Previous |  Up |  Next


realization; nonlinear systems; algebraic methods
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.
[1] Aström, K. J., Murray, R. M.: Feedback Systems: An Introduction for Scientists and Engineers. Princeton University Press, 2008. DOI 10.1086/596297 | MR 2400446
[2] Bartosiewicz, Z., Kotta, Ü., Tõnso, M., Wyrwas, M.: Accessibility conditions of MIMO nonlinear control systems on homogeneous time scales. Math. Control Relat. Fields 6 (2016), 217-250. DOI 10.3934/mcrf.2016002 | MR 3510298
[3] Belikov, J., Kotta, P., Kotta, Ü., Tõnso, M.: 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] Belikov, J., Kotta, Ü., Tõnso, M.: Adjoint polynomial formulas for nonlinear state-space realization. IEEE Trans. Automat. Control 59 (2014), 256-261. DOI 10.1109/cdc.2012.6427109 | MR 3163347
[5] Belikov, J., Kotta, Ü., Tõnso, M.: Realization of nonlinear MIMO system on homogeneous time scales. Eur. J. Control 23 (2015), 48-54. DOI 10.1016/j.ejcon.2015.01.006 | MR 3339645
[6] Bronstein, M., Petkovsek, M.: An introduction to pseudo-linear algebra. Theoret. Comput. Sci. 157 (1996), 3-33. DOI 10.1016/0304-3975(95)00173-5 | MR 1383396
[7] Conte, G., Moog, C. H., Perdon, A. M.: Algebraic Methods for Nonlinear Control Systems. Theory and Applications. Springer, London 2007. DOI 10.1007/978-1-84628-595-0 | MR 2305378
[8] Crouch, P. E., Lamnabhi-Lagarrigue, F.: 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 | MR 0956266
[9] Delaleau, E., Respondek, W.: Lowering the orders of derivatives of controls in generalized state space systems. J. Math. Systems, Estimation, Control 5 (1995), 1-27. MR 1651823 | Zbl 0852.93016
[10] Halas, M., Kawano, Y., Moog, C. H., Ohtsuka, T.: 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] Halas, M., Kotta, Ü.: A transfer function approach to the realisation problem of nonlinear systems. Int. J. Control 85 (2012), 320-331. DOI 10.1080/00207179.2011.651748 | MR 2881269
[12] Kolchin, E. R.: Differential Algebra and Algebraic Groups. Academic Press, New York 1973. MR 0568864
[13] Kotta, Ü., Mullari, T.: Equivalence of realizability conditions for nonlinear control systems. Proc. Est. Acad. Sci. Physics. Math. 55 (2006), 24-42. MR 2211488
[14] Kotta, Ü., Sadegh, N.: 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.
[15] Morales, V. L., Plestan, F., Glumineau, A.: Linearization by completely generalized input-output injection. Kybernetika 35 (1999), 793-802. MR 1747977
[16] Plestan, F., Glumineau, A.: Linearization by generalized input-output injection. Systems Control Lett. 31 (1997), 115-128. DOI 10.1016/s0167-6911(97)00025-x | MR 1461807
[17] Sontag, E. D.: Mathematical Control Theory. Springer-Verlag, New York 1998. DOI 10.1007/978-1-4612-0577-7 | MR 1640001 | Zbl 0945.93001
[18] Tõnso, M., Kotta, Ü.: 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] 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
[20] 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
Partner of
EuDML logo