Robust Finite-Time 𝐻 ∞ Control for Nonlinear Markovian Jump Systems with Time Delay under Partially Known Transition Probabilities

This paper is concerned with the problem of robust finite-time 𝐻 ∞ control for a class of nonlinear Markovian jump systems with time delay under partially known transition probabilities. Firstly, for the nominal nonlinear Markovian jump systems, sufficient conditions are proposed to ensure finite-time boundedness, 𝐻 ∞ finite-time boundedness, and finite-time 𝐻 ∞ state feedback stabilization,respectively.Then,arobustfinite-time 𝐻 ∞ statefeedbackcontrollerisdesigned,which,foralladmissibleuncertainties, guarantees the 𝐻 ∞ finite-time boundedness of the corresponding closed-loop system. All the conditions are presented in terms of strict linear matrix inequalities. Finally a numerical example is provided to demonstrate the effectiveness of all the results.


Introduction
Markovian jump systems, a class of hybrid dynamical systems, which consists of an indexed family of continuous or discrete-time subsystems and a set of Markovian chain that orchestrates the switching between them at stochastic time instants, have received extensive attention over the past few decades [1,2]. Many real world processes, such as economic systems [3], manufacturing systems [4], electric power systems [5], and communication systems [6], may be modeled as Markovian jump systems when any malfunction of sensors or actuators cause a jump behavior in process performance. Recently, nonlinear Markovian jump systems have been extensively applied and developed in various disciplines of science and engineering, and a great number of excellent works have been developed [7][8][9].
Generally speaking, the behavior of nonlinear Markovian jump systems is determined by the transition probabilities in the jumping process. Usually, it is assumed that the information on transition probabilities was completely known. However, transition probabilities may be partially known for some real systems. For example, the networked control systems can be modeled by nonlinear Markovian jump systems with partially known transition probabilities when the packet dropouts or channel delays occur [10]. In addition, there are few results about the known bounds of transition probability rates or the fixed connection weighting matrices [11,12]. Therefore, it is reasonable to study Markovian jump systems with partially known transition probabilities, especially, when it is difficult to measure the bounds of transition probability rates. It stimulates the research interests of the author.
Uncertainties and time delay frequently occur in various engineering systems, which usually is a source of instability and often causes undesirable performance and even makes the system out of control [14,15]. Therefore, time delay systems with robustness have received an increasing attention among the control community [16][17][18]. On the other hand, one may be interested in not only system stability but also a bound of system trajectories over a fixed short time [19]. For instance, for the problem of robot arm control [7], when the robot works under different environmental conditions with changing payloads, it requests that the angle position of the arm should not exceed some threshold in a prescribed time interval. Meanwhile, the scholars attach more importance to the ∞ control problem, which is to find a stable controller such that the disturbance attenuation level is below a prescribed level. There are a great number of useful and interesting results about ∞ control problem for linear and nonlinear Markovian jump systems in the literature [20][21][22][23][24][25]. To the best of our knowledge, the synthesis issue of 2 Abstract and Applied Analysis robust finite-time ∞ control for nonlinear Markovian jump systems with time delay under partially known transition probabilities has not been fully investigated until now, which motivates us to carry out the present study.
In this paper, we investigate the problem of robust finitetime ∞ control for nonlinear Markovian jump systems with time delay under partially known transition probabilities. The main contributions lie in the fact that some tractable sufficient conditions are provided to ensure ∞ finite-time boundedness or finite-time ∞ state feedback stabilization. A robust finite-time ∞ state feedback controller is designed, which guarantees the ∞ finite-time boundedness of the closed-loop system. Seeking computational convenience, all the conditions are cast in the format of linear matrix inequalities. Finally, a numerical example is provided to demonstrate the effectiveness of the main results.
Notations. Throughout this paper, the notations used are fairly standard. For real symmetric matrices and , the notation ≥ (resp., > ) means that the matrix -is positive semi-definite (resp., positive definite). represents the transpose matrix of , and −1 represents the inverse matrix of . max ( ) ( min ( )) is the maximum (resp., minimum) eigenvalue of a matrix . diag{ } represents the block diagonal matrix of and . is the unit matrix with appropriate dimensions, and the term of symmetry is stated by the asterisk * in a matrix. R stands for the -dimensional Euclidean space, R × is the set of all × real matrices, and M = {1, 2, . . . , } means a set of positive numbers. ‖ * ‖ denotes the Euclidean norm of vectors. E{⋅} denotes the mathematical expectation of the stochastic process or vector. 2 [0, +∞) is the space of -dimensional square integrable function vector over [0, +∞).

