# Extended Dissipative Control for Markovian Jump Time-Delayed Systems with Bounded Disturbances

Authors: T. Poongodi; T. Saravanakumar; Prem Prakash Mishra; Quanxin Zhu
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

In this work, the result of reachable set bounding and extended dissipative control synthesis of the Markovian jump time-delayed system is studied subject to stochastic actuator failures and partially known transition probabilities. Specifically, a novel actuator fault model is designed, in which the actuator fault matrix satisfies a certain probabilistic condition. Under the construction of an appropriate Lyapunov–Krasovskii functional (LKF), as well as reciprocal convex approach, Jensen’s integral inequality, and reachable set lemma, delay-dependent sufficient criteria are obtained in terms of linear matrix inequalities (LMIs) for finding an ellipsoid to bound the reachable sets of the Markovian jump time-delayed system with bounded disturbances. Finally, two numerical examples are provided to validate the effectiveness of the proposed strategy.


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## Body

## 1. Introduction

Markovian jump systems are a kind of stochastic switched systems, in which the switching between multiple modes is governed by a stochastic process described by a Markov chain. Furthermore, Markovian jump systems are appropriate model systems with random abrupt variations, which may be due to malfunction of machine in manufacturing systems, fluctuations in operating points, and environmental disturbances [1–5]. Therefore, a remarkable progress has been made in the study of Markovian jump systems and has proved its successful applications in chemical systems, economic systems, power systems, electrical circuit systems, and so on [6–9]. A new feedback controller is designed in [10] to ensure the finite-time mean square stability of the nonlinear Markovian jump systems, in which the nonlinear element satisfies the prescribed conic sector condition. Very recently, the problem of model reduction for nonhomogeneous Markovian jump systems is studied in [11], in which asynchronization is designed by a hidden Markov model with Markovian jump parameters. Xia et al. [12] reported the result of the dissipative sampled-data scheme for fuzzy Markovian jump systems. However, it is mentioned that the abovementioned studies are based on a particular ideal assumption that the information of each element in the transition probabilities matrix is available. In fact, it is hard to measure or determine the knowledge of all the elements, which motivates further study on Markovian jump systems with partly known transition probabilities that are more general [13–15].It is known that reachable set estimation is used to find an appropriate domain, which can bound all the reachable states of a dynamical system with respect to input disturbances and zero initial conditions. During the past few years, reachable set estimation of dynamical systems has various practical applications in aircraft collision avoidance, peak-to-peak gain minimization, and control synthesis. Therefore, it is necessary to study the problem of reachable set estimation with time-delay dynamical systems, and a great number of results have been paid much attention in past few years [16–18]. For instance, the problem of reachable set estimation of singular systems is studied subjected to time delay and bounded peak inputs [19]. Based on the novel LKF approach, sufficient conditions are derived in terms of LMIs to ensure the reachable set of the linear systems bounded in [20]. Feng et.al. [21] reported the reachable set estimation of discrete-time fuzzy systems subjected to time delay and external disturbance.In practical control systems, there often exist faults due to the reasons of actuators aging, zero shift, electromagnetic interference, and nonlinear amplification in different frequency field. During the system operation, faults may affect the sensors, the actuators, and the system components, which may lead to intolerable performance of the closed-loop dynamical system. Therefore, from a safety as well as performance point of view, it is required to design a fault-tolerant controller, which can tolerate actuator or sensor faults and guarantee the stability and performance of resulting closed-loop systems [22–25]. For example [26], a fuzzy-based fault-tolerant control approach is designed for nontriangular nonlinear systems with respect to immeasurable states and actuator failures. Sun et.al. [27] studied the composite reliable control problem for stochastic systems with disturbance observer. Based on the Lyapunov–Krasovskii functional together with partially known transition conditions, a new set of sufficient condition is derived in [28] for the nonlinear time-delay systems subjected to random fluctuations.On the other side, the concept of extended dissipative was firstly considered in [29] where some well-known performance indices, namely, H∞ performance, passivity performance, mixed H∞ passivity performance, L2−L∞ performance, and Q,S,R-dissipative performance can be obtained by adjusting weighting matrices of extended dissipative [29–31]. Very recently, the concept of extended dissipative has been effectively applied to several dynamical systems in reports [32–34]. Zhu et.al. [35] reported the problem of finite-time extended dissipative-based control of neutral switched systems with stochastic actuator failures. Under the extended dissipative analysis with an average dwell-time approach, an explicit expression of the desired Markov switching estimator is presented in [36]. To the best of authors’ knowledge, the extended dissipative concept together with reachable set estimation has not been applied to Markovian jump time-delayed systems subjected to probabilistic faults yet in the existing literatures, which motivates our present study. The main notable contributions of this paper can be highlighted as follows:(i)
