Control of rational systems using linear-fractional representations and linear matrix inequalities

Abstract

Every system of the form x ̇ = f(x, u) , y = g( x, u), where f and g are rational functions of the state x and linear functions of the input u, possesses a linear-fractional representation (LFR). In this LFR, the system is viewed as an LTI system, connected with a diagonal feedback element linear in the state. We devise an algorithm for computing LFRs. Based on this construction, we give sufficient conditions for various properties to hold for the open-loop system. These include checking whether a given polytope is stable, finding a lower bound on the decay rate of trajectories initiating in this polytope, computing an upper bound on the L 2 gain, etc. All these conditions are obtained by analyzing the properties of a differential inclusion related to the LFR, and given as convex optimization problems over linear matrix inequalities (LMIs). We show how to use this approach for static state-feedback synthesis. We then generalize the results to dynamic output-feedback synthesis, in the case when f and g are linear in every state coordinate that is not measured. Extensions towards a class of nonrational and uncertain nonliner systems are discussed.