimensions.Table 1
Symbol notations.
NotationDenotesℝnn-dimensional Euclidean spaceℝn×mn×m real matrices∗symmetric matrixℑVThe weak infinitesimal operator ofVIUnit matrixATMatrix transposeA−1Matrix inverse0Zero matrix∗Euclidean vector normHerAA+ATλmaxPMaximum eigenvalue ofPλminPMinimum eigenvalue ofPdiagABThe block-diagonal matrix ofA and BΓThe sample spaceΔThe algebra of eventsℙ·The probability measure which defined onΔ


## 2. System Formulation

### 2.1. System Description

We consider the nonlinear parameter varying hidden Markov jump systems (PV-HMJSs) defined on the probability spaceΓ,Δ,Prob·:(1)x˙tyt=fHtptxt,ωt+gHtptut,xt=x0,pt=p0,Ht=H0,t=0,where Ht=h1t,h2t and pt=p1t,p2t,…,pmtT. The notations of the PV-HMJSs (1) are given in Table 2.Table 2
The notations of the PV-HMJSs (1).
NotationDenotesxt∈ℝnThe systems stateyt∈ℝmThe controlled outputut∈ℝpThe controlled inputωt∈ℝqThe disturbance inputptThe time-varying parameter matrixHtA right-continuous hidden Markov chainx0The initial state of the systemsp0The initial time-varying parameter matrixH0The initial hidden Markov processThe right-continuous hidden Markov chainHt is composed of the hidden state h1t in the finite set ℳ:=1,2,…,M and the observation state h2t in the finite set M:=1,2,…,N. Furthermore, the hidden Markov chain Ht∈ℳ×M can be seen as a homogeneous Markov process which satisfies(2)ℙHt+τ=J,L|Ht=I,K=RI,K,J,Lτ+oτ,I,K≠J,L,1+RI,K,J,Lτ+oτ,I,K=J,L,where RI,K,J,L≥0,∀I,K≠J,L is the transition rate with ∑ℐ,K≠J,ℒℛℐ,K,Jℒ=−rℐ,K and for any L,K∈M,I,J∈ℳ, which satisfies(3)RI,K,I,L=αJLKπII,I≠I,qKLI,L≠K,J=I,πJJ+qKKJ,J=I,K=L0,otherwise,in which ∑L=1NαJLK=1, Π=πIJ is the transition rate matrix and satisfies πIJ≥0 for ∀I≠J, ℵℐ=qKLI is the Π-dependent conditional probability matrix and satisfies qKLI≥0 for ∀K≠L, πII=−∑I≠JπIJ, and qKK=−∑L≠KqKLI.Remark 1.
From relation (3), we can summarize that the same state will be visited of the hidden state h1t and the observation state h2t if ℳ=M, αJLK=0 with J≠L, and αJJK=1 with J=L. In this case, the full information of the hidden state h1t will be provided for the detector. If αJLK=1 and αJLL=0 with K≠L, only one jump will occur between the hidden state h1t and the observation state h2t. Moreover, the observer will not provide any information if M=1, qKLI=0, and αI11=1 and the observation state is h2t=1. For more specific details of the relationship between the hidden state h1t and the observation state h2t, we can refer to [3, 6, 35, 36].
In actual engineering applications, we often encounter situations where the system mode information accessible by the controller/observer is usually inaccurate. In other words, the actual model of the system cannot be observed by the controller, which makes the information between the controller mode and the system mode asynchronous. Therefore, a new controller modeh2t related to the system mode h1t needs to be introduced.
The PV-HMJSs (1) can be rewritten as the following PV-HMJSs:(4)x˙tyt=Fh1tptGxt,ut,ωt,xt=x0,pt=p0,h1t=h10,t=0,h1t∈ℳ,where(5)Fh1tpt=Ah1tptBh1tptW1h1tptFh1tptCh1tptDh1tptW2h1tpt0.
Gxt,ut,ωt=colxt,ut,ωt,fxt,t, AHtpt is the time-varying parameter-depende nt matrix, and fxt,t is the unknown nonlinear function.
We useAIpt, BIpt, W1Ipt, FIpt, CIpt, DIpt, and W2Ipt to denote Ah1tpt, Bh1tpt, W1h1tpt, Fh1tpt, Ch1tpt, Dh1tpt, and W2h1tpt, when h1t=I, I∈ℳ, respectively.
For anyn=1,2,…,m, the time-varying parameter matrix pt and its variation rate p˙t are both assumed bounded, i.e., pnt∈p¯1p¯1 and p˙nt∈p¯2p¯2. Moreover, the time-varying parameter matrix pt is also measurable and affine parameter dependent in real time. That means the following equations hold:(6)AIp=A0+∑I=1mAIpI,BIp=B0+∑I=1mBIpI,W1Ip=W10+∑I=1mW1IpI,CIp=C0+∑I=1mCIpI,DIp=D0+∑I=1mDIpI,W2Ip=W20+∑ℐ=1mW2IpI,FIp=F0+∑I=1mFIpI,where AI, BI, W1I, FI, CI, DI, and W2I with I=0,1,…,m are known matrices.
We suppose the system states of the PV-HMJSs (9) be available, we design the following h2t-dependent asynchronous controller:(7)ut=Kh2tptxt,h2t∈M,where Kh2tpt is the h2-dependent controller gain which will be solved in Theorem 3.
Submitting theh2t-dependent controller (7) into the PV-HMJSs (5), we can get the following closed-loop PV-HMJSs if h1t=I and h2t=K, I∈ℳ, K∈M:(8)x˙tyt=A¯IKpW1IpFIpC¯IKpW2Ip0xtωtfxt,t,x0=x0,I0=I0,K0=K,0,where A¯IKp=AIp+BIpKKp and C¯IKp=CIp+DIpKKp.


