**An Optimal Instrumental Variable Identification Approach for Left Matrix Fraction Description Models**

**Mohamed AKROUM**, **Kamel HARICHE**

Department of Electrical and Electronic Engineering

M’hamed Bougara University, 35000 Avenue de l’independence, Boumerdes, Algeria

**Abstract**: The main contribution of this paper is the extension of the Simplified Refined Instrumental Variable (SRIV) identification algorithm for SISO systems to the identification of MIMO systems described by a Left Matrix Fraction Description (LMFD). The performance of the extended algorithm is compared to the well-known MIMO four-step instrumental variable (IV4) algorithm. Monte Carlo simulations for different signal to noise ratios are conducted to assess the performance of the algorithm.

**Keywords**: Multivariable System Identification, SRIV, LMFD, IV4, Steiglitz-McBride.

**Mohamed Akroum** received his Electrical Engineer degree in 1995 from M’hamed Bougara University and his Magister degree in 1998 from Houari Boumedienne University of Sciences and Technology. He worked in the Advanced Technologies Development Center in Algiers for two years and He is currently a Lecturer at M’hamed Bougara University and he is preparaing a Ph.D degree since 2003. His current research interests include MIMO system identification and multivariable automatic control.

**Kamel Hariche** received his M.Sc. and PhD. degrees from the University of Houston (USA) in 1978 and 1987 respectively. He is currently a Professor at M’hamed Bougara University. His research is related to linear and nonlinear systems as well as Automatic control of MIMO systems.

**>>Full text**

**CITE THIS PAPER AS**:

Mohamed AKROUM, Kamel HARICHE, **An Optimal Instrumental Variable Identification Approach for Left Matrix Fraction Description Models**, *Studies in Informatics and Control*, ISSN 1220-1766, vol. 17 (4), pp. 361-372, 2008.

**1. Introduction**

Advanced engineering applications require suitable mathematical models structures. These model structures are either obtained mathematically using physical laws or experimentally using system identification techniques.

Basically, System identification deals with the problem of obtaining “approximate” models of dynamic systems from measured input-output data.

Many different identification methods have been proposed for both SISO and MIMO systems. Among these we can mention the PEM and n4sid [1] for the identification of state space models, and the ARX, IVX and IV4 methods [1] for systems modeled by a Left Matrix Fraction Description. These methods have been implemented and are available in the Matlab System identification toolbox [2] .

An interesting identification algorithm was proposed by Young [3] [4] and is referred to as the Simplified Refined Instrumental Variable (SRIV). It is an optimal instrumental variable algorithm proposed for the identification of noisy SISO systems.

It is the purpose of this paper to extend the algorithm for the identification of noisy MIMO systems described by a Left Matrix Fraction Description. The performance of the extended algorithm is then compared to that of the MIMO IV4 algorithm used as a benchmark.

In this paper the m-input p-output noisy multivariable system is assumed to be modeled in matrix fraction description form as:

y[k] = A^{-1}(q^{-1})B(q^{-1})u[k]+e[k] (1)

A(q^{-1})=I_{p}+A_{1}q^{-1}+…+A_{na}q^{-na}

where

B(q^{-1})=B_{1}q^{-1}+…+B_{nb}q^{-nb}

e[k] is a white noise vector and q is the shift operator

**5. Conclusion**

This paper has presented an extension of the SRIV algorithm to MIMO systems described by a Left Matrix Fraction Description using the Kronicker product. Block filtering of the input/output as well as iterativity are the main features of the algorithm.

A simulations example illustrated the superiority of the MIMO SRIV algorithm over the MIMO IV4 and the MIMO least squares algorithms.

**REFERENCES**

- LJUNG, L.,
**System Identification: Theory for the user**, Prentice Hall, 1999 . - LJUNG, L.,
**System identification toolbox for Matlab**, Version, 6.1.2, www.mathworks.com, 2005. - YOUNG, P., A.J. JAKEMAN,
**Refined instrumental variable methods of time-series analysis**, International Journal of Control 29, 1979, pp. 621-644. - YOUNG, P., An instrumental variable approach to ARMA model identification and estimation, Proceedings of the 14th IFAC Symposium on System identification (SYSID’2006 Newcastle, Australia), 2006.
- STEIGLITZ, K., L.E. MCBRIDE,
**A technique for the identification of linear systems**, IEEE Transactions on Automatic Control, 10, 1965, pp. 461-464. - STOICA, P., T. SODERSTROM, 1981,
**The Steiglitz -McBride identification algorithm revisited**, IEEE Transactions on Automatic Control, 29, pp. 712-719. - FASSOIS, S.D.,
**MIMO LMS-ARMAX identification of vibrating structures**, Mechanical Systems and Signal Processing, 15, 2001, pp. 723-735. - NEHORAI, A., M. MORF,
**Recursive identification algorithms for Right Matrix Fraction Description models**, IEEE Transactions on Automatic Control, 29, 1984. pp. 1103-1106. - KAILATH, T.,
**Linear Systems**, Prentice Hall, 1980. - ZABOT, H., K. HARICHE,
**On solvents-based model reduction of MIMO systems**, International journal of systems science, 28, 1997, pp. 499-505.