Numerical Discretization Methods for the Discounted Linear Quadratic Control Problem

TL;DR

This paper develops and compares three numerical discretization methods for solving the discounted linear-quadratic control problem with time delays, highlighting the efficiency of the step-doubling method.

Contribution

It introduces a novel formulation of the discrete system matrices as differential equations and compares three numerical methods for their solution.

Findings

All three methods accurately solve the differential equations.

The step-doubling method is the fastest while maintaining accuracy.

The methods effectively handle the time-delayed discounted LQ-OCP.

Abstract

This study focuses on the numerical discretization methods for the continuous-time discounted linear-quadratic optimal control problem (LQ-OCP) with time delays. By assuming piecewise constant inputs, we formulate the discrete system matrices of the discounted LQ-OCPs into systems of differential equations. Subsequently, we derive the discrete-time equivalent of the discounted LQ-OCP by solving these systems. This paper presents three numerical methods for solving the proposed differential equations systems: the fixed-time-step ordinary differential equation (ODE) method, the step-doubling method, and the matrix exponential method. Our numerical experiment demonstrates that all three methods accurately solve the differential equation systems. Interestingly, the step-doubling method emerges as the fastest among them while maintaining the same level of accuracy as the fixed-time-step ODE method.