A Polynomial Chaos Approach to Stochastic LQ Optimal Control: Error Bounds and Infinite-Horizon Results

TL;DR

This paper addresses stochastic LQR problems under non-Gaussian disturbances by introducing a Polynomial Chaos Expansion (PCE) framework to propagate uncertainty through discrete-time dynamics. By decoupling uncertainty sources into independent PCE components, it derives finite-horizon solutions via a set of decoupled subproblems and provides constructive error bounds for moving-horizon truncations, as well as an analysis of infinite-horizon asymptotics. It shows that the infinite-horizon solution converges to a unique stationary pair in the sense of probability measures and that a finite-dimensional approximation can achieve any prescribed accuracy in the Wasserstein distance. A numerical example demonstrates the approach on a non-Gaussian disturbance-prone system, validating both the theoretical results and the practical relevance of the truncation and stationary analyses.

Abstract

The stochastic linear--quadratic regulator problem subject to Gaussian disturbances is well known and usually addressed via a moment-based reformulation. Here, we leverage polynomial chaos expansions, which model random variables via series expansions in a suitable $\mathcal{L}^2$ probability space, to tackle the non-Gaussian case. We present the optimal solutions for finite and infinite horizons and we analyze the infinite-horizon asymptotics. We show that the limit of the optimal state-input trajectory is the unique solution to a corresponding stochastic stationary optimization problem in the sense of probability measures. Moreover, we provide a constructive error analysis for finite-dimensional polynomial chaos approximations of the optimal solutions and of the optimal stationary pair in non-Gaussian settings. A numerical example illustrates our findings.

Figures (4)

• Figure 1: Optimal trajectories of OCP \ref{['eq:StochPCEj']} in PCE coefficients $\IfNoValueTF{-NoValue-} {\textsf{x}} {\textsf{x}^{-NoValue-}}^{j,\star}$, $j\in\mathbb{I}_{[-2,N-1]}$.
• Figure 2: Truncation of PCE coefficients.
• Figure 3: Trajectories of PCE coefficients $\{( \IfNoValueTF{-NoValue-} {\textsf{x}} {\textsf{x}^{-NoValue-}}_k^{j,\star}, \IfNoValueTF{-NoValue-} {\textsf{u}} {\textsf{u}^{-NoValue-}}_k^{j,\star})\}_{k=0}^{29}$, $j\in\mathbb{I}_{[-2,29]}$. Red-dashed line: Expectation of $(\mu_X^\diamond,\mu_U^\diamond)$.
• Figure 4: Approximation of $(\bar{X}^\star,\bar{U}^\star)$ via truncated PCEs for different $p$. Left: PDFs of $(\bar{X}^\star,\bar{U}^\star)$ and of $(\bar{X}^{\text{trun},\star},\bar{U}^{\text{trun},\star})$; right: PDFs of $(\Delta\bar{X}^{\star},\Delta\bar{U}^{\star})$.

Theorems & Definitions (25)

• Definition 1: Polynomial chaos expansion
• Definition 2: Exact PCE representation
• Proposition 1: Exact uncertainty propagation
• Definition 3: Overtaking optimality
• Lemma 1: Optimal solution via PCE
• proof
• Proposition 2: PCE coefficient trajectories
• proof
• Lemma 2: Quantification of truncation errors
• proof