Improved robust H ∞ exponential mean square stabilization for uncertain Markov jump delay systems based on memory-state feedback control

This paper investigates the problem of robust H ∞ exponential mean square stabilisation for uncertain Markov jump systems with time-varying and mode-dependent delays based on memory-state feedback control. Attention is focused on the design of memory-state feedback controller for uncertain Markov jump systems with time-varying and mode-dependent delays such that the closed-loop uncertain Markov jump systems satisﬁes robust H ∞ exponential mean square stability. The improved conditions for the solvability of the memory-state feedback control problem are obtained via designing mode-dependent and delay-dependent L-K functional. The desired memory-state feedback controller is given by using linear matrix inequalities. Two simulation examples including a numerical example and a practical example of the industrial non-isothermal continuous stirred tank reactor are used to demonstrate the effectiveness and usefulness of the delayed feedback control technique that this paper proposes.


INTRODUCTION
In the past many years, the study of Markov jump theory has attracted more and more attention due to the wide range of applications of Markov jump systems (MJSs) in the fields of chemical plants, robots, airplanes, and so on [1][2][3][4][5]. MJSs are a kind of hybrid system, which combine continuous and discrete values and are widely used in many physics and engineering such as network control systems and fault-tolerant control systems, where the environmental changes, system failures and maintenance can be reflected effectively [6][7][8][9][10]. With the rapid development of MJSs, stability analysis and control synthesis have been extensively researched in the past years, and some important results in feedback control and filtering have been obtained [11][12][13][14][15][16][17].
As an inevitable natural phenomenon, time delays often appear in various dynamic systems, which can easily lead to design problem for MJSs with time delays was solved in [39], where delay-independent results were presented in the form of linear matrix inequalities (LMIs).
It is worth noting that exponential stability, mean square stability and mean square asymptotic stability for plenty of dynamic systems have been investigated in [31,40,41]. As the combination of exponential stability and mean square stability, exponential mean square stability (EMSS) has been paid much attention due to its faster convergence rate [1,42]. However, the EMSS has not been fully studied for uncertain delayed MJSs.
On the other hand, feedback control of various dynamic systems can effectively guarantee system stability and improve the system performance [43][44][45][46][47][48][49]. However, in practice, the efficiency of the closed-loop system can be reduced due to time delays. How to overcome time delays has been the focus of attention over the past few decades. Recently, many famous scholars have conducted in-depth research on memory-state feedback control, where the delayed state information was considered in the controller design [50,51]. Yet, as far as we know, robust H ∞ EMSS problem and memory-state feedback control for MJSs have not been adequately studied, which are full of challenging due to EMSS index and time-varying delay characters in feedback controller.
The intervention of exponential function for EMSS and mode-dependent time-varying delay makes the solution to weak infinitesimal operator intricate [1,42]. In addition, the memory-state feedback controller itself contains modedependent time-varying delays, which makes the solution even more complicated [50,51]. How to deal with robust H ∞ EMSS for delayed MJS based on memory-state feedback control deserves further study. This is our research motivation in this work.
In this work, robust H ∞ exponential mean square stabilisation for UMJSs with mode-dependent and time-varying delays based on memory-state feedback control will be investigated. The design of the memory-state feedback controller and the robust H ∞ stabilisation problem will be the focus of this paper, and the closed-loop system will be EMSS under all allowable uncertainties, mode-dependent time-varying delays. Apart from the above requirements, H ∞ performance level will be satisfied. Furthermore, the improved conditions of solvable problems can be obtained by using LMIs. In the end, a numerical example and the industrial non-isothermal continuous stirred tank reactor (INCSTR) [52] will be employed to demonstrate the effectiveness and usefulness of the method that this work brings up. The main contributions of this paper are summarised as follows: (1) The mode-independent and mode-dependent time-varying delays are considered both in the memory-state feedback controller and in the UMJS; (2) On the premise of achieving EMSS of UMJS, robust H ∞ exponential mean-square stabilisation for closed-loop UMJS is achieved based on a memory-state feedback controller; (3) The mode-dependent memory-state feedback control can be extended to asynchronous feedback control and can be reduced to the special memoryless-state feedback control via the proposed method in this work.

