Covariance Steering With Optimal Risk Allocation

Abstract

This article extends the optimal covariance steering (CS) problem for linear stochastic systems subject to chance constraints so as to account for an optimal allocation of the risk. Previous works have assumed a <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">uniform</i> risk allocation in order to cast the optimal control problem as a semidefinite program, which can be solved efficiently using standard SDP solvers. An iterative risk allocation (IRA) formalism is used to solve the optimal risk allocation problem for CS using a two-stage approach. The upper-stage of IRA optimizes the risk, which is a convex problem, while the lower-stage optimizes the controller with the new constraints. The process is applied iteratively until the optimal risk allocation that achieves the lowest total cost is obtained. The proposed framework results in solutions that tend to maximize the terminal covariance, while still satisfying the chance constraints, thus leading to less conservative solutions than previous methodologies. In this article, we consider both polyhedral and cone state chance constraints. Finally, we demonstrate the approach to a spacecraft rendezvous problem and compare the results with other competing approaches.