### 2.2. Main Definitions and Lemmas

Definition 1 (see [32]).
Given two positive constants0<J1<J2, a weighting matrix RIK>0, and a finite-time interval 0T, the closed-loop PV-HMJSs (8) with ωt≡0 is SFTS within J1J2TRIK if, for any initial condition x0TRIKx0≤J1, we have(9)ExtTRIKxt≤J2,t∈0T.Definition 2 (see [32]).
Given two positive constants0<J1<J2, a weighting matrix RIK>0, and a finite-time interval 0T, the closed-loop PV-HMJSs (12) is stochastically finite-time bounded (SFTB) within J1J2TRIK if, for any initial condition x0TRIKx0≤J1, we have ExtTRIKxt≤J2,t∈0T.Definition 3 (see [6]).
Under zero initial condition, theh2t-dependent controller (9) is said to be a SFTB-H∞ controller of the PV-HMJSs (7) if there exists a h2t-dependent controller gain KKp,K∈M such that the closed-loop PV-HMJSs (10) be SFTB and satisfies the following H∞-gain performance index:(10)E∫0TyTτyτdτ<φE∫0TωTτωτdτ.Lemma 1 (see [22]).
We have2ETXYF⩽βETXTXE+β−1FTYTYF for any given proper dimensional matrices E and F, positive scalar β>0, and matrices X and Y.Assumption 1.
For any given positive scalarsT and ϖ, the disturbance input ωt is bounded with ∫0TωTτωτdτ≤ϖ.Assumption 2.
For a given positive scalarϑ>0, the unknown state-dependent nonlinear function fxt,t satisfies fxt,t⩽ϑxt.Remark 2.
In this paper, the transient character in a given finite-time interval is considered for the stochastic linear parameter varying hidden HMJSs. For the given hidden HMJSs with initial conditions (or bounded disturbance input), we said such systems is stochastically finite-time stable (SFTS) (or stochastically finite-time bounded (SFTB)) if the systems’ state does not exceed a given limit within a finite-time interval. The specific definitions of the SFTS and SFTB are shown in Definitions1 and 2. For more details of the SFTS and SFTB, we can refer to [31–33].


