Bifurcation analysis for non-local design of a hybrid observer for the impulsive Goodwin’s oscillator

Abstract

AbstractThe impulsive Goodwin’s oscillator is a mathematical model capturing the dynamics arising in a closed-loop system, where a third-order linear time-invariant plant is subject to an intrinsic pulse-modulated feedback. Originally, the model was motivated by pulsatile regulation in endocrine systems but also has other potential applications. The asymptotic estimation of the hybrid state of the impulsive Goodwin’s oscillator is considered in this paper. A hybrid observer makes use of the continuous plant output to correct the estimates of the state vector through two output error feedbacks: a continuous and a discrete one. When the hybrid state estimation error is zero, the observer is in a synchronous mode characterized by the firings of the impulses in the observer feedback and those of the plant occurring simultaneously. The synchronous mode thus corresponds to an equilibrium point of the hybrid state error dynamics. To guarantee asymptotic convergence of the observer to the synchronous mode, the basin of attraction of the equilibrium has to include all feasible initial deviations of the state estimates. To guarantee the above properties, a numerical observer design approach based on bifurcation analysis of a discrete map capturing the observer state transitions from one impulse firing to another is proposed and its efficacy is demonstrated in simulation.