Abstract
This paper applies the Thomas decomposition technique to nonlinear control systems, in particular to the study of the dependence of the system behavior on parameters. Thomas’ algorithm is a symbolic method which splits a given system of nonlinear partial differential equations into a finite family of so-called simple systems which are formally integrable and define a partition of the solution set of the original differential system. Different simple systems of a Thomas decomposition describe different structural behavior of the control system in general. The paper gives an introduction to the Thomas decomposition method and shows how notions such as invertibility, observability and flat outputs can be studied. A Maple implementation of Thomas’ algorithm is used to illustrate the techniques on explicit examples.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Aubry, P., Lazard, D., Moreno Maza, M.: On the theories of triangular sets. J. Symb. Comput. 28(1–2), 105–124 (1999)
Avanessoff, D., Pomet, J.-B.: Flatness and Monge parameterization of two-input systems, control-affine with 4 states or general with 3 states. ESAIM Control Optim. Calc. Var. 13(2), 237–264 (2007)
Bächler, T.: Counting solutions of algebraic systems via triangular decomposition. PhD thesis, RWTH Aachen University, Germany (2014). http://publications.rwth-aachen.de/record/444946?ln=en
Bächler, T., Gerdt, V.P., Lange-Hegermann, M., Robertz, D.: Algorithmic Thomas decomposition of algebraic and differential systems. J. Symb. Comput. 47(10), 1233–1266 (2012)
Bächler, T., Lange-Hegermann, M.: Algebraic Thomas and Differential Thomas: Thomas decomposition of algebraic and differential systems. http://wwwb.math.rwth-aachen.de/thomasdecomposition
Blinkov, Y.A., Cid, C.F., Gerdt, V.P., Plesken, W., Robertz, D.: The MAPLE package “Janet”: I. Polynomial systems. II. Linear partial differential equations. In: Ganzha, V.G., Mayr, E.W., Vorozhtsov, E.V. (eds.) Proceedings of the 6th International Workshop on Computer Algebra in Scientific Computing, Passau, Germany, pp. 31–40 resp. pp. 41–54 (2003). http://wwwb.math.rwth-aachen.de/Janet
Boulier, F., Lazard, D., Ollivier, F., Petitot, M.: Computing representations for radicals of finitely generated differential ideals. Appl. Algebra Eng. Commun. Comput. 20(1), 73–121 (2009)
Cohn, R.M.: Difference Algebra. Wiley-Interscience, New York (1965)
Conte, G., Moog, C.H., Perdon, A.M.: Nonlinear Control Systems. Lecture Notes in Control and Information Sciences, vol. 242. Springer, London (1999)
Diop, S.: Differential-algebraic decision methods and some applications to system theory. Theor. Comput. Sci. 98(1), 137–161 (1992)
Diop, S.: Elimination in control theory. Math. Control. Signals Syst. 4(1), 17–32 (1991)
Eisenbud, D.: Commutative Algebra – with a View Toward Algebraic Geometry. Graduate Texts in Mathematics, vol. 150. Springer, New York (1995)
Fliess, M., Glad, S.T.: An algebraic approach to linear and nonlinear control. In: Trentelman, H.L., Willems, J.C. (eds.) Essays on Control: Perspectives in the Theory and its Applications, pp. 223–267. Birkhäuser, Boston (1993)
Fliess, M., Lévine, J., Martin, P., Rouchon, P.: Flatness and defect of non-linear systems: introductory theory and examples. Int. J. Control. 61(6), 1327–1361 (1995)
Gao, X.-S., Luo, Y., Yuan, C.M.: A characteristic set method for ordinary difference polynomial systems. J. Symb. Comput. 44(3), 242–260 (2009)
Gao, X.-S., van der Hoeven, J., Yuan, C.M., Zhang, G.L.: Characteristic set method for differential-difference polynomial systems. J. Symb. Comput. 44(9), 1137–1163 (2009)
Gerdt, V.P.: On decomposition of algebraic PDE systems into simple subsystems. Acta Appl. Math. 101(1–3), 39–51 (2008)
Gerdt, V.P., Blinkov, Y.A.: Involutive bases of polynomial ideals. Math. Comput. Simul. 45, 519–541 (1998)
Gerdt, V.P., Lange-Hegermann, M., Robertz, D.: The Maple package TDDS for computing Thomas decompositions of systems of nonlinear PDEs. Comput. Phys. Commun. 234, 202–215 (2019)
Glad., S.T.: Differential algebraic modelling of nonlinear systems. In: Kaashoek, M.A., van Schuppen, J.H., Ran, A.C.M. (eds.) Realization and Modelling in System Theory, pp. 97–105. Birkhäuser, Boston (1989)
Hubert, E.: Notes on triangular sets and triangulation-decomposition algorithms. I. Polynomial systems. II. Differential systems. In: Winkler, F., Langer, U. (eds.) Symbolic and Numerical Scientific Computation, Hagenberg (2001), pp. 1–39 resp. 40–87. Lecture Notes in Computer Science, vol. 2630. Springer, Berlin (2003)