## 2.1. System Description

We consider the nonlinear parameter varying hidden Markov jump systems (PV-HMJSs) defined on the probability spaceΓ,Δ,Prob·:(1)x˙tyt=fHtptxt,ωt+gHtptut,xt=x0,pt=p0,Ht=H0,t=0,where Ht=h1t,h2t and pt=p1t,p2t,…,pmtT. The notations of the PV-HMJSs (1) are given in Table 2.Table 2
The notations of the PV-HMJSs (1).
NotationDenotesxt∈ℝnThe systems stateyt∈ℝmThe controlled outputut∈ℝpThe controlled inputωt∈ℝqThe disturbance inputptThe time-varying parameter matrixHtA right-continuous hidden Markov chainx0The initial state of the systemsp0The initial time-varying parameter matrixH0The initial hidden Markov processThe right-continuous hidden Markov chainHt is composed of the hidden state h1t in the finite set ℳ:=1,2,…,M and the observation state h2t in the finite set M:=1,2,…,N. Furthermore, the hidden Markov chain Ht∈ℳ×M can be seen as a homogeneous Markov process which satisfies(2)ℙHt+τ=J,L|Ht=I,K=RI,K,J,Lτ+oτ,I,K≠J,L,1+RI,K,J,Lτ+oτ,I,K=J,L,where RI,K,J,L≥0,∀I,K≠J,L is the transition rate with ∑ℐ,K≠J,ℒℛℐ,K,Jℒ=−rℐ,K and for any L,K∈M,I,J∈ℳ, which satisfies(3)RI,K,I,L=αJLKπII,I≠I,qKLI,L≠K,J=I,πJJ+qKKJ,J=I,K=L0,otherwise,in which ∑L=1NαJLK=1, Π=πIJ is the transition rate matrix and satisfies πIJ≥0 for ∀I≠J, ℵℐ=qKLI is the Π-dependent conditional probability matrix and satisfies qKLI≥0 for ∀K≠L, πII=−∑I≠JπIJ, and qKK=−∑L≠KqKLI.Remark 1.
From relation (3), we can summarize that the same state will be visited of the hidden state h1t and the observation state h2t if ℳ=M, αJLK=0 with J≠L, and αJJK=1 with J=L. In this case, the full information of the hidden state h1t will be provided for the detector. If αJLK=1 and αJLL=0 with K≠L, only one jump will occur between the hidden state h1t and the observation state h2t. Moreover, the observer will not provide any information if M=1, qKLI=0, and αI11=1 and the observation state is h2t=1. For more specific details of the relationship between the hidden state h1t and the observation state h2t, we can refer to [3, 6, 35, 36].
In actual engineering applications, we often encounter situations where the system mode information accessible by the controller/observer is usually inaccurate. In other words, the actual model of the system cannot be observed by the controller, which makes the information between the controller mode and the system mode asynchronous. Therefore, a new controller modeh2t related to the system mode h1t needs to be introduced.
The PV-HMJSs (1) can be rewritten as the following PV-HMJSs:(4)x˙tyt=Fh1tptGxt,ut,ωt,xt=x0,pt=p0,h1t=h10,t=0,h1t∈ℳ,where(5)Fh1tpt=Ah1tptBh1tptW1h1tptFh1tptCh1tptDh1tptW2h1tpt0.
Gxt,ut,ωt=colxt,ut,ωt,fxt,t, AHtpt is the time-varying parameter-depende nt matrix, and fxt,t is the unknown nonlinear function.
We useAIpt, BIpt, W1Ipt, FIpt, CIpt, DIpt, and W2Ipt to denote Ah1tpt, Bh1tpt, W1h1tpt, Fh1tpt, Ch1tpt, Dh1tpt, and W2h1tpt, when h1t=I, I∈ℳ, respectively.
For anyn=1,2,…,m, the time-varying parameter matrix pt and its variation rate p˙t are both assumed bounded, i.e., pnt∈p¯1p¯1 and p˙nt∈p¯2p¯2. Moreover, the time-varying parameter matrix pt is also measurable and affine parameter dependent in real time. That means the following equations hold:(6)AIp=A0+∑I=1mAIpI,BIp=B0+∑I=1mBIpI,W1Ip=W10+∑I=1mW1IpI,CIp=C0+∑I=1mCIpI,DIp=D0+∑I=1mDIpI,W2Ip=W20+∑ℐ=1mW2IpI,FIp=F0+∑I=1mFIpI,where AI, BI, W1I, FI, CI, DI, and W2I with I=0,1,…,m are known matrices.
We suppose the system states of the PV-HMJSs (9) be available, we design the following h2t-dependent asynchronous controller:(7)ut=Kh2tptxt,h2t∈M,where Kh2tpt is the h2-dependent controller gain which will be solved in Theorem 3.
Submitting theh2t-dependent controller (7) into the PV-HMJSs (5), we can get the following closed-loop PV-HMJSs if h1t=I and h2t=K, I∈ℳ, K∈M:(8)x˙tyt=A¯IKpW1IpFIpC¯IKpW2Ip0xtωtfxt,t,x0=x0,I0=I0,K0=K,0,where A¯IKp=AIp+BIpKKp and C¯IKp=CIp+DIpKKp.


