Uppsala University
Department of Information Technology

### Licentiate thesis 2000-011

# Model Reduction for Diffusion Systems

### Bharath Bhikkaji

December 2000

**Abstract:**
Diffusion phenomena has been studied with a lot of interest, for a long time, due to its historical and practical significance. In the recent days it has thrown a lot of interest among control engineers, as more and more practical systems, varying from stock markets to environmental pollution, have been observed to involve diffusion.

Diffusion systems are normally modeled by linear partial differential equations (LPDE's) of the form

*partial T(x,t)/partial x = ***L**T(x,t) (eq1.0)

where **L** is a second order linear spatial differential operator and * T(x,t) * is the physical quantity, whose variations in the spatial domain cause diffusion. To characterise diffusion phenomena, one has to obtain the solution of (eq1.0) either analytically or numerically. Note that, since (eq1.0) involves a second order spatial operator and a first order time derivative, one needs at least two boundary conditions in the spatial domain, *x*, and an initial condition at time *t=0*, for determining * T(x,t) *.
LPDE's of the type (eq1.0) can be interpreted as infinite order linear time invariant systems (LTI systems) with inputs as boundary conditions. To compute the solution of (eq1.0) numerically, one has to approximate, explicitly or implicitly, the underlying infinite order system by a finite order system. Any numerical scheme, which computes the solution of (eq1.0), essentially approximates the underlying infinite order LTI system by a finite order LTI system. The efficiency of the approximation, for a given problem, varies for the different numerical schemes.

In this thesis, we make an attempt to explore more about diffusion systems in general. As a starting point, we consider a simple case of one dimensional heat diffusion across a homogeneous region. The resulting LPDE is first shown explicitly to be an infinite order dynamical system. An approximate solution is computed from a finite order approximation of the true infinite order dynamical system. In this thesis, we first construct the finite order approximations using certain standard PDE solvers based on Chebyshev polynomials. From these finite order approximations we choose the best one, from a model reduction perspective, and use it as a benchmark model. We later construct two more approximate models, by exploiting the given structure of the problem and we show by simulations that these models perform better than the chosen benchmark.

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