Operator theoretic framework for optimal placement of sensors and actuators for control of nonequilibrium dynamics

Abstract

In this paper, we present a novel operator theoretic framework for optimal placement of actuators and sensors in nonlinear systems. The problem is motivated by its application to control of nonequilibrium dynamics in the form of temperature in building systems and control of oil spill in oceanographic flow. The controlled evolution of a passive scalar field, modeling the temperature distribution or density of oil dispersant, is governed by a linear advection partial differential equation (PDE) with spatially located actuators and sensors. Spatial locations of actuators and sensors are optimized to maximize the controllability and observability regions of the linear advection PDE. Linear transfer Perron–Frobenius and Koopman operators, associated with the advective velocity field, are used to provide an analytical characterization for the controllable and observable spaces of the advection PDE. Set-oriented numerical methods are proposed for the finite dimensional approximation of the linear transfer operators. The finite dimensional approximation is shown to introduce weaker notion of controllability and observability, referred to as coarse controllability and observability. The finite dimensional approximation is used to formulate the optimization problem for the optimal placement of sensors and actuators. The optimal placement problem is a combinatorial optimization problem. However, the positivity property of the linear transfer operator is exploited to provide an exact solution to the optimal placement problem using greedy algorithm. Application of the framework is demonstrated for the placement of sensors in a building system for the detection of contaminants and for optimal release of dispersant location for control of contaminant in a Double Gyre velocity field. Simulation results reveal interesting connections between the optimal location of actuators and sensors, maximizing the controllability and observability regions respectively, and the coherent structures in the fluid flow.