A Nonsmooth Dynamics Perspective on State Observer Design for Mechanical Systems with Unilateral Constraints

Pascal V. Preiswerk, R. Leine
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Abstract

One of the main difficulties in the state observer design for impulsive mechanical systems is the so-called peaking phenomenon: even for an arbitrarily small pre-impact estimation error, a slight mismatch between the impact time instants of the observer and the observed system can lead to large post-impact estimation error. Therefore, Lyapunov’s stability theorems cannot be directly applied. For linear mechanical systems with unilateral constraints, we propose to take a Nonsmooth Dynamics perspective on the problem, which allows to sidestep the main difficulties by transforming and approximating the original continuous-time system by a discrete linear complementarity system through the use of the Schatzman-Paoli scheme. The discretization acts as a regularization, i.e. the impacts take place over two consecutive time steps. Furthermore, it involves force and impact laws on position-level with the favorable property of maximal monotonicity. Finally, a passivity-based observer design for discrete linear complementarity systems can be applied.
单侧约束机械系统状态观测器设计的非光滑动力学视角
脉冲机械系统状态观测器设计的主要困难之一是所谓的峰值现象:即使对于任意小的冲击前估计误差,观测器的冲击时间瞬间与被观察系统之间的轻微不匹配也会导致很大的冲击后估计误差。因此,李雅普诺夫稳定性定理不能直接应用。对于具有单边约束的线性机械系统,我们建议采用非光滑动力学的观点来解决问题,通过使用Schatzman-Paoli格式将原始连续时间系统转换和近似为离散线性互补系统,从而避免了主要困难。离散化作为正则化,即影响发生在两个连续的时间步长。此外,它还涉及位置水平上的力和冲击规律,具有极大单调性的良好性质。最后,将基于无源性的观测器设计应用于离散线性互补系统。
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