Technical Report 2000-019

Reduced order models for diffusion systems

Torsten Söderström and Bharath Bhikkaji

August 2000


Mathematical models for diffusion processes like heat propagation, dispersion of pollutants etc., are normally partial differential equations which involve certain unknown parameters. To use these mathematical models as the substitutes of the true system, one has to determine these parameters.

Partial differential equations (PDE) of the form beapartial u(x,t)/partial t = L u(x,t)  (eq1.1)eea where L is a linear differential (spatial) operator, describe infinite dimensional dynamical systems. To compute a numerical solution for such partial differential equations, one has to approximate the underlying system by a finite order one. By using this finite order approximation, one then computes an approximate numerical solution for the PDE.

We consider a simple case of heat propagation in a homogeneous wall. The resulting partial differential equation, which is of the form (eq1.1), is approximated by finite order models by using certain existing numerical techniques like Galerkin and Collocation etc. These reduced order models are used to estimate the unknown parameters involved in the PDE, by using the well developed tools of system identification.

In this paper we concentrate more on the model reduction aspects of the problem. In particular, we examine the model order reduction capabilities of the Chebyshev polynomial methods used for solving partial differential equation.

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