Finite-time stability for impulsive stochastic nonlinear system with uncertain parameter

This paper considered the finite-time mean square stability for impulsive stochastic nonlinear system(ISNS) with uncertain parameter. By selecting the modal dependent Lyapunov functional, using average impulsive interval approach and combining Linear Matrix Inequality (LMI) method, the correlation stability criterion of ISNS was obtained. A gain matrix of the feedback controller is obtained on the basis of this sufficient condition. Finally, the LMI toolbox in Matlab was used for data simulation, and the relevant state trajectory demonstrates the availability of the proposed theory.


Introduction
Impulsive disturbance phenomenon cannot be avoided in many engineering systems and is one of the main causes of system instability.For example, [1] studied the exponential stability problem for stochastic impulsive system with Markov switching.Reference [2] studied the influence of the stabilizing and anti-stabilizing impulse on the linear stochastic time-delay system.
However, most of literature studied the stability is based on Lyapunov stability.The notion of finite-time stability differs from Lyapunov stability.Lyapunov stability focuses on the stable operation in a system rather than the transient process.Compared to Lyapunov stability, finite-time stability is a much more common process.The advantages of finite-time stability is that system state trajectory does not surpass a given interval during a predetermined time range.In Reference [3], one researched the impulsive sample control problem and obtained the corresponding finite-time stability criteria.Reference [4] focused on homogeneous systems, obtained some new finite-time stability criterion on negative degrees of homogeneity.
In order to describe dynamic systems in which the structure or parameters change abruptly due to disturbances, Markov systems have received much attention in recent years [5][6][7].For example, reference [6] studied the fault observer design problem of Markov jump systems.Reference [7] focused on the stability criteria for nonhomogeneous linear Markov jump systems.Up to now, there have been few works on finite-time stability for ISNS.
Inspired by the above literature, we researched ISNS and considered the case when time-varying delays appeared in the system.Through selecting a modal dependent Lyapunov functional and using LMI, the correlation stability criterion was obtained.In the end, we show the corresponding stable picture through exemplifying an example with ISNS.The initial value is satisfying with the condition 0 ( ) is the Markov process, which is defined on S with transition probability For simplicity, let ( ) , matrices with appropriate dimension.Let uncertainties matrices , , , where  , ( ,0) 0, ( ,0,0) 0 The state feedback controller of ISNS is denoted as where i J is the gain matrix of the system.Substituting (2) into system(1), one can obtain Before proving the following theorems, we will introduce the relevant definitions and lemmas of finite-time stability.Definition 2.1 [8] For given positive scalar 1 2 , , system (1)  , 0 then the average impulsive interval is called a  , the occurrences of the impulses on the interval ( , ] s t is standed by ( , ) N t s .
Lemma 2.3 [9] For given matrices , For appropriate dimensions matrices , , X Y F , and T F F I  , suppose there is positive scalar Lemma 2.5 [10] For real valued matrices A, B with appropriate dimensions, suppose there exists positive definite matrix G and the below inequality holds

Main Results
We focused on the stability criterion for ISNS in this section.By using LMI, the existence of the feedback controller is obtained.
Theorem 3.1.Assume that the impulsive sequence satisfies (5).Given positive numbers 1 2 , , c c T T T 1 4 4 where is finite-time mean-square stable with respect to 1 2 ( , , ) Proof.Let ( ) , t i i S    , consider the Lyapunov functional of the following form ) When Taking mathematical expectation, one has E ( ) E ( ), N.
By replacing the term i A in (6) , and adding 1 , we obtain According to the proof of Theorem 1, system(3) is shown to be finite-time stable in regard to 1 2 ( , , ) c c T .Multiplying (24) left and right by diag 1 1 { , } , , , , respectively.Combining with Schur complement, (27) can be obtained.Since  , the following inequality holds: Therefore, it can be obtained from (26) that

Numerical Example
Select a class of ISNS (1) with two subsystem, the data is given as follows 1.1 2 0.5 1.1 0.3 , , , 0.3 0.9 0.8 1.9 0.9 The corresponding controller gains is obtained:  

. 1 {
Multiply the above equation left and right by 1 i L  , respectively, then (8) can be obtained.Accordingly,(7) can be obtained by multiplying (25) both sides with 1

2 (
the system(1) is shown in Fig.1.Applying the LMI Toolbox to solve the LMIs (6)-(8), There is no feasible solution.Fig.2shows the trajectory of open-loop system(1).It indicates that nolinear system (1) is not mean-square finite-time stable in regard to 1

Figure 2 .
Figure 2. State trajectory of system(1) Using LMI Toolbox to deal with the LMIs conditions (24)-(26), we can obtain the below feasible solution:

Figure 2 .
Figure 2. State trajectory of the system(3) under feedback control5.ConclusionWe researched the finite-time stability problem for ISNS.By selecting the modal dependent Lyapunov functional, combining LMI method.We constructed the correlation criterion of open loop ISNS.Through numerical simulations, we obtained the simulation pictures that satisfy the stability in the definition, and verified the validity of the proposed theorem.In future work, the research can be broadened to semi-Markov switched impulsive stochastic linear or nonlinear systems.