Optimal steering of a linear stochastic system to a final probability distribution, part II

TL;DR

This paper develops methods for optimally steering linear stochastic systems to desired Gaussian distributions over finite and infinite horizons, using Riccati equations and semi-definite programming, with practical examples.

Contribution

It introduces new conditions for optimality and feasibility in steering stochastic systems to Gaussian distributions, including convex optimization approaches.

Findings

Finite-horizon steering to any Gaussian distribution is always feasible for controllable systems.

Stationary Gaussian distributions are characterized by a Lyapunov-like equation.

Optimal controls can be computed via semi-definite programs.

Abstract

We consider the problem of minimum energy steering of a linear stochastic system to a final prescribed distribution over a finite horizon and to maintain a stationary distribution over an infinite horizon. We present sufficient conditions for optimality in terms of a system of dynamically coupled Riccati equations in the finite horizon case and algebraic in the stationary case. We then address the question of feasibility for both problems. For the finite-horizon case, provided the system is controllable, we prove that without any restriction on the directionality of the stochastic disturbance it is always possible to steer the state to any arbitrary Gaussian distribution over any specified finite time-interval. For the stationary infinite horizon case, it is not always possible to maintain the state at an arbitrary Gaussian distribution through constant state-feedback. It is shown that covariances of admissible stationary Gaussian distributions are characterized by a certain Lyapunov-like equation. We finally present an alternative to solving the system of coupled Riccati equations, by expressing the optimal controls in the form of solutions to (convex) semi-definite programs for both cases. We conclude with an example to steer the state covariance of the distribution of inertial particles to an admissible stationary Gaussian distribution over a finite interval, to be maintained at that stationary distribution thereafter by constant-gain state-feedback control.