The concept of extended dissipative and reachable set estimation is successfully first time applied to Markovian jump time-delayed systems with partially known transition probabilities and bounded disturbances(ii)
We have investigated the result of the extended dissipative analysis and stochastic fault-tolerant control design for a class of Markovian jump systems where the transition rate matrix satisfies the partially known probability conditions(iii)
Under the appropriate LKF, Jensen’s integral inequality and the reciprocal convex approach, a new set of sufficient conditions is obtained in the form of LMIs to guarantee the Markovian jump time-delayed system bounded by an ellipsoid and satisfies the extended dissipative performance index(iv)
Two numerical simulation demonstrate the feasibility and effectiveness of the proposed approachThe remaining part of this work is constructed as follows. The system description and problem formulation are provided in Section2. The main results on extended dissipative-based stochastic fault-tolerant control for Markovian jump time-delayed systems with partially known transition probabilities and bounded disturbance are given in Section 3. Two examples are presented in Section 4 to demonstrate the effectiveness and feasibility of the obtained results. Finally, conclusions and the direction of future research are made in Section 5.Notations 1.
The superscripts “T” and “−1” stand for matrix transposition and matrix inverse, respectively; ℝn and ℝn×m denote the n−dimensional Euclidean space and the set of all n×m real matrices, respectively; the notation P>0≥0 means that P is real, symmetric, and positive definite (positive semi-definite); I is the identity matrix of appropriate dimension; the notation E⋅ stands for the expectation operator; and “∗” is used to represent a term that is induced by symmetry.


## 2. Problem Formulation and Preliminaries

Consider the following Markovian jump systems with time-varying delay defined on a complete probability spaceΩ,ℱ,ℙ and described as follows:(1)x˙t=Artxt+Adrtxt−dt+BrtuFrtt+Drtωt,zt=C1rtxt+C2rtωt,xt=ϕt,t∈−d,0,where xt∈ℝn is the state vector; uFrtt∈ℝn is the control input with respect to actuator fault; ωt∈ℝm is the disturbance; zt represents system output; ϕt is the initial condition of the state; and rt,t≥0 is the Markov process and takes values in a finite set ℕ=1,2,…,N, and it satisfies the following transition probability conditions:(2)Probrt+Δt=jrt=i=πijΔt+oΔt,ifi≠j,1+πiiΔt+oΔt,ifi=j,where Δt>0 and limΔt⟶0OΔt/Δt=0; πij≥0 is the transition rate from mode i at time t to mode j at time t+Δt if i≠j and πii=−∑j=1,i≠jNπij; the matrices Art, Adrt, Brt, Drt, C1rt, and C2rt are known matrices of compatible dimensions. The set ℕ comprises the various operation modes of system (1), and for each possible value of rt=i∈ℕ, the system matrices of the ith mode are denoted by Ai, Adi, BiDi, C1i, and C2i. Further, the time-varying delay dt satisfies(3)0≤dt≤d,d˙t≤μ<1,where d and μ are the given constants. The disturbance term ω¯t is assumed to be bounded:(4)ωTtωt≤ω¯2,where ω¯ is a positive scalar.On the other side, the transition rate matricesΠ for system (1) with N operation modes are defined by(5)Π=π11?π13…π1Nπ21??…?⋮⋮⋮⋱⋮?πN2πN3…πNN,where “?” represents the unknown transition rate. For the simplicity of presentation, we introduce the following notations. For i∈ℕ, let T=TKi+TUKi with(6)TKi=j:ifπijare known,TUKi=j:ifπijare unknown.The reachable set of system (1) is denoted as follows:(7)ℜx=xt∈ℝnxtandωtsatisy1and4.For a positive-definite symmetric matrixPi>0, we define an ellipsoid εPi bounding the reachable set (7) as follows:(8)εPi=xxTPix≤1,x∈ℝn.We consider the following static feedback controller for system (1):(9)uit=Kixt,in which Ki represent the controller gain matrices to be determined. Then, the following stochastic fault model is adopted for this study:(10)uiFt=Ξuit,=∑j=1mξjHjKixt,where Ξ=diagξ1,…,ξm with ξjj=1,…,m are the m unrelated random variables. It is assumed that ξj is with mathematical expectation μj and variance σj2, respectively, and Hj=diag0,…,0︸j−i,1,0,…,0︸m−j.Also, we defineΞ¯=diagμ1,…,μm and Δ=diagσ1…σm.Combining (1) and (10), we obtain the following closed-loop systems:(11)x˙t=Ai+BiΞ¯Kixt+BiΞ−Ξ¯Kixt+Adixt−τt+Diωt,zt=C1ixt+C2iωt.Our main aim of this study is to derive an ellipsoidζ, as small as possible, that bounds the reachable set ℜx of system (1) subject to time-varying delays and bounded disturbances (4). Then, we will design a stochastic fault-tolerant controller (10), such that the reachable set of the closed-loop system (11) is bounded and extended dissipative by a given ellipsoid εPi. Before proving our main results, the following lemmas, assumption, and definition are introduced.Lemma 1 (see [35]).
