We propose an offline algorithm that simultaneously estimates discrete and continuous components of a hybrid system’s state. We formulate state estimation as a continuous optimization problem by relaxing the discrete component and using a robust loss function to accommodate large changes in the continuous component during switching events. Subsequently, we develop a novel nonsmooth variable projection algorithm with Gauss–Newton updates to solve the state estimation problem and prove the algorithm’s global convergence to stationary points. We demonstrate the effectiveness of our approach by comparing it to a state-of-the-art filter bank method, and by applying it to simple piecewise-linear and -nonlinear mechanical systems undergoing intermittent impact.

我们提出了一种离线算法，该算法可同时估计混合系统状态的离散和连续分量。我们通过放宽离散分量并使用健壮的损失函数来适应开关事件期间连续分量的较大变化，将状态估计公式化为连续优化问题。随后，我们开发了一种具有高斯-牛顿更新的新颖的非光滑变量投影算法，以解决状态估计问题并证明该算法在平稳点上的全局收敛性。通过将其与最新的滤波器组方法进行比较，并将其应用于经受间歇冲击的简单分段线性和非线性机械系统，我们证明了该方法的有效性。