The model of closed-loop control by thermostats: properties and numerical simulations

TL;DR

This paper introduces a mathematical model for closed-loop control of reaction-diffusion processes using multiple sensors and controllers, analyzing its properties, stability, and behavior under perturbations through theoretical and numerical methods.

Contribution

It develops a new model for multi-device closed-loop control, analyzing existence, uniqueness, and robustness, with numerical simulations illustrating its properties.

Findings

The model ensures existence and uniqueness of solutions.

Control system stability depends on weight configurations.

Numerical simulations demonstrate effective regulation near reference states.

Abstract

A closed-loop control of a reaction-diffusion type process is introduced. The control system consist of a finite number of control and measurement devices. The measurement devices collect information about the current state of the process. The control devices influence the process, responding to data obtained from the measurement devices. Each control device takes into account the data from all measurement devices. The rule of accounting the data from measurement devices by a single control device involves defining suitable weights for each pair of one control device and one measurement device. A weight reflects how important is a given measurement device to a given control device. The aim of this control system is to bring the process possibly close to a user defined reference state or trajectory. We are interested in a situation where the user can adjust the control system by choice of the control and measurement devices and the weights. For this reason, one of the aims of the preset work is to study the behavior under perturbations of these elements for the mathematical model realizing the above control concept. Moreover, we formulate and justify results concerning existence and uniqueness of solutions of the investigated model. Finally, numerical prototypes illustrating properties of the model are presented.