Convexified Open-Loop Stochastic Optimal Control for Linear Non-Gaussian Systems

Abstract

We consider stochastic optimal control of linear dynamical systems with additive non-Gaussian disturbance. We propose a novel, sampling-free approach, based on Fourier transformations and convex optimization, to cast the stochastic optimal control problem as a difference-of-convex program. In contrast to existing moment based approaches, our approach invokes higher moments, resulting in less conservatism. We employ piecewise affine approximations and the well-known convexconcave procedure, to efficiently solve the resulting optimization problem via standard conic solvers. We demonstrate that the proposed approach is computationally faster than existing particle based and moment based approaches, without compromising probabilistic safety constraints.

Approximate Quantiles for Stochastic Optimal Control of LTI Systems with Arbitrary Disturbances

“…Moment based approaches [3][4][5] require analytic expressions for computational parameters that may only exist conditionally [6,7], or introduce conservative reformulations of constraints through Boole's inequality [8][9][10]. Fourier transforms have been used to bypass the quadrature computation required to evaluate probability integrals [11], and in combination with piecewise affine approximations, have been used to evaluate chance constraints [12,13] for linear time-invariant (LTI) systems with noise processes that elicit log-concave probability density functions (pdf). Sample based approaches have been employed for systems with known disturbances [14,15].…”

Approximate Quantiles for Stochastic Optimal Control of LTI Systems with Arbitrary Disturbances

“…Approaches that rely upon moments [7], [8], [9] may create excessive conservativism, and typically require an iterative approach to controller synthesis and risk allocation, to circumvent non-convexity that arises in the process of separating joint chance constraints into individual chance constraints via Boole's inequality [10], [11], [12]. Recent work has employed Fourier transforms in combination with piecewise affine approximations [13], [14], to evaluate chance constraints without quadrature for linear time-invariant (LTI) systems with disturbance processes that have log-concave probability distribution functions.…”