New approaches to stability analysis for time-varying delay systems

Abstract

In this paper, two new estimation approaches namely delay-dependent-matrix-based (DDMB) reciprocally convex inequality approach and DDMB estimation approach, are introduced for stability analysis of time-varying delay systems. Different from existing estimation techniques with constant matrices, the estimation approaches are with delay-dependent matrices, which can employ more free matrices and utilize more information of both time delay and its derivative. Based on the estimation approaches, less conservative stability criteria with lower computational complexity are derived in the form of linear matrix inequalities (LMIs). Finally, two numerical examples are given to illustrate the advantages of the proposed methods.

Introduction

Time delay, as a natural phenomenon of real world, is inevitably encountered in many fields such as ecological groups, Internet environment, industry processes, and so on [1], [2], [3], [4], [5], [6], [7], [8]. It has been shown that time delay may lead to undesired dynamic behavior such as oscillation, performance degradation, or even instability [9], [10], [11], [12], [13], [14], [15], [16]. So, stability analysis, as the prerequisite in many applications, is strongly required for time-delay systems.

Up to now, it has been one of the hot topics to derive less conservative stability criteria for time-delay systems. For this purpose, various approaches have been proposed. Among the approaches developed for this topic, the Lyapunov-Krasovskii functional (LKF) method is the most popular one. With the Lyapunov stability theory, the improvement for stability criteria of time-delay systems can be achieved mainly from the two phases: (1) constructing appropriate LKFs and (2) estimating the derivatives of the LKFs as tight as possible.

For phase (1), various types of the LKFs have been constructed, such as augmented LKFs [17], [18], [19], [20], delay-partitioning LKFs [21], [22], weighting-delay-based LKFs [23], and LKFs with triple integral terms [24]. Most recently, by modifying the free-matrix-based integral inequality [25], new LKFs have been constructed in [26], [27], [28]. It is noted that more slack matrices are employed in the LKFs of [26], [27], [28], which effectively improved the feasible regions of the stability conditions.

For phase (2), to obtain accurate upper bounds of the derivative of the LKFs, the main difficulty lies in getting tighter lower bounds for $L(b,a,\mathcal{Y})={\int }_a^b{\dot{\omega }}^T\left(s\right)\mathcal{Y}\dot{\omega }\left(s\right)ds$. To handle such kind of integrals, some estimation techniques have been reported such as free-weighting matrix technique [29], Jesen inequality [30], Wirtinger-based integral inequality [31], free-matrix-based integral inequality [25], reciprocally convex inequality [32], and improved reciprocally convex inequality [33]. These estimation techniques play key roles in reducing the conservatism for time-delay systems. Recently, by introducing an augmented LKF and applying Wirtinger-based integral inequality, authors in [17] have derived delay-dependent stability criteria for time-varying delay systems. In [25], by free-matrix-based integral inequality, less conservative stability criteria for time-varying delay systems have been obtained. In [26], improved results have been established for time-varying delay systems by the modified free-matrix-based integral inequality. However, in the aforementioned works, only constant matrices are employed in the estimation techniques. Estimation approaches with delay-dependent matrices have not been considered, which leaves room for improvement.

In this study, two new estimation approaches, delay-dependent-matrix-based (DDMB) reciprocally convex inequality approach and DDMB estimation approach, are proposed. In the estimation approaches, delay-dependent matrices are introduced, which are associated with both time delay and its derivative. Compared with the existing estimation techniques with constant matrices, the estimation approaches in this paper employ more free matrices and can make full use of the time delay information. Based on the two estimation approaches, improved stability criteria are presented for time-varying delay systems. Finally, the effectiveness and superiorities of the established criteria are demonstrated by two numerical examples.

Notations: col{⋅⋅⋅} and diag{⋅⋅⋅} represent, respectively, a column vector and a block-diagonal matrix. $\text{Sym}\left\{X\right\}=X+X^T$. ℜⁿ and ℜ^n × n denote the n-dimensional Euclidean space and the set of all n × n real matrices, respectively. I_n, 0_n, and 0_n,m stand for n × n identity matrix, n × n, and n × m zero matrices, respectively. For real symmetric matrices X and Y, the notation X > Y means that the matrix $X-Y$ is positive definite. The symmetric term in a matrix is denoted by *.