Robust Stabilization for a Class of Descriptor Systems under Input Delays: The Attractive Invariant Ellipsoid Approach

Abstract

The paper deals with the robust control design for descriptor (implicit) linear systems governed by a semi-explicit Differential Algebraic Equation (DAE) under two conditions: the unknown time-varying input delay and the unbounded exogenous perturbation presence. In order to provide numeric stabilizing conditions of the descriptor system solution, a linear feedback control and a Luenberguer-like observer are designed by the application of the conventional Attractive Invariant Ellipsoid Method (AIEM). The AIEM ensures both practical stability and robustness properties of the descriptor system solution. Practical stability is associated with the implicit system solution convergence into a specific geometrical region delimited by an Ellipsoid. Robustness properties are described in the sense of the exogenous perturbation rejection. The AIEM is an effective and well-known approach to ensure stability and robustness of dynamical systems governed by ordinary differential equations (ODE) but not for implicit systems governed by DAE. The main contribution on this work is the use of AIEM under unknown time-varying input delay presence. From the principles of AIEM for time-delay systems we propose an adequate linear feedback control and a Lyapunov-Krasovskii functional associated to a convergence zone in form of an ellipsoid. Minimization of the attractive ellipsoid is numerically described by the solution of a Bilinear Matrix Inequiality (BMI). Finally, an academic example supports the theoretical results.