Linear Quadratic Gaussian Synthesis for a Heated/Cooled Rod Using Point Actuation and Point Sensing

TL;DR

This paper develops an LQG control strategy for a heated/cooled rod with point actuators and sensors, explicitly solving for optimal feedback and state estimation despite limited measurements.

Contribution

It introduces an explicit LQG synthesis for the heat equation with point actuation and sensing, combining a finite-dimensional Kalman filter with a linear quadratic regulator.

Findings

Explicit solution for optimal feedback control.

Construction of an infinite-dimensional Kalman filter.

Effective stabilization of the temperature profile.

Abstract

We consider a rod that is heated/cooled and sensed at multiple point locations. To stabilize it to a constant temperature we set up a Linear Quadratic Regulator that we explicitly solve by the method of completing the square to find the optimal linear state feedback for the point actuators. But we don't assume that the whole state is mesureable so we construct an infinite dimensional Kalman filter to estimate the whole state from a finite number of noisy point measurements. These two components yield a Linear Quadratic Gaussian (LQG) Synthesis for the heat equation under point actuation and point sensing.