Problem Formulation and Preliminaries
Give a probability space (Ω, F, P), where Ω is the sample space, F is the algebra of events, and P is the probability measure defined on F. The random process { , ≥ 0} is a Markovian stochastic process taking values in a finite set M = {1, 2, . . . , } with the transition probability rate matrix Π = { }, , ∈ M, and the transition probability from mode at time to mode at time + Δ is expressed as with the transition probability rates ≥ 0, for , ∈ M, ̸ = , and ∑ =1, ̸ = = − , where Δ > 0, and lim Δ → 0 ( (Δ )/Δ ) = 0.
Consider the following nonlinear Markovian jump system with time delay in the probability space (Ω, F, P): where ( ) ∈ R is the state vector, ( ) ∈ R is the control input, ( ) ∈ 2 [0, +∞) is an arbitrary external disturbance, ( ) ∈ R is the control output, ( ) represents a vectorvalued initial function, and ∈ R + is the constant delay.
(⋅, ⋅, ⋅): M × R × R → R is an unknown nonlinear function. ( ), ( ), ( ), ( ), ( ), ( ), ( ), and ( ) are known mode-dependent constant matrices with appropriate dimensions. Δ ( ), Δ ( ), and Δ ( ) are unknown matrices, denoting the uncertainties in the system, and the uncertainties are time-varying but norm bounded uncertainties satisfying Consider the following state feedback controller: where ( ) and ( ) are the state feedback gains to be designed. Then the closed-loop system is as follows:   In addition, the transition probability rates are considered to be partially known; that is, some elements in matrix Π = { } are unknown. For instance, for system (2) with four subsystems, the transition probability rate matrix Π may be as 11 12 ? ? ? ? 23 24 31 ? 33 ? 41 ? ? ?
where "?" represents the unknown transition probability rate. ∀ ∈ M, we denote M = ∪ , and Moreover, if ̸ = 0, it is further described as where ∈ M represents the th known transition probability rate of the set in the th row of the transition probability rate matrix Π.

Remark 1. When
= 0, = M, it is reduced to the case where the transition probability rates of the Markovian jump process { , ≥ 0} are completely known. When = 0, = M, it means that the transition probability rates of the Markovian jump process { , ≥ 0} are completely unknown. Mixing the above two aspects, here, a general form is considered.
In this paper, the following assumptions, definitions, and lemmas play an important role in our later development.
Definition 8 (see [26]). In the Euclidean space {R ×M×R + }, introduce the stochastic Lyapunov function for system (2) as ( ( ), ), and the weak infinitesimal operator satisfies Remark 9. It easily follows from (12) that 11 ≥ 0, 22 ≥ 0. So 11 and 22 can be decomposed as Remark 10. It is noticed that finite-time stability can be regarded as a particular case of finite-time boundedness by setting ( ) = 0. That is, finite-time boundedness implies finite-time stability, but the converse is not true.
Lemma 11 (see [27]). Let , , , and be real matrices of appropriate dimensions with ≤ ; then for a positive scalar > 0, there holds: The aim in this paper is to find a tractable solution to the problem of finite-time ∞ state feedback stabilization.