For any symmetric positive definite matrixH∈ℝn×n and a scalar h>0, such that the following integrations are well-defined, then(12)h∫t−htxTsHxsds≥∫t−htxsdsTH∫t−htxTs.Lemma 2 (see [35]).
Given appropriate dimensioned matricesS11, S12, and S22 with S11T=S11, S22T=S22, then the inequality S11+S12TS22−1S12<0 holds if and only if S11S12T∗−S22<0or−S22S12∗S11<0.Lemma 3 (see [18]).
Letf1,f2,…,fN:ℝm⟶ℝ have positive values in an open subset D of ℝm. Then, the reciprocally convex combination of fi over D satisfies(13)minαi|αi>0,∑αi=1∑i1αifit=∑ifit+maxgi,jt∑i≠jgi,jt,subjected to(14)gi,j:ℝm⟶ℝ,gj,it=gi,jt,fitgi,jtgi,jtfjt≥0.Lemma 4 (see [18]).
LetVxt be a Lyapunov function for system (11), Vxt0≤βω¯2/α, t≥0, α>0, β>0, and the constant ω¯ is mentioned in (3). If V˙xt+αVxt−βωTtωt≤0, then Vxt≤βω¯2/α∀t≥t0.Assumption 1.
(see [35]). For given real symmetric matrices Φ1,Φ2,Φ3, and Φ4, the following conditions hold:(1)
Φ1≤0, Φ3>0, and Φ4≥0(2)
D2i.Φ4=0, for all i∈ℐ(3)
Φ1+Φ2.Φ4=0(4)
D2iTΦ1D2i+D2iTΦ2+Φ2TD2i+Φ3>0, for all i∈ℐDefinition 1.
(see [35]). For given matrices Φ1,Φ2,Φ3, and Φ4, the considered closed-loop system (8) is said to be the extended dissipative if there exists a scalar ρ such that the upcoming inequality holds for all Tf≥0 and wt∈ℒ20,∞:(15)E∫0TfZtdt≥sup0≤t≤TfEzTtΦ4zt≥ρ,where Zt=zTtΦ1zt+2zTtΦ2wt+wTtΦ3wt.


## 3. Main Results

In this section, the first reachable set estimation problem for Markovian jump time-delayed systems with partially known transition probabilities is considered based on Jensen’s integral equality, novel reachable set lemma, and improved reciprocally convex combination method. Next, under extended dissipative performance index, a novel stochastic fault-tolerant controller is proposed and a sufficient condition is given.Theorem 1.
For any given scalarsd≥0, μ, μl, σjj=1,…,m, ω¯, and the matrix Ki, if there exists appropriately dimensioned matrices Pi>0, Qi>0, Q>0, R>0, S, M>0, and N>0, symmetric matrices E, E1, Wi, and Zi, and a scalar α>0, the following LMIs hold for i∈ℕ:(16)Θi=Θ6×6AiTℋiTN∗−d2N−10∗∗−H−1<0,(17)∑j∈Tkiπije−αdQj−Zi≤0,(18)Pj−Wi≤0,j∈Tuki,j≠i,(19)e−αdQj−Zi≤0,j∈Tuki,j≠i,(20)Pj−Wi≥0,j∈Tuki,j=i,(21)Qj−Zi≥0,j∈Tuki,j=i,(22)ME∗M≥0,NE1∗N≥0,where(23)Θ1,1=2PiAi+BiΞKi+∑j∈TkiπijPj−Wi+S+αPi+Qi+R+dM−e−αdN,Θ1,2=PiAdi+e−αdN−E1,Θ1,3=e−αdE1,Θ1,6=PiDi,Θ2,2=e−αd1−μS+E1+E1T−2N,Θ2,3=e−αdN−E1,Θ3,3=−e−αdR+Qi+N,Θ4,4=−e−αddM,Θ4,5=−e−αddE,Θ5,5=−e−αddM,Θ6,6=−αω¯2,Ai=Ai+BiΞ¯KiNAdiN000DiN,ℋi=σ1BiH1Ki⋮σiBiHkKi⋮σmBiHmKimn×n0mn,n0mn,n0mn,n,H=diagN,…,N︸m.Then, the reachable set of system (11) can be bounded by the ellipsoid εPi.Proof.
