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

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.


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][2][3][4][5].
erefore, 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][7][8][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][14][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, peakto-peak gain minimization, and control synthesis. erefore, 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][17][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 closedloop dynamical system. erefore, from a safety as well as performance point of view, it is required to design a faulttolerant controller, which can tolerate actuator or sensor faults and guarantee the stability and performance of resulting closed-loop systems [22][23][24][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, L 2 − L ∞ performance, and (Q, S, R)-dissipative performance can be obtained by adjusting weighting matrices of extended dissipative [29][30][31]. Very recently, the concept of extended dissipative has been effectively applied to several dynamical systems in reports [32][33][34]. Zhu et.al. [35] reported the problem of finite-time extended dissipativebased 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 timedelayed systems subjected to probabilistic faults yet in the existing literatures, which motivates our present study. e main notable contributions of this paper can be highlighted as follows: (i) e 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 timedelayed system bounded by an ellipsoid and satisfies the extended dissipative performance index (iv) Two numerical simulation demonstrate the feasibility and effectiveness of the proposed approach e remaining part of this work is constructed as follows. e system description and problem formulation are provided in Section 2. e main results on extended dissipativebased 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.
e superscripts "T" and "(− 1)" stand for matrix transposition and matrix inverse, respectively; R n and R n×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.

Problem Formulation and Preliminaries
Consider the following Markovian jump systems with timevarying delay defined on a complete probability space (Ω, F, P) and described as follows: where x(t) ∈ R n is the state vector; u F (r t )(t) ∈ R n is the control input with respect to actuator fault; ω(t) ∈ R m is the disturbance; z(t) represents system output; ϕ(t) is the initial condition of the state; and r t , 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: where Δ(t) > 0 and lim Δ(t)⟶0 O(Δ(t))/Δ(t) � 0; π ij ≥ 0 is the transition rate from mode i at time t to mode j at time t + Δ(t) if i ≠ j and π ii � − N j�1, i ≠ j π ij ; the matrices A(r t ), A d (r t ), B(r t ), D(r t ), C 1 (r t ), and C 2 (r t ) are known matrices of compatible dimensions. e set N comprises the various operation modes of system (1), and for each possible value of 2 Mathematical Problems in Engineering r t � i ∈ N, the system matrices of the ith mode are denoted by A i , A di , B i D i , C 1i , and C 2i . Further, the time-varying delay where d and μ are the given constants. e disturbance term ω(t) is assumed to be bounded: where ω is a positive scalar.
where "?" represents the unknown transition rate. For the simplicity of presentation, we introduce the following no- e reachable set of system (1) is denoted as follows: and ω(t) satisy (1) and (4) . (7) For a positive-definite symmetric matrix P i > 0, we define an ellipsoid ε(P i ) bounding the reachable set (7) as follows: We consider the following static feedback controller for system (1): in which K i represent the controller gain matrices to be determined. en, the following stochastic fault model is adopted for this study: where Ξ � diag ξ 1 , . . . , ξ m with ξ j (j � 1, . . . , m) are the m unrelated random variables. It is assumed that ξ j is with mathematical expectation μ j and variance σ 2 j , respectively, Combining (1) and (10), we obtain the following closedloop systems: Our main aim of this study is to derive an ellipsoid ζ, as small as possible, that bounds the reachable set R x of system (1) subject to time-varying delays and bounded disturbances (4). en, 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 ε(P i ). Before proving our main results, the following lemmas, assumption, and definition are introduced.

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 scalars
and a scalar α > 0, the following LMIs hold for i ∈ N: where en, the reachable set of system (11) can be bounded by the ellipsoid ε(P i ).
Proof. We consider the following Lyapunov-Krasovskii functional candidate for systems (11) as where Define the weak infinitesimal operator E for the Lyapunov functional together with Markovian jump system (11) as Furthermore, we have

