Stability Analysis of Pulse-Width-Modulated Feedback Systems with Time-Varying Delays

The stability problem of pulse-width-modulated feedback systems with time-varying delays and stochastic perturbations is studied. With the help of an improved functional construction method, we establish a new Lyapunov-Krasovskii functional and derive several stability criteria about th moment exponential stability.


Introduction
PWM has been widely used in many fields, such as attitude control systems, adaptive control systems, signal processing, and modeling of neuron behavior [1][2][3].In actual progress, it has always operated in all kinds of disturbances.At the same time, time-varying delays inevitably occur owing to the unavoidable finite switching speed of amplifiers.For some systems, the effect of time-varying delays can be ignored.How to keep the scheduled operation or work of the state running well, especially in engineering applications, is becoming more and more significant.
A growing number of scholars are devoting time to the PWM feedback systems; meanwhile a set of stability results has been established by a variety of methods [4][5][6][7][8][9][10][11][12].Also, lots of scholars have researched some systems with delays [7,[13][14][15][16][17][18][19].In [15], the authors investigated robust exponential stability and delayed-state-feedback stabilization of uncertain impulsive stochastic systems with time-varying delays.Sun and Cao [16] gave some definitions on the pth moment exponential stability in mean and established several pth moment globally stability criteria in mean.By using the Lyapunov technique and Razumikhin method, the authors in [17] investigated impulsive effects on stability analysis of high-order BAM neural networks with time delays.And in [18], they established the global exponential stability of the neural networks with its estimated exponential convergence rate.The authors in [19] gave a time-varying delay-dependent criterion for impulsive synchronization to ensure the delayed discrete complex networks switching topology tending to a synchronous state.
To the best of the author's knowledge, there are a few (if any) results for the stability analysis of the critical case of PWM systems with time-varying delays and stochastic perturbations, most of the existing work only considers one condition [7,8,15].In the present paper, we try to make a contribution to this issue.It is noted that the linear plant in this paper has one and only one pole at origin, and the rest of the poles are in the left side of the complex plane, which is more representative and more universal.Based on the references [7,13], the present paper will further study the stability of PWM feedback systems with timevarying delays and establish several new stability criteria.In Section 2, we give some definitions and lemmas.In Section 3, firstly, a criterion on mean square exponential stability of stochastic feedback systems with time-varying delays is given.Secondly, by introducing new variables, we will establish a new Lyapunov-Krasovskii functional with the help of an improved functional construction method.Then, associating with linear matrix inequalities, we will establish criteria for the pth moment exponential stability and the pth moment exponential asymptotic stability.Finally, we demonstrate the applicability of our results by means of an example.

Notations and Definitions
A pulse-width modulator is described by with where () = () − (), () is the external input and () is the system output and   is the pulse-width for  = 0, 1, 2, . ... The sampling period , the amplitude of the pulse , and  are all assumed to be constants.Consider the following stochastic PWM feedback system with time-varying delays: where  ∈   , ,  ∈ ,  is the output of the pulsewidth modulator,  is the nonlinear stochastic perturbations, ,   , ,  are the matrices of appropriate dimensions,  is the pulse-width control matrix, 0 <  ≤   , and   is a scalar wiener process which is defined on the probability space (Ω, , ).
Definition 5 (see [20,21]).A stochastic system is pth moment exponential asymptotic stability.If for any fixed initial condition, then lim Especially, when  = 2, we called the system mean square exponential asymptotic stability.

Stability Analysis
The block diagram of the PWM feedback system (3) is shown in Figure 1.Now, consider the following stochastic feedback system with time-varying delays: where  ∈   , () is a function of (), ( − ),  is the nonlinear stochastic perturbations, ,   ,  are matrices of appropriate dimensions, 0 <  ≤   , and   is a scalar wiener process which is defined on the probability space (Ω, , ).
For system (11), we let which satisfies that where  1 ,  2 are constant matrices of appropriate dimensions.
Proof.Using the Itô isometry, we get infinitesimal generation operators of functional (20).Consider Based on (22) By Definition 5, the system (3) is pth moment exponential asymptotic stability.Then, we discuss the functional (24) based on its parameter set.

Figure 1 :
Figure 1: Block diagram of PWM feedback systems subjected to multiplicative disturbances.
(3) + )  −1 .Remark 14.For system(3), only  can be controlled easily in reality.So we may let  =  and  satisfies formula (38).Then we get another corollary which can be applied in reality. Fr time-varying delay  > 0, the system (3) is pth moment exponential stability and pth moment exponential asymptotic stability if there exist scalars  > 0,  > 0, and the parameter set Φ 1 such that the following linear matrix inequality holds:   −     +   * − +     2  2  +  2  2