Joint chance constrained input shaping

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.