We consider the following Lyapunov–Krasovskii functional candidate for systems (11) as(24)Vxt,rt,t=∑m=15Vmxt,rt,t,where(25)V1xt,rt,t=xTtPrtxt,V2xt,rt,t=∫t−dteαt−sxTsQrtxsds+∫t−dtteαt−sxTsSxsds,V3xt,rt,t=∫t−dteαt−sxTsRxsds,V4xt,rt,t=∫t−dteαt−sd−t+sxTsMxsds,V5xt,rt,t=d∫−d0∫t+θteαt−sx˙TsNx˙sdsdθ.
Define the weak infinitesimal operatorE for the Lyapunov functional together with Markovian jump system (11) as(26)£Vxt,rt,t=limΔt⟶01ΔtEVxt+Δt,rt+Δt,t+Δtxt,rt=i−Vxt,rt,t.
Furthermore, we have(27)£V1xt,i,t=2xTtPix˙t+xTt∑j=1NπijPjxt−αV1xt,i,t+αV1xt,i,t,(28)£V2xt,i,t=xTtQixt−e−αdxTt−dQixt−d+∫t−dteαt−sxTs∑j=1NπijQjxsds+xTtSxt−1−μe−αdtxTt−dtSxt−dt−αV2xt,i,t,(29)£V3xt,rt,t=xTtRxt−e−αdxTt−dRxt−d−αV3xt,i,t,(30)£V4xt,i,t=dxTtMxt−∫t−dteαt−sxTsMxsds−αV4xt,i,t,(31)£V5xt,i,t=d2x˙TtNx˙t−d∫t−dteαt−sx˙TsNx˙sds−αV5xt,i,t.
Considering the circumstance that the information of transition probabilities is not completely known, for any arbitrary matricesWi=WiT and Zi=ZiT, the following zero equations hold due to ∑j=1Nπij=0:(32)−xTt∑j=1NπijWixt=0,(33)−∫t−dteαt−sxTs∑j=1NπijZixsds=0.
Using Lemmas1 and 3 to the integral terms in equations (30) and (31), we have(34)−∫t−dteαt−sxTsMxsds≤−e−αd∫t−dtxTsMxsds=−e−αd∫t−dttxTsMxsds+∫t−dt−dtxTsMxsds=−e−αddddt∫t−dtxsdsTM∫t−dtxsds−e−αdddd−dt∫t−dt−dtxsdsTM∫t−dt−dtxsds,≤−e−αdd∫t−dttxsds∫t−dt−dtxsdsTME∗M∫t−dttxsds∫t−dt−dtxsds,where the matrices M and E satisfy ME∗M≥0.
Similarly, we have(35)−d∫t−dteαt−sx˙TsNx˙sds≤−e−αd∫t−dttx˙sds∫t−dt−dtx˙sdsTNE1∗N∫t−dttx˙sds∫t−dt−dtx˙sds,(36)≤−e−αdxt−xt−dtxt−dt−xt−dTNE1∗Nxt−xt−dtxt−dt−xt−d,where the matrices N and E1 satisfy NE1∗N≥0.
Combining (22) to (35), we have(37)£Vxt,i,t+αVxt,i,t−αω¯2ωTtωt≤2xTtPix˙t+xTtαPi+Qi+S+R+dMxt+xTt−dQi−e−αdRxt−d−1−μe−αdtxTt−dtSxt−dt+ξTtAiTNAiξt+xTt∑j=1mσjBiHjKiTNσjBiHjKixt+xt∑j∈TkiπijPj−Wixt+xt∑j∈TukiπijPj−Wixt+e−αd∫t−dtxTs∑j∈TkiπijQj−Zixsds+e−αd∫t−dtxTs∑j∈TukiπijQj−Zixsds−e−αdd∫t−dttxsds∫t−dt−dtxsdsTME∗M∫t−dttxsds∫t−dt−dtxsds−e−αdxt−xt−dtxt−dt−xt−dTNE1∗N×xt−xt−dtxt−dt−xt−d−αω¯2ωTtωt,where ξTt=xTtxTt−dtxTt−d∫t−dttxTsds∫t−dt−dtxTsdsωTs. Furthermore, it is noted that πii=−∑j=1Nπij and πij≥0 for all j≠i, which implies πii<0 for all i∈ℕ. Taking mathematical expectation, the above inequality can be rewritten as(38)E£Vxt,i,t+αVxt,i,t−αω¯2ωTtωt<0.
