State-feedback control of rational systems using linear-fractional representations and LMIs

Considers a time-invariant, continuous system x˙=f(x,u), where f is a rational function of the state x, linear in the input u. The author introduces a linear-fractional representation (LFR) for the system, which consists of viewing it as an LTI system, connected with a diagonal feedback operator linear in the state. Using this representation, the authors devise 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 on this polytope, etc. All these conditions are obtained by analyzing the properties of a related differential inclusion, and checked using convex optimization over linear matrix inequalities (LMIs). The method extends to (static) state-feedback synthesis.