Square Root-Factorized Covariance Steering

TL;DR

This work introduces a square-root covariance steering method for finite-horizon, discrete-time linear time-varying systems with Gaussian noise and chance constraints. By propagating the Cholesky factor $S_k$ of the state covariance and employing a QR-based update, the method yields a numerically stable and scalable optimization, solved via sequential convex programming with a trust region. It proves global optimality for the unconstrained, expectation-of-quadratic case and local optimality under chance constraints, with equivalence of optimality under a Cholesky diffeomorphism to full-covariance formulations. Numerical experiments demonstrate favorable scalability, competitive performance under CCs, and superior numerical reliability in covariance propagation compared with full-covariance approaches. The framework offers flexible cost formulations and robust performance for stochastic control problems with tight uncertainty demands, such as spacecraft rendezvous and obstacle-rich planning.

Abstract

Covariance steering (CS) synthesizes a control policy which drives the state's mean and covariance matrix towards desired values. Offering tractable computation of a closed-loop policy which can obey chance constraints in uncertain environments, application to many real-world control problems have been proposed. We consider the chance-constrained, discrete-time, linear time-varying CS with Gaussian noise. The contribution of this paper is a novel solution method for this problem, explicitly writing the propagation equations of the Cholesky factor of the state covariance matrix by using the QR decomposition. The use of the square-root form of covariance matrices brings two key benefits over other existing methods: (i) computational scalability and (ii) numerical reliability. (i) Compared to solution methods that require large block matrix formulations, the proposed method scales better with the growth in horizon length, shows better optimality, and uses memoryless state feedback. (ii) Compared to another class of methods that explicitly define the covariance matrix as variables, the proposed method allows flexible cost formulations and shows better numerical reliability when uncertainty terms are smaller than the mean. On the other hand, these benefits come with a minor drawback: the propagation equation of covariance square roots is non-convex, necessitating sequential convex programming to solve. However, this paper proves the global optimality of the proposed approach for CS without chance constraints. When chance constraints are present, the existing optimal CS formulation is also non-convex, and we prove that the proposed approach shares the same local minima. We verify the mathematical arguments via extensive numerical simulations.

Figures (5)

• Figure 1: Comparisons for the unconstrained case. (Left) Comparison of the average (in solid lines) and the min-max range (in shaded regions) of the solution times between liuOptimalCovarianceSteering2025okamotoOptimalStochasticVehicle2019, and the proposed method (Right) Comparison of the costs for okamotoOptimalStochasticVehicle2019 and the proposed method, relative to liuOptimalCovarianceSteering2025
• Figure 2: Comparisons for the obstacle-avoidance case. (Left) Average (in solid lines) and the min-max range (in shaded regions) of the solution times. (Right) Objective function divided by the horizon length $N$
• Figure 3: Obstacle avoidance with Liu's formulation with Penalty CCP (left) and proposed method (right) for the case with $N=40$
• Figure 4: Projection of trajectory onto the $x$-$y$ plane for the spacecraft rendezvous problem, with $3\sigma$ covariance ellipses around the mean trajectory
• Figure 5: Covariance propagation loss for the spacecraft rendezvous problem

Theorems & Definitions (17)

• Lemma 1: Ch. 3.8 of ipsenNumericalMatrixAnalysis2009
• Lemma 2
• Lemma 3
• Lemma 4
• proof
• Lemma 5: Closed-Loop Square Root Covariance Propagation
• proof
• Remark 1
• Proposition 1: Invariance of Local Optimality via a Diffeomorphism
• proof