The 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.

脉冲式古德温振荡器是一个数学模型，可捕获闭环系统中产生的动力学，在该系统中，三阶线性时不变设备要经受固有的脉冲调制反馈。最初，该模型是由内分泌系统中的脉动调节驱动的，但也具有其他潜在应用。本文考虑了脉冲古德温振荡器的混合状态的渐近估计。混合观测器利用连续的工厂输​​出通过两个输出误差反馈来校正状态向量的估计值：连续的和离散的反馈。当混合状态估计误差为零时，观察者处于同步模式，其特征在于观察者反馈中的脉冲点火和同时发生的植物的脉冲点火。因此，同步模式对应于混合状态误差动力学的平衡点。为了保证观察者渐近收敛到同步模式，平衡的吸引盆必须包括状态估计的所有可行的初始偏差。为了保证上述特性，提出了一种基于离散图分叉分析的数值观测器设计方法，该方法捕获了从一个脉冲激发到另一个脉冲激发的观测器状态转换，并在仿真中证明了其有效性。