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Keywords:
nonlinear control systems; input-output models; realization; pseudo-linear algebra
nonlinear control systems; input-output models; realization; pseudo-linear algebra
Summary:
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}.
References:
[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. MR 2353693 | Zbl 1136.93026
[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.: An introduction to pseudo-linear algebra. Theoret. Comput. Sci. 157 (1996), 3-33. DOI 10.1016/0304-3975(95)00173-5 | MR 1383396 | Zbl 0868.34004
[4] Choquet-Bruhat, Y., DeWitt-Morette, C., Dillard-Bleick, M.: Analysis, Manifolds and Physics, Part I: Basics. North-Holland, Amsterdam 1982. MR 0685274
[5] Cohn, R. M.: Difference Algebra. Wiley-Interscience, New York 1965. MR 0205987 | Zbl 0127.26402
[6] Conte, G., Moog, C. H., Perdon, A. M.: Algebraic Mehtods for Nonlinear Control Systems. Springer-Verlag, London 2007. MR 2305378
[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. MR 1651823 | Zbl 0852.93016
[8] Grizzle, J. W.: A linear algebraic framework for the analysis of discrete-time nonlinear systems. SIAM J. Control Optim. 31 (1991), 1026-1044. DOI 10.1137/0331046 | MR 1227545 | Zbl 0785.93036
[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. MR 2742768 | Zbl 1237.93043
[10] Hauser, J., Sastry, S., Kokotović, P.: Nonlinear control via approximate input-output linearization: The ball and beam example. IEEE Trans. Automat. Control 37 (1992), 392-398. DOI 10.1109/9.119645 | MR 1148727
[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. MR 2778925 | Zbl 1205.93027
[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. MR 1951488 | Zbl 1076.93011
[14] Kotta, Ü., Tõnso, M.: Realization of discrete-time nonlinear input-output equations: Polynomial approach. Automatica 48 (2012), 255-262. DOI 10.1016/j.automatica.2011.07.010 | MR 2889418
[15] McConnell, J. C., Robson, J. C.: Noncommutative Noetherian Rings. John Wiley and Sons, New York 1987. MR 0934572 | Zbl 0980.16019
[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.: State maps for linear systems. SIAM J. Control Optim. 35 (1997), 1053-1091. DOI 10.1137/S0363012994268412 | MR 1444349 | Zbl 0884.93006
[18] Schaft, A. J. van der: On realization of nonlinear systems described by higher-order differential equations. Math. Systems Theory 19 (1987), 239-275. DOI 10.1007/BF01704916 | MR 0871787
[19] Sontag, E. D.: On the observability of polynomial systems, I: Finite-time problems. SIAM J. Control Optim. 17 (1979), 139-151. DOI 10.1137/0317011 | MR 0516861 | Zbl 0409.93013
[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. MR 2604250 | Zbl 1179.93079
[22] Yuz, J. I., Goodwin, G. C.: On sampled-data models for nonlinear systems. IEEE Trans. Automat. Control 50 (2005), 1477-1489. DOI 10.1109/TAC.2005.856640 | MR 2171147
[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. MR 2778926 | Zbl 1205.93030
[24] Institute of Cybernetics at Tallinn University of Technology: The nonlinear control webpage. Website, http://nlcontrol.ioc.ee (2012).
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