## 2.2. Main Definitions and Lemmas

Definition 1 (see [32]).
Given two positive constants0<J1<J2, a weighting matrix RIK>0, and a finite-time interval 0T, the closed-loop PV-HMJSs (8) with ωt≡0 is SFTS within J1J2TRIK if, for any initial condition x0TRIKx0≤J1, we have(9)ExtTRIKxt≤J2,t∈0T.Definition 2 (see [32]).
Given two positive constants0<J1<J2, a weighting matrix RIK>0, and a finite-time interval 0T, the closed-loop PV-HMJSs (12) is stochastically finite-time bounded (SFTB) within J1J2TRIK if, for any initial condition x0TRIKx0≤J1, we have ExtTRIKxt≤J2,t∈0T.Definition 3 (see [6]).
Under zero initial condition, theh2t-dependent controller (9) is said to be a SFTB-H∞ controller of the PV-HMJSs (7) if there exists a h2t-dependent controller gain KKp,K∈M such that the closed-loop PV-HMJSs (10) be SFTB and satisfies the following H∞-gain performance index:(10)E∫0TyTτyτdτ<φE∫0TωTτωτdτ.Lemma 1 (see [22]).
We have2ETXYF⩽βETXTXE+β−1FTYTYF for any given proper dimensional matrices E and F, positive scalar β>0, and matrices X and Y.Assumption 1.
For any given positive scalarsT and ϖ, the disturbance input ωt is bounded with ∫0TωTτωτdτ≤ϖ.Assumption 2.
For a given positive scalarϑ>0, the unknown state-dependent nonlinear function fxt,t satisfies fxt,t⩽ϑxt.Remark 2.
In this paper, the transient character in a given finite-time interval is considered for the stochastic linear parameter varying hidden HMJSs. For the given hidden HMJSs with initial conditions (or bounded disturbance input), we said such systems is stochastically finite-time stable (SFTS) (or stochastically finite-time bounded (SFTB)) if the systems’ state does not exceed a given limit within a finite-time interval. The specific definitions of the SFTS and SFTB are shown in Definitions1 and 2. For more details of the SFTS and SFTB, we can refer to [31–33].


