Improved Non-Fragile State Feedback Control for Stochastic Jump Systems With Uncertain Parameters and Mode-Dependent Time-Varying Delays

This paper reports the investigation on non-fragile state feedback control for stochastic Markovian jump systems with uncertain parameters and mode-dependent time-varying delays. The resulted closed-loop system is stochastic stabilization by virtue of an improved L-K functional. By free-weight-matrix technique, the non-fragile state feedback controller is designed and novel conditions for robust stochastic stabilization are acquired in the form of linear matrix inequalities. Two examples including a take-off and landing guidance system are employed to show the effectiveness and validity of the proposed approach.


I. INTRODUCTION
As a very significant type of hybrid systems, Markovian jump systems (MJSs) have been paid more and more concern for many years. MJSs have always been investigated due to which can describe the dynamic systems with abrupt variations, as well as the important significance of practice and application in control and communication fields; see, [1]- [7]. MJSs are also deemed to be a special type of switched systems with continuous time and discrete modes. Jump switching phenomenon among the subsystems of MJSs are based on the modes, which are decided by Markovian process [8]- [13].
The stability analysis and control synthesis for MJSs have got extensive attention for a long time. A mass of satisfactory results have been obtained with the further study of MJSs. For example, by using the stochastic linear copositive Lyapunov function method, [14] studied the stochastic stability and stabilization of autonomous and non-autonomous MJSs, The associate editor coordinating the review of this manuscript and approving it for publication was Zhiguang Feng . respectively; [15] addressed the stability and stabilization for stochastic MJSs with random switching signals in the form of linear matrix inequalities (LMIs), and then extended the results to uncertain and partially unknown transition rate matrices case.
Recently, many researchers focus on MJSs with uncertain parameters, which exist in the state or input. For instance, the passivity of uncertain MJSs was analyzed in [4]; improved delay-dependent stochastic stability for MJSs was proved by applying slack matrix variable method in [16]. In addition, time-delay phenomenon is almost ubiquitous in dynamic systems, which often leads to poor performance and instability of the systems; see, [17]- [22]. Specifically, [1] designed a controller of the MJSs with uncertain parameters and multiple time delays; a new criteria was established to show the stability of uncertain MJSs with polytopic parameter uncertainties and delays in [23]; the issue of stability analysis for uncertain time-delay MJSs was taken into account in [24].
It is worth pointing out that many practical application systems are affected by stochastic disturbance, such as the amplitude and frequency of the power supply voltage, the ambient temperature, humidity and air pressure and the variation of load [25]- [27]. Therefore, it is essential to study the issues related to stochastic systems. The problems of filters for some stochastic systems were considered and some novel stability criteria were obtained in [28]. Beyond that, the topic of controllers of stochastic systems is often mentioned, as pointed out in [29]- [31]. MJSs combined with stochastic phenomenon, which called the stochastic MJSs, were considered in [32], [33]. It is worth mentioning that the H ∞ controller of uncertain stochastic systems was designed by LMI approach in [34].
Noting that feedback control can effectively realize the stabilization of various kinds of dynamic systems [35]- [46]. In fact, there exists quite small uncertainties in controller implementation, which may cause the concerned closed-loop system efficiency reduction even unstabitily. Therefore, so far, numerous results about non-fragile controller designing were reported [47]- [50]. To mention a few, non-fragile controller was discussed for repeated scalar non-linearities time-varying delays systems in [51]; non-fragile stochastic stabilization for uncertain stochastic time-delay systems with time-dependent parameters was solved in [52]; non-fragile stabilization for MJSs were considered in [53]- [57]; nonfragile H ∞ state feedback controller was mentioned for singular fuzzy MJS with a interval delay in [58]. According to our limited knowledge, the issue of non-fragile control for stochastic MJSs with uncertain parameters and mode-dependent time-varying delays is rarely studied, which stimulates the current research. This paper will address the issue of the non-fragile state feedback controller for stochastic MJSs with uncertain parameters and time-varying mode-dependent delays. The parameter uncertainty will be assumed to be norm bounded. Improved L-K functional will be constructed to demonstrate the closed-loop system is robust stochastic stabilization. By free-weight-matrix (FWM) technique, novel sufficient conditions for stochastic stability will be acquired in terms of LMIs. Two examples will be presented to attest the validity of the proposed approach.
The main contributions of this paper are summed up as follows: (1) the mode-dependent and time-varying delays τ r t (t) are simultaneously considered for stochastic MJSs; (2) time-varying delays derivative constraint condition oḟ τ r t (t) ≤ h < 1 is extended toτ r t (t) ≤ h based on FWM technique, thus, the constraint condition is more general than the most existed results; (3) non-fragile controller is considered and improved mode-dependent and delay-dependent L-K functional is constructed in this paper.
Notation. E(·) represents the expectation operator; | · | represents the Euclidean vector norm; Matrix Q > 0(Q ≥ 0) refers to Q is positive definite( positive semidefinite ). * expresses a term that is induced by symmetry in symmetric block matrices or long matrix expressions.

