An algebraic framework generalizing the concept of transfer functions to nonlinear systems.

*(English)*Zbl 1283.93077Summary: An algebraic point of view in nonlinear control systems, built up by introducing a differential field of meromorphic functions and a vector space of differential forms, is in this paper extended by introducing skew polynomials. Such polynomials act as operators over a vector space of differential forms. The left skew polynomial ring defined over the field of meromorphic functions is embedded to its quotient field to provide a basis for a symbolic computation of nonlinear systems. Members of such a quotient field are suggested as the transfer functions of nonlinear systems whereby the concept of the transfer functions is generalized to nonlinear systems. The theory is applied to nonlinear control systems and argues the invariance of introduced transfer functions of nonlinear systems to static state transformations, the existence of input–output descriptions for state space representations, availability of transfer function algebra for nonlinear systems, etc.

##### Keywords:

non-commutative rings; nonlinear systems; pseudo-linear algebra; skew polynomials; transfer functions##### Software:

GTF_Tools
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##### References:

[1] | Aranda-Bricaire, E.; Kotta, Ü; Moog, C., Linearization of discrete-time systems, SIAM journal of control optimization, 34, 1999-2023, (1996) · Zbl 0863.93014 |

[2] | Bronstein, M.; Petkovšek, M., An introduction to pseudo-linear algebra, Theoretical computer science, 157, 3-33, (1996) · Zbl 0868.34004 |

[3] | Chyzak, F.; Salvy, B., Non-commutative elimination in ore algebras proves multivariate identities, Journal of symbolic computation, 26, 187-227, (1998) · Zbl 0944.05006 |

[4] | Conte, G.; Moog, C.H.; Perdon, A.M., Nonlinear control systems: an algebraic setting, (1999), Springer London · Zbl 0920.93002 |

[5] | Grizzle, J.W., A linear algebraic framework for the analysis of discrete-time nonlinear systems, SIAM journal of control optimization, 31, 1026-1044, (1993) · Zbl 0785.93036 |

[6] | Halás, M. (2006a). Quotients of noncommutative polynomials in nonlinear control systems. In 18th European meeting on cyberntetics and systems research, Vienna, Austria |

[7] | Halás, M. (2006b). Symbolic computation for nonlinear systems. Ph.D. thesis, Department of Automation and Control, Faculty of Electrical Engineering and Information Technology, Slovak University of Technology, Bratislava, Slovakia (in Slovak) |

[8] | Halás, M. (2007). Ore algebras: A polynomial approach to nonlinear time-delay systems. In Ninth IFAC workshop on time-delay systems, Nantes, France |

[9] | Halás, M., & Huba, M. (2006). Symbolic computation for nonlinear systems using quotients over skew polynomial ring. In 14th Mediterranean conference on control and automation, Ancona, Italy |

[10] | Halás, M., & Kotta, Ü. (2007a). Extension of the concept of transfer function to discrete-time nonlinear control systems. In European control conference, Kos, Greece |

[11] | Halás, M., & Kotta, Ü. (2007b). Pseudo-linear algebra: A powerful tool in unification of the study of nonlinear control systems. In Seventh IFAC symposium NOLCOS, Pretoria, South Africa |

[12] | Isidori, A., Nonlinear systems, (1989), Springer New York · Zbl 0714.93021 |

[13] | Ježek, J., Non-commutative rings of fractions in algebraical approach to control theory, Kybernetika, 32, 81-94, (1996) · Zbl 0874.16023 |

[14] | Ježek, J., Rings of skew polynomials in algebraical approach to control theory, Kybernetika, 32, 63-80, (1996) · Zbl 0874.16022 |

[15] | Kotta, Ü. (2000). Irreducibility conditions for nonlinear input-output difference equations. In 39th IEEE conference on decision and control, Sydney, Australia |

[16] | Leith, D.J.; Leithead, W.E., Gain-scheduled and nonlinear systems: dynamic analysis by velocity-based linearization families, International journal of control, 70, 289-317, (1998) · Zbl 0930.93018 |

[17] | Leith, D.J.; Leithead, W.E., Gain-scheduled controller design: an analytic framework directly incorporating non-equilibrium plant dynamics, International journal of control, 70, 249-269, (1998) · Zbl 0979.93517 |

[18] | Ondera, M. \(G T F \_ T o o l s\): A Maple-based package for transfer functions of nonlinear control systems.〈http://www.kar.elf.stuba.sk/ mondera/GTF_Tools 〉 |

[19] | Ore, O., Linear equations in non-commutative fields, Annals of mathematics, 32, 463-477, (1931) · JFM 57.0166.01 |

[20] | Ore, O., Theory of non-commutative polynomials, Annals of mathematics, 34, 480-508, (1933) · JFM 59.0925.01 |

[21] | Xia, X.; Márquez-Martínez, L.A.; Zagalak, P.; Moog, C.H., Analysis of nonlinear time-delay systems using modules over non-commutative rings, Automatica, 38, 1549-1555, (2002) · Zbl 1017.93031 |

[22] | Zheng, Y.; Cao, L., Transfer function description for nonlinear systems, Journal of east China normal university (natural science), 2, 15-26, (1995) |

[23] | Zheng, Y., Willems, J., & Zhang, C. (1997). Common factors and controllability of nonlinear systems. In 36th IEEE conference on decision and control, San Diego, CA, USA |

[24] | Zheng, Y.; Willems, J.; Zhang, C., A polynomial approach to nonlinear system controllability, IEEE transactions on automatic control, 46, 1782-1788, (2001) · Zbl 1175.93045 |

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