##
Svein Hovland, Tommy Gravdahl

### Stabilizing a CFD model of an unstable system through model reduction

In this paper we consider stabilization of a computational fluid dynamics CFD model of an unstable system. As a case study, cooling of a two-dimensional plate with an inner heat source with positive feedback from the temperature is considered. The plate temperature is originally described by a partial differential equation PDE. In order to simulate and design a controller for the system, the PDE is discretized using computational fluid dynamics.

The open-loop CFD model has unstable eigenvalues, and it is demonstrated that without control, the plate temperature is increasing above desired limits due to the inner heat source. By introducing feedback control through cooling on the boundaries of the plate, we wish to control the plate temperature to reach a desired uniform distribution.

Although acceptable for simulation purposes, the CFD model is of such high order that it is infeasible to use it in control design. This is often the case in computational fluid dynamics, necessitating model order reduction tools.

Based on simulation data and the CFD model equations, the order of the model is reduced by proper orthogonal decomposition POD. POD, also known as the Karhunen-Lograveve Expansion, is a method for reducing the complexity of a model while keeping the essential dynamics of the original model, in order to perform the control design. The resulting reduced order model is unstable, but it has a state space structure, which is convenient for controller synthesis.

A stabilizing controller based on pole placement is designed for the reduced order model. To account for stationary errors and the deviation between the full CFD model and the reduced order model, integral action is included to enhance performance. The controller is then extended to the full CFD model, where it is shown through simulations to stabilize the system around the desired temperature distribution.

The demonstrated procedure makes it possible to analyze stability properties of a class of systems that would otherwise be very computationally demanding.