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

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 L2 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.

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