# Model reduction and identification of heat diffusion systems

Diffusion is a phenomenon in which particles move from regions of higher density to regions of lower density. Many physical systems, in fields as diverse as plant biology and finance, are known to involve diffusion phenomena. Typically, diffusion systems are modeled by partial differential equations (PDEs), which include certain parameters. These parameters characterize a given diffusion system. Therefore, for both modeling and simulation of a diffusion system, one has to either know or determine these parameters. Moreover, as PDEs are infinite order dynamic systems, for computational purposes one has to approximate them by a finite order model. In this project we have investigated the two issues of model reduction and parameter estimation by considering certain specific cases of heat diffusion.

Most of the results obtained are summarized in the theses [1], [2], [11].

A one-dimensional heat diffusion model has been used for several aims concerning model reduction and parameter estimation. It describes heat diffusion across a homogenous wall. The dynamics and the boundary conditions are given by

Note that *q_i (t)* is the heat supplied from the interior, *T_e(t)* is exterior temperature, and *a* and *k* are the unknown parameters. More specifically, *a* is the diffusion coefficient and *k* is the thermal conductivity. We regard *q_i(t)* and *T_e(t)* as input signals and *T_i(t)* as an output signal.

We have constructed several approximate models for the one dimensional PDE model using various techniques. In particular, we use numerical schemes such as Chebyshev-Galerkin, Chebyshev-Tau, Chebyshev-Interpolation and Chebyshev-Collocation to construct approximate models, [1], [6], [12]. In [11] and [13], we use the finite difference approximation to construct the approximate models, and in [7] we have used the fact that the given PDE model involves dynamics with varying time constants to construct approximate models.

We have extended part of the work in [11] to a two-dimensional case, see [3]. The model and its boundary conditions read

Parameter estimation for the one-dimensional model has been investigated in [13], [8]. We have examined convergence properties and derived an approximate asymptotic covariance matrix of the parameter estimates.

Recursive identification schemes have been derived. In [9] we adopt the standard recursive prediction error method (RPEM). Further, in [4], [5], [10] we derived and analysed a recursive algorithm based on a sliding time window and a frequency domain approach.

Current and former researchers:

Bharath Bhikkaji, Kaushik Mahata, Susanne Remle and Torsten Söderström.

At an early stage the project was supported by the Swedish Research Council for Engineering Science. A previous project summary is available.

## References

[1] B. Bhikkaji. Reduced Order Models for Diffusion Systems. Licentiate thesis, Department of Information Technology, Uppsala University, Uppsala, 2000.

[2] B. Bhikkaji. Model reduction and parameter estimation for diffusion systems. PhD thesis, (Comprehensive summaries of Uppsala Dissertations from the Faculty of Science and Technology, 974), Uppsala University, 2004.

[3] B. Bhikkaji, K. Mahata and T. Söderström. Reduced order models for a two-dimensional diffusion system. To appear in *International Journal of Control*.

[4] Bhikkaji, K. Mahata and T. Söderström. Recursive algorithm for estimating parameters in a one-dimensional heat diffusion system: derivation and implementation. Submitted for publication.

[5] Bhikkaji, K. Mahata and T. Söderström. Recursive algorithm for estimating parameters in a one-dimensional heat diffusion system: analysis. Submitted for publication.

[6] B. Bhikkaji and T. Söderström. Reduced order models for diffusion systems. *International Journal of Control*, vol 74, no 15, pp 1543-1577, October 2001.

[7] B. Bhikkaji and T. Söderström. Reduced order models for diffusion systems using singular perturbations. *Journal of Energy and Buildings*, vol 33, no 8, 769-781, October 2001.

[8] B. Bhikkaji and T. Söderström. Bias and variance of parameter estimates for a one dimensional heat diffusion system. *Proc. of 15th IFAC World Congress*, Barcelona, Spain, 25-26 July 2002.

[9] B. Bhikkaji and T. Söderström. Recursive algorithm for estimating parameters in a one-dimensional diffusion system. National Control Conference (Reglermöte 2002), Linköping, May 2002.

[10] Bhikkaji, T. Söderström and K. Mahata. Recursive algorithm for estimating parameters in a one-dimensional heat diffusion system: derivation and implementation. *Proc. of 13th IFAC Symposium on System Identification*, Rotterdam, The Netherlands, August 2003.

[11] S. Remle. Modeling and Parameter Estimation of The Diffusion Equation. Licentiate thesis, Department of Information Technology, Uppsala University, Uppsala, 2000.

[12] T. Söderström and B. Bhikkaji. Reduced order models for diffusion systems via collocation methods. *Proc of 12th IFAC Symposium on System Identification*, Santa Barbara, CA, 21-23 June 2000.

[13] T. Söderström and S. Remle. Parameter estimation for diffusion models. *Proc of 12th IFAC Symposium on System Identification*, Santa Barbara, CA, 21-23 June 2000.