离散力学系统分析与最优控制的结构线性化
Structured Linearization of Discrete Mechanical Systems for Analysis and Optimal Control
摘要：
变分积分器非常适合于力学系统的模拟，因为它们保留了系统的力学量，例如动量，或者如果涉及外部强迫的变化，以及完整约束。虽然它们不是能量保存的，但它们确实表现出长期稳定的能量行为。然而，变分积分器经常通过在每个时间步求解隐式差分方程来模拟机械系统动力学，该隐式差分方程此外纯粹根据不同时间步的配置来表达。本文制定了一阶和二阶线性化的变分积分器的方式，是服从控制分析和合成，创建一个桥梁之间的现有分析和最优控制工具的离散动态系统和变分积分器的机械系统在广义坐标强迫和完整约束。强迫摆是用来说明技术。第二个例子解决了离散LQR问题，找到一个局部稳定控制器的40自由度系统与6个约束。
英文摘要：
Variational integrators are well-suited for simulation of mechanical systems because they preserve mechanical quantities about a system such as momentum, or its change if external forcing is involved, and holonomic constraints. While they are not energy-preserving they do exhibit long-time stable energy behavior. However, variational integrators often simulate mechanical system dynamics by solving an implicit difference equation at each time step, one that is moreover expressed purely in terms of configurations at different time steps. This paper formulates the first- and second-order linearizations of a variational integrator in a manner that is amenable to control analysis and synthesis, creating a bridge between existing analysis and optimal control tools for discrete dynamic systems and variational integrators for mechanical systems in generalized coordinates with forcing and holonomic constraints. The forced pendulum is used to illustrate the technique. A second example solves the discrete LQR problem to find a locally stabilizing controller for a 40 DOF system with 6 constraints.
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