LMI and SDP technique for stability analysis of nonlinear delay systems subject to constraints

• Published in: Optimization letters Vol. 13; no. 8; pp. 1937 - 1952
• Main Authors: Sedova, N., Pertseva, I.
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.11.2019
• Subjects: Computational Intelligence; Mathematics; Mathematics and Statistics; Numerical and Computational Physics; Operations Research/Decision Theory; Optimization; Original Paper; Simulation; System with constraints; Stability; Nonlinear delay system; Razumikhin technique; LMI; SDP; Takagi–Sugeno system
• ISSN: 1862-4472; 1862-4480
• DOI: 10.1007/s11590-018-1352-9

The aim of the study is to present an estimation of the domain of attraction (DA) for nonlinear delay systems. The proposed approach is based on the assumption that on a subset of instant states the system is represented by a continuous-time Takagi–Sugeno (TS) system with delay. Because of the constraints defining the subset under consideration, an important issue is to estimate the set of initial points such that the trajectories starting at those points remain to satisfy the constraints. Using Lyapunov functions with the Razumikhin technique, we describe the problem of constructing an inner estimation of the DA in terms of invariant ellipsoids. As the result, the problem reduces to linear matrix inequalities and semidefinite programs which can be solved numerically. The method is also applied to constructing some outer estimate for the attractor of a nonlinear delay system subject to asymptotic stability for an approximating TS system. The resulting ellipsoidal estimate for the attractor depends on the estimate of the approximation error.