Optimal Sensor Placement for Observer Design

Abstract

This paper is motivated by sensor location problems for infinite dimensional linear distributed parameter systems (DPS) defined by partial differential equations (PDEs). In this setting the term “sensor location” usually refers to a spatial location. However, the problem is also meaningful in finite dimensional models and results in a parameterization of the system output matrix. The problem of “optimal” sensor (and actuator) location has a long history and plays an important role in controller design, system identification and state estimation and, depending on the application, can be formulated in many ways. Any optimal location problem implies that there is a cost function to be minimized and the choice of this cost function places specific requirements on the computational methods needed to solve the problem. In addition, as shown in the papers Yang and Morris [2014, 2015] choices such as “optimizing observability” can lead to ill-posed problems and result in non convergence of suboptimal placements. On the other hand, minimum error variance methods (Kalman filtering) can be shown be well-posed. Efficient computational methods for solving the corresponding Riccati partial differential equation have been developed and are readily available. In the classic paper by Luenberger [1964], he introduced the term “observer” for deterministic state estimation to distinguish it from the Kalman filter. In this paper we focus on state estimators based on the Luenberger observer and suggest some natural cost functions for sensor location. The framework is fairly general and applies to both finite and infinite dimensional systems. Examples are given to illustrate the ideas.