Stabilization of “linear with varying coefficients” systems

Abstract

This section is concerned with the application of vector norm based stability analysis techniques developed in another chapter (Borne et al., 1993) of this book. For pedagogical reasons, this paper is restrained to a single class of problems and to a particular class of systems which leads to simple developments. More general processes can be further investigated in the same way but with a heavier technical part (Meizel, 1984) The proposed method is essentially a parametric optimization one. Typically, the considered process model depends upon a time-varying perturbation input and for a constant value of this one, the process model is supposed to be almost linear stationary. The considered stability property is the exponential stability of an equilibrium point. Regarding this property, the “non linear” characteristics stem both from the fact that considered processes are not linear stationary and from the fact that the stability analysis tool gives sufficient conditions which can be more or less restrictive with respect to the intrinsic — but unknown — stability properties of the studied equilibrium. For an a priori chosen control structure, the problem consists in tuning the control parameters and simultaneously in adapting the stability analysis tool so that the stability conditions are the least restrictive possible. This last feature is achieved by using some parametrizations that are obviously efficient on the well-known case of linear stationary processes while keeping the validity of the method for nonlinear systems. This parametrization heuristic leads to the definition of a scalar criterion, almost everywhere differentiable, whose iterative minimization is the design method.