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Optimal Field Reconstruction of Distributed Process Systems from Partial Measurements

Míriam R. García, Carlos Vilas, Julio R. Banga, and Antonio A. Alonso. Process Engineering, IIM-CSIC, Eduardo Cabello 6, Vigo, Spain

Monitoring and control of chemical processes are extremely conditioned by the on-line measurements available. Aspects such as the type and number of hardware sensors are at a high extend dependent by the cost and reliability of the process variables under consideration, what calls for software sensors capable of efficiently reconstruct the current state of the process variables. In fact, these aspects are especially relevant when dealing with processes exhibiting complex nonlinear distributed dynamics as it is the case of pattern formation in fluid dynamics or reaction-diffusion systems, and extremely important when taking place in 2D or 3D spatial domains. In that context, capturing the essential dynamic features of the field (without hardware sensors covering the whole domain) demands systematic methodologies for efficient field reconstruction from partial measurements optimally placement in the spatial domain.

Early approaches, starting back in the 70’s, combine classical discretization techniques, such as finite differences method (FDM) or finite element method (FEM), to approximate the partial differential equations (PDEs) by ordinary differential equations (ODEs) with criteria to place sensors so to maintain the covariance matrix well conditioned. However, these approaches present two important disadvantages: the use of exhaustive search procedures for the systematic selection of optimal sensor location (useful only for a few number of sensors) and field reconstruction from local basis, such as those used in FEM and FDM, which requires the solution of a large number of ODEs.

Alternative approaches include the one proposed by Antoniades and Christofides [1] to solve the placement problem by standard unconstrained optimization taking advantage of the time scale separation properties of transport-reaction systems. The approach, although elegant, requires the process to be under control and restricts the number of sensors to the dimension of the slow dynamics. In [2],[3] the optimal sensor selection problem was formulated for large number of sensors as that of minimizing the distance between the measurement subspace and the subspace which capture systems dynamics using guided search algorithms.

In this contribution, we make use of the dissipative nature of diffusion-convection systems to produce reduced order approximations expressed in terms of globally defined basis functions such as those obtained by spectral methods or by the so-called Proper Orthogonal Decomposition (POD) method. However, efficient computation of these global functions involves the solution of an eigenvalue problem associated with operators such as spatial integrals. For that propose, we exploit the underlying structure of the FEM to identify the algebraic equivalents of the corresponding infinite dimensional operators. Such algebraic structure is also employed to extend the classical sensor placement problem by searching among regions or subdomains of measurements (partitions of the whole spatial domain suitable for placing sensors arrays) thus ensuring its solvability using guided search methods as the ones developed by [2],[3]. The ideas proposed in this work have been illustrated on reconstruction temperature and velocity field on the well known natural convection Rayleigh-Bénard Problem.

[1] C. Antoniades and P.D. Christofides. Integrating nonlinear output feedback control and optimal actuator/sensor placement for transport-reaction processes. Chem. Eng. Sci., 56(15): 4517-4535, 2001.

[2] A.A. Alonso, C.E. Frouzakis, and I.G. Kevrekidis. Optimal sensor placement for state reconstruction of distributed process systems. AIChE Journal, 50: 1438-1452, 2004.

[3] A.A. Alonso, I.G. Kevrekidis, J.R. Banga, and C.E. Frouzakis. Optimal sensor location and reduced order observer design for distributed process systems. Comput. Chem. Eng., 28(1-2): 27-35, 2004.