(589c) Improved Model Reduction Approach for Output Feedback Control of Fast Evolving Spatially Distributed Processes

In recent years the interest in control of spatially distributed systems has significantly increased due to the need to synthesize controllers for complex transport-reaction processes that are characterized by the coupling of chemical reaction with significant convection, diffusion, and dispersion phenomena. Such processes as exemplified by catalytic reactions, polymerization processes, plasma etching and semiconductor manufacturing, exhibit spatial variations that need to be explicitly accounted by the controller.

An important observation is that the long term behavior of the above chemical processes can be captured by a finite number of degrees of freedom thus the infinite dimensional partial differential equation (PDE) description can be approximated by reduced order models (ROMs) in the form of ordinary differential equations (ODEs) [3]. A widely used method to compute the ROM, is the method of weighted residuals. The prerequisite basis functions can be obtained either analytically or using statistical methods such as proper orthogonal decomposition (POD). POD generates optimal orthogonal empirical eigenfunctions that capture the energy of a data set to describe the data trends. The POD approach assumes a priori availability of a sufficiently large ensemble of PDE solution data which excites the most prevalent spatial modes. Such an ensemble is necessary to characterize the behavior of the system dynamics and compute empirical eigenfunctions in this method. However, in practice, it is difficult to generate such an ensemble so that all possible dominant spatial modes are appreciably contained within the corresponding snapshots. The resulting eigenfunctions, therefore, are representative of the corresponding ensemble only. During closed-loop simulation, situations may arise when the existing eigenfunctions fail to accurately represent the dynamics of the PDE system.

A solution is to initially compute the eigenfunctions using the available ensemble of snapshots, and then to recompute the eigenfunctions as needed when more information from the system becomes available. However, this would require the solution of the eigenvalue-eigenvector problem every time, which may become computationally expensive, and hence, unsuitable for online computations as the process evolves. To reduce the computational load and to circumvent the limitation of sufficiently large ensemble of profile data, the recursive computation of eigenfunctions, known as adaptive proper orthogonal decomposition (APOD), could be used as additional data from the process becomes available, also designing tailored feedback control structures [4]. The requirements on continuous measurement sensors were then reduced using APOD-based dynamic observers [1, 2]. The performance of the controller structure hinges on the frequency of the sampling which must be of the same order as the frequency of the appearance of new trends.

In this paper, we focus on output feedback control of fast evolving distributed parameter systems based on continuous point measurements available from limited number of sensors and infrequent distributed snapshots. The developed methodology for robust output feedback control is based on the successful integration of dynamic observers with static controllers. A refined ensembling approach is used in APOD to recursively update the eigenfunctions as the closed-loop process evolves through different regions of the state space based on maximizing retained information that is received from the infrequent distributed sensor measurements. The proposed controller is illustrated on the Kuramoto-Sivashinksy equation with and without uncertainty in the presence of highly nonlinear dynamics and chaotic behavior, where they are called to stabilize the system at an open-loop unstable system steady state. The original and the modified ensemble construction approaches for APOD are compared in different conditions and the robustness of modified APOD with respect to uncertainty, number of snapshots and number of continuous point sensors and their locations is illustrated.

[1] D. Babaei Pourkargar and A. Armaou. Control of dissipative partial differential equation systems using APOD-based dynamic observer designs. In Proceedings of the American Control Conference, Washington, DC, in press, 2013.

[2] D. Babaei Pourkargar and A. Armaou. Output feedback control of distributed parameter systems using APOD-based dynamic observer designs. Automatica, submitted, 2013.

[3] P. D. Christofides. Nonlinear and robust control of PDE systems. Birkh ¨ auser, New York, 2000.

[4] S. Pitchaiah and 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.