## 3. Main Results

In this section, we will give some sufficient conditions to solve the SFTB-H∞ observation-mode-based asynchronous controller (11) and obtain the h2t-dependent controller gain KKp,K∈M. We aim the closed-loop PV-HMJSs (12) to fulfill the SFTB-H∞ condition under the action of the observation-mode-based asynchronous controller (11).Theorem 1.
Given four positive constantsω>0, J1>0, ϖ>0, and T>0, the closed-loop PV-HMJSs (12) is SFTB within J1J2TRIKϖ if there exist positive constants 0<J1<J2, λ¯PIK′>0, λ¯PIK′>0, φ>0, and Ht-dependent positive-definite symmetric matrices PIK and RIK, where I,K∈ℳ×M, such that(11)ΥPIKW1IpFIpTPIKΘIK∗−φI00∗∗−β−1I0∗∗∗−ΨIK<0,(12)J1λ¯PIK′+φϖeωT<J2λ¯PIK′,
where(13)Υ=HerPIKA¯IKp+rI,KI,K−ωPIK+β−1ϑI,PIK′=RIK−1/2PIKRIK−1/2,λ¯PIK′=maxI,K∈ℛλmaxPIK′,λ¯PIK′=minI,K∈ℛλminPIK′,ΨIK=diagPνI,K1−1,PνI,K2−1,…,PνI,KhI,K−1,ΘIK=ℛI,KνI,K1I,…,ℛI,KνI,KhI,KI.Proof.
Select a SLKF candidate as follows:(14)Vxt,Ht,t=xTtPIKxt.
Recalling to the closed-loop PV-HMJSs (8), we can get the following weak infinitesimal operator of Vxt,Ht,t:(15)ℑVxt,Ht,t=limτ⟶01τEVxt+τ,Ht+τ,t+τ|xt,Ht−Vxt,Ht,t=xTtP1xt+HerxTtPIKW1Ipωt+HerxTtPIKFIpfxt,t,where(16)P1=HerPIKA¯IKp+∑J,L∈ℛℛI,Kj,lPJL,in which(17)∑J,L∈ℛℛI,KJ,LPJL=∑J,Lℛ′ℛI,KJ,LPJL+rIKIKPIK=ΘIKΨIK−1ΘIKT+rIKIKPIK,with(18)ℛ′=ℛI,KJ,L≠0,J,L∈ℛ,I,K≠J,L=νI,K1rI,K,…,νI,KhI,KℛI,K,ΨI,K=diagPνI,K1−1,PνI,K2−1,…,PνI,KhI,K−1,ΘI,K=rI,KνI,K1I,…,rI,KνI,KhI,KI.
Recalling to Lemma1 and Assumption 2, we know that(19)ℑVxt,Ht,t<xTtP2xt+HerxTtPIKW1ipωt,where(20)P2=HerPIKA¯IKp+∑J,L∈rrI,KJ,LPJL+βFIpTPIKPIKFIp+β−1ϑI.
FromℑVxt,Ht,t<0, it follows that(21)xtωtTψPIKW1Ip∗0xtωt<0,where(22)ψ=HerPIKA¯IKp+∑J,L∈rrI,KJ,LPJL+βFIpTPIKPIKFIp+β−1ϑI.
Recalling to inequalities (11) and (18)–(21), we have the following inequality with positive constant ω>0:(23)EℑVxt,Ht,t<ωVxt,Ht,t+φωTtωt.
Then, we have the following relation by integrating both left and right sides of inequality (23) from 0 to t for ∀t∈0T:(24)EVxt,Ht,t−EVx0,H0<φ∫0tωTτωτdτ+ω∫0tEVxt,Ht,τdτ.
Thus,(25)EVxt,Ht,t<J1λ¯PIK′+φϖ+ω∫0tEVxt,Ht,τdτ.
The above inequality can be rewritten as (26) by considering the Gronwall inequality:(26)EVxt,Ht,t<J1λ¯PIK′+φϖeωt<J1λ¯PIK′+φϖeωT.
From Rayleigh inequality, we know thatEVxt,Ht,t>λ¯PIK′ExtTRIKxt. Then, we have(27)ExtTRIKxt<J1λ¯PIK′+φϖeωTλ¯PIK′.
Thus, theExtTRIKxt<J2 for ∀t∈0T holds by inequality (12). This completes the proof.
In Theorem1, we give some sufficient conditions to ensure the SFTB of the closed-loop PV-HMJSs (12). Then, the SFTB-H∞ condition will be given in the following Theorem 2 on the basis of Theorem 1.Theorem 2.
Given four positive constantsω>0, J1>0, T>0, and ϖ>0 with ∫0tωTτωτdτ≤ϖ, the SFTB-H∞ condition (10) of the closed-loop PV-HMJSs (8) will be satisfied within J1J2TRIKϖ if there exist positive constants 0<J1<J1, λ¯PIK′>0, λ¯PIK′>0, φ>0, and Ht-dependent positive-definite symmetric matrices PIK and RIK, where I,K∈ℳ×M, such that inequalities (12) and (28) hold:(28)ϰPIKW1Ip+C¯IKpTW2IpFIpTPIKΘIK∗W2IpTW2Ip−φI00∗∗−β−1I0∗∗∗−ΨIK<0,where ϰ=Υ+eωTC¯IKpTC¯IKp.Proof.
