A novel hybrid observer that estimates the states of an oscillating system composed of a linear chain structure and an intrinsic pulse-modulated feedback is considered. This particular type of plant model appears in e.g. endocrine systems with pulsatile hormone secretion. The observer reconstructs the continuous states of the model as well as the firing times and weights of the feedback impulses. Since the pulse-modulated feedback is intrinsic, no measurements of the discrete part of the plant are available to the observer. For a periodical plant solution, to reconstruct the hybrid state, the impulses in the observer have to be synchronized with those in the plant. The observer is equipped with two feedback loops driven by the output estimation error. One of these is utilized to correct the estimates of the continuous states. In contrast with previous observer designs, the estimate of the next impulse firing time is implemented by means of a finite-memory convolution operator. A pointwise mapping capturing the propagation of the continuous plant and observer states through the discrete cumulative sequence of the feedback firing instants is derived. Local stability properties of the synchronous mode are related to the spectral radius of the Jacobian of the pointwise mapping. The observer design is based on assigning a guaranteed convergence rate to the local dynamics of a synchronous mode through the output error feedback gains to the continuous and discrete part of the observer. The observation of a stable $m$-cycle in the plant is treated to establish a general scenario, whereas the special case of 1-cycle is worked out in detail as the most common one. A numerical example illustrates the observer performance in the case of periodic modes of low multiplicity in an impulsive model of testosterone regulation in the male. Despite the local nature of the design approach, convergence to a synchronous mode is observed for a wide range of initial conditions for the discrete state estimate in the observer.

考虑一种新颖的混合观测器，该观测器可估计由线性链结构和固有脉冲调制反馈组成的振荡系统的状态。这种特定类型的植物模型出现在例如具有搏动激素分泌的内分泌系统中。观察者重建模型的连续状态以及触发时间和反馈脉冲的权重。由于脉冲调制反馈是固有的，因此观察者无法获得植物离散部分的测量结果。对于周期性的工厂解决方案，要重建混合状态，必须将观察者中的脉冲与工厂中的脉冲同步。观察者配备了两个由输出估计误差驱动的反馈回路。其中之一被用来校正连续状态的估计。与先前的观察者设计相比，下一次脉冲触发时间的估计是通过有限内存卷积算子实现的。得出了一个点状映射，它捕获了通过反馈触发瞬间的离散累积序列来连续植物和观察者状态的传播。同步模式的局部稳定性与点状映射的雅可比谱谱半径有关。观察者的设计基于通过对观察者的连续和离散部分的输出误差反馈增益，将保证的收敛速度分配给同步模式的局部动力学。观察稳定 得出了一个点状映射，它捕获了通过反馈触发瞬间的离散累积序列来连续植物和观察者状态的传播。同步模式的局部稳定性与点状映射的雅可比谱谱半径有关。观察者的设计基于通过对观察者的连续和离散部分的输出误差反馈增益，将保证的收敛速度分配给同步模式的局部动力学。观察稳定 得出了一个点状映射，它捕获了通过反馈触发瞬间的离散累积序列来连续植物和观察者状态的传播。同步模式的局部稳定性与点状映射的雅可比谱谱半径有关。观察者的设计基于通过对观察者的连续和离散部分的输出误差反馈增益，将保证的收敛速度分配给同步模式的局部动力学。观察稳定 观察者的设计基于通过对观察者的连续和离散部分的输出误差反馈增益，将保证的收敛速度分配给同步模式的局部动力学。观察稳定 观察者的设计基于通过对观察者的连续和离散部分的输出误差反馈增益，将保证的收敛速度分配给同步模式的局部动力学。观察稳定$米$处理工厂中的“一周期”可建立一般情况，而将“一周期”的特殊情况作为最常见的一例进行详细研究。数值示例说明了在男性睾丸激素调节冲动模型的低多重性周期性模式下观察者的表现。尽管设计方法具有局部性质，但对于观察者中离散状态估计的各种初始条件，仍可以观察到收敛到同步模式。