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.
This work is supported by the NSF grant IIS-0347456 and the research startup funds from UIUC ECE Dept. and Hopkins WSE. The authors thank Prof. R. Fossum for valuable comments and Prof. A. Juloski for providing datasets for the experiments.
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References
Alessandri, A., Coletta, P.: Design of Luenberger observers for a class of hybrid linear systems. In: Hybrid Systems: Computation and Control, pp. 7–18 (2001)
Anderson, B.D.O., Johnson, C.R.: Exponential convergence of adaptive identification and control algorithms. Automatica 18(1), 1–13 (1982)
Balluchi, A., Benvenuti, L., Di Benedetto, M., Sangiovanni-Vincentelli, A.: Design of observers for hybrid systems. In: Tomlin, C.J., Greenstreet, M.R. (eds.) HSCC 2002. LNCS, vol. 2289, pp. 76–89. Springer, Heidelberg (2002)
Bemporad, A., Ferrari, G., Morari, M.: Observability and controllability of piece- wise affine and hybrid systems. IEEE Trans. on Aut. Cont. 45(10), 1864–1876 (2000)
Bemporad, A., Garulli, A., Paoletti, S., Vicino, A.: A greedy approach to identification of piecewise affine models. In: Hybrid Systems: Computation and Control. LNCS, pp. 97–112. Springer, Heidelberg (2003)
Bemporad, A., Roll, J., Ljung, L.: Identification of hybrid systems via mixed-integer programming. In: IEEE Conf. on Decision & Control, pp. 786–792 (2001)
Doucet, A., Logothetis, A., Krishnamurthy, V.: Stochastic sampling algorithms for state estimation of jump Markov linear systems. IEEE Transactions on Automatic Control 45(1), 188–202 (2000)
Ferrari-Trecate, G., Mignone, D., Morari, M.: Moving horizon estimation for hybrid systems. IEEE Transactions on Automatic Control 47(10), 1663–1676 (2002)
Ferrari-Trecate, G., Muselli, M., Liberati, D., Morari, M.: A clustering technique for the identification of piecewise affine systems. Automatica 39(2), 205–217 (2003)
Harris, J.: Algebraic Geometry: A First Course. Springer, Heidelberg (1992)
Juloski, A., Heemels, W., Ferrari-Trecate, G.: Data-based hybrid modelling of the component placement process in pick-and-place machines. In: Control Engineeting Practice (to appear)
Juloski, A., Weiland, S., Heemels, M.: A Bayesian approach to identification of hybrid systems. In: IEEE Conf. on Decision & Control (2004)
Kanatani, K., Matsunaga, C.: Estimating the number of independent motions for multibody motion segmentation. In: Asian Conf. on Computer Vision (2002)
Niessen, H., Juloski, A.: Comparison of three procedures for identification of hybrid systems. In: Conference on Control Applications (2004)
Pavlovic, V., Rehg, J.M., Cham, T.J., Murphy, K.P.: A dynamic Bayesian network approach to figure tracking using learned dynamic models. In: Proc. of the Intl. Conf. on Comp. Vision, pp. 94–101 (1999)
Tugnait, J.K.: Detection and estimation for abruptly changing systems. Automatica 18(5), 607–615 (1982)
Del Vecchio, D., Murray, R.: Observers for a class of hybrid systems on a lattice. In: Hybrid Systems: Computation and Control (2004)
Verriest, E.I., De Moor, B.: Multi-mode system identification. In: European Control Conference (1999)
Vidal, R.: Identification of PWARX hybrid models with unknown and possibly different orders. In: IEEE Conf. on Decision & Control (2004)
Vidal, R., Anderson, B.D.O.: Recursive identification of switched ARX hybrid models: Exponential convergence and persistence of excitation. In: CDC (2004)
Vidal, R., Chiuso, A., Soatto, S.: Observability and identifiability of jump linear systems. In: IEEE Conf. on Decision & Control, pp. 3614–3619 (2002)
Vidal, R., Chiuso, A., Soatto, S., Sastry, S.: Observability of linear hybrid systems. In: Hybrid Systems: Computation and Control, pp. 526–539 (2003)
Vidal, R., Soatto, S., Ma, Y., Sastry, S.: An algebraic geometric approach to the identification of a class of linear hybrid systems. In: Proceedings of CDC (2003)
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Ma, Y., Vidal, R. (2005). Identification of Deterministic Switched ARX Systems via Identification of Algebraic Varieties. In: Morari, M., Thiele, L. (eds) Hybrid Systems: Computation and Control. HSCC 2005. Lecture Notes in Computer Science, vol 3414. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-31954-2_29
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DOI: https://doi.org/10.1007/978-3-540-31954-2_29
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