SYSTEM DESCRIPTION
Given a completed probability space (Ω, , ), we consider the following UMJS with time-varying delays: where In the system (1), for r t = i ∈ , the time-varying delays (t ) and i (t ) satisfying where , s i and s are given scalars. A(r t ), A a (r t ), A d (r t ), B(r t ), D(r t ), E (r t ) are known real constant matrices with compatible dimensions. A(t, r t ), A a (t, r t ), A d (t, r t ), B(t, r t ), D(t, r t ) are rewritten as △A(t, r t ), △A a (t, r t ), △A d (t, r t ), △B(t, r t ), △D(t, r t ) represent unknown matrices with time-varying parameter uncertainty, where then G (r t ), M 1 (r t ), M 2 (r t ), M 3 (r t ), M 4 (r t ), M 5 (r t ) are known real constant matrices for all r t ∈ , and F (t, r t ) are the uncertain time-varying matrices satisfying Now, the memory-state (time delays) feedback controller is designed as follows: where K 1 (r t ), K 2 (r t ), K 3 (r t ) ∈ ℝ p×n are the parameters of the memory-state feedback controller with the appropriate dimension to be determined. In order to facilitate the following research, for each r t = i ∈ , A(t, r t ) is rewritten as A i (t ) etc.

Remark 1.
Feedback control of various dynamic systems can effectively guarantee system stability and improve the system performance [42][43][44][45][46][47][48][49]. However, in practice, due to time delays, the efficiency of the closed-loop system can be reduced. How to overcome the delays has been paid much attention during the past decades. The state feedback control can be a better and more effective control strategy to make the system work steadily and normally. Recently, memory-state feedback control has received much attention, where the delayed state information is considered in the controller design [50,51]. In this work, the mode-independent and mode-dependent time-varying delays are considered both in the memory-state feedback controller and in the UMJS.
In order to effectively design the mode-dependent memorystate feedback controller, the following definitions are given firstly.

MAIN RESULTS
First of all, we consider the following UMJS with modeindependent and mode-dependent time-varying delays as follows: Theorem 1. The unforced MJS (6) achieves EMSS and satisfies robust H ∞ disturbance attenuation if there exist matrices P 1 > 0, P 2 > 0, … , P  > 0, Q > 0 and Q i > 0 such that the following matrix inequalities hold for i=1, 2 ,\ldots ,  : where Proof. First, set a new process {(x t , r t ), t ≥ 0} by Define a L-K functional candidate for the unforced MJS (6) as: Set £ be the weak infinitesimal generator of the stochastic process {x t , r t }. Then, for each r t = i ∈ S and any scalar u > 0, we have Noting that i j ≥ 0, for each j ≠ i and ii ≤ 0, we have □ Observing (8), it can be shown that Noting that and observing (2), (12) and (13), for each r t = i ∈ S , we can deduce that and we can get Using the Schur complement formula to (17), we have where Then, there exists constant > 0 such that From the above, we can get and By using Dynkin's formula, for any T > 0, > 0, we have And noting that therefore, for any scalar > 0, we have Choose a scalar > 0 such that then Noting that we have Hence, the unforced MJS (6) achieves EMSS.
Then, for (t ) ≠ 0 ∈  2 [0, ∞) and the zero initial condition, we have Observing (10) and by calculating, we can get Observing Γ i < 0, we can find that the MJS (6) with modedependent time-varying delays satisfies robust H ∞ exponential mean square stabilisation with H ∞ disturbance attenuation . Now, we consider the UMJS (1) with mode-dependent and time-varying delays based on memory-state feedback con-troller (3). To this end, the closed-loop UMJS is descried as follows: (8) and the following matrix inequalities hold for i=1, 2 ,\ldots ,  :

Theorem 2. The closed-loop UMJS (37) with mode-dependent timevarying delays and memory-state feedback controller (3) achieves EMSS and robust H
where , Proof. Using the same L-K functional as Theorem 1 and the similar proof method of Theorem 1, we can get Applying the Schur complement to (40), we can get where Then, there exists constant̂> 0 such that Thus, where sup □ Hence, closed-loop UMJS (37) with mode-dependent timevarying delays and memory-state feedback controller (3) satisfies robust exponential mean square stabilisation.

Theorem 3. The closed-loop UMJS (37) with mode-dependent time-varying delays and memory-state feedback controller (3) achieves
EMSS and H ∞ performance if there exist matrices N 1 > 0, N 2 > 0, … , N  > 0,Q > 0,Q i > 0 and scalars > 0, k > 0 such that (8) and the following LMIs hold for i=1, 2 ,\ldots ,  : Then, the appropriate memory-state feedback controller can be realised by Proof. Noting that △A(t, r t ), △A a (t, r t ), △A d (t, r t ), △B(t, r t ), △D(t, r t ) are unknown matrices representing time-varying parameter uncertainties, there exist > 0 and k > 0, such that and then, (38) in Theorem 2 can be rewritten aŝ □ And then, define wherẽ Pre-and post-multiplying (52) Then, let and applying Schur complement formula to (53), we can acquire Ξ i < 0, then Theorem 3 is satisfied based on Theorem 2. The desired memory-delayed feedback controller is given by:  (48).
Based on the inequality relation in (54), the conditions of Theorem 3 are sufficiently guaranteed to be LMIs, which enable us to obtain the expected memory-state feedback controller.

