(375k) Feedback Control of Distributed Processes Using Adaptive Proper Orthogonal Decomposition with Partial Information; The Gappy APOD Method

Authors: 
Pitchaiah, S. - Presenter, The Pennsylvania State University
Armaou, A. - Presenter, Pennsylvania State University


The problem of feedback control of spatially distributed processes described by dissipative partial differential equations(PDEs) is considered. Typically, this problem is addressed through model reduction where finite dimensional approximations to the original PDE system are derived. An usual approach used in the formulation of the aforementioned reduced order model is the proper orthogonal decomposition (POD) along with method of snapshots. This methodology, however, requires an availability of large representative ensemble of snapshots. Generating such an ensemble is not straightforward (experimentally infeasible) as it necessitates using a suitably designed input to excite all the modes. To address these concerns, we have formulated the adaptive proper orthogonal decomposition methodology (APOD) [5], [6], wherein initially the basis functions required for the construction of the reduced order model were derived from a small ensemble of snapshots that does not represent the entire process. These basis functions were updated, in a computationally efficient way, as new data from the process becomes available.

An unavoidable assumption of the above data driven model reduction methodologies for optimization and controller synthesis is the necessity for ?complete? snapshots (in the sense the profiles must span the whole process domain). As expected, even though APOD offers a computationally feasible methodology for the implementation of online feedback control in distributed process, it is based on a data driven methodology and consequently it requires complete information of the snapshot over the spatial domain of the process. However, obtaining such information might be infeasible owing to high sensor costs and limited availability of sensors [2].

In this work, we extend the applicability of APOD to situations wherein the availability of large number of sensors is restricted. We thus develop a new methodology which we name adaptive gappy proper orthogonal decomposition (AGPOD). Inspired by the results on Gappy POD [4], [7], in AGPOD we reconstruct the snapshots from the available partial information, obtained from restricted point measurements; we thus avoid the necessity for large number of sensors. To demonstrate the effectiveness of AGPOD, we apply the proposed methodology to stabilize an unstable uniform steady state of Kuramoto-Sivashinksy equation (KSE) [3], [1], a dissipative PDE model that describes incipient instabilities in a variety of physical and chemical systems, using limited information available from restricted number of point measurement sensors.

REFERENCES [1] A. ADROVER AND M. GIONA, Modal reduction of PDE models by means of snapshot archetypes, Physica D, 182 (2003), pp. 23?45. [2] 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 (2004), pp. 27?45. [3] A. ARMAOU AND P. D. CHRISTOFIDES, Dynamic optimization of dissipative PDE systems using nonlinear order reduction, Chem. Eng. Sci., 57 (2002), pp. 5083?5114. [4] R. EVERSON AND L. SIROVICH, The karhunen loeve procedure for gappy data, J. Opt. Soc. Am, 12 (1995), pp. 1657?1664. [5] S. PITCHAIAH AND A. ARMAOU, Output feedback control of distributed parameter systems using adaptive proper orthogonal decomposition, Ind. & Eng. Chem. Res., accepted (2010). [6] A. VARSHNEY, S. PITCHAIAH, AND A. ARMAOU, Feedback control of dissipative distributed parameter systems using adaptive model reduction, AICHE J., 55 (2008), pp. 906?918. [7] K. WILLCOX, Unsteady flow sensing and estimation via the gappy proper orthogonal decomposition, Computers & Fluids, 35 (2006), pp. 208?226.

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