On the Nonlinear Observability of Polynomial Dynamical Systems

Daniel Gerbet; Klaus Röbenack

doi:10.52846/stccj.2021.1.1.15

Authors

Daniel Gerbet Institute of Control Theory, TU Dresden
Klaus Röbenack Technische Universität Dresden

DOI:

https://doi.org/10.52846/stccj.2021.1.1.15

Keywords:

Nonlinear systems, observability, ideals, varieties

Abstract

Controllability and observability are important system properties in control theory. These properties cannot be easily checked for general nonlinear systems. This paper addresses the local and global observability as well as the decomposition with respect to observability of polynomial dynamical systems embedded in a higher-dimensional state-space. These criteria are applied on some example system.

References

V. I. Arnold, Ordinary Differential Equations. Springer, 1992.

Z. Bartosiewicz, “Local observability of nonlinear systems,” Systems & Control Letters, vol. 25, no. 4, pp. 295–298, 1995.

——, “Algebraic criteria of global observability of polynomial systems,” Automatica, vol. 69, pp. 210–213, 2016.

T. Becker and V. Weispfenning, Gr¨obner Bases, 2nd ed. New York: Springer-Verlag, 1998.

J. Bochnak, M. Coste, and M. Roy, Real Algebraic Geometry. Berlin: Springer-Verlag, 1998.

D. A. Cox, J. Little, and D. O’Shea, Ideals, Varieties, and Algorithms, 4th ed. Switzerland: Springer International Publishing, 2015.

D. Gerbet and K. R¨obenack, “Nonlinear observability for polynomial systems: Computation and examples,” in International Conference on System Theory, Control and Computing (ICSTCC 2020), Sinaia, Romania, Oct. 2020, pp. 425–432.

D. Gerbet and K. R¨obenack, “On global and local observability of nonlinear polynomial systems: A decidable criterion,” at – Automatisierungstechnik, vol. 68, no. 6, pp. 395–409, 2020.

——, “Proving asymptotic stability with LaSalle’s invariance principle: On the automatic computation of invariant sets using quantifier elimination,” in International Conference on Control, Decition and Information Technologies (CoDIT’20), Prague, Czech Republic, 2020, accepted for publication.

M. L. J. Hautus, “Controllability and observability conditions for linear autonomous systems,” Ned. Akad. Wetenschappen, Proc. Ser. A, vol. 72, pp. 443–448, 1969.

R. Hermann and A. J. Krener, “Nonlinear controllability and observability,” IEEE Trans. on Automatic Control, vol. 22, no. 5, pp. 728–740, 1977.

R. E. Kalman, “On the general theory of control systems,” in First International Congress of the International Federation of Automatic Control (IFAC), Moscow, 1960.

Y. Kawano and T. Ohtsuka, “Observability at an initial state for polynomial systems,” Automatica, vol. 49, no. 5, pp. 1126–1136, 2013.

——, “Observability conditions by polynomial ideals,” Asian Journal of Control, vol. 19, no. 3, pp. 821–831, 2017.

——, “Global observability of polynomial systems,” in SICE Annual Conference 2010, Proceedings of, 2010, pp. 2038–2041.

H. G. Kwatny and G. L. Blankenship, Nonlinear Control and Analytical Mechanics: A Computational Approach. Boston: Birkh¨auser, 2000.

J. M. Lee, Introduction to Smooth Manifolds, ser. Graduate Texts in Mathematics. New York: Springer, 2006, vol. 218.

H. Nijmeijer and A. J. van der Schaft, Nonlinear Dynamical Control systems. New York: Springer-Verlag, 1990.

T. Paradowski, B. Tibken, and R. Swiatlak, “An approach to determine observability of nonlinear systems using interval analysis,” in Proc. American Control Conference (ACC), Seattle, USA, May 2017, pp. 3932–3937.

T. Paradowski, S. Lerch, M. Damaszek, R. Dehnert, and B. Tibken, “Observability of uncertain nonlinear systems using interval analysis,” Algorithms, vol. 13, no. 3, p. 66, 2020.

K. R¨obenack, “Computation of multiple Lie derivatives by algorithmic differentiation,” J. of Computational and Applied Mathematics, vol. 213, no. 2, pp. 454–464, 2008.

——, Nichtlineare Regelungssysteme: Theorie und Anwendung der exakten Linearisierung. Berlin, Heidelberg: Springer Vieweg, 2017.

K. R¨obenack and K. J. Reinschke, “An efficient method to compute Lie derivatives and the observability matrix for nonlinear systems,” in Proc. Int. Symposium on Nonlinear Theory and its Applications (NOLTA), vol. 2, Dresden, Sep. 2000, pp. 625–628.

K. R¨obenack and R. Voßwinkel, “Formal verification of local and global observability of polynomial systems using quantifier elimination,” in International Conference on System Theory, Control and Computing (ICSTCC 2019), Sinaia, Romania, Oct. 2019, pp. 314–319.

——, “L¨osung regelungstechnischer Problemstellungen mittels Quantorenelimination,” Automatisierungstechnik, vol. 67, no. 9, pp. 714–726, 2019.

E. D. Sontag, “A concept of local observability,” Systems & Control Letters, vol. 5, pp. 41–47, 1984.

R. Thomas, “Deterministic chaos seen in terms of feedback circuits: Analysis, synthesis, ’labyrinth chaos’,” International Journal of Bifurcation and Chaos, vol. 9, no. 10, pp. 1889–1905, 1999.

B. Tibken, “Observability of nonlinear systems — an algebraic approach,” in Proc. IEEE Conf. on Decision and Control (CDC), vol. 5, Nassau, Bahamas, Dec. 2004, pp. 4824–4825.