» Download Article

Authors
Lauer, Fabien; Vidal, Rene; Bloch, Gerard

Identifier
10.3182/20090706-3-FR-2004.00093

Index Terms
Hybrid and Distributed System Identification; Machine Learning and Data Mining

Abstract
We propose a general framework for identification of linear discrete-time hybrid systems in which arbitrary loss functions can be easily included. Our framework includes the algebraic (Vidal et al., 2003) and support vector regression (Lauer and Bloch, 2008a,b) methods as particular cases. Inspired by these approaches, we then propose an optimization framework that relies on the minimization of a product of loss functions. Here, the identification problem is recast as a nonlinear and non-convex, though continuous, optimization program that involves only the model parameters as variables. As a result, its complexity scales linearly with the number of data and it can easily be solved using standard global optimization methods. Moreover, we show that by choosing a saturated loss function, such as HampelÂ’s loss function, the algorithm can efficiently deal with noise and outliers in the data. The final result is a general framework for linear hybrid system identification that can deal efficiently with noise, outliers, and large data sets. Numerical experiments demonstrate the efficiency and robustness of the proposed approach.

References

A. Bemporad, A. Garulli, S. Paoletti, and A. Vicino. A bounded-error approach to piecewise affine system identification. IEEE Trans. on Automatic Control}, 50(10):1567--1580, 2005.

A. Cichocki and R. Unbehauen. Neural Networks for Optimization and Signal Processing. Wiley, 1993.

G. Ferrari-Trecate, M. Muselli, D. Liberati, and M. Morari. A clustering technique for the identification of piecewise affine systems. Automatica, 39(2):205--217, 2003.

W. Huyer and A. Neumaier. Global optimization by multilevel coordinate search. Journal of Global Optimization, 14(4):331--355, 1999.

A. L. Juloski, W. Heemels, G. Ferrari-Trecate, R. Vidal, S. Paoletti, and J. H. G. Niessen. Comparison of four procedures for the identification of hybrid systems. In Proc. of the 8th Int. Conf. on Hybrid Systems: Computation and Control (HSCC), Zurich, Switzerland, volume 3414 of LNCS, pages
  354--369, 2005a.
  
A. L. Juloski, S. Weiland, and W. Heemels. A Bayesian approach to identification of hybrid systems. IEEE Trans. on Automatic Control, 50(10):1520--1533, 2005b.

F. Lauer. From Support Vector Machines to Hybrid System Identification. PhD thesis, UniversitŽ Henri PoincarŽ Nancy 1, 2008.

F. Lauer and G. Bloch. A new hybrid system identification algorithm with automatic tuning. In Proc. of the 17th IFAC World Congress, Seoul, Korea, pages 10207--10212, 2008.

F. Lauer and G. Bloch. Switched and piecewise nonlinear hybrid system identification. In Proc. of the 11th Int. Conf. on Hybrid Systems: Computation and Control (HSCC), St. Louis, MO, USA, volume 4981 of LNCS, pages 330--343, 2008.

Y. Ma and R. Vidal. Identification of deterministic switched ARX systems via identification of algebraic varieties. In Proc. of the 8th Int. Conf. on Hybrid Systems: Computation and Control (HSCC), Zurich, Switzerland, volume 3414 of LNCS, pages 449--465, 2005.

H. Nakada, K. Takaba, and T. Katayama. Identification of piecewise affine systems based on statistical clustering technique. Automatica, 41(5):905--913, 2005.

S. Paoletti, A. L. Juloski, G. Ferrari-Trecate, and R. Vidal. Identification of hybrid systems: a tutorial. European Journal of Control, 13(2-3):242--262, 2007.

J. Roll, A. Bemporad, and L. Ljung. Identification of piecewise affine systems via mixed-integer  programming. Automatica, 40(1):37--50, 2004.

V. N. Vapnik. The Nature of Statistical Learning Theory. Springer-Verlag, New York, NY, USA, 1995.

R. Vidal. Recursive identification of switched ARX systems. Automatica, 44(9):2274--2287, 2008.

R. Vidal, S. Soatto, Y. Ma, and S. Sastry. An algebraic geometric approach to the identification of a class of linear hybrid systems. In Proc. of the 42nd IEEE Conf. on Decision and Control (CDC), Maui, Hawai, USA, pages 167--172, 2003.