Finite-Time Boundedness Analysis.
In this subsection, we will consider the problem of finite-time boundedness for the nominal system of nonlinear Markovian jump system (2) with ( , ) = 0 for all ≥ 0; that is, Under the controller (5), the closed-loop system iṡ Theorem 12. Given > 0, if there exist positive constants and , symmetric positive definite matrices ∈ R × , ∈ R × and ∈ R × , and symmetric matrices then system (18) ( = 0) under partially known transition probabilities is finite-time bounded with respect to ( 1 , 2 , , , ), where Proof. For system (18) ( = 0), choose a Lyapunov function candidate where > 0. Then by Definition 8, we get Based on Lemma 11, there exist scalars such that Substituting (27) into (26) yields It is easy to obtain that Abstract and Applied Analysis 5 From (28) and (29), the following holds: ( ) Due to the fact that ∑ =1 = 0 for arbitrary symmetric matrices , (30) can be written as Noticing that ≥ 0 for all ̸ = and for all ∈ M, if ∈ (the elements of the diagonal are known), by inequalities (20) and (21), the following inequalities hold: If ∈ (the elements of the diagonal are unknown), according to the inequalities (20)- (22), inequality (32) holds. Multiplying (32) by − yields Applying Dynkin's formula for (33), we obtain which shows This together with̃= Considering that and combining (36) and (37), it follows that The proof is complete.

6
Abstract and Applied Analysis Corollary 13. Given > 0, if there exist positive constants , , and , symmetric positive definite matrices ∈ R × , and ∈ R × , and symmetric matrices then system (18) ( = 0) under partially known transition probabilities is finite-time bounded with respect to ( 1 , 2 , , , ), where 3.2. Finite-Time ∞ Performance Analysis. In this subsection, based on Corollary 13, some sufficient conditions will be provided ensuring the ∞ finite-time boundedness of system (18) and the ∞ finite-time stabilization of system (19). ∈ R × and ∈ R × , and symmetric matrices where This together with (49) for any symmetric matrices .
Abstract and Applied Analysis Under zero initial condition, using Dynkin's formula yields Further, it implies that Therefore expression (14) holds with = √ . The proof is complete.
It is clear that (54) is a nonlinear matrix inequality due to the existence of the nonlinear terms , , , and . In order to solve the desired controller , we give the following result. ( 1 , 2 , , , ), if there exist positive scalars , , , 1 , and 2 , symmetric positive definite matrices ∈ R × , symmetric matrices W ∈ R × , and matrices ∈ R × and
Proof. It is clear that system (18) where Since < 0, ∀ ∈ M, inequality (67) is discussed in the following two cases. Case 1. When ∈ , the left side of (67) becomes where Applying Schur complement lemma to (69), then (59) easily follows.

Numerical Examples
This section considers the following four-mode uncertain nonlinear Markovian jump systems with time delay as follows.    The trajectory of x(t) with w(t) = 0 (case I) The four cases for the transition probability matrix considered in Table 1.
Solving the LMIs (77)-(82) in Theorem 17, the robust finite-time ∞ state feedback controller gains of are given by Table 2. Figures 1, 2, and 3 are presented. For every figure, the four different transition probability matrices cases are included, which can be better to demonstrate the effectiveness of the design method. Figure 1 depicts the trajectories of system state ( ) and the corresponding switching signal. It can be seen that system (6) is robust finite-time stable, which implies that system (2) is robust finite-time ∞ state feedback stabilizable via the designed state feedback controller (5). Figure 2 depicts the trajectories of system state ( ) with ( ) ̸ = 0 and the corresponding switching signal. It can be seen that system (6) is robust finite-time bounded. The trajectory of the output ( ) is described in Figure 3, which further shows the effectiveness of the designed controller (5).

Conclusions
In this paper, we have dealt with the problem of robust finitetime ∞ control for a class of nonlinear Markovian jump systems with time delay under partially known transition probabilities. Based on the free-weighting matrices approach, all sufficient conditions have been firstly proposed to ensure finite-time boundedness, ∞ finite-time boundedness, and finite-time ∞ state feedback stabilization for the given system. We have also designed a robust finite-time ∞ state feedback controller, which guarantees the ∞ finite-time boundedness of the closed-loop system. All the conditions have been presented in terms of strict linear matrix inequalities. Finally, a numerical example has been provided to demonstrate the effectiveness of all the results.