On robustness of systems with structured uncertainties

Abstract

We consider the robust stability problem for nominally linear system with nonlinear, structured perturbations. The system is of the form

$$\dot x = A_N x + \sum\limits_{j = 1}^q {p_j A_j x} $$

The Lyapunov direct method has been often utilized to determine the bounds for nonlinear, time-dependent functions p_j which can be tolerated by a stable nominal system. In most cases quadratic forms are used either as components of vector Lyapunov function or as a function itself. The resulting estimates are usually conservative. As it is known, often the conservatism is due to the fact that a quadratic form is used as a Lyapunov function. To reduce the conservatism of the bounds we propose to use a piecewise quadratic Lyapunov function. An example demonstrates application of the proposed method.