Abstract
We present realization theory for a class of autonomous discrete-time hybrid systems called semi-algebraic hybrid systems. These are systems in which the state and output equations associated with each discrete state are defined by polynomial equalities and inequalities. We first show that these systems generate the same output as semi-algebraic systems and implicit polynomial systems. We then derive necessary and almost sufficient conditions for existence of an implicit polynomial system realizing a given time-series data. We also provide a characterization of the dimension of a minimal realization as well as an algorithm for computing a realization from a given time-series data.
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References
Isidori, A.: Nonlinear Control Systems. Springer, Heidelberg (1989)
Sontag, E.D.: Polynomial Response Maps. Lecture Notes in Control and Information Sciences, vol. 13. Springer, Heidelberg (1979)
Sontag, E.D.: Realization theory of discrete-time nonlinear systems: Part I – the bounded case. IEEE Transaction on Circuits and Systems CAS-26(4) (1979)
Fliess, M.: Matrices de Hankel. J. Math. Pures Appl. (23), 197–224 (1973)
Sussmann, H.: Existence and uniqueness of minimal realizations of nonlinear systems. Mathematical Systems Theory 10, 263–284 (1977)
Jakubczyk, B.: Realization theory for nonlinear systems, three approaches. In: Fliess, M., Hazewinkel, M., (eds.) Algebraic and Geometric Methods in Nonlinear Control Theory, pp. 3–32. D. Reidel Publishing Company (1986)
Wang, Y., Sontag, E.: Generating series and nonlinear systems: analytic aspects, local realizability and I/O representations. Forum Mathematicum (4), 299–322 (1992)
Bartosiewicz, Z.: Realizations of polynomial systems. In: Algebraic and geometric methods in nonlinear control theory., Math. Appl., vol. 29, pp. 45–54. Dordrecht, Reidel (1986)
Wang, Y., Sontag, E.: Algebraic differential equations and rational control systems. SIAM Journal on Control and Optimization (30), 1126–1149 (1992)
Grossman, R., Larson, R.: An algebraic approach to hybrid systems. Theoretical Computer Science 138, 101–112 (1995)
Petreczky, M.: Realization theory for linear switched systems: Formal power series approach. Systems and Control Letters 56(9-10), 588–595 (2007)
Petreczky, M.: Realization theory for bilinear switched systems: A formal power series approach. In: Proc. of 44th IEEE Conference on Decision and Control, pp. 690–695 (2005)
Petreczky, M.: Realization Theory of Hybrid Systems. PhD thesis, Vrije Universiteit, Amsterdam (2006), http://www.cwi.nl/~mpetrec
Petreczky, M.: Hybrid formal power series and their application to realization theory of hybrid systems. In: Proc. 17th International Symposium on Mathematical Theory of Networks and Systems (2006)
Petreczky, M., Pomet, J.B.: Realization theory of nonlinear hybrid systems. In: Proceedings of CTS-HYCON Workshop on Hybrid and Nonlinear Control Systems (2006)
Petreczky, M.: Realization theory for discrete-time piecewise-affine hybrid systems. In: Proc 17th Internation Symposium on Mathematical Theory of Networks and Systems (2006)
Petreczky, M., Vidal, R.: Metrics and topology for nonlinear and hybrid systems. In: Bemporad, A., Bicchi, A., Buttazzo, G. (eds.) HSCC 2007. LNCS, vol. 4416, pp. 459–472. Springer, Heidelberg (2007)
Petreczky, M., Vidal, R.: Realization theory of stochastic jump-Markov linear systems. In: Proceedings 46th IEEE Conference on Decision and Control (2007)
Ma, Y., Vidal, R.: Identification of deterministic switched ARX systems via identification of algebraic varieties. In: Morari, M., Thiele, L. (eds.) HSCC 2005. LNCS, vol. 3414, pp. 449–465. Springer, Heidelberg (2005)
Vidal, R.S.S., Sastry, S.: An algebraic geometric approach to the identification of linear hybrid systems. In: IEEE Conference on Decision and Control, pp. 167–172 (2003)
Vidal, R.: Identification of PWARX hybrid models with unknown and possibly different orders. In: Proceedings of the IEEE American Conference on Control, pp. 547–552 (2004)
Kunz, E.: Introduction to commutative algebra and algebraic geometry. Birkhaeuser, Stuttgard (1985)
Cox, D., Little, J., O’Shea, D.: Ideal, varieties, and algorithms. Springer, New York (1997)
Brumfiel, G.W.: Partialy Ordered Rings and Semi-Algebraic Geometry. Cambridge University Press, Cambridge (1979)
Bochnak, J., Coste, M., Roy, M.F.: Real Algebraic Geometry. Springer, Heidelberg (1998)
Basu, S., Pollack, R., Roy, M.F.: Algorithms in Real Algebraic Geometry. Springer, Heidelberg (2003)
Collins, P., van Schuppen, J.H.: Observability of hybrid systems and Turing machines. In: Proceedings of the 43rd IEEE Conference on Decision and Control, pp. 7–12 (2004)
Ma, Y., Yang, A., Derksen, H., Fossum, R.: Estimation of subspace arrangements with applications in modeling and segmenting mixed data. SIAM Review (to appear, 2007)
Ho, B.L., Kalman, R.E.: Effective construction of linear state-variable models from input/output data. In: Proc. 3rd Allerton Conf. on Circuit and System Theory, pp. 449–459 (1965)
Caines, P.: Linear Stochastic Systems. John Wiley and Sons, New-York (1988)
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Petreczky, M., Vidal, R. (2008). Realization Theory for Discrete-Time Semi-algebraic Hybrid Systems. In: Egerstedt, M., Mishra, B. (eds) Hybrid Systems: Computation and Control. HSCC 2008. Lecture Notes in Computer Science, vol 4981. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-78929-1_28
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DOI: https://doi.org/10.1007/978-3-540-78929-1_28
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