Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Optimal Covariance Steering for Discrete-Time Linear Stochastic Systems

Fengjiao Liu, George Rapakoulias, Panagiotis Tsiotras

TL;DR

This work tackles the finite-horizon covariance steering problem for discrete-time linear stochastic systems under a quadratic cost. It derives a rigorous existence/uniqueness theory for the optimal control, proves that mean and covariance steering can be separated under no chance constraints, and demonstrates that the optimal policy can be computed exactly via a semidefinite program, with an alternative Newton-based method for solving the associated Riccati difference equations. The approach provides exact, scalable computation of the optimal controller and delivers insights into the structure of the Riccati-like equations that arise in covariance steering. The results have practical implications for uncertainty-aware trajectory planning and control in engineering systems where Gaussian state distributions are natural models. The paper also outlines directions for extending the framework to multiplicative noise, chance constraints, and data-driven noise-covariance estimation.

Abstract

In this paper, we study the optimal control problem for steering the state covariance of a discrete-time linear stochastic system over a finite time horizon. First, we establish the existence and uniqueness of the optimal control law for a quadratic cost function. Then, we show the separation of the optimal mean and the covariance steering problems. We also develop efficient computational methods to solve for the optimal control law, which is identified as the solution to a semi-definite program. The effectiveness of the proposed approach is demonstrated through numerical examples. In the process, we also obtain some novel theoretical results for a matrix Riccati difference equation, which may be of independent interest.

Optimal Covariance Steering for Discrete-Time Linear Stochastic Systems

TL;DR

This work tackles the finite-horizon covariance steering problem for discrete-time linear stochastic systems under a quadratic cost. It derives a rigorous existence/uniqueness theory for the optimal control, proves that mean and covariance steering can be separated under no chance constraints, and demonstrates that the optimal policy can be computed exactly via a semidefinite program, with an alternative Newton-based method for solving the associated Riccati difference equations. The approach provides exact, scalable computation of the optimal controller and delivers insights into the structure of the Riccati-like equations that arise in covariance steering. The results have practical implications for uncertainty-aware trajectory planning and control in engineering systems where Gaussian state distributions are natural models. The paper also outlines directions for extending the framework to multiplicative noise, chance constraints, and data-driven noise-covariance estimation.

Abstract

In this paper, we study the optimal control problem for steering the state covariance of a discrete-time linear stochastic system over a finite time horizon. First, we establish the existence and uniqueness of the optimal control law for a quadratic cost function. Then, we show the separation of the optimal mean and the covariance steering problems. We also develop efficient computational methods to solve for the optimal control law, which is identified as the solution to a semi-definite program. The effectiveness of the proposed approach is demonstrated through numerical examples. In the process, we also obtain some novel theoretical results for a matrix Riccati difference equation, which may be of independent interest.
Paper Structure (12 sections, 24 theorems, 117 equations, 1 figure)

This paper contains 12 sections, 24 theorems, 117 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Existence & Uniqueness of the Optimal Control
  4. Candidate Optimal Control
  5. Riccati Difference Equation
  6. Existence and Uniqueness of the Solution
  7. Separation of Mean and Covariance Steering
  8. Computation of the Optimal Control
  9. Newton's Method
  10. Semi-Definite Program
  11. Numerical Example
  12. Concluding Remarks

Key Result

Theorem 1

Let $\bar{\Sigma}_{0} \succ 0$ and $\bar{\Sigma}_{N} \succ D_{N-1} D_{N-1}^{\hbox{\tiny\sf T}}$. Under Assumptions asm:A-inv-asm:G-D, the unique optimal control law that solves the covariance steering problem for system sys:disc-time is given, for all $k = 0, 1, \ldots, N-1$, by where $\Pi_{k}$ is the unique solution of the coupled matrix difference equations Furthermore, for all $k = 0, 1, \ldo

Figures (1)

  • Figure 1: Ten sample paths with the optimal state mean and three-standard-deviation tolerance region.

Theorems & Definitions (53)

  • Theorem 1
  • Remark 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Proposition 1
  • Proposition 2
  • proof
  • Theorem 2
  • proof
  • ...and 43 more