Finite-Time Control for Markovian Jump Systems with Polytopic Uncertain Transition Description and Actuator Saturation

and Applied Analysis 3 0 5 10 15 20 25 30 −0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 t (s) Sy ste m st at ex 1 Figure 2: Response of the system state x 1 . 0 5 10 15 20 25 30 0.2


Introduction
In the aspect of modeling practical systems with abrupt random changes, such as manufacturing system, telecommunication, and economic systems, MJS have powerful ability.MJS have been extensively studied during the past decades and many systematic results have been obtained [1][2][3].The peak-to-peak filtering problem was studied for a class of Markov jump systems with uncertain parameters in [4].A robust  2 state feedback controller for continuoustime Markov jump linear systems subject to polytopic-type parameter uncertainty was designed in [5].In [6], the authors address the stabilization problem for single-input Markov jump linear systems via mode-dependent quantized state feedback for control.
Actuator saturation which can lead to poor performance of the closed-loop system is another active research area.In practical situations, it may be encountered sometimes.How to preserve the closed-loop system performance in the case of actuator saturation would be more meaningful.In [7], the  ∞ control problem for discrete-time singular Markov jump systems with actuator saturation was considered.In [8] the stochastic stabilization problem for a class of Markov jump linear systems subject to actuator saturation was considered.
In some practical applications, the behavior of the system over a finite-time interval is mainly considered.Finite-time stable (FTS) and Lyapunov asymptotic stability are independent concepts.The concept of FTS was first introduced in [9].A system is said to be finite-time stable if, given a bound on the initial condition, its state does not exceed a certain threshold during a specified time interval.FTS of linear timevarying systems was considered in [10].Sufficient conditions for the solvability of both the state and the output feedback problems are stated.Amato [11] provided a necessary and sufficient condition for the FTS of linear-varying systems with jumps.Recently, robust finite-time  ∞ control of jump systems was dealt with in [12][13][14].In [15], the problems of finite-time stability analysis were investigated for a class of Markovian switching stochastic systems.To the best of authors' knowledge, however, the problem of finite-time  2 - ∞ performance for discrete-time MJS with imprecise transition probabilities and time-varying delays has not been well addressed, which motivates our work.
This paper deals with this problem.More specifically, the actuator is saturation.By using the Lyapunov-Krasovskii functional, a new sufficient condition for stochastic asymptotic stability with finite-time  2 - ∞ performance is derived in terms of LMI.Based on this, the existence condition of the desired performance which guarantees finite-time stability and an  2 - ∞ performance of the MJS is presented.A numerical example is provided to show the effectiveness of the proposed results.
Throughout the paper, if not explicitly stated, matrices are assumed to have compatible dimensions.The notation  > (≥, <, ≤)0 is used to denote a symmetric positive definite (positive semidefinite, negative, negative semidefinite) matrix. min (⋅) and  max (⋅) represent the minimum and maximum eigenvalues of the corresponding matrix, respectively. is the identity matrix with compatible dimensions.‖ ⋅ ‖ refers to the Euclidean norm of vectors and [⋅] stands for the mathematical expectation.For a symmetric block matrix, " * " is used as an ellipsis for the terms that are obtained by symmetry.
In general, FTB and FTS are different.If there is external disturbance in systems, the concept of FTB is used.Conversely, FTS is addressed.
The objective of this paper is to design a delayed feedback controller which satisfies the given attenuation level of system (1).The design procedure is given in the next section.

Main Results
In this section, firstly stochastic FTB analysis of nominal time-delay MJS (1) is provided.Then, these results will be extended to the MJS (1) with actuator saturation and uncertain transition probability.LMI conditions are established.
In the second part, stochastic FTB is established: On the other hand, From Definition 2, we have By ( 23) and (25), we know This completes the proof.
Subsequently, to establish the energy-to-peak performance for the system (1), assume that the initial values for the plant are zeros and consider the following function: For any nonzero   ∈  2 [0, ∞) and  > 0, it follows from (18) that It follows from (27) that {()} < ∑ −1 =0      .
For all the time instants  > 0, the expectation of the output can be evaluated as (31) Applying Definition 4, the statement of Theorem 6 is true.

Numeral Example
To illustrate the proposed results, a numerical example is considered for finite-time  2 - ∞ control.The system is described by (1) and assumed to have two modes; Ω = {1, 2}.The mode switching is governed by a Markov chain that has the following transition probability matrix: The system matrices are as follows:

Conclusion
The problem of finite-time  2 - ∞ control for MJS has been studied.By using the Lyapunov functional approach, a sufficient condition is derived such that the closed-loop MJS are stochastic FTB and satisfy the given level.The controller can be obtained by using the exiting LMI optimization techniques.Finally, numerical and simulation results demonstrate the effectiveness of the results of the paper.

Assume 𝐿 2 -
∞ performance of level  = 0.3; by applying Theorem 7, we can explicitly compute the optimally achievable closed-loop  2 - ∞ performance  from Theorem 7 as  = 0.2056.Response of the system state is depicted in Figures2 and 3.

Theorem 7 .
Consider the uncertain time-delay system (1); there exists a state feedback controller (  ()) such that the uncertain time-delay system (1) is finite-time  2 - ∞ control with respect to (ℎ 1 ℎ 2    ), if the following LMIs hold: