Kybernetika 48 no. 6, 1100-1113, 2012

State-space realization of nonlinear control systems: unification and extension via pseudo-linear algebra

Juri Belikov, Ülle Kotta and Maris Tõnso


In this paper the tools of pseudo-linear algebra are applied to the realization problem, allowing to unify the study of the continuous- and discrete-time nonlinear control systems under a single algebraic framework. The realization of nonlinear input-output equation, defined in terms of the pseudo-linear operator, in the classical state-space form is addressed by the polynomial approach in which the system is described by two polynomials from the non-commutative ring of skew polynomials. This allows to simplify the existing step-by-step algorithm-based solution. The paper presents explicit formulas to compute the differentials of the state coordinates directly from the polynomial description of the nonlinear system. The method is straight-forward and better suited for implementation in different computer algebra packages such as \textit{Mathematica} or \textit{Maple}.


realization, nonlinear control systems, input-output models, pseudo-linear algebra


93E12, 62A10


  1. Z. Bartosiewicz, {Ü}. Kotta, E. Pawłuszewicz and M. Wyrwas: Algebraic formalism of differential one-forms for nonlinear control systems on time scales. Proc. Estonian Acad. Sci. 56 (2007), 264-282.   CrossRef
  2. J. Belikov, {Ü}. Kotta and M. T{õ}nso: An explicit formula for computation of the state coordinates for nonlinear i/o equation. In: 18th IFAC World Congress, Milano 2011, pp. 7221-7226.   CrossRef
  3. M. Bronstein and M. Petkovšek: An introduction to pseudo-linear algebra. Theoret. Comput. Sci. 157 (1996), 3-33.   CrossRef
  4. Y. Choquet-Bruhat, C. DeWitt-Morette and M. Dillard-Bleick: Analysis, Manifolds and Physics, Part I: Basics. North-Holland, Amsterdam 1982.   CrossRef
  5. R. M. Cohn: Difference Algebra. Wiley-Interscience, New York 1965.   CrossRef
  6. G. Conte, C. H. Moog and A. M. Perdon: Algebraic Mehtods for Nonlinear Control Systems. Springer-Verlag, London 2007.   CrossRef
  7. E. Delaleau and W. Respondek: Lowering the orders of derivatives of controls in generalized state space systems. J. Math. Syst., Estim. Control 5 (1995), 1-27.   CrossRef
  8. J. W. Grizzle: A linear algebraic framework for the analysis of discrete-time nonlinear systems. SIAM J. Control Optim. 31 (1991), 1026-1044.   CrossRef
  9. M. Halás, {Ü}. Kotta, Z. Li, H. Wang and C. Yuan: Submersive rational difference systems and their accessibility. In: International Symposium on Symbolic and Algebraic Computation, Seoul 2009, pp. 175-182.   CrossRef
  10. J. Hauser, S. Sastry and P. Kokotović: Nonlinear control via approximate input-output linearization: The ball and beam example. IEEE Trans. Automat. Control 37 (1992), 392-398.   CrossRef
  11. {Ü}. Kotta, P. Kotta and M. Halás: Reduction and transfer equivalence of nonlinear control systems: Unification and extension via pseudo-linear algebra. Kybernetika 46 (2010), 831-849.   CrossRef
  12. {Ü}. Kotta, P. Kotta, M. T{õ}nso and M. Halás: State-space realization of nonlinear input-output equations: Unification and extension via pseudo-linear algebra. In: 9th International Conference on Control and Automation, Santiago, Chile 2011, pp. 354-359.   CrossRef
  13. {Ü}. Kotta and M. T{õ}nso: Removing or lowering the orders of input shifts in discrete-time generalized state-space systems with Mathematica. Proc. Estonian Acad. Sci. 51 (2002), 238-254.   CrossRef
  14. {Ü}. Kotta and M. T{õ}nso: Realization of discrete-time nonlinear input-output equations: Polynomial approach. Automatica 48 (2012), 255-262.   CrossRef
  15. J. C. McConnell and J. C. Robson: Noncommutative Noetherian Rings. John Wiley and Sons, New York 1987.   CrossRef
  16. Z. H. Pang, G. Zheng and C.X. Luo: Augmented state estimation and LQR control for a ball and beam system. In: 6th IEEE Conference on Industrial Electronics and Applications, Beijing 2011, pp. 1328-1332.   CrossRef
  17. P. Rapisarda and J. C. Willems: State maps for linear systems. SIAM J. Control Optim. 35 (1997), 1053-1091.   CrossRef
  18. A. J. van der Schaft: On realization of nonlinear systems described by higher-order differential equations. Math. Systems Theory 19 (1987), 239-275.   CrossRef
  19. E. D. Sontag: On the observability of polynomial systems, I: Finite-time problems. SIAM J. Control Optim. 17 (1979), 139-151.   CrossRef
  20. M. T{õ}nso and {Ü}. Kotta: Realization of continuous-time nonlinear input-output equations: Polynomial approach. Lecture Notes in Control and Inform. Sci., Springer Berlin / Heidelberg 2009, pp. 633-640.   CrossRef
  21. M. T{õ}nso, H. Rennik and {Ü}. Kotta: WebMathematica-based tools for discrete-time nonlinear control systems. Proc. Estonian Acad. Sci. 58 (2009), 224-240.   CrossRef
  22. J. I. Yuz and G. C. Goodwin: On sampled-data models for nonlinear systems. IEEE Trans. Automat. Control 50 (2005), 1477-1489.   CrossRef
  23. J. Zhang, C. H. Moog and X. Xia: Realization of multivariable nonlinear systems via the approaches of differential forms and differential algebra. Kybernetika 46 (2010), 799-830.   CrossRef
  24. Institute of Cybernetics at Tallinn University of Technology: The nonlinear control webpage. Website, \url{} (2012).   CrossRef