Estimating stable delay intervals with a discretized Lyapunov–Krasovskii functional formulation

• DOI: 10.1016/j.automatica.2014.04.019
• Publication Date: 2014-05-09T16:35:02Z
• Authors: Yongmin Li; Keqin Gu; Jianping Zhou; Shengyuan Xu
• Keywords: Stability theory; Matrix (chemical analysis); Delay differential equation

Abstract In general, a system with time delay may have multiple stable delay intervals. Especially, a stable delay interval does not always contain zero. Asymptotically accurate stability conditions such as discretized Lyapunov–Krasovskii functional (DLF) method and sum-of-square (SOS) method are especially effective for such systems. In this article, a DLF-based method is proposed to estimate the maximal stable delay interval accurately without using bisection when one point in this interval is given. The method is formulated as a generalized eigenvalue problem (GEVP) of linear matrix inequalities (LMIs), and an accurate estimate may be reached by iteration either in a finite number of steps or asymptotically. The coupled differential–difference equation formulation is used to illustrate the method. However, the idea can be easily adapted to the traditional differential–difference equation setting.