Using Lemma2 in (38), we obtain the inequalities (16)–(22). Based on the inequality in (27) and Lemma 4, for any initial point t0, if Vx0,i,t0≤1, then Vxt,i,t≤1 for any t≥t0. Under the construction of the LKF in (17), we have xTtPixt≤Vxt,i,t≤1, which means that the reachable set ℜx of the considered system (11) is bounded via the ellipsoid εPi. Hence, this proof is finished.
Now, we are going to solve the delay-dependent extended dissipative-based control for Markovian jump time-delayed systems with stochastic actuator failures.Theorem 2.
For any given scalarsd≥0, μ, μl, σjj=1,…,m, and ω¯ and the constant matrices Φ1,Φ2,Φ3, and Φ4, if there exists appropriately dimensioned matrices Xi>0, Q^ij>0, Q^i, R^i>0, S^i, M^i>0, and V>0, symmetric matrices E^i, E^1i, W^i, and Z^i, and a scalar α>0, the following LMIs hold for i,j∈ℕ:(39)Θ˜6×6A˜iTℋ˜iTΛ1iΣ˜T∗−d2V000∗∗−H˜00∗∗∗−ϒ1i0∗∗∗∗−I<0,i∈Tki,(40)Θ˜6×6A˜iTℋ˜iTΛ1iΣ˜T∗−d2V000∗∗−H˜00∗∗∗−ϒ1i0∗∗∗∗−I<0,i∈Tuki,(41)∑j∈Tkiπije−αdQ^ij−Z^i≤0,(42)−W^iXi∗−Xj≤0,j∈Tuki,j≠i,(43)e−αdQ^ij−Z^i≤0,j∈Tuki,j≠i,(44)Xj−W^i≥0,j∈Tuki,j=i,(45)e−αdQ^ij−Z^i≥0,j∈Tuki,j=i,(46)M^iE^i∗M^i≥0,N˜E^1i∗N˜≥0,where(47)Θ˜1,1=2AiXi+BiΞ¯Yi−∑j∈TkiπijW^i+πiiXi+Q^i+S^i+αXi+R^i+dM^i−e−αdN˜,Θ˜1,2=AdiXi+e−αdN˜−E^1i,Θ1,3=e−αdE1i^,Θ˜1,6=XiC1iTΦ2+XiDi,Θ˜2,2=e−αd−1−μS^i+E^1i+E^1iT−2N˜,Θ˜2,3=e−αdN˜−E^1i,Θ˜3,3=−e−αdR^i+Qi^+N˜,Θ˜4,4=−e−αddM^i,Θ˜4,5=−e−αddE^i,Θ˜5,5=−e−αddM^i,Θ˜6,5=C2iTΦ2−αω¯2,A˜i=Ai+BiΞ¯YiAdiXi000Di,ℋ˜i=σ1BiH1Ki⋮σiBiHkYi⋮σmBiHmYimn×n0mn,n0mn,n0mn,n,H˜=diagV,…,V︸m,N˜=V−2Xi,Σ˜=XiC1iTΦ10000XiC2iTΦ1,Λ1i=πik1iXiπik2iXi…πikr−1iXiπikr+1iXi…πikmiXi,ϒ1i=diagXk1i,Xk2i,…,Xkr−1i,Xkr+1i,…,Xkmi,Λ2i=πik1iXiπik2iXi…πikmiXi,Υ2i=diagXk1i,Xk2i,…,Xkmi.Then, the reachable set of the closed-loop system (1) is bounded by the prescribed ellipsoid in εXi, and the desired state-feedback controller can be obtained as Ki=YiXi−1.Proof.
Next, we design a dissipative-based stochastic fault-tolerant controllerKi, such that the reachable set of system (11) is bounded by a given ellipsoid εPi. This theorem is followed from Theorem 1 and the same line as in previous theorem; then, we have(48)£Vxt,i,t+αVxt,i,t−αω¯2ωTtωt−Zt<EξTtΘ^iξt,where(49)Θ^i=ΘiΣT∗−I<0.
