Towards rigorous robust optimal control via generalized high-order moment expansion

This paper is concerned with the rigorous solution of worst-case robust optimal control problems having bounded time-varying uncertainty and nonlinear dynamics with affine uncertainty dependence. We propose an algorithm that combines existing uncertainty set-propagation and moment-expansion approaches. Specifically, we consider a high-order moment expansion of the time-varying uncertainty, and we bound the effect of the infinite-dimensional remainder term on the system state, in a rigorous manner, using ellipsoidal calculus. We prove that the error introduced by the expansion converges to zero as more moments are added. Moreover, we describe a methodology to construct a conservative, yet more computationally tractable, robust optimization problem, whose solution values are also shown to converge to those of the original robust optimal control problem. We illustrate the applicability and accuracy of this approach with the robust time- optimal control of a motorized robot arm