Ince, E.L.: Ordinary Differential Equations. Dover Publications, New York (1956)
Janet, M.: Leçons sur les systèmes d’équations aux dérivées partielles. Cahiers Scientifiques IV. Gauthiers-Villars, Paris (1929)
Kolchin, E.R.: Differential Algebra and Algebraic Groups. Pure and Applied Mathematics, vol. 54. Academic, New York (1973)
Kwakernaak, H., Sivan, R.: Linear Optimal Control Systems. Wiley-Interscience, New York (1972)
Lange-Hegermann, M.: Counting solutions of differential equations. PhD thesis, RWTH Aachen University, Germany (2014). http://publications.rwth-aachen.de/record/229056?ln=en
Lange-Hegermann, M.: The differential counting polynomial. Found. Comput. Math. 18(2), 291–308 (2018)
Lange-Hegermann, M., Robertz, D.: Thomas decompositions of parametric nonlinear control systems. In: Proceedings of the 5th Symposium on System Structure and Control, Grenoble, France, pp. 291–296 (2013)
Lemaire, F., Moreno Maza, M., Xie, Y.: The RegularChains library in MAPLE. SIGSAM Bull. 39, 96–97 (2005). September
Levandovskyy, V., Zerz, E.: Obstructions to genericity in study of parametric problems in control theory. In: Park, H., Regensburger, G. (eds.) Gröbner Bases in Control Theory and Signal Processing. Radon Series on Computational and Applied Mathematics, vol. 3, pp. 127–149. Walter de Gruyter, Berlin (2007)
Lévine, J.: On necessary and sufficient conditions for differential flatness. Appl. Algebra Eng. Commun. Comput. 22(1), 47–90 (2011)
Mishra, B.: Algorithmic Algebra. Texts and Monographs in Computer Science. Springer, New York (1993)
Nijmeijer, H., van der Schaft, A.: Nonlinear Dynamical Control Systems. Springer, New York (1990)
Picó-Marco, E.: Differential algebra for control systems design: constructive computation of canonical forms. IEEE Control Syst. Mag. 33(2), 52–62 (2013)
Plesken, W.: Counting solutions of polynomial systems via iterated fibrations. Arch. Math. (Basel) 92(1), 44–56 (2009)
Pommaret, J.-F.: Partial Differential Equations and Group Theory. Mathematics and Its Applications, vol. 293. Kluwer Academic Publishers Group, Dordrecht (1994)
Pommaret, J.-F.: Partial Differential Control Theory. Mathematics and Its Applications, vol. 530. Kluwer Academic Publishers Group, Dordrecht (2001)
Pommaret, J.-F., Quadrat, A.: Formal obstructions to the controllability of partial differential control systems. In: Proceedings of IMACS, Berlin, Germany, vol. 5, pp. 209–214 (1997)
Riquier, C.: Les systèmes d’équations aux dérivées partielles. Gauthiers-Villars, Paris (1910)
Ritt, J.F.: Differential Algebra. American Mathematical Society Colloquium Publications, vol. XXXIII. American Mathematical Society, New York (1950)
Robertz, D.: Formal Algorithmic Elimination for PDEs. Lecture Notes in Mathematics, vol. 2121. Springer, Cham (2014)
Robertz, D.: Recent progress in an algebraic analysis approach to linear systems. Multidimens. Syst. Signal Process. 26(2), 349–388 (2015)
Seidenberg, A.: An elimination theory for differential algebra. Univ. California Publ. Math. (N.S.) 3, 31–65 (1956)
Thomas, J. M.: Differential Systems. American Mathematical Society Colloquium Publications, vol. XXI. American Mathematical Society, New York (1937)
Wang, D.: Decomposing polynomial systems into simple systems. J. Symb. Comput. 25(3), 295–314 (1998)
Wang, D.: Elimination Methods. Texts and Monographs in Symbolic Computation. Springer, Vienna (2001)
Wu, W.T.: Mathematics Mechanization. Mathematics and Its Applications, vol. 489. Kluwer Academic Publishers Group, Dordrecht; Science Press, Beijing (2000)
Acknowledgements
The first author was partially supported by Schwerpunkt SPP 1489 of the Deutsche Forschungsgemeinschaft. The authors would like to thank an anonymous referee for several useful remarks. They would also like to thank S. L. Rueda for pointing out reference [34].
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2020 Springer Nature Switzerland AG
About this chapter
Cite this chapter
Lange-Hegermann, M., Robertz, D. (2020). Thomas Decomposition and Nonlinear Control Systems. In: Quadrat, A., Zerz, E. (eds) Algebraic and Symbolic Computation Methods in Dynamical Systems. Advances in Delays and Dynamics, vol 9. Springer, Cham. https://doi.org/10.1007/978-3-030-38356-5_4
Download citation
DOI: https://doi.org/10.1007/978-3-030-38356-5_4
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-38355-8
Online ISBN: 978-3-030-38356-5
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)