Σ=C1iTΦ10000C2iTΦ1, Θ1,6=C1iTΦ2+Di,Θ6,6=C2iTΦ2−α/ω¯2, and the rest of Θi are the same in Theorem 1. Defining Xi=Pi−1 and applying the congruence transformation diagXi,Xi,Xi,Xi,Xi,I,N,Xi, where Xi=N,…,N︸m to (49) and setting Q^i=XiQXi, S^i=XiSXi, R^i=XiRXi, M^i=XiMXi, N−1=V, and Ki=YiXi−1, we obtain the LMIs in Theorem 2. Then, under zero initial condition,(50)£Vxt,i,t+αVxt,i,t−αω¯2ωTtωt−Zt≤0.
Applying Dynkin’s formula, we have(51)∫0TfZsds≥EVxt,i,t+eαTfEVx0,i,0−αω¯2ωTtωt,≥xTtPixt+ρ.
To prove extended dissipative, we consider two casesΦ4=0 and Φ4>0, respectively. First, if Φ4=0, then (51) follows for any Tf≥0 that(52)∫0TfZsds≥0.
This satisfies the definitionΦ4=0. Furthermore, when Φ4>0 as mentioned in Assumption 1, we have the matrices Φ1=0,Φ2=0, and Φ4>0. For any 0≤t≤tf, we have E∫0TfZtdt≥E∫0TfZtdt+ρ. Then, there exist a scalar 0<b<1 s.t. E∫0TfZtdt≥xTtΘ^ixt+ρ. Then, by (51), we have(53)zTtΦ4zt=bxTtC1iTΦ4C1ixt≤bxTtPixt.
It is mentioned that for any0≤t≤Tf,(54)E∫0TfJtdt≥zTtΦ4zt+ρ.
Thus, inequality (54) holds for all Tf≥0. This completes the proof.Remark 1.
It is noted that the condition in equations (39)–(46) are LMIs with respect to all the matrix variables. In order to obtain an ellipsoid with the shortest major principal axis for the reachable set of closed-loop system (11), we can maximise a positive scalar δ with respect to LMIs (39) and(55)Xi−1≥δI.
We use the following approximation of the above inequality:
minimiseδ subject to δIXiXiXi≥0, subject to the LMIs (39)–(46).Remark 2.
It is noted that most of the existing studies based on Markovian jump systems subject to fully known transition probabilities. However, in practice, it is not an easy way to scale all transition probabilities because its cost is very expensive. Therefore, the considered problems based on Markovian jump time-delayed systems with partially known probabilities is more convenient than existing works [13, 24, 29] during the jump process.Remark 3.
It is noted that our considered control model (10) is indeed effective in improving the performances of dynamical time-delay systems. Moreover, compared with the existing feedback reliable control approaches [25–27], our stochastic fault-tolerant controller under actuator saturation achieves better performance when information of the random variable is available. Furthermore, existing fault-tolerant control approaches are only designed under the assumption of actuator fault matrix which is partially failure or fully failure. Nevertheless, in practice, the actuators may not be fully failures under the cases, namely, electromagnetic inferences, nonlinear amplification, and cipher deviation. Based on the above facts, the actuator fault matrix need to satisfy a convinced random circulation in the probability time bound values between zero and one. Hence, it is necessary and important to take such nondeterministic fluctuating term in our proposed method.
Now, consider the following time-delay system:(56)x˙t=Axt+Adxt−dt+Dωt,xt=ϕt,t∈−d,0.
Next, we provide a corollary for finding an ellipsoid to bound the reachable set of the time-delay system (56).Corollary 1.
For some scalarsd≥0, μ, α, and ω¯, if there exist appropriately dimensioned matrices P>0, Q>0, R>0, S>0, M>0, and N>0, any matrices E and E1 such that the following LMIs hold:(57)Γ11Γ12Γ1300PDdATN∗Γ22Γ23000dAdTN∗∗Γ330000∗∗∗Γ44Γ4500∗∗∗∗Γ5500∗∗∗∗∗−αω¯2dDTN∗∗∗∗∗∗−N<0,(58)ME∗M≥0,NE1∗N≥0,where Γ11=2PA+S+αP+Q+dM−e−αdN, Γ12=PAd+e−αdN−E1, Γ13=e−αdE1, Γ22=e−αd−1−μS+E1+E1T−2N, Γ23=e−αdN−E1, Γ33=−e−αdQ+N, Γ44=−e−αd/dM, and Γ45=−e−αd/dE, Γ55=−e−αd/dM; then, the reachable sets of system (56) can be bounded by the ellipsoid εPi.Proof.
