LQR for Systems with Probabilistic Parametric Uncertainties: A Gradient Method

TL;DR

This paper introduces a gradient-based approach using polynomial chaos theory to solve the LQR problem for systems with probabilistic uncertainties, ensuring robustness and computational efficiency.

Contribution

It develops a novel gradient descent method that directly optimizes structured feedback gains for uncertain systems, with proven convergence and error decay rates.

Findings

The method achieves higher computational efficiency than BMI-based approaches.

Gradient descent converges linearly for the proposed problem.

Approximation error decays algebraically with polynomial order.

Abstract

A gradient-based method is proposed for solving the linear quadratic regulator (LQR) problem for linear systems with nonlinear dependence on time-invariant probabilistic parametric uncertainties. The approach explicitly accounts for model uncertainty and ensures robust performance. By leveraging polynomial chaos theory (PCT) in conjunction with policy optimization techniques, the original stochastic system is lifted into a high-dimensional linear time-invariant (LTI) system with structured state-feedback control. A first-order gradient descent algorithm is then developed to directly optimize the structured feedback gain and iteratively minimize the LQR cost. We rigorously establish linear convergence of the gradient descent algorithm and show that the PCT-based approximation error decays algebraically at a rate $O(N^{-p})$ for any positive integer $p$, where $N$ denotes the order of the polynomials. Numerical examples demonstrate that the proposed method achieves significantly higher computational efficiency than conventional bilinear matrix inequality (BMI)-based approaches.