II. PROBLEM FORMULATION AND PRELIMINARIES
We consider the following stochastic MJSs with uncertainties and time-varying mode-dependent delays for giving a probability space ( , F, P): where x(t) ∈ R n is the state; ψ(t) is a initial function; u(t) ∈ R m is the control input; z(t) ∈ R q is control output; ω(t) is a scalar Brownian motion. τ r t (t) denotes the time-varying mode-dependent delay satisfying where scalars τ > 0 and h are known. Continuous time Markovian process r t takes values in a finite set S := {1, 2, . . . , s}, := [π ij ] s×s is the transition rate matrix which satisfies the common conditions in [16], [53], [54], [56], etc. For simplicity, X r t (t) will be expressed as X i (t) for each B 1i (t) and B 2i (t) stand for the uncertain parameters, which are unknown matrices and listed below: where M i , N li , i ∈ S, l = 1, . . . , 6 are known matrices, F i (t) is uncertain matrix function satisfying Now, the mode-dependent non-fragile state feedback controller is designed as: the following form of controller gain perturbations K i (t) is considered: where H 1i and E 1i are known and real matrices. Unknown matrix function F 1i (t) satisfying Remark 1: It is noted that the time-delay τ r t (t) of system (1) is time-varying and mode-dependent, the derivative constraint condition ofτ r t (t) ≤ h < 1 is extended toτ r t (t) ≤ h based on FWM technique. To the best of our knowledge, VOLUME 8, 2020 time-varying delays associated with the modes are seldom taken into account and the derivative constraint condition of delays are usually written asτ (t) ≤ h < 1 in the existed literature. This paper gives a more general condition than the most existed results.

III. MAIN RESULTS
For the sake of simplicity, we define the drift term f (t) of the unforced system as (10) and the diffusion term g(t) as In the sequel, we put forward a stochastic stability criterion for the mentioned system (1).
Theorem 1: Unforced stochastic MJS (1) is stochastically stable, if for all i ∈ S and j = 1, . . . , 6, there exist matrices Proof: We choose the following L-K functional candidate: where The weak infinitesimal generator L of the random process where , x T (s)Qx(s)ds 97678 VOLUME 8, 2020 By the Newton-Leibnitz formula, we have 2ξ It is easy to get from (10): , together with (16)- (18) to (14), which signify that Now, let max |π ij | = η, we educe that (22) From Lemma 1 and lemmas in [26], there exists a matrix and Then, the following inequalities can be hold: Thus, using (20)- (25) to (19): Consequently, where By Schur complement, imply that ϒ i < 0, thus Moreover, there exists scalar γ > 0, then Applying Dynkin's formula, we obtain and which means Thus, by Definition 1, closed-loop system (1) is stochastically stable. Remark 2: Significantly, the improved delay-dependent and mode-dependent Lyapunov-Krasovskii functional are constructed for the designing of the desired controller of the stochastic MJS with the time-varying mode-dependent delays.
We focus on the closed-loop stochastic MJSs (1) with (5). The drift termf (t) and the diffusion termĝ(t) are represented as:f Theorem 2: Stochastic MJSs (1) is stochastically stable, if for all i ∈ S and j = 1, . . . , 6, there exist matrices Q > 0, R > 0, S > 0, Z > 0, P i > 0, T and U , and scalars j > 0, τ > 0 and h, such that: Now, together with (15)- (17) and (37), we find that By (40), (41), we can get the following inequalities: Thus, Consequently, By Schur complement, we acquireΥ i < 0, which means Similar to Theorem 1, the considered system (1) is stochastically stable. Theorem 3: Stochastic MJS (1) is stochastically stable via the non-fragile state feedback controller (5), if for all i ∈ S and j = 1, . . . , 6, there exist matrices Q > 0, R > 0, S > 0, Z > 0, P i > 0, T i and U i , and scalars j > 0, such that the following LMIs are satisfied: Then, the desired controller (5) can be realized by Proof: Along the line of the proof of Theorem 2, we pre-multiply and post-multiply diag(P i , I , I , I ) to˜ i , the inequalities in Theorem 3 are educed via Schur complement, wherẽ Therefore, by Theorem 2, we can easily finish the proof. Remark 3: Noticing that the appropriate dimensions    Figure 1 shows the jump modes of Example 1. Figure 2 depicts the considered system is unstable without VOLUME 8, 2020  state feedback controller. The trajectories of the system state with non-fragile controller (5) are given in Figure 3.

Example 1: System (1) is considered with parameters
Example 2: Consider the vertical take-off and landing helicopter system (VTOLHS). In this practical example, the system parameters are changed due to time-varying environment such as wind speed, so the model belongs to Markovian jump system. This system can be described by (1) and x 1 (t) is the velocity of horizontal, x 2 (t) is velocity of vertical, x 3 (t) is velocity of pitch, x 4 (t) is angle of pitch and ω(t) is airflow interference. Consult the parameters in [56] as follows: where α 32 (i), α 34 (i) and β 21 (i) describe the most significant characteristics impacted by the airspeed changes. The switching is assumed to follow a Markovion process between three modes with the transition rate matrix given as follows:  The trajectories of the VTOLHS state with the non-fragile controller (5) are given in Figure 6. These simulation figures VOLUME 8, 2020 demonstrate that the designed controller can effectively guarantee the stochastic stability.

V. CONCLUSION
This paper has reported the investigation on non-fragile control for stochastic MJSs with uncertain parameters and mode-dependent time-varying delays. The robust stochastic stabilization of closed-loop stochastic MJSs has been realized by virtue of the improved Lyapunov-Krasovskii functional. By FWM technique, the non-fragile state feedback controller has been designed and novel robust stochastic stabilization conditions have been acquired in terms of LMIs. Two examples including the VTOLHS have proved the effectiveness and validity of the proposed approach. In the future works, we will find more effective ways to solve the problem of non-fragile feedback control for fuzzy systems, switching systems, singular systems, etc. His research interests include Markov jump systems, time-delay systems, and feedback control and filtering.