Robust Control in ℓ₁-Formulation: Model Validation and Perturbation Weight Estimation

Abstract

In robust system analysis and robust controller design, the parameters of the nominal system and admissible perturbation classes are usually assumed to be known. In many cases, these classes are described by constraints on perturbation norms and weight filters. Model validation consists in verifying whether the nominal model and the a priori assumptions on measurement perturbations are consistent. A far more difficult problem is the estimation of norms or weights of perturbations from measurement data. In this paper, model verification and perturbation weight estimation are investigated within the framework the ℓ₁-theory of robust control corresponding to the fundamental signal space ℓ_∞. For any model in the form of a linear stationary discrete system with structured uncertainty, model validation is reduced to verifying whether a system of inequalities generated by measurements holds or not. For a model with unstructured uncertainty and diagonal weighted perturbations, optimization of perturbation weights is reduced a fractional quadratic programming problem. For a model with perturbations of irreducible multipliers of the transfer matrix of the nominal system, optimization of perturbation weights is reduced a linear programming problem.