Actuator and sensor placement in linear advection PDE with building system applications

TL;DR

This paper develops a gramian-based method for optimal actuator and sensor placement in linear advection PDEs, with applications to building systems for temperature and contaminant control, exploiting the PDE's structure for explicit formulas.

Contribution

It introduces an explicit gramian formula for actuator and sensor placement in advection PDEs, leveraging flow properties for improved placement strategies.

Findings

Explicit gramian formulas for placement criteria

Flow stability influences sensor and actuator locations

Simulation validates theoretical placement strategies

Abstract

We study the problem of actuator and sensor placement in a linear advection partial differential equation (PDE). The problem is motivated by its application to actuator and sensor placement in building systems for the control and detection of a scalar quantity such as temperature and contaminants. We propose a gramian based approach to the problem of actuator and sensor placement. The special structure of the advection PDE is exploited to provide an explicit formula for the controllability and observability gramian in the form of a multiplication operator. The explicit formula for the gramian, as a function of actuator and sensor location, is used to provide test criteria for the suitability of a given sensor and actuator location. Furthermore, the solution obtained using gramian based criteria is interpreted in terms of the flow of the advective vector field. In particular, the almost everywhere stability property of the advective vector field is shown to play a crucial role in deciding the location of actuators and sensors. Simulation results are performed to support the main results of this paper.