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

Juri Belikov; Ülle Kotta; Maris Tõnso

Kybernetika (2012)

  • Volume: 48, Issue: 6, page 1100-1113
  • ISSN: 0023-5954

Abstract

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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 Mathematica or Maple.

How to cite

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Belikov, Juri, Kotta, Ülle, and Tõnso, Maris. "State-space realization of nonlinear control systems: unification and extension via pseudo-linear algebra." Kybernetika 48.6 (2012): 1100-1113. <http://eudml.org/doc/251428>.

@article{Belikov2012,
abstract = {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 Mathematica or Maple.},
author = {Belikov, Juri, Kotta, Ülle, Tõnso, Maris},
journal = {Kybernetika},
keywords = {nonlinear control systems; input-output models; realization; pseudo-linear algebra; nonlinear control systems; input-output models; realization; pseudo-linear algebra; ring of skew polynomials; single algebraic framework},
language = {eng},
number = {6},
pages = {1100-1113},
publisher = {Institute of Information Theory and Automation AS CR},
title = {State-space realization of nonlinear control systems: unification and extension via pseudo-linear algebra},
url = {http://eudml.org/doc/251428},
volume = {48},
year = {2012},
}

TY - JOUR
AU - Belikov, Juri
AU - Kotta, Ülle
AU - Tõnso, Maris
TI - State-space realization of nonlinear control systems: unification and extension via pseudo-linear algebra
JO - Kybernetika
PY - 2012
PB - Institute of Information Theory and Automation AS CR
VL - 48
IS - 6
SP - 1100
EP - 1113
AB - 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 Mathematica or Maple.
LA - eng
KW - nonlinear control systems; input-output models; realization; pseudo-linear algebra; nonlinear control systems; input-output models; realization; pseudo-linear algebra; ring of skew polynomials; single algebraic framework
UR - http://eudml.org/doc/251428
ER -

References

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  1. Bartosiewicz, Z., Kotta, Ü., Pawłuszewicz, E., Wyrwas, M., Algebraic formalism of differential one-forms for nonlinear control systems on time scales., Proc. Estonian Acad. Sci. 56 (2007), 264-282. Zbl1136.93026MR2353693
  2. Belikov, J., Kotta, Ü., Tõnso, M., An explicit formula for computation of the state coordinates for nonlinear i/o equation., In: 18th IFAC World Congress, Milano 2011, pp. 7221-7226. 
  3. Bronstein, M., Petkovšek, M., 10.1016/0304-3975(95)00173-5, Theoret. Comput. Sci. 157 (1996), 3-33. Zbl0868.34004MR1383396DOI10.1016/0304-3975(95)00173-5
  4. Choquet-Bruhat, Y., DeWitt-Morette, C., Dillard-Bleick, M., Analysis, Manifolds and Physics, Part I: Basics., North-Holland, Amsterdam 1982. MR0685274
  5. Cohn, R. M., Difference Algebra., Wiley-Interscience, New York 1965. Zbl0127.26402MR0205987
  6. Conte, G., Moog, C. H., Perdon, A. M., Algebraic Mehtods for Nonlinear Control Systems., Springer-Verlag, London 2007. MR2305378
  7. Delaleau, E., Respondek, W., Lowering the orders of derivatives of controls in generalized state space systems., J. Math. Syst., Estim. Control 5 (1995), 1-27. Zbl0852.93016MR1651823
  8. Grizzle, J. W., 10.1137/0331046, SIAM J. Control Optim. 31 (1991), 1026-1044. Zbl0785.93036MR1227545DOI10.1137/0331046
  9. Halás, M., Kotta, Ü., Li, Z., Wang, H., Yuan, C., Submersive rational difference systems and their accessibility., In: International Symposium on Symbolic and Algebraic Computation, Seoul 2009, pp. 175-182. Zbl1237.93043MR2742768
  10. Hauser, J., Sastry, S., Kokotović, P., 10.1109/9.119645, IEEE Trans. Automat. Control 37 (1992), 392-398. MR1148727DOI10.1109/9.119645
  11. Kotta, Ü., Kotta, P., Halás, M., Reduction and transfer equivalence of nonlinear control systems: Unification and extension via pseudo-linear algebra., Kybernetika 46 (2010), 831-849. Zbl1205.93027MR2778925
  12. Kotta, Ü., Kotta, P., Tõnso, M., Halás, M., 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. 
  13. Kotta, Ü., Tõnso, M., 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. Zbl1076.93011MR1951488
  14. Kotta, Ü., Tõnso, M., 10.1016/j.automatica.2011.07.010, Automatica 48 (2012), 255-262. MR2889418DOI10.1016/j.automatica.2011.07.010
  15. McConnell, J. C., Robson, J. C., Noncommutative Noetherian Rings., John Wiley and Sons, New York 1987. Zbl0980.16019MR0934572
  16. Pang, Z. H., Zheng, G., Luo, C.X., 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. 
  17. Rapisarda, P., Willems, J. C., 10.1137/S0363012994268412, SIAM J. Control Optim. 35 (1997), 1053-1091. Zbl0884.93006MR1444349DOI10.1137/S0363012994268412
  18. Schaft, A. J. van der, 10.1007/BF01704916, Math. Systems Theory 19 (1987), 239-275. MR0871787DOI10.1007/BF01704916
  19. Sontag, E. D., 10.1137/0317011, SIAM J. Control Optim. 17 (1979), 139-151. Zbl0409.93013MR0516861DOI10.1137/0317011
  20. Tõnso, M., Kotta, Ü., Realization of continuous-time nonlinear input-output equations: Polynomial approach., Lecture Notes in Control and Inform. Sci., Springer Berlin / Heidelberg 2009, pp. 633-640. 
  21. Tõnso, M., Rennik, H., Kotta, Ü., WebMathematica-based tools for discrete-time nonlinear control systems., Proc. Estonian Acad. Sci. 58 (2009), 224-240. Zbl1179.93079MR2604250
  22. Yuz, J. I., Goodwin, G. C., 10.1109/TAC.2005.856640, IEEE Trans. Automat. Control 50 (2005), 1477-1489. MR2171147DOI10.1109/TAC.2005.856640
  23. Zhang, J., Moog, C. H., Xia, X., Realization of multivariable nonlinear systems via the approaches of differential forms and differential algebra., Kybernetika 46 (2010), 799-830. Zbl1205.93030MR2778926
  24. Institute of Cybernetics at Tallinn University of Technology, The nonlinear control webpage., Website, http://nlcontrol.ioc.ee (2012). 

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