LQR for Systems with Probabilistic Parametric Uncertainties: A Gradient Method
This work addresses robust control design for uncertain systems, offering an incremental improvement in computational efficiency for domain-specific applications.
The paper tackles the LQR problem for linear systems with probabilistic parametric uncertainties by proposing a gradient-based method that leverages polynomial chaos theory and policy optimization, achieving significantly higher computational efficiency than conventional BMI-based approaches.
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.