Remark 3. Robust H ∞ EMSS of open-loop MJSs with time-varying delays is acquired firstly. Then, on the premise of achieving EMSS, robust H ∞ exponential mean-square stabilisation for closed-loop UMJS (37) is achieved based on memory-state feedback controller.
It should be pointed out that the computation is complicated in the proposed approach. When the size of LMI gets bigger, the calculation process will be more complicated and the solvability of the LMIs will be affected. Therefore, the complicated positive definite matrices are mentioned in this work, and the strict LMIs have been employed to improve the accuracy of numerical calculations.

Remark 4. Theorem 1 illustrates that the unforced MJS achieves EMSS and satisfies robust H ∞ disturbance attenuation via linear matrix inequality; Theorem 2 shows that the closed-loop UMJS with mode-dependent time-varying delays and memory-state feedback controller achieves EMSS and robust H ∞ performance via matrix inequalities; Theorem 3 states that the closed-loop UMJS with mode-dependent timevarying delays and memory-state feedback controller achieves EMSS and H ∞ performance via linear matrix inequality. Furthermore, we construct and implement the expected memory-state feedback controller in Theorem 3.
Note the controller (3) can be simplified as Here, we give the first two forms of state feedback controller based on Theorem 3. The corresponding corollaries are as follows: (37) with mode-dependent timevarying delays and memory-state feedback controller (56) achieves exponential mean square stabilisation and H ∞ performance if there exist matrices N 1 > 0, N 2 > 0, … , N  > 0,Q > 0,Q i > 0 and scalars > 0, k > 0 such that (8) and the following LMIs hold for i = 1, 2, …,

Corollary 1. The closed-loop UMJS
In this case, the appropriate memory-state feedback controller can be realised by Corollary 2. The closed-loop UMJS (37) with mode-dependent timevarying delays and memory-state feedback controller (57) achieves exponential mean square stabilisation and H ∞ performance if there exist matrices N 1 > 0, N 2 > 0, … , N  > 0,Q > 0,Q i > 0 and scalars > 0 and k > 0 such that (8) and the following LMIs hold for i=1, 2 ,\ldots ,  :Ξ In this case, the appropriate memory-state feedback controller can be realised by    H ∞ performance indices of memory-state feedback control (Theorem 3 ) are compared with that of memoryless-state feedback control (Corollary 1) in Table 1: [52], where the reactor parameters will change because of the internal and external environment, so the reactor can be considered as an UMJS. Let x 1 (t ) and x 2 (t ) represent the reactant and reaction temperature, respectively. External interference Time (s) (t ) = [ 1 (t ) 2 (t )] T represents the inlet temperature,  (t ) represents the cooling medium, and the INCSTR can be represented by UMJS (1) with three modes and the following parameters:    (48) and (59), we obtain H ∞ disturbance attenuation = 5.3848 and the following parameters:  Figure 4 depicts closed-loop UMJS state x(t ) with memory-state feedback controller containing jump modes, Figure 5 describes closed-loop UMJS state x(t ) with memoryless-state feedback controller containing jump modes and Figure 6 shows estimated output z(t ) containing jump modes.
H ∞ performance indices of memory-state feedback control (Theorem 3) are compared with that of memoryless-state feedback control (Corollary 1) in

CONCLUSION
This paper has investigated the problem of robust H ∞ EMSS for UMJSs with time-varying and mode-dependent delays based on memory-state feedback control. Attention has been focused on the design of memory-state feedback controller for UMJSs with time-varying and mode-dependent delays such that the closed-loop UMJS satisfies robust H ∞ EMSS. The improved conditions for the solvability of the memory-state feedback control problem has been obtained via designing mode-dependent and delay-dependent L-K functional. The desired memory-state feedback controller has been given by using LMIs. Two simulation examples including the INCSTR have been used to demonstrate the effectiveness and usefulness of the delayed feedback control technique that this work proposes.
In the future work, we will deeply research the asynchronous tracking and non-fragile memory-state feedback control for delayed UMJSs based on sampled-data and event-triggered mechanisms. The aforesaid recommendations leave a good prospect for future research.