The same SLKF is selected as in Theorem1 and we introduce(29)EℑVxt,Ht,t−ωVxt,Ht,t<φωTtωt−eωtEyTtyt.
Inequality (28) guarantees that inequality (29) is established. Then, we use e−ωt to multiply inequality (29) and integrate such inequality from 0 to T under zero initial condition, and it yields(30)e−ωtEVxt,Ht,t<E∫0Tφe−ωτωTτωτ−yTτyτdτ.
We can rewrite inequality (30) as the following inequality by considering EVxt,Ht,t>0:(31)E∫0TyTτyτdτ<E∫0Tφe−ωτdTτdτdτ<φE∫0TωTτωτdτ.
Thus, we can obtain(32)E∫0TyTτyτdτ<φE∫0TωTτωτdτ.
Recalling Definition3, we can get the SFTB-H∞ condition (10) of the closed-loop PV-HMJSs (8). This completes the proof.
Through the analysis of Theorem2, we know that the state feedback controller gain matrix Kh2tpt cannot be solved by Matlab LMI tools due to the nonlinear terms in inequality (28). In the following Theorem 3, some sufficient conditions will be given to obtain finite-time asynchronous controller gain Kh2tpt.Theorem 3.
Given four positive constantsω>0, J1>0, T>0, and ϖ>0 with ∫0tωTτωτdτ≤ϖ, there exist a finite-time H∞-gain asynchronous controller with h2-dependent state feedback gain KKp=ZKpSK−1 such that the SFTB-H∞ condition (10) of the closed-loop PV-HMJSs (12) will be satisfied within J1J2TRIKϖ if there exist positive scalars 0<J1<J2, λ¯1>0, φ>0, β>0, μℐ>0, φ>0, Ht-dependent positive-definite symmetric matrices XIK, RIK, where I∈ℳ, K∈M, h2-dependent positive-definite symmetric matrix SK, and h2-dependent matrix ZKp, such that(33)L1IKp+B1IK<0,(34)λ¯1RIK−1<XIK<RIK−1,(35)φϖ−J2e−ωTλ¯1J1J1−λ¯1<0,where(36)L1IKp=HerAIpLISK+BIpZKp00CIpLISK+DIpZKpTeωTCIpLISK+DIpZKpT0μII00000T,B1IK=B1IK′B2IK′∗R3IK′,B1IK′=O1IKW1IpXIKXIKFIpT∗Ξ5I0∗∗−β−1I,B2IK′=00XIKΘIK0XIKW2IpT0000,R3IK′=diag−ξ−1I,−ξI,−ΨIK′,O1IK=ℛi,kI,K−ωXIK+β−1ϑQIK,Ξ5I=W2IpTW2Ip−φI,ΘIK=rI,KνI,K1I,…,rI,KνI,KhI,KI,ΨIK′=diagXℛI,K1−1,XℛI,K2−1,…,XℛI,KhI,K−1.Proof.
We substituteA¯IKp and C¯IKp into inequality (34) and yields(37)Ξ1IK′Ξ4IKFIpTPIKΘIK∗Ξ5I00∗∗−β−1I0∗∗∗−ΨIK<0,where(38)Ξ1IK′=Ξ1IK+Ξ2IK+Ξ3IK,Ξ1IK=HerPIKAIp+BIpKKp,Ξ2IK=eωTCIp+DIpKKpTCIp+DIpKKp,Ξ3IK=ℛI,KI,K−ωPIK+β−1ϑI,Ξ4IK=PIKW1Ip+CIp+DIpKKpTW2Ip,Ξ5I=W2IpTW2Ip−φI.
We usediagPIK−1,I,II to pre- and postmultiply inequality (37), and let XIK=PIK−1, YIKp=KKpXIK, and QIK=XIKXIK, and we use the Schur complement lemma. It can be seen that the following inequality satisfies with positive scalar ξ>0:(39)R1IK′R2IK′∗R3IK′<0,where(40)R1IK′=Ξ1IK′W1IpXIKXIKFIpT∗Ξ5i0∗∗−β−1I,R2IK′=Ξ2IK′Ξ3IK′XIKΘIK0XIKW2IpT0000,R3IK′=diag−ξ−1,−ξ,−ΨIK′,ΨIK′=diagXℛI,K1−1,XℛI,K2−1,…,XℛI,KhI,K−1,Ξ1IK′=HerAIpXIK+BIpYIKp+rI,KI,K−ωXIK+β−1ϑQIK,Ξ2IK′=CIpXIK+DIpYIKpT,Ξ3IK′=eωTCIpXIK+DIpYIKpT.
