State Estimation for Switched Time-Varying Systems with Delay and Nonlinear Disturbance : An Integral Inequality Method

This paper focuses on the problem of state estimation for certain switched time-varying systems with time-varying delay and nonlinear disturbance. By using an integral inequality technique and amethod used in positive systems, we have established several explicit criteria for state estimation of the system, which reduce to stability criteria for some particular cases.The involved nonlinear disturbance of the system takes more general form including both the internal disturbance and the external disturbance. Three numerical examples are also given to verify the validity of the obtained theoretical results.


Introduction
Switched system is a particular hybrid system containing a number of subsystems and a switching signal.Each subsystem is usually described by a definite differential equation or difference equation.For switched systems, the issue of stability plays a key role in system analysis.As a result, stability of switched systems has received considerable attention during the past several decades owing to its extensive applications in automotive engine control system [1], chemical process control system [2], multiagent systems [3,4], and so on.Some basic problems in stability and design of switching systems were put forward by Liberzon and Morse [5].Later, there are several very important monographs devoted to the stability analysis and design of switched systems; e.g., see Liberzon [6] and Sun and Ge [7].There are also many interesting results for stability of switched systems in [8][9][10][11][12][13][14][15].In most of the existing references, the Lyapunov-Krasovskii functional method was most commonly used for switched time-invariant systems.It seems to us that little attention has been paid to the stability of switched time-varying systems.Recently, by using a positive system method, exponential stability of switched time-varying systems with delay and nonlinear disturbance was investigated in [16,17].
Integral inequality plays an important role in qualitative analysis of delay systems [18][19][20][21][22][22][23][24][25].For example, stabilization of switched systems with impulsive effects and disturbances was studied in [18] by using the Gronwall integral inequality.By introducing a generalized Gronwall-Bellman inequality, the authors established stability criteria under arbitrary switching for switched systems with general nonlinear disturbances in [21].Later, the main results in [21] were extended to switched delay systems with nonlinear disturbances in [25].The same method was also applied to study a class of switched delay systems in [23], where global exponential stability criteria for the system were established.
Note that time delay has attracted much attention in the theory analysis of switched systems due to its detrimental effects on system performance such as oscillation [24,[26][27][28][29][30] and stability [31][32][33][34][35][36].Inspired by the work in [18,25], we will use a delay integral inequality technique and a method developed in positive systems [37,38] to study the problem of state estimation for a class of switched time-varying systems with time-varying delay and nonlinear disturbance.The main contributions of this paper are as follows: (1) unlike most existing results in the literature, all the subsystems considered in this paper are time-varying; (2) explicit global (local) state estimation criteria will be established for the cases when the nonlinear disturbance satisfies linear and nonlinear growth conditions, respectively; (3) the nonlinear disturbance of the system has more general form which contains both the internal state disturbance and the external input disturbance, and hence it contains some cases in the literature.
The rest of this paper is organized as follows.In Section 2, we introduce some notations and preliminaries that are essential for deriving the main results of this paper.Section 3 then focuses on establishing explicit state estimation criteria for the system.Simulations are given to illustrate the main results in Section 4. Finally, conclusions are drawn in Section 5.
We need the following assumptions for establishing the main results of this paper.
The following two lemmas play a crucial role in the state estimation of system (1).

Main Results
We first study the case of  = 1 in Assumption ( 2 ).
Theorem 3. Assume that ( 1 ) and ( 2 ) with  = 1 hold.If there exists an -dimensional vector  = (  ) ≻ 0 such that and then all solutions of system ( ) satisfy where are the th entry of vectors     ,     ,     , and     , respectively.
Without loss of generality, assume that () = .Denote by  + (()) the right derivative of (()) along the trajectory of system (1).We get from assumptions ( Therefore, According to definitions of   for  = 1, 2, 3, 4, we derive from the above inequality that Multiply the above inequality by  ∫  0 [ 1 ()− 3 λ()] and let Then we have Integrating it from 0 to , we obtain Set where k1 is defined as in Theorem Consequently, Combining this and Lemma 1, it implies that By using the definition of (), we have Therefore, (10) holds.This completes the proof of Theorem 3.
, we have that condition (27) holds and the corresponding solutions of system (1) satisfy That is, system (1) is locally exponentially stable.
Consequently, all solutions of system (1) are bounded if

Numerical Examples
In this section, three examples are given to illustrate the main results.

Conclusions
The problem of state estimation for switched time-varying systems with time-varying delay and nonlinear disturbance has been discussed in this paper.When the nonlinear disturbance satisfies linear and nonlinear growth conditions, explicit global (local) state estimation criteria have been established.For some particular cases, exponential stability and boundness of the system are taken into consideration.The method used in this paper is mainly based on the integral inequality technique.Finally, three numerical examples demonstrate the effectiveness of our main results.