Mathematical Problems in Engineering
Considering the circumstance that the information of transition probabilities is not completely known, for any arbitrary matrices W i � W T i and Z i � Z T i , the following zero equations hold due to N j�1 π ij � 0: Using Lemmas 1 and 3 to the integral terms in equations (30) and (31), we have where the matrices M and E satisfy where the matrices N and E 1 satisfy Combining (22) to (35), we have Furthermore, it is noted that π ii � − N j�1 π ij and π ij ≥ 0 for all j ≠ i, which implies π ii < 0 for all i ∈ N. Taking mathematical expectation, the above inequality can be rewritten as Using Lemma 2 in (38), we obtain the inequalities (16)- (22). Based on the inequality in (27) and Lemma 4, for any initial point t 0 , if V(x 0 , i, t 0 ) ≤ 1, then V(x t , i, t) ≤ 1 for Mathematical Problems in Engineering any t ≥ t 0 . Under the construction of the LKF in (17), we have x T (t)P i x(t) ≤ V(x t , i, t) ≤ 1, which means that the reachable set R x of the considered system (11) is bounded via the ellipsoid ε(P i ). Hence, this proof is finished. Now, we are going to solve the delay-dependent extended dissipative-based control for Markovian jump timedelayed systems with stochastic actuator failures. □ Theorem 2. For any given scalars d ≥ 0, μ, μ l , σ j (j � 1, . . . , m), and ω and the constant matrices and Z i , and a scalar α > 0, the following LMIs hold for i, j ∈ N: where (47) en, the reachable set of the closed-loop system (1) is bounded by the prescribed ellipsoid in ε(X i ), and the desired state-feedback controller can be obtained as Proof. Next, we design a dissipative-based stochastic faulttolerant controller K i , such that the reachable set of system 8 Mathematical Problems in Engineering (11) is bounded by a given ellipsoid ε(P i ). is theorem is followed from eorem 1 and the same line as in previous theorem; then, we have where 2i Φ 2 − α/ω 2 , and the rest of Θ i are the same in eorem 1. Defining X i � P − 1 i and applying the congruence Applying Dynkin's formula, we have To prove extended dissipative, we consider two cases ‖Φ 4 ‖ � 0 and ‖Φ 4 ‖ > 0, respectively. First, if ‖Φ 4 ‖ � 0, then (51) follows for any T f ≥ 0 that is satisfies the definition ‖Φ 4 ‖ � 0. Furthermore, when ‖Φ 4 ‖ > 0 as mentioned in Assumption 1, we have the ma- It is mentioned that for any 0 ≤ t ≤ T f , us, inequality (54) holds for all T f ≥ 0. is 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 We use the following approximation of the above inequality: minimise δ subject to δI X i X i X i ≥ 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. erefore, 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][26][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: Next, we provide a corollary for finding an ellipsoid to bound the reachable set of the time-delay system (56).

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: 10 Mathematical Problems in Engineering Output matrices are taken as follows: C 11 � C 13 � 0.01 0.15 , C 12 � C 14 � 0.13 0.12 , We assume that the partially known transition rate matrix is given by Here, we consider two cases: Case 1. Let us take that the actuator is normal. Given Ξ � 1I, α � 0.2, μ � 0.5, d � 2.6, and ω � 1 and variances σ 1 � 0, 3I. e 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 eorem 2 with disturbance input ω(t) � 0.03 cos(t) and initial condition x(0) � [− 1, 1.5], we obtain a set of feasible solutions. e controller gain matrices are displaced as follows: 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 ε(X − 1 i ). Also, it is observed from Figure 3 that the state feedback controller stabilizes system (11) in the presence of partially known transition probabilities.  we can get feasible solutions, which are not given here. Moreover, the corresponding gain matrices are calculated as follows:  Figure 4 that the reachable set of system (11) is contained in the ellipsoid ε(X − 1 i ). 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 ε(X − 1 i ) when α � 0.25, which is shown in Figure 6. Figure 7 shows the mode transition rates of the closedloop 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.
Example 2. In this numerical result, we consider an electrical circuit as given in Figure 6 [29] where i L (t) and u c (t) represent the currents passing through the inductances and the voltage across the capacitor, respectively. en, by using the Kirchhoff laws, we obtain for r(t) � 1, 2, .

(66)
Now, we set x 1 (t) � u c (t) and x 2 (t) � i L (t), and assume that there exists the noise signal w(t) ∈ R n , which is assumed to be an arbitrary signal in L 2 [0, ∞), then the electrical circuit of Figure 6 [29] can be arranged as the Markovian jump system with the following parameters: where C 1 � 0.5F, C 2 � 0.8F, R � 0.01Ω, L 1 � 4H, L 2 � 8H, and the remaining matrices of system (11) are taken as zero. Taken, the other positive values are Ξ � 0.6I, μ 1 � 0.3, μ 2 � 0.2 and σ 1 � 0.37 and σ 2 � 0.57, with the transition probability matrix given in (47), and solving the LMIs in eorem 2 with help of MATLAB LMI tool box, we obtain feasible solutions, and controller gain matrices are displayed as follows: We consider the initial condition ϕ(t) � 0.5e t e t/0.3 T , t ∈ [− 0.9, 0]. Based on the gain    matrices, the state responses and control curve of the closedloop system (11) are presented in Figure 8. Furthermore, Figure 9 represents the mode transition rates r(t) 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.

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. en, 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.
en, 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.

Data Availability
e data used to support the findings of this study are included within the article.