Optimal sensor placement for state reconstruction of distributed process systems

Abstract

In this contribution we propose a systematic approach to field reconstruction of distributed process systems from a limited and usually reduced number of measurements. The method exploits the time scale separation property of dissipative processes and concepts derived from principal angles between subspaces, to optimally placing a given number of sensors in the spatial domain. Basic ingredients of the approach include the identification of a low-dimensional subspace capturing most of the relevant dynamic features of the distributed system, and the solution of a max–min optimization problem through a guided search technique. The low-dimensional subspace can be defined either through a spectral basis (eigenfunctions of a linear or linearized part of the operator) or through a semiempirical expansion known in the engineering literature as the Proper Orthogonal Decomposition (POD) or Karhunen–Loeve expansion. For both cases, the optimal sensor placement problem will be solved by taking advantage of the underlying algebraic structure of the low-dimensional subspace. The implications of this approach for dynamic observer design will be discussed together with examples illustrating the proposed methodology.