(568y) Estimation of Distributed Parameter Systems Using Recursively Updated Empirical Basis Functions

While many aspects of state estimation and monitoring have extensively been studied for lumped processes described by ordinary differential equations (ODEs), little work can be found for distributed parameters systems described by partial differential equations (PDEs). The infinite dimensional representation of such processes exemplified by microelectronic fabrications can be approximated using a finite number of ODEs. Such ODE system is constructed by discretization of the PDEs using Galerkin’s method [1]. The set of basis functions needed to discretize the system is initially computed and then recursively updated using adaptive proper orthogonal decomposition (APOD) [2], [3]. Thus the estimation problem of distributed parameter system with infinite points inside the domain is reduced to computing the states of reduced order model described by a set of ODE.

Initially, a static observer based on the continuous point measurements is used to estimate the states of ROM. Static observers have two major issues; numerous measurement sensors are required otherwise the static observer gives erroneous estimates, also the static observer existence depends on the location and shape of the measurement sensors. To overcome these issues, we employed a dynamic observer that conceptually needs only one point measurement to predict the dynamic behavior of the modes. Assuming the periodic availability of the process snapshots and continuous availability of point measurements from a restricted number of sensors an APOD-based Luenberger-like dynamic observer is synthesized.

An unavoidable assumption of previous APOD-based observer that employ data driven model reduction technique is the necessity for complete snapshots (in the sense the profiles must span the whole process domain). As expected, even though APOD relaxes the requirement for representative snapshot ensemble, it requires “spatially complete” snapshots. However, obtaining such information might not be feasible owing to high sensor costs and limited availability of sensors [4]. In recent years there has been considerable effort to relax this assumption in the field of model reduction; substituting it with the assumption that periodically all the regions of the process domain are sampled. The basic premise of the research work is that a set of mobile sensors achieve better estimation performance than a set of immobile sensors. To enhance the performance of the state estimator, a network of sensors that are capable of moving within the spatial domain is utilized [5]. Specifically, such an estimation process is achieved by using a set of spatially distributed mobile sensors. The objective is to provide mobile sensor control policies that aim to improve the state estimate. The metric for such an estimate improvement is taken to be the expected state estimation error. The effectiveness of the proposed estimation methods is successfully illustrated on monitoring of spatiotemporal temperature dynamics in a catalytic reactor.

 

[1] P. D. CHRISTOFIDES, Nonlinear and robust control of PDE systems, Birkhauser, New York, 2000.  

[2] D. BABAEI POURKARGAR and A. ARMAOU, Design of APOD-based switching dynamic observers and output feedback control for a class of nonlinear distributed parameter systems, J. Proc. Cont., to appear (2014).

[3] D. BABAEI POURKARGAR and A. ARMAOU, Modification to adaptive model reduction for regulation of distributed parameter systems with fast transients, AIChE J., 59(12) (2013), pp. 4595–4611.

[4] 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. Comp. & Chem. Eng., 28:27–45, 2004.

[5] M. A. DEMETRIOU and I. I. HUSSEIN, Estimation of spatially distributed processes using mobile spatially distributed sensor network. SIAM J. Control Optim., 48(1):266-291, 2009.