Numerical optimal control for delay differential equations: A simultaneous approach based on linearization of the delayed state

TL;DR

This paper introduces a novel numerical optimal control method for delay differential equations by linearizing delayed states and discretizing with implicit Euler, demonstrated on a nuclear reactor example.

Contribution

It proposes a simultaneous control approach based on linearization of delays and implicit discretization, addressing challenges in controlling systems with time delays.

Findings

Linearization simplifies delay differential equations for control.

The implicit Euler method provides stable discretization.

Numerical example demonstrates practical applicability.

Abstract

Time delays are ubiquitous in industry, and they must be accounted for when designing control strategies. However, numerical optimal control (NOC) of delay differential equations (DDEs) is challenging because it requires specialized discretization methods and the time delays may depend on the manipulated inputs or state variables. Therefore, in this work, we propose to linearize the delayed states around the current time. This results in a set of implicit differential equations, and we compare the steady states and the corresponding stability criteria of the DDEs and the approximate system. Furthermore, we propose a simultaneous approach for NOC of DDEs based on the linearization, and we discretize the approximate system using Euler's implicit method. Finally, we present a numerical example involving a molten salt nuclear fission reactor.