System identification of rhythmic hybrid dynamical systems via discrete time harmonic transfer functions

Few tools exist for identifying the dynamics of rhythmic systems from input-output data. This paper investigates the system identification of stable, rhythmic hybrid dynamical systems, i.e. systems possessing a stable limit cycle but that can be perturbed away from the limit cycle by a set of external inputs, and measured at a set of system outputs. By choosing a set of Poincaré sections, we show that such a system can be (locally) approximated as a linear discrete-time periodic system. To perform input-output system identification, we transform the system into the frequency domain using discrete-time harmonic transfer functions. Using this formulation, we present a set of stimuli and analysis techniques to recover the components of the HTFs nonparametrically. We demonstrate the framework using a hybrid spring-mass hopper. Finally, we fit a parametric approximation to the fundamental harmonic transfer function and show that the poles coincide with the eigenvalues of the Poincaré return map.