Robust Stability Analysis of Linear Parameter Varying Systems

Abstract

The robust stability analysis of nonlinear systems is a complex and challenging task. A well-known method to analyse them is the Monte-Carlo simulation. However, analysing a system in such a statistical manner can be very time consuming and there is no guarantee that the worst-case is actually found. An alternative approach is to represent the nonlinear system in the framework of linear parameter varying (LPV) systems. The present work studies the stability of LPV systems with a focus on time-variant parameters with bounded variation rates. Based on the existence of parameter dependent Lyapunov functions, a concept of analysing LPV systems is presented. Unlike the Monte-Carlo simulation, this concept provides sufficient conditions for stability. As part of this concept, an approach based on the Full Block S-Procedure is introduced. With this procedure, it is possible to reduce the model complexity at the cost of a more conservative system description. The Full Block S-Procedure requires a linear fractional representation of the LPV model and leads to a feasibility problem with linear matrix inequalities. The resulting stability test allows to specify a trade-off between conservatism and computational effort. This trade-off is examined on two examples. The first is a controlled axis-1 model of a standard six axes industrial manipulator and the second is a model of a heaving and pitching airfoil.