Even though the state estimation and monitoring problems have reached certain maturity for lumped dynamic processes described by ordinary differential equations (ODEs), they remain challenging tasks for spatially distributed processes (SDPs). SDPs can be mathematically modeled by a set of partial differential equations (PDEs), frequently arise in chemical and advanced materials industries can be exemplified by packed and fluidized bed reactors, polymerization and crystallization processes, chemical vapor deposition processes, plasma discharge systems and microelectronic fabrication processes. The infinite dimensional representation of such systems can be approximated by a finite number of ODEs because a time-scale separation can in practice be identified between the finite-dimensional slow (and possibly unstable) and infinite-dimensional fast (and stable) subsystems . The resulting system approximations in the form of low-dimensional ODEs can be applied as the basis for Luenberger-type dynamic observer designs. Practically, such set of ODEs are constructed by discretizing governing PDEs using Galerkin’s method. However, in general the required basis functions by Galerkin’s method cannot be computed by analytical methods, specifically when the describing PDEs are nonlinear and/or the process operates over irregular domains.
To extend the applicability of model reduction approaches to general SDPs, statistical approaches such as proper orthogonal decomposition (POD) may be applied to compute an optimal set of empirical basis functions. The essential requirement of the POD-like approaches is the a-priori availability of solution profiles of the governing PDEs which present the global spatiotemporal dynamics of the studied system. To overcome such vital requirement we employ adaptive proper orthogonal decomposition (APOD) to recursively compute this set of required empirical basis functions as needed during the process evolution [2, 3]. An unavoidable assumption of previously developed methodologies for dynamic observer synthesis that employ data driven model reduction techniques like APOD is that the complete snapshots which span the whole process domain are accessible as time evolves. Obtaining such information might not be feasible owing to high sensor costs and limited availability of sensors which sample all the regions of the process domain.
We propose two methods to estimate the required snapshots as a demarcation point to develop process estimation methodologies which relax both the complete snapshot assumption and the representative ensemble requirement; (1) using a PDE-based Luenberger-type dynamic observer which employs moving sensors information, (2) a combination of scanning sensor networks and APOD methodology. To extend the applicability of APOD to situations when the availability of distributed sensing is limited, we base the proposed work on Gappy-APOD . A small numbers of moving sensors that provide continuous local measurements are employed to recursively construct the approximate reduced order model.
The basic premise of the proposed method 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. 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 approaches are then successfully implemented on temperature estimation in a tubular reactor where the thermal spatiotemporal dynamics can be mathematically modeled by a PDE. Beyond the obvious savings on communication costs and sensor network size, the study of this problem rovides a tool for the assessment of the robustness of the synthesized dynamic observers and will allow us to identify the fundamental limits on the monitoring structure tolerance to communication suspension.
 P.D. CHRISTOFIDES, Nonlinear and robust control of PDE systems, Birkhauser, New York, 2000.
 D. BABAEI POURKARGAR, A. ARMAOU, Design of APOD-based switching dynamic observers and output feedback control for a class of nonlinear distributed parameter systems, Chem. Eng. Sci., DOI:10.1016/j.ces.2015.02.032, 2015.
 D. BABAEI POURKARGAR, A. ARMAOU, Modification to adaptive model reduction for regulation of distributed parameter systems with fast transients, AIChE J., 59(12): 4595–4611, 2013.
 S. PITCHAIAH, A. ARMAOU, Output feedback control of dissipative PDE systems with partial sensor information based on adaptive model reduction, AIChE J., 59(3), 747–760, 2013.
 M.A. DEMETRIOU, I.I. HUSSEIN, Estimation of spatially distributed processes using mobile spatially distributed sensor network. SIAM J. Control Optim., 48(1):266-291, 2009.
See more of this Group/Topical: Computing and Systems Technology Division