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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 …