Considering the asynchronous of (39), we rewrite inequality (39) as follows:(41)B1IK+B2IK<0,where(42)B1IK=B1IK′B2IK′∗R3IK′,B1ik′=O1IKW1IpXIKXIKFIpT∗Ξ5I0∗∗−β−1I,B2IK′=00XIKΘIK0XIKW2IpT0000,B2IK=HerAIpXIK+BIpYIKp00CIpXIK+DIpYIKpTeωTCIpXIK+DIpYIKpT0I00000T,O1IK=rI,KI,K−ωXIK+β−1ϑQIK.
ForB2IK, we define XIK=μILISK, where LI is a nonsingular unit matrix for ∀i∈ℳ. We can obtain(43)AIpXIK+BIpYIKp=μIAIp+BIpKKpLISK,CIpXIK+DIpYIKp=μICIp+DIpKKpLISK.
SinceLI is a nonsingular unit matrix, we can get inequality (41) by inequality (33) with ZIKp=KKpLISK. Moreover, we know that HerLISK>0, i.e., HerSK>0 from inequality (33). Thus, ZKp is also a nonsingular matrix, which means that KKp=ZKpSK−1 can be defined and solved.
DefiningX¯IK=RIK−1/2XIKRIK−1/2, λ¯X¯IK=maxI,K∈rλmaxX¯IK, and λ¯X¯IK=minI,K∈rλminX¯IK, letting λ¯1≤λ¯X¯IK and λ¯1≥λ¯X¯IK, and considering λ¯X¯IK=1/λ¯PIK′ and λ¯X¯ℐK=1/λ¯PℐK′    , inequality (12) can be rewritten as(44)J1λ¯1+φϖ<J2e−ωTλ¯1.
By considering the eigenvalue conversion method and using the Schur complement lemma, inequalities (34) and (35) can be obtained. This completes the proof.
However, we also cannot get the state feedback controller gain matrixKh2tpt by Matlab LMI tools in Theorem 3 due to the dependence of the parameters. Next, the methods of approximate basis function and gridding technique will be used to deparameterize the parameter-dependent matrices in Theorem 3. The specific deparameterization process is as follows:(45)Ap=A0p+∑I=1mAIpI<0,∀p∈ℨ,(46)ℨ=pt∈ℝp:p¯1I⩽pIt⩽p¯1I,∀I∈1,2,…,m,where pt is the time-varying parameter. From inequalities (45) and (46), we know that ℨ denotes an LMI and Ap has infinite number of LMIs. Then, we select the following basis function FIptJ=1nf to solve inequality (45):(47)AIpt=∑J=1nfFIptAIJ.
Furthermore, the infinite LMIs of theAp can be transformed into the finite ones if we divide the space of parameter changes into finite-dimensional grids, which means the following relation satisfied for each set of parameters on the grid for ∀pt∈ℨ:(48)Ap=∑J=1nfFJptA0+∑J=1nf∑i=1mFJptAIJ<0.Remark 3.
In Theorem3, the difficulty of calculation will be increased because of the asynchronous characteristic between the controller modes and the system modes. In order to solve such difficulty, the controller modes h2t is converted h1t. Moreover, some sufficient conditions are obtained to make the closed-loop HMJSs with linear parameter varying be SFTB-H∞ by introduced auxiliary variables.Remark 4.
In addition, we have introduced the stochastic Lyapunov–Krasovskii functional methods in Theorem3, which will induce somewhat conservatism of the main results. In future work, we can reduce the impact of conservativeness through replacing the quasi-one-sided Lipschitz condition or one-sided Lipschitz condition with local Lipschitz condition in Assumption 2.


