Attractivity of the synchronous mode in hybrid observers for the impulsive Goodwin's oscillator subject to harmonic exogenous excitation

The paper deals with basin-of-attraction analysis of two observers estimating the states of a hybrid version of Goodwin's oscillator forced by a continuous exogenous signal. Under harmonic exogenous excitation, the impulsive Goodwin's oscillator exhibits bistability, which phenomenon significantly complicates observer design. The attractivity of a null solution of the hybrid state error estimation dynamics (termed as synchronous mode) has to be maximized in order to cover all possible initial state estimate deviations. A detailed analysis of the synchronous mode for a previously considered observer reveals a considerable asymmetricity in the basin of attraction relative to a jump instant and susceptibility to overjumps. As a result, for some initial conditions, the observer may converge to a stable stationary mode distinct from the synchronous one. To circumvent this, an improved hybrid observer with a more elaborate frequency modulation law in the output error feedback structure is proposed. Numerical analysis suggests that the new observer is able to approach the synchronous mode from all admissible initial estimates.