Nonlinear dynamics of a positive hybrid observer for the impulsive Goodwin’s oscillator: a design study

The impulsive Goodwin oscillator (IGO) is nowadays an established mathematical model of pulsatile regulation that is suitable for e.g. capturing non-basal regulation of testosterone, cortisol, and growth hormone. The model consists of a continuous linear time-invariant block closed by a nonlinear pulse-modulated feedback. The hybrid closed-loop dynamics are highly nonlinear. The endocrine feedback is biologically implemented by the bursts of a release hormone secreted by the hypothalamus and not accessible for measurement. This poses a particular state estimation problem, where both the continuous states of the IGO and the firings of the impulsive feedback have to be reconstructed from the continuous outputs, i.e. the hormone concentrations measurable in the blood stream. A hybrid observer with two output error feedback loops, one for the continuous state estimates and another for the discrete one, is considered. Positivity of the observer estimates is demonstrated. The observer design problem at hand is, for all feasible initial conditions, to guarantee the asymptotic convergence of the observer estimates at highest possible rate to the state vector of the IGO. To solve the design problem, bifurcation analysis of the observer dynamics is performed and the basin of attraction for the stationary solution with a zero state estimation error is evaluated. The observer convergence rate is evaluated through the largest Lyapunov exponent. The efficacy of the design approach is confirmed by simulation.