LQR for Systems with Probabilistic Parametric Uncertainties: A Gradient Method

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.

💡 Research Summary

This paper addresses the linear‑quadratic regulator (LQR) problem for linear time‑invariant systems whose dynamics depend nonlinearly on time‑invariant random parameters. Classical robust designs treat uncertainties in a worst‑case sense, which is often overly conservative and computationally demanding, especially when the uncertain parameters appear non‑affinely. The authors propose a fundamentally different approach: they model the uncertain parameters probabilistically and employ polynomial chaos theory (PCT) to obtain a deterministic surrogate model of finite dimension.

Using an orthonormal polynomial basis {ϕ_i(ξ)} associated with the probability distribution of the random vector ξ, the state x(t,ξ) is expanded as x(t,ξ)≈Z_N(t)ϕ_N(ξ). Galerkin projection yields a lifted linear system
  ẋ̂_N(t)= (A_N – B_N (I_{N+1}⊗K)) x̂_N(t) ,
where A_N=E