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Generation of stable limit cycles with prescribed frequency and amplitude via polynomial feedback

We consider a controllable linear time invariant model in state space of dimension n which might be the Jacobian linearization of a nonlinear model. Alternatively it may arise from a preceding input-output or input-state linearization. The usual objective for such systems is to stabilize an equilibrium. However, it might as well be interesting to have a stable limit cycle around the equilibrium. So far, limit cycles are often studied in the context of nonsmooth dynamics. In contrast, our approach results in a smooth and simple feedback. The first step is to impose a pair of purely imaginary eigenvalues to the system while the second one is to construct a bilinear form with which the resulting oscillations can be stabilized at a given amplitude.