(555f) Adaptive Control of Dissipative Distributed Parameter Systems Via Recursively Updated Reduced Order Models

In dissipative distributed parameter systems (DPSs) with underlying physical and chemical spatiotemporal phenomena such as chemical reactions, diffusion, convection and dispersion, physical properties and reaction parameters are often unknown. Such systems exemplified by flow reactors, chemical vapor deposition systems and microelectronic fabrication processes, can be mathematically modeled by dissipative partial differential equations (PDEs) with unknown parameters [1]. Thus, we may need to identify the unknown parameters during process evolution to design the control structure for such systems.

Adaptive control and system identification methods have been used extensively in estimation and control of linear DPSs over regular domains [2], [3]. One of the limitations of current techniques is that they can not directly be applied to neither general linear DPSs over irregular domains nor nonlinear DPSs. Such limitations may be circumvented via model reduction. In this work, we focus on dissipative systems whose infinite dimensional representation of the PDE dynamics can be approximated by reduced order finite dimensional models [1]. The recursively updated reduced order model (ROM) of the system is constructed by discretization of the PDEs using Galerkin’s method. The set of basis functions needed for discretizing the system is initially computed and then recursively updated using adaptive proper orthogonal decomposition [4], [5]. Then the resulting recursively updated ROM that includes unknown parameters due to unknown reaction and diffusion terms in the PDE system, is considered as the basis for Lyapunov-based adaptive control design. The effectiveness of the proposed control structure is successfully illustrated on regulation of the spatiotemporal dynamic of temperature in a catalytic reactor with unknown reaction and diffusion parameters.

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

[2] M. BOHM, M. A. DEMETRIOU, S. REICH, AND I. G. ROSEN, Model reference adaptive control of distributed parameter systems, SIAM Journal on Control and Optimization, 36 (1998), pp. 33–81.

[3] A. SMYSHLYAEV AND M. KRSTIC, Adaptive control of parabolic PDEs, Princeton Univ. Press, Princeton, NJ, 2010.

[4] 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).

[5] 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.