Covariance Steering with Optimal Risk Allocation

TL;DR

This paper presents an iterative risk allocation method for covariance steering in linear stochastic systems, optimizing risk distribution to improve control performance under chance constraints.

Contribution

It introduces an IRA-based framework for optimal risk allocation in covariance steering, leading to less conservative solutions and novel convex relaxations for chance constraints.

Findings

IRA optimizes risk for lower total cost

Framework maximizes terminal covariance while satisfying constraints

Demonstrated effectiveness in spacecraft rendezvous scenario

Abstract

This paper extends the optimal covariance steering problem for linear stochastic systems subject to chance constraints to account for optimal risk allocation. Previous works have assumed a uniform risk allocation to cast the optimal control problem as a semi-definite program (SDP), which can be solved efficiently using standard SDP solvers. We adopt an Iterative Risk Allocation (IRA) formalism, which uses a two-stage approach to solve the optimal risk allocation problem for covariance steering. The upper-stage of IRA optimizes the risk, which is proved to be a convex problem, while the lower-stage optimizes the controller with the new constraints. This is done iteratively so as to find the optimal risk allocation that achieves the lowest total cost. 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. We also introduce two novel convex relaxation methods to approximate quadratic chance constraints as second-order cone constraints. We finally demonstrate the approach to a spacecraft rendezvous problem and compare the results.