Numerical Discretization Methods for Linear Quadratic Control Problems with Time Delays
Zhanhao Zhang, Steen Hørsholt, John Bagterp Jørgensen
TL;DR
This work addresses the discretization of continuous-time linear-quadratic optimal control problems with time delays by modeling the weight matrices as differential equations and solving them with three numerical schemes: an ordinary differential equation (ODE) method, a matrix-exponential method, and a step-doubling method. The methods yield discrete-time equivalents of the LQ-OCP, enabling a CT-MPC implementation tested on a simulated cement mill system. Among the approaches, step-doubling delivers the fastest computation with accuracy comparable to the ODE method, while the matrix-exponential approach provides a reference solution for error assessment. The results demonstrate that the CT-MPC constructed from these discretizations can stabilize and effectively control the cement mill under disturbances and reference changes, highlighting the practical viability of LQ discretization for time-delayed control problems.
Abstract
This paper presents the numerical discretization methods of the continuous-time linear-quadratic optimal control problems (LQ-OCPs) with time delays. We describe the weight matrices of the LQ-OCPs as differential equations systems, allowing us to derive the discrete equivalent of the continuous-time LQ-OCPs. Three numerical methods are introduced for solving proposed differential equations systems: 1) the ordinary differential equation (ODE) method, 2) the matrix exponential method, and 3) the step-doubling method. We implement a continuous-time model predictive control (CT-MPC) on a simulated cement mill system, and the objective function of the CT-MPC is discretized using the proposed LQ discretization scheme. The closed-loop results indicate that the CT-MPC successfully stabilizes and controls the simulated cement mill system, ensuring the viability and effectiveness of LQ discretization.