Choose the Lyapunov functional candidate as follows:(59)Vt,xt=∑i=15Vit,xt,where(60)V1t,xt=xTtPxt,V2t,xt=∫t−dteαt−sxTsQxsds+∫t−dtteαt−sxTsSxsds,V3t,xt=∫t−dteαt−sd−t+sxTsMxsds,V4t,xt=d∫−d0∫t+θteαt−sx˙TsNx˙sdsdθ.
By following the similar steps as in Theorem1 with small changes, the proof is omitted here.Remark 4.
It is noted that the dissipative performance of this paper consists of some well-known control-based performances. For instance, by settingΨ1=0,Ψ2=0,Ψ3=γ2I, inequality (15) becomes the L2−L∞ performance. If we fix Ψ1=−I,Ψ2=0,Ψ3=γ2I,andΨ4=0, inequality (15) reduces to the H∞ case. By choosing Ψ1=0,Ψ2=I,Ψ3=γI, and Ψ4=0, inequality (15) becomes a passivity-based control. When we take Ψ1=−γ−1θ,Ψ2=1−θ,Ψ3=γθ, and Ψ4=0, inequality (15) corresponds to mixed H∞ and passivity case. Furthermore, when we set Ψ4=0,Ψ1=Q,Ψ2=S,andΨ3=ℛ−mI, inequality (15) changes to the Q,S,ℛ-dissipative performance.


## 4. Numerical Simulation

In this section, numerical examples are provided to demonstrate the feasibility of the approaches proposed in this paper.Example 1.
Consider Markovian jump time-delayed system (11) with four modes and the following parameters:(61)A1=0001,Ad1=−10−10.9,B1=−1−1−1.50,D1=1−0.3,A2=0001.2,Ad2=−1.3−0.90−1.2,B2=−1.2−1.11.30.3,D2=1−0.35,A3=0000.9,Ad3=−1.3−1.10−1.1,B3=−1.1−1.21.10.2,D3=1−0.44,A4=0001.1,Ad4=−0.9−0.90−1.1,B4=−0.9−0.70.90.1,D4=1−0.5.
Output matrices are taken as follows:(62)C11=C13=0.010.15,C12=C14=0.130.12,C21=C23=0.7,C22=C24=0.5.
We assume that the partially known transition rate matrix is given by(63)−1.50.2????0.30.40.6?−1.5???0.4?.
Here, we consider two cases:Case 1.
Let us take that the actuator is normal. GivenΞ=1I, α=0.2, μ=0.5, d=2.6, and ω¯=1 and variances σ1=0, σ2=0, Φ1=0.4I,Φ2=0.6I,Φ3=1.5I,andΦ4=0.3I. The state responses of the open-loop system are given in Figure 1, and it is seen that the open-loop system is unstable. By solving the LMI conditions in Theorem 2 with disturbance input ωt=0.03cost and initial condition x0=−1,1.5, we obtain a set of feasible solutions. The controller gain matrices are displaced as follows:(64)K1=−0.06012.76572.7150−3.2046,K2=−1.0415−3.28473.18513.5214,K3=−0.7894−2.84992.44352.5206,K4=−0.03322.55302.5321−2.9344.
Based on the controller gain matrices, the state trajectories and bounding reachable set of system (11) are shown in Figures 2 and 3, respectively. Moreover, Figure 2 represents the reachable set of system (11) bounded by the ellipsoid εXi−1. Also, it is observed from Figure 3 that the state feedback controller stabilizes system (11) in the presence of partially known transition probabilities.Figure 1
State responses of open-loop system (11).Figure 2
Bounding ellipsoid and reachable set of system (11) via a traditional feedback controller.Figure 3
State responses of system (11) via a traditional feedback controller.Case 2.
Consider the stochastic actuators failures withΞ=0.7I, μ1=0.2, and μ2=0.3 and σ1=0.5 and σ2=0.6. Choose the values α=0.2, μ=0.5, d=2.70, ω¯=1, Φ1=0.4I,Φ2=0.6I,Φ3=1.5I,Φ4=0.3I together with the transition probability matrix given in (47), and solving the LMIs in Theorem 2 with aid of the MATLAB LMI tool box, we can get feasible solutions, which are not given here. Moreover, the corresponding gain matrices are calculated as follows:(65)K1=0.22513.12030.3003−0.0232,K2=0.0825−0.24930.1575−0.0852,K3=0.1208−0.88410.2189−0.1490,K4=0.14800.24500.2295−0.0019.
