This paper addresses the problem of robust input shaping for rest to rest maneuvers of linear systems with parametric uncertainties. A stochastic optimization problem with a quadratic cost function is posed which probabilistically penalizes excursions of terminal time state values from desired values due to uncertainties. This quadratic cost represented by a hyper-sphere, is approximated by a hyper-polygon to permit a convex problem formulation, where the joint chance constraints are represented using statistics of the uncertain terminal states. Polynomial Chaos is used as an uncertainty quantification tool to estimate the first two moments of the stochastic state variables necessary for the implementation of the chance constraints. The solution to the optimization problem yields the desired input shaper. Several analytical methods of dealing with the joint chance constraints are investigated and compared on illustrative benchmark examples. The framework presented permits the users to trade-off performance for robustness to any desired level.

本文解决了具有参数不确定性的线性系统的静息操纵的鲁棒输入整形问题。提出了具有二次成本函数的随机优化问题，由于不确定性，该优化问题可能会从期望值中惩罚终端时间状态值的偏移。由超球面表示的二次成本由超多边形近似，以允许凸问题公式化，其中联合机会约束使用不确定终端状态的统计表示。多项式混沌被用作不确定性量化工具，以估计随机机会变量的实施所需的随机状态变量的前两个时刻。优化问题的解决方案将产生所需的输入整形器。研究了处理联合机会约束的几种分析方法，并在示例性基准示例上进行了比较。提出的框架允许用户在性能方面进行权衡，以使鲁棒性达到任何期望的水平。