Article

Identification of deterministic switched ARX systems via identification of algebraic varieties

Authors:

René VidalAuthors Info & Claims

HSCC'05: Proceedings of the 8th international conference on Hybrid Systems: computation and control

Pages 449 - 465

https://doi.org/10.1007/978-3-540-31954-2_29

Published: 09 March 2005 Publication History

Abstract

We present a closed-form (linear-algebraic) solution to the identification of deterministic switched ARX systems and characterize conditions that guarantee the uniqueness of the solution. We show that the simultaneous identification of the number of ARX systems, the (possibly different) model orders, the ARX model parameters, and the switching sequence is equivalent to the identification and decomposition of a projective algebraic variety whose degree and dimension depend on the number of ARX systems and the model orders, respectively. Given an upper bound for the number of systems, our algorithm identifies the variety and the maximum orders by fitting a polynomial to the data, and the number of systems, the model parameters, and the switching sequence by differentiating this polynomial. Our method is provably correct in the deterministic case, provides a good sub-optimal solution in the stochastic case, and can handle large low-dimensional data sets (up to tens of thousands points) in a batch fashion.

References

[1]

A. Alessandri and P. Coletta. Design of Luenberger observers for a class of hybrid linear systems. In Hybrid Systems: Computation and Control, pages 7-18. 2001.

Digital Library

[2]

B.D.O. Anderson and C.R. Johnson. Exponential convergence of adaptive identification and control algorithms. Automatica, 18(1):1-13, 1982.

Digital Library

[3]

A. Balluchi, L. Benvenuti, M. Di Benedetto, and A. Sangiovanni-Vincentelli. Design of observers for hybrid systems. In Hybrid Systems: Computation and Control, volume 2289 of LNCS, pages 76-89. Springer Verlag, 2002.

Digital Library

[4]

A. Bemporad, G. Ferrari, and M. Morari. Observability and controllability of piecewise affine and hybrid systems. IEEE Trans. on Aut. Cont., 45(10):1864-76, 2000.

[5]

A. Bemporad, A. Garulli, S. Paoletti, and A. Vicino. A greedy approach to identification of piecewise affine models. In Hybrid Systems: Computation and Control, LNCS, pages 97-112. Springer Verlag, 2003.

Digital Library

[6]

A. Bemporad, J. Roll, and L. Ljung. Identification of hybrid systems via mixed-integer programming. In IEEE Conf. on Decision & Control, pages 786-792, 2001.

[7]

A. Doucet, A. Logothetis, and V. Krishnamurthy. Stochastic sampling algorithms for state estimation of jump Markov linear systems. IEEE Transactions on Automatic Control, 45(1):188-202, 2000.

[8]

G. Ferrari-Trecate, D. Mignone, and M. Morari. Moving horizon estimation for hybrid systems. IEEE Transactions on Automatic Control, 47(10):1663-1676, 2002.

[9]

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.

Digital Library

[10]

J. Harris. Algebraic Geometry: A First Course. Springer-Verlag, 1992.

[11]

A. Juloski, W. Heemels, and G. Ferrari-Trecate. Data-based hybrid modelling of the component placement process in pick-and-place machines. In Control Engineeting Practice. To appear.

[12]

A. Juloski, S. Weiland, and M. Heemels. A Bayesian approach to identification of hybrid systems. In IEEE Conf. on Decision & Control, 2004.

[13]

K. Kanatani and C. Matsunaga. Estimating the number of independent motions for multibody motion segmentation. In Asian Conf. on Computer Vision, 2002.

[14]

H. Niessen and A.Juloski. Comparison of three procedures for identification of hybrid systems. In Conference on Control Applications, 2004.

[15]

V. Pavlovic, J. M. Rehg, T. J. Cham, and K. P. Murphy. A dynamic Bayesian network approach to figure tracking using learned dynamic models. In Proc. of the Intl. Conf. on Comp. Vision, pages 94-101, 1999.

