Multi-Objective Optimal Control with Arbitrary Additive and Multiplicative Noise

TL;DR

This paper develops a convex semidefinite programming approach for multi-objective optimal control of systems with additive and multiplicative noise, providing conditions for stability and optimal linear controllers.

Contribution

It introduces a convex reformulation of the control problem with indefinite quadratic constraints and characterizes optimal controllers as affine or linear depending on convexity.

Findings

Optimal controllers depend on the covariance matrix of state and control.

The problem is convex and solvable via semidefinite programming.

Provides necessary and sufficient conditions for mean square stability.

Abstract

In this paper, we consider the problem of multi-objective optimal control of a dynamical system with additive and multiplicative noises with given second moments and arbitrary probability distributions. The objectives are given by quadratic constraints in the state and controller, where the quadratic forms maybe indefinite and thus not necessarily convex. We show that the problem can be transformed to a semidefinite program and hence convex. The optimization problem is to be optimized with respect to a certain variable serving as the covariance matrix of the state and the controller. We show that affine controllers are optimal and depend on the optimal covariance matrix. Furthermore, we show that optimal controllers are linear if all the quadratic forms are convex in the control variable. The solutions are presented for both the finite and infinite horizon cases. We give a necessary and sufficient condition for mean square stabilizability of the dynamical system with additive and multiplicative noises. The condition is a Lyapunov-like condition whose solution is again given by the covariance matrix of the state and the control variable. The results are illustrated with an example.