Kybernetika 54 no. 4, 736-747, 2018

Realization of nonlinear input-output equations in controller canonical form

Arvo Kaldmäe and Ülle KottaDOI: 10.14736/kyb-2018-4-0736

Abstract:

In this paper necessary and sufficient conditions are given which guarantee that there exists a realization of a set of nonlinear higher order differential input-output equations in the controller canonical form. Two cases are studied, corresponding respectively to linear and nonlinear output functions. The conditions are formulated in terms of certain sequence of vector spaces of differential 1-forms. The proofs suggest how to construct the transformations, necessary to obtain the specific state space realizations. Multiple examples are added, which describe different scenarios.

Keywords:

nonlinear systems, realization, algebraic methods

Classification:

93B11, 93C10, 93B25

paper.pdf

References:

  1. K. J. Aström and R. M. Murray: Feedback Systems: An Introduction for Scientists and Engineers. Princeton University Press, 2008.   DOI:10.1086/596297
  2. Z. Bartosiewicz, Ü. Kotta, M. Tõnso and M. Wyrwas: Accessibility conditions of MIMO nonlinear control systems on homogeneous time scales. Math. Control Relat. Fields 6 (2016), 217-250.   DOI:10.3934/mcrf.2016002
  3. J. Belikov, P. Kotta, Ü. Kotta and M. Tõnso: Practical polynomial formulas in MIMO nonlinear realization problem. In: 51st IEEE Conference on Decision and Control, Hawaii 2012, pp. 1253-1258.   DOI:10.1109/cdc.2012.6427109
  4. J. Belikov, Ü. Kotta and M. Tõnso: Adjoint polynomial formulas for nonlinear state-space realization. IEEE Trans. Automat. Control 59 (2014), 256-261.   DOI:10.1109/cdc.2012.6427109
  5. J. Belikov, Ü. Kotta and M. Tõnso: Realization of nonlinear MIMO system on homogeneous time scales. Eur. J. Control 23 (2015), 48-54.   DOI:10.1016/j.ejcon.2015.01.006
  6. M. Bronstein and M. Petkovsek: An introduction to pseudo-linear algebra. Theoret. Comput. Sci. 157 (1996), 3-33.   DOI:10.1016/0304-3975(95)00173-5
  7. G. Conte, C. H. Moog and A. M. Perdon: Algebraic Methods for Nonlinear Control Systems. Theory and Applications. Springer, London 2007.   DOI:10.1007/978-1-84628-595-0
  8. P. E. Crouch and F. Lamnabhi-Lagarrigue: State space realizations of nonlinear systems defined by input-output differential equations. In: Analysis and Optimization of Systems, Springer, Berlin, Heidelberg 1988, pp 138-149.   DOI:10.1007/bfb0042209
  9. E. Delaleau and W. Respondek: Lowering the orders of derivatives of controls in generalized state space systems. J. Math. Systems, Estimation, Control 5 (1995), 1-27.   CrossRef
  10. M. Halas, Y. Kawano, C. H. Moog and T. Ohtsuka: Realization of a nonlinear system in the feedforward form: a polynomial approach. In: 19th IFAC World Congress, Cape Town 2014, pp. 9480-9485.   DOI:10.3182/20140824-6-za-1003.00990
  11. M. Halas and Ü. Kotta: A transfer function approach to the realisation problem of nonlinear systems. Int. J. Control 85 (2012), 320-331.   DOI:10.1080/00207179.2011.651748
  12. E. R. Kolchin: Differential Algebra and Algebraic Groups. Academic Press, New York 1973.   CrossRef
  13. Ü. Kotta and T. Mullari: Equivalence of realizability conditions for nonlinear control systems. Proc. Est. Acad. Sci. Physics. Math. 55 (2006), 24-42.   CrossRef
  14. Ü. Kotta and N. Sadegh: Two approaches for state space realization of NARMA models: bridging the gap. Math. Comput. Model. Dyn. Syst. 8 (2002), 21-32.   DOI:10.1076/mcmd.8.1.21.8340
  15. V. L. Morales, F. Plestan and A. Glumineau: Linearization by completely generalized input-output injection. Kybernetika 35 (1999), 793-802.   CrossRef
  16. F. Plestan and A. Glumineau: Linearization by generalized input-output injection. Systems Control Lett. 31 (1997), 115-128.   DOI:10.1016/s0167-6911(97)00025-x
  17. E. D. Sontag: Mathematical Control Theory. Springer-Verlag, New York 1998.   DOI:10.1007/978-1-4612-0577-7
  18. M. Tõnso and Ü. Kotta: Realization of continuous-time nonlinear input-output equations: polynomial approach. In: 12th International Conference on Computer Aided Systems Theory, Gran Canaria 2009, pp. 633-640.   DOI:10.1007/978-3-642-04772-5\_82
  19. A. J. van der Schaft: On realization of nonlinear systems described by higher-order differential equations. Math. Systems Theory 19 (1987), 239-275.   DOI:10.1007/bf01704916
  20. 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