Optimal control of the state statistics for a linear stochastic system

TL;DR

This paper develops optimal control strategies for linear stochastic systems focusing on steering and maintaining state distributions, with explicit solutions for finite and infinite horizon cases under Gaussian noise.

Contribution

It introduces a novel approach to control based on terminal state distributions, extending classical LQG by replacing endpoint penalties with distribution specifications.

Findings

Steering to any Gaussian distribution over finite time is possible with controllability.

Explicit covariance conditions for stationary distributions are derived.

Closed-form solutions are available when noise and control input channels are identical.

Abstract

We consider a variant of the classical linear quadratic Gaussian regulator (LQG) in which penalties on the endpoint state are replaced by the specification of the terminal state distribution. The resulting theory considerably differs from LQG as well as from formulations that bound the probability of violating state constraints. We develop results for optimal state-feedback control in the two cases where i) steering of the state distribution is to take place over a finite window of time with minimum energy, and ii) the goal is to maintain the state at a stationary distribution over an infinite horizon with minimum power. For both problems the distribution of noise and state are Gaussian. In the first case, we show that provided the system is controllable, the state can be steered to any terminal Gaussian distribution over any specified finite time-interval. In the second case, we characterize explicitly the covariance of admissible stationary state distributions that can be maintained with constant state-feedback control. The conditions for optimality are expressed in terms of a system of dynamically coupled Riccati equations in the finite horizon case and in terms of algebraic conditions for the stationary case. In the case where the noise and control share identical input channels, the Riccati equations for finite-horizon steering become homogeneous and can be solved in closed form. The present paper is largely based on our recent work in arxiv.org/abs/1408.2222, arxiv.org/abs/1410.3447 and presents an overview of certain key results.