# American Institute of Mathematical Sciences

June  2016, 6(2): 217-250. doi: 10.3934/mcrf.2016002

## Accessibility conditions of MIMO nonlinear control systems on homogeneous time scales

 1 Faculty of Computer Science, Bialystok University of Technology, Wiejska 45A, 15-351 Białystok, Poland 2 Institute of Cybernetics at Tallinn University of Technology, Akadeemia tee 21, 12618 Tallinn, Estonia, Estonia 3 Faculty of Computer Science, Białystok University of Technology, Wiejska 45A, 15-351 Białystok, Poland

Received  March 2015 Revised  February 2016 Published  April 2016

A necessary and sufficient accessibility condition for the set of nonlinear higher order input-output (i/o) delta differential equations is presented. The accessibility definition is based on the concept of an autonomous element that is specified to the multi-input multi-output systems. The condition is presented in terms of the greatest common left divisor of two left differential polynomial matrices associated with the system of the i/o delta-differential equations defined on a homogenous time scale which serves as a model of time and unifies the continuous and discrete time. We associate the subspace $\mathcal{H}_{\infty}$ of the vector space of differential one-forms with the considered system. This subspace is invariant with respect to taking delta derivatives. The relation between $\mathcal{H}_\infty$ and the element of a left free module over the ring of left differential polynomials is presented. The presented accessibility condition provides a basis for system reduction, i.e. for finding the transfer equivalent minimal accessible representation of the set of the i/o equations which is a suitable starting point for constructing an observable and accessible state space realization. Moreover, the condition allows to check the transfer equivalence of nonlinear systems, defined on homogeneous time scales.
Citation: Zbigniew Bartosiewicz, Ülle Kotta, Maris Tőnso, Małgorzata Wyrwas. Accessibility conditions of MIMO nonlinear control systems on homogeneous time scales. Mathematical Control & Related Fields, 2016, 6 (2) : 217-250. doi: 10.3934/mcrf.2016002
##### References:
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Robson, Noncommutative Noetherian Rings, Graduate Studies in Mathematics, 30. American Mathematical Society, Providence, RI, 2001. doi: 10.1090/gsm/030.  Google Scholar [20] M. Ondera, Computer-Aided Design of Nonlinear Systems and their Generalized Transfer Functions, PhD thesis, Slovak University of Technology in Bratislava, 2008. Google Scholar [21] O. Ore, Theory of non-commutative polynomials, Annals of Mathematics, 34 (1933), 480-508. doi: 10.2307/1968173.  Google Scholar [22] J.-F. Pommaret, Partial Differential Control Theory. Vol. I. Mathematical Tools; Vol. II Control Systems, Mathematics and Its Applications 530, Kluwer Academic Publishers, Dordrecht, 2001. doi: 10.1007/978-94-010-0854-9.  Google Scholar [23] V. M. Popov, Some properties of the control systems with irreducible matrix-transfer functions, Differential Equations and Dynamical Systems, II (Univ. Maryland, College Park, Md., 1969), 169-180. Lecture Notes in Math., 144, Springer, Berlin, 1970.  Google Scholar [24] A. J. van der Schaft, On realization of nonlinear systems described by higher-order differential equations, Mathematical Systems Theory, 19 (1987), 239-275. doi: 10.1007/BF01704916.  Google Scholar [25] J. C. Willems, The behavioral approach to open and interconnected systems, IEEE Control Systems Magazine, 27 (2007), 46-99. doi: 10.1109/MCS.2007.906923.  Google Scholar

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##### References:
 [1] E. Aranda-Bricaire, Ü. Kotta and C. Moog, Linearization of discrete-time systems, SIAM J. Contr. Optim., 34 (1996), 1999-2023. doi: 10.1137/S0363012994267315.  Google Scholar [2] E. Artin, Geometric Algebra, Interscience Publishers, Inc., New York-London, 1957. doi: 10.1002/9781118164518.  Google Scholar [3] Z. Bartosiewicz, Ü. Kotta, E. Pawłuszewicz, M. Tőnso and M. Wyrwas, Algebraic formalism of differential $p$-forms and vector fields for nonlinear control systems on homogeneous time scales, Proc. Estonian Acad. Sci., 62 (2013), 215-226. doi: 10.3176/proc.2013.4.02.  Google Scholar [4] Z. Bartosiewicz, Ü. Kotta, E. Pawłuszewicz and M. Wyrwas, Algebraic formalism of differential one-forms for nonlinear control systems on time scales, Proc. Estonian Acad. of Sci. Phys. Math., 56 (2007), 264-282.  Google Scholar [5] J. Belikov, V. Kaparin, Ü. Kotta and M. Tőnso, NLControl website,, 2014. Available from: , ().   Google Scholar [6] J. Belikov, Ü. Kotta and M. Tőnso, Realization of nonlinear MIMO system on homogeneous time scales, European Journal of Control, 23 (2015), 48-54. doi: 10.1016/j.ejcon.2015.01.006.  Google Scholar [7] M. Bohner and A. Peterson, Dynamic Equations on Time Scales. An Introduction with Applications., Birkhäuser, Boston, 2001. doi: 10.1007/978-1-4612-0201-1.  Google Scholar [8] M. Bronstein and M. Petkovšek, An introduction to pseudo-linear algebra, Theoretical Computer Science, 157 (1996), 3-33. doi: 10.1016/0304-3975(95)00173-5.  Google Scholar [9] R. L. Bryant, S. S. Chern, R. B. Gardner, H. L. Goldschmitt and P. A. Griffiