Stability analysis of systems with time-varying delays using overlapped switching Lyapunov Krasovskii functional

Abstract

This paper presents a new stability analysis approach for time-varying delay continuous systems which formulates the stability problem by presenting an overlapped switching Lyapunov Krasovskii functional (OSLKF). An exponential stability assessment method is proposed which considers two sets of conditions; local and discontinuity conditions. The local conditions establish the stability of the system based on the assumption that the delay parameters belong to a subdomain of the original domain. The discontinuity conditions are used to deal with the discontinuity issue of the switching Lyapunov functions at their switching points. These conditions have two major properties; it is not required that the time derivative of the delay to be less than unity and that the global exponential stability analysis of the model is possible. To illustrate the performance of the proposed method, two numerical examples are presented which demonstrate the effectiveness of the proposed approach as compared to the existing methodologies.

Introduction

The existence of time delays in the natural dynamic systems of real and practical process is a known fact. This natural phenomenon can critically affect the performance of the model. For instance, it causes the oscillation, performance deterioration, limit cycles and even instability in the model [1]. Therefore, the stability analysis of the time-varying delay systems has attracted enormous interest in the literatures [2], [3], [4], [5], [6], [7], [8], [9], [10].

The problem of the stability analysis of delay systems can be rationally divided into two categories namely constant and time-varying delay systems. The stability analysis of the systems with constant delays are basically investigated in [11] and the references there in. This study proposes necessary and sufficient conditions for this type of stability problem.

The basic approach in this topic is the Lyapunov-Krasovskii functional. This idea has been used in many different ways. Indeed, different types of the Lyapunov-Krasovskii functions are used in previous studies [12], [13], [14], [15], [16], [17], [18], [19]. The main purpose of these studies is to reduce the conservativeness of the stability conditions measured by the maximal allowable upper bound. Thus, researchers have focused on developing a set of stability conditions with the largest maximal allowable upper bound for the delay.

The conservativeness of the Lyapunov-Krasovskii functional methods have two major reasons namely the Lyapunov function structure and the range of its time-varying delays [19]. Previous studies have tried to develop a well-defined Lyapunov-Krasovskii functional in which the required conditions are less conservative as compared to the past studies. Usually, these studies used a set of various concepts such as Jensen inequality [20], Gu's inequality [21], augmented models [7], polynomial-based integral inequalities [22], [23] and free-matrix based integral inequalities [24].

Park et al. developed a method based on the augmented Lyapunov-Krasovskii functional [25] in which the dynamics of the model are added to the pre-determined stability conditions. This idea is extended until the free-matrix based integral inequalities idea was proposed in [19], in which some of the candidate matrices in the augmented Lyapunov-Krasovskii functional do not need to be positive semi-definite. Although, this idea reduces the conservativeness, the dimension of the matrix inequalities increases and it causes computational burden and practical complexity [1].

Recently, novel inequalities have been exploited in this analysis problem to reduce the conservativeness of the stability conditions of conventional Lyapunov-Krasovskii functional. For instance, the Bessel-Legendre inequality has been encountered in recent approaches [26], [27]. In [27], the inequalities are derived by choosing polynomials as a group of auxiliary functions. This inequality is amenable to the stability analysis problem of time varying delay system which results in less conservative stability criteria than the conventional form of Lyapunov–Krasovskii functional approaches. In [27], the mentioned inequality has been successively used by constructing a proper quadratic functional based on some integral based term of system's states. These approaches assuredly reduce the conservativeness while the proposed conditions are more complicated than those for the previous methods.

To reduce problem conservativeness and complexities, this paper presents a novel method to investigate the exponential stability of continuous models in the presence of a time-varying delay via exploiting OSLKF which switches based on the delay parameter. First, the method assumes that the time varying delay domain is divided into small subdomains and proposes a local condition for each subdomain which results in a more feasible problem. Then, the method is extended to include the time delay in the original domain using OSLKF. This procedure imposes the discontinuity of the Lyapunov functions at their switching points. To cope with this issue, a set of discontinuity conditions have been proposed and proved by the theorems.

The paper presents three theorems. The first theorem investigates the system stability where the time-varying delay belongs to a pre-defined subdomain. Clearly, this feasibility problem will be less conservative than that pertaining the whole domain. This theorem is developed based on the augmented Lyapunov-Krasovskii functional. However, the proposed theorem investigates the global exponential stability of the delay model when the range of the delay is restricted.

Then, a novel minimum-type OSLKF is developed to investigate the stability of the model over the whole range of time-varying delay (Theorems 2 and 3). In each subdomain of the time-varying delay, the Lyapunov function is the minimum of two Lyapunov-Krasovskii functional which corresponds to the corners of this subdomain. Additionally, this idea proposes a set of LMI conditions that guarantee the stability of model in each subdomain of the original domain.

The simulation result section contains two popular examples which are frequently encountered in the previous studies. The results reveal the superiority of the proposed method as compared to the previous studies.

Notations. I_n denotes the nth dimensional identity matrix, 0_n denotes the nth dimensional square matrix with zero elements, $\mathbb{R}$ represents the set of real numbers and ${\mathbb{R}}_{+}$ is the set of positive real numbers, * in the matrix inequalities shows the symmetry of matrix, X^T presents the transpose of the matrix X, $He\left\{X\right\}=X+X^T$, X^⊥ denotes a basis for the null-space of X (i.e. $X{X}^{\perp }=0$)