The obtained ellipsoidεXi−1 and the state trajectory of system (11) with the proposed controller are shown in Figures 4 and 5. It is seen from Figure 4 that the reachable set of system (11) is contained in the ellipsoid εXi−1. Also, it is observed from Figure 5 that when actuator failures occur, system (11) with the proposed reliable controller operates well and maintains the system stability. Next, the reachable set of system (11) is contained in εXi−1 when α=0.25, which is shown in Figure 6. Figure 7 shows the mode transition rates of the closed-loop system (11). Furthermore, in Table 1, we estimate the values of δ¯min for different values of d. From the simulation analysis, it is easy to understand that the considered Markovian jump system (11) is bounded and extended dissipative even in the occurrence of partially known transition probabilities, time delay, stochastic faults, uncertain parameters, and external disturbances.Figure 4
Bounding ellipsoid and reachable set of system (11) with the stochastic actuator fault.Figure 5
State responses of system (11) with stochastic actuator fault.Figure 6
Bounding ellipsoid and reachable set of the system whenα=0.25.Figure 7
Mode transitionsrt.Table 1
Calculated minimumδ for various values of d.
d0.40.81.21.622.4δ¯min9.64308.14217.00035.92793.90141.0217Example 2.
In this numerical result, we consider an electrical circuit as given in Figure6 [29] where iLt and uct represent the currents passing through the inductances and the voltage across the capacitor, respectively. Then, by using the Kirchhoff laws, we obtain for rt=1,2,(66)diLtdt=uLtLrt=u−uct−iLtLrt,ductdt=iLtCrt.
Now, we setx1t=uct and x2t=iLt, and assume that there exists the noise signal wt∈ℝn, which is assumed to be an arbitrary signal in ℒ20,∞, then the electrical circuit of Figure 6 [29] can be arranged as the Markovian jump system with the following parameters:(67)A1=01C1−1L1−RL1,A2=01C2−1L1−RL1,B11=01L1,B12=01L2,where C1=0.5F,C2=0.8F,R=0.01Ω,L1=4H,L2=8H, and the remaining matrices of system (11) are taken as zero. Taken, the other positive values are Ξ=0.6I, μ1=0.3, μ2=0.2 and σ1=0.37 and σ2=0.57, with the transition probability matrix given in (47), and solving the LMIs in Theorem 2 with help of MATLAB LMI tool box, we obtain feasible solutions, and controller gain matrices are displayed as follows:(68)K1=−0.0867−1.4364,K2=−0.4654−2.9528.
We consider the initial conditionϕt=0.5etet/0.3T,t∈−0.9,0. Based on the gain matrices, the state responses and control curve of the closed-loop system (11) are presented in Figure 8. Furthermore, Figure 9 represents the mode transition rates rt of the closed-loop system (11). Under the simulation result, it is concluded that proposed system (11) is stochastically stable and extended dissipative even in the presence of stochastic fault, uncertain parameters, and gain fluctuations. Hence, our proposed controller is an active one to stabilize Markovian jump system (11) with respect to stochastic fault.Figure 8
Simulation results of the closed-loop system (11).
(a)(b)Figure 9
Mode transition ratesrt.


## 5. Conclusion

In this work, we have investigated the reachable set bounding and extended dissipative controller synthesis of time-delayed Markovian jump systems with partially known transition probabilities and stochastic actuator failures and bounded disturbances. A new practical actuator fault model is designed to achieve a less conservative result, not only when the system is operating properly but also in the presence of certain actuator failures. Based on the LKF, reciprocal convex approach, Jensen’s integral inequality, and novel reachable set lemma, a new set of delay-dependent LMI conditions is constructed in the form of LMIs for estimating an ellipsoid to bound the reachable sets of Markovian jump time-delayed systems with disturbances. Then, this result is extended to the control design of considered systems with extended dissipative performance indices. Two numerical simulation examples have been provided to demonstrate the effectiveness of the proposed methods. Then, the problem of the event-triggered control for a class of T-S fuzzy semi-Markovian jump systems with stochastic cyberattacks randomly gain variations, and bounded disturbances are an untreated work, which will be the topic of our future work.



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*Source: 5685324-2020-07-06.xml*