[16]

J. K. Tugnait. Detection and estimation for abruptly changing systems. Automatica, 18(5):607-615, 1982.

Digital Library

[17]

D. Del Vecchio and R. Murray. Observers for a class of hybrid systems on a lattice. In Hybrid Systems: Computation and Control. 2004.

[18]

E.I. Verriest and B. De Moor. Multi-mode system identification. In European Control Conference, 1999.

[19]

R. Vidal. Identification of PWARX hybrid models with unknown and possibly different orders. In IEEE Conf. on Decision & Control, 2004.

[20]

R. Vidal and B.D.O. Anderson. Recursive identification of switched ARX hybrid models: Exponential convergence and persistence of excitation. In CDC, 2004.

[21]

R. Vidal, A. Chiuso, and S. Soatto. Observability and identifiability of jump linear systems. In IEEE Conf. on Decision & Control, pages 3614-3619, 2002.

[22]

R. Vidal, A. Chiuso, S. Soatto, and S. Sastry. Observability of linear hybrid systems. In Hybrid Systems: Computation and Control, pages 526-539. 2003.

Digital Library

[23]

R. Vidal, S. Soatto, Y. Ma, and S. Sastry. An algebraic geometric approach to the identification of a class of linear hybrid systems. In Proceedings of CDC, 2003.

Cited By

Nazari SRashidi BZhao QHuang B(2016)An Iterative Algebraic Geometric Approach for Identification of Switched ARX Models with NoiseAsian Journal of Control10.1002/asjc.127718:5(1655-1667)Online publication date: 1-Sep-2016
https://dl.acm.org/doi/10.1002/asjc.1277
Ozay NLagoa CSznaier M(2015)Set membership identification of switched linear systems with known number of subsystemsAutomatica (Journal of IFAC)10.1016/j.automatica.2014.10.10151:C(180-191)Online publication date: 1-Jan-2015
https://dl.acm.org/doi/10.1016/j.automatica.2014.10.101
Millérioux GDaafouz J(2014)On persistent excitations for the identification of switched linear dynamical systems over finite fieldsAutomatica (Journal of IFAC)10.1016/j.automatica.2014.10.05050:12(3246-3252)Online publication date: 1-Dec-2014
https://dl.acm.org/doi/10.1016/j.automatica.2014.10.050
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Published In

cover image Guide Proceedings

HSCC'05: Proceedings of the 8th international conference on Hybrid Systems: computation and control

March 2005

45 pages

ISBN:3540251081

Editors:
Manfred Morari
Automatic Control Laboratory, ETH Zurich, Zurich, Switzerland
,
Lothar Thiele
Computer Engineering and Networks Laboratory, ETH Zurich, Zurich, Switzerland

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HYCON
ARTIST

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Springer-Verlag

Berlin, Heidelberg

Publication History

Published: 09 March 2005

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Cited By

Nazari SRashidi BZhao QHuang B(2016)An Iterative Algebraic Geometric Approach for Identification of Switched ARX Models with NoiseAsian Journal of Control10.1002/asjc.127718:5(1655-1667)Online publication date: 1-Sep-2016
https://dl.acm.org/doi/10.1002/asjc.1277
Ozay NLagoa CSznaier M(2015)Set membership identification of switched linear systems with known number of subsystemsAutomatica (Journal of IFAC)10.1016/j.automatica.2014.10.10151:C(180-191)Online publication date: 1-Jan-2015
https://dl.acm.org/doi/10.1016/j.automatica.2014.10.101
Millérioux GDaafouz J(2014)On persistent excitations for the identification of switched linear dynamical systems over finite fieldsAutomatica (Journal of IFAC)10.1016/j.automatica.2014.10.05050:12(3246-3252)Online publication date: 1-Dec-2014
https://dl.acm.org/doi/10.1016/j.automatica.2014.10.050
Petreczky MBako Lvan Schuppen JJohansson KYi W(2010)Identifiability of discrete-time linear switched systemsProceedings of the 13th ACM international conference on Hybrid systems: computation and control10.1145/1755952.1755973(141-150)Online publication date: 12-Apr-2010
https://dl.acm.org/doi/10.1145/1755952.1755973

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