Nonlinear uncertain systems and robustness optimization

An approach to (adaptive) stabilization is developed for a class of uncertain nonlinearly perturbed linear dynamical systems modelled by a controlled differential inclusion on Rⁿ. The essential features are nonlinear feedback and optimization of robustness with respect to unmatched uncertainty. The approach is geometric: (i) a family ?? of proper subspaces V ? Rⁿ is introduced with the property that {0} ? V is an asymptotically stable equilibrium for the projection, onto V, of the unperturbed linear flow; (ii) for the perturbed system, it is then shown that each V ? ?? can be rendered attractive and (almost) invariant by some choice of feedback determined by the nature of the perturbation and other a priori system information; specifically, under differing hypotheses, we can render V ? ?? almost invariant and attractive, or invariant and finite-time attractive, or adaptively attractive via, respectively, linear high-gain feedback or discontinuous feedback, or adaptive feedback. Associated with each V ? ?? is a measure of robustness with respect to unmatched uncertainty (a stability radius), quantified by a real parameter η. The problem of optimizing this measure is formulated as an optimization problem over ??, solvable (in a supremal sense of η-maximizing sequences (V_j) ??) via solutions of particular algebraic Riccati equations of order k ≤ (n - m), where m ≤ n is the control dimension; typically k = n - m..