Mean and variance of the LQG cost function

TL;DR

This paper derives analytical formulas for the mean and variance of the LQG cost function, enhancing understanding of its statistical properties and enabling better risk management in control systems.

Contribution

It provides the first explicit expressions for the mean and variance of the LQG cost function using two different mathematical methods.

Findings

Derived formulas for discounted and non-discounted costs

Applicable to finite-time and infinite-time horizons

Successfully applied to reduce probability of high costs

Abstract

Linear Quadratic Gaussian (LQG) systems are well-understood and methods to minimize the expected cost are readily available. Less is known about the statistical properties of the resulting cost function. The contribution of this paper is a set of analytic expressions for the mean and variance of the LQG cost function. These expressions are derived using two different methods, one using solutions to Lyapunov equations and the other using only matrix exponentials. Both the discounted and the non-discounted cost function are considered, as well as the finite-time and the infinite-time cost function. The derived expressions are successfully applied to an example system to reduce the probability of the cost exceeding a given threshold.