An MDADT-Based Approach for L 2-Gain Analysis of Discrete-Time Switched Delay Systems

We study the L 2 -gain analysis problem for a class of discrete-time switched systems with time-varying delays. A mode-dependent average dwell time (MDADT) approach is applied to analyze the L 2 -gain performance for these discrete-time switched delay systems. Combining a multiple Lyapunov functional method with the MDADT approach, sufficient conditions expressed in form of a set of feasible linear matrix inequalities (LMIs) are established to guarantee the L 2 -gain performance. Finally, a numerical example will be provided to demonstrate the validity and usefulness of the obtained results.


Introduction
Switched systems consist of a finite number of subsystems and a logical law which orchestrates the switching behaviors between these subsystems.These dynamical systems can mathematically model many practical engineering applications with switching characteristics in a variety of disciplines; see, for example, [1][2][3][4][5][6][7].
A constrained switching signal can be regarded as a powerful tool to stabilize and control these switched systems [8][9][10].Among them, the average dwell time (ADT) switching is the most common and typical one.It guarantees that the number of types of switching in a finite interval be bounded and the average time between any two types of consecutive switching not be less than a positive constant [11,12].In recent years, it has been recognised that ADT is flexible and efficient for dynamics analysis of many switched systems [8,[13][14][15][16].However, the ADT switching's property that the average time interval between any two types of consecutive switching should be greater than a positive number   makes the dwell time independent of the system modes.Hence whether the dwelling at some classes of subsystems will deteriorate the disturbance attenuation cannot be predicted.
As shown in [17], the minimum of admissible ADT is computed by two mode-independent parameters: the increase coefficient of the Lyapunov-like function and the decay rate of the Lyapunov function, which will cause certain conservativeness.To solve the problem, more recently, a new mode-dependent ADT concept has been introduced in [18].Two mode-independent parameters can be set in a modedependent manner, which will reduce the conservativeness.
Even though stability analysis for the switched systems with MDADT has been investigated extensively (see, e.g., [17,18]), how to solve the  2 -gain problem of the switched systems with MDADT is interesting and worthwhile to study.This has motivated our study in this paper.
The rest of the paper is as follows.In Section 2, we introduce the class of discrete-time switched system, some necessary definitions, and lemmas.In Section 3, sufficient conditions for ensuring  2 -gain for the discrete-time switched delay system are constructed.In Section 4, a numerical example is presented to illustrate the obtained results.Conclusion remarks are given in Section 5.

Preliminaries and Problem Statement
Consider a discrete-time switched system with a timevarying delay: where () ∈   is the system state, () ∈  To proceed, we need the following definitions and lemmas.
Definition 1 (see [11]).For any  2 >  1 ≥ 0 and any switching signal ,  1 ≤  <  2 , let   ( 1 ,  2 ) denote the number of types of switching of  over ( 1 ,  2 ).If   ( 1 ,  2 ) ≤  0 +  2 −  1 /  holds for  0 ≥ 0 and   > 0, then   is the average dwell time and  0 is the chatter bound.Without loss of generality, we choose  0 = 0. Definition 2 (see [18]).For a switching signal  and any  ≥  ≥ 0, let   (, ) be the switching numbers in which the th subsystem is activated over the interval [, ] and let   (, ) denote the total running time of the th subsystem over the interval [, ],  ∈ .We say that  has a modedependent average dwell time (MDADT)   if there exist positive numbers   and   such that and we call   the mode-dependent chatter bounds.Here, we choose   = 0 as well.

𝐿 2 -Gain Analysis
Firstly, we will introduce two important lemmas for the  2gain analysis of the switched delay system (1).The first lemma will provide the decay estimation of the Lyapunov functional   () along the trajectory of the switched delay system without disturbances.

Mathematical Problems in Engineering
Proof.Using Lemma 8 and (1), we have Based on Lemmas 4 and 5, it holds that Then, it follows from (35) and (38) that Combining (32), ( 33) with (34) will lead to This completes the proof.Now, our  2 -gain analysis results can be presented as follows.
It can be seen from Figure 1 that under the designed MDADT switching signals the switched delay system can achieve better dynamics performance and disturbance tolerance capability, which shows the potentiality of our results in practice.

Conclusions
In this paper, the problem of  2 -gain analysis for discretetime switched systems with MDADT switching has been investigated.By combining with the multiple Lyapunov function method, sufficient conditions are established to ensure  2 -gain performance for discrete-time switched delay system, and the admissible MDADT switching signals are also designed accordingly.Finally, a numerical example is given to demonstrate the usefulness of the obtained results.
is the controlled output, () is a vector-valued initial function,  0 is the initial time, and () is the disturbance input which belongs to  2 [0, +∞).() is the time-varying delay and satisfies 0 <   < () ≤   , where   and   denote the upper and the lower bounds of the delays. is the switching signal, which takes its values in the finite set  = {1, . . .