Realization of nonlinear input-output equations in controller canonical form

Kaldmäe, Arvo ; Kotta, Ülle

Kybernetika, Tome 54 (2018) no. 4, pp. 736-747

Abstract (VO)

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.

Détail

DOI : 10.14736/kyb-2018-4-0736

Classification : 93B11, 93B25, 93C10
Keywords: realization; nonlinear systems; algebraic methods

Bibliographie

[1] Aström, K. J., Murray, R. M.: Feedback Systems: An Introduction for Scientists and Engineers. Princeton University Press, 2008. | DOI | MR

[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 | MR

[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

[4] Belikov, J., Kotta, Ü., Tõnso, M.: Adjoint polynomial formulas for nonlinear state-space realization. IEEE Trans. Automat. Control 59 (2014), 256-261. | DOI | MR

[5] Belikov, J., Kotta, Ü., Tõnso, M.: Realization of nonlinear MIMO system on homogeneous time scales. Eur. J. Control 23 (2015), 48-54. | DOI | MR

[6] Bronstein, M., Petkovsek, M.: An introduction to pseudo-linear algebra. Theoret. Comput. Sci. 157 (1996), 3-33. | DOI | MR

[7] Conte, G., Moog, C. H., Perdon, A. M.: Algebraic Methods for Nonlinear Control Systems. Theory and Applications. Springer, London 2007. | DOI | MR

[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 | MR

[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 | Zbl

[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

[11] Halas, M., Kotta, Ü.: A transfer function approach to the realisation problem of nonlinear systems. Int. J. Control 85 (2012), 320-331. | DOI | MR

[12] Kolchin, E. R.: Differential Algebra and Algebraic Groups. Academic Press, New York 1973. | MR

[13] Kotta, Ü., Mullari, T.: Equivalence of realizability conditions for nonlinear control systems. Proc. Est. Acad. Sci. Physics. Math. 55 (2006), 24-42. | MR

[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

[15] Morales, V. L., Plestan, F., Glumineau, A.: Linearization by completely generalized input-output injection. Kybernetika 35 (1999), 793-802. | MR

[16] Plestan, F., Glumineau, A.: Linearization by generalized input-output injection. Systems Control Lett. 31 (1997), 115-128. | DOI | MR

[17] Sontag, E. D.: Mathematical Control Theory. Springer-Verlag, New York 1998. | DOI | MR | Zbl

[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

[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 | MR

[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 | Zbl