Regions of stability for limit cycle oscillations in piecewise linear systems

Oscillations appear in numerous applications from biology to technology. However, besides local results, rigorous stability and robustness analysis of oscillations are rarely done due to their intrinsic nonlinear behavior. Poincare´ maps associated with the system cannot typically be found explicitly and stability is estimated using extensive simulations and experiments. This paper gives conditions in the form of linear matrix inequalities (LMIs) that guarantee asymptotic stability in a reasonably large region around a limit cycle for a class of systems known as piecewise linear systems (PLS). Such conditions, based on recent results on impact maps and surface Lyapunov functions (SuLF), allow a systematic and efficient analysis of oscillations of PLS or arbitrarily close approximations of nonlinear systems by PLS. The methodology applies to any locally stable limit cycle of a PLS, regardless of the dimension and the number of switching surfaces of the system, and is illustrated with a biological application: a fourth-order neural oscillator, also used in many robotics applications such as juggling and locomotion.