## 4. Numerical Example

In this section, we consider a class of PV-HMJSs with two subsystems.Subsystem 1:(49)A1pt=−16.567−12.1+0.05000.05sint+0.02000.02cost,B1pt=11+0.2sint0.1cost,W11pt=0.5sint+0.2cost,F1pt=0.4sint+0.1cost,W21pt=0.30.10.20.2+0.01000.01sint+0.02000.02cost,C1pt=−3.3−0.20.1−2.8+−0.100−0.2sint+−0.300−0.2cost,D1pt=0.30.2+0.1sint0.02cost.Subsystem 2:(50)A2pt=−12.384−11.2+0.02000.01sint+0.01000.01cost,B2pt=22+0.4sint0.3cost,W12pt=0.4sint+0.2cost,F2pt=0.3sint+0.1cost,W22pt=0.10.20.20.1+0.01000.01sint+0.02000.02cost,C2pt=−2.8−0.10.2−3.1+−0.200−0.2sint+−0.300−0.1cost,D2pt=0.30.2+0.2sint0.01cost,where sint and cost are time-varying parameters and bounded with −11. We assume the weighted matrix R=I, the initial consideration J1=0.8, the external disturbance ωt=0.1sin22t0.2sin2t, and the unknown state-dependent nonlinear function fxt,t=0.3x1t0.1x2t in finite-time interval T=5. The other values of the constants are given as ϖ=0.3, λ¯1=5.089, λ¯1=0.5, μ1=1.2, μ2=1.5, ω=0.3, β=0.3, ϑ=1, and ξ=0.2.We assume theh2t-dependent controller gain KKp with two control schemes for two different jump modes, i.e., ℳ∈1,2 and M∈1,2. The Π-dependent conditional probability matrices are ℵ1=−0.60.60.4−0.4 and ℵ2=0.3−0.3−0.10.1. The parameters of αjlk are α1=αjl1=0.30.70.40.6 and α2=αjl2=0.20.80.50.5.Solving LMIs (33)–(35), we can get the following finite-time state feedback gain matrices:(51)K1p=2.9251−1.1431+−1.1633sint0.3333cost,K2p=3.0817−0.7694+−0.5792sint−0.171cost.Meanwhile, the other relevant solutions are given asJ2=5.0894 and φ=9.8. Then, the simulation results are shown in Figures 1–3.Figure 1
The jump modeHt of the PV-HMJSs. (a) The jump mode of the hidden state h1(t). (b) The jump mode of the observation state h2(t).
(a)(b)Figure 2
Thext of the closed-loop PV-HMJSs.Figure 3
Theyt of the closed-loop PV-HMJSs.The jump modes ofh1t and h2t are shown in Figure 1. The state trajectory xt and yt of the closed-loop PV-HMJSs (8) are shown in Figures 2 and 3. From Figures 2 and 3, we can seen the SFTB-H∞ condition is satisfied with ExtTRIKxt<J2 in which J2=5.0894 and φ=9.8.Remark 5.
Similar to some results of reinforcement learning methods [41–44], this paper also considered the problem of stabilization of stochastic Markov jump systems. Moreover, we also use the hidden Markov model to denote the asynchronous characteristic between system modes and controller modes. Different from some existing results on the finite-time control problem [35, 36], the problem of SFTB-H∞ asynchronous control is studied for continuous-time HMJSs via parameter varying in this paper. Compared with the existing results of asynchronous control for discrete-time HMJSs [37–40], this paper firstly considers the asynchronous stabilization problem for the continuous-time HMJSs with parameter varying models.
For the future work, we can learn from the reinforcement learning methods to study the problem of finite-time asynchronous online control for the continuous-time PV-HMJSs.


## 5. Conclusion

This paper studied the SFTB-H∞ asynchronous control problem for continuous-time PV-HMJSs. Some sufficient conditions are given to solve the SFTB-H∞ asynchronous control gain by considering the methods of SLKF and LMIs. The designed SFTB-H∞ asynchronous controller makes the closed-loop PV-HMJSs satisfy the SFTB-H∞ condition. Finally, we use a numerical example to show the validity of the main results of this paper. For the future research work, the asynchronous control problem for fuzzy PV-HMJSs apply approximation method will be considered.



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