State Feedback Optimal L₂-Induced Control of Nonlinear Systems Utilizing Universal Approximation

This paper presents an optimal L2-induced control problem for systems with multiple sector-bounded nonlinearities. Sufficient boundedness conditions for the L2-induced norm are derived in terms of a specific system of linear matrix inequalities (LMIs). Based on these conditions, an optimal state feedback control problem is then formulated and solved for the considered class of nonlinear systems. A procedure to reduce the conservatism of the derived conditions is also provided. The proposed formulation, which explicitly considers multiple sector-bounded nonlinearities, is useful because it enables optimal L2-control problems for a much wider class of nonlinearities. Indeed, by invoking the universal approximation theorem, one may represent nonlinearities that do not satisfy sector-bounded conditions as a weighted sum of sector-bounded sigmoid functions. The theoretical and procedural developments are illustrated by a numerical example consisting of the state feedback optimal L2-induced control of a forced van der Pol oscillator.