  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.
