# (747e) Control of Unstable Heat Equation On Two-Dimensional Time-Varying Domain Using Empirical Eigenfunctions

- Conference: AIChE Annual Meeting
- Year: 2012
- Proceeding: 2012 AIChE Annual Meeting
- Group: Computing and Systems Technology Division
- Session:
- Time:
Thursday, November 1, 2012 - 4:35pm-4:55pm

Many industrial transport-reaction nonlinear processes

involve the change in the shape of the material and domain of analysis as a

result of phase change, chemical reaction, external forces, and mass transfer.

The mathematical models of such processes are obtained from conservation laws,

such as mass, momentum and/or energy, and usually form nonlinear partial

differential equations (PDEs).

The approach of modal decomposition, which implies that the

dominant behavior of PDE systems can be approximately described by

finite-dimensional systems, is extensively used for the controller synthesis of

PDE systems. A well-known methodology in the extraction of eigenfunctions of

nonlinear PDEs is the use of Karhunen-Loeve (KL)

decomposition on an ensemble of solutions obtained from numerical or

experimental resolution of the system. These modes, known as empirical

eigenfunctions, are used in the derivation of accurate nonlinear reduced-order

approximations of many diffusion-reaction systems and fluid flows.

Compared to the extensive research efforts on the

order-reduction of distributed parameter systems modelled

by PDEs, there are only few studies to address model-reduction and control of

PDE systems with spatially time-varying domain. Assuming that the evolution of

domain is known a priori, which can be measured in many processes, KL

decomposition cannot be directly applied to the solutions of PDEs with

time-varying domain. Armaou and Christofides

used a mathematical transformation to represent the nonlinear PDE on an

appropriate time-invariant domain and applied KL decomposition to obtain the

set of eigenfunctions on the fixed domain [1,2]. In the study of the internal

combustion engine flows by Fogleman et al., the velocity

fields are stretched in one dimension to obtain data on a fixed grid such that

the divergence of the original velocity field (continuity) is preserved [3].

Following these contributions, one way to deal with the aforementioned problem

is to map the set of the solutions on a time-invariant domain and then apply KL

decomposition, however, different mappings could change the energy content of

the solutions. The idea here is to map the solutions of the PDE system on a fixed reference geometry while preserving the invariance

of physical properties (energy) of the solutions.

To find the control law to stabilize the unstable

steady-state of a nonlinear heat-equation on a two-dimensional spatially

time-varying domain, we first map a set of the solutions of the nonlinear PDE

describing the system behavior to a reference configuration while preserving

the invariant property of thermal energy. A basis can be found by using the KL

decomposition on the mapped solutions, and by applying the inverse mapping, a

set of time-varying empirical eigenfunctions are obtained that capture the most

energy of the system. Subsequently, the empirical eigenfunctions are used as a

basis for Galerkin method to derive the reduced-order

ODE model that accurately captures the dominant dynamics of the PDE system.

Finally, the ODE system is used for the synthesis of nonlinear controller.

**References:**

1. A. Armaou, P. D. Christofides, Nonlinear Feedback Control

of Parabolic Partial Differential Equation Systems with Time-dependent Spatial

Domains, J. Math. Anal. Appl. 239 (1), 1999, 124-157.

2. A. Armaou, P.

D. Christofides, Finite-Dimensional Control of

Nonlinear Parabolic PDE Systems With Time-Dependent Spatial Domains Using

Empirical Eigenfunctions, Int. J. Appl. Math. Comput.

Sci. 11 (2), 2001, 287-317.

3. M. Fogleman, J. Lumley, D. Rempfer, D. Haworth, Application of the Proper Orthogonal Decomposition to

Datasets of Internal Combustion Engine Flows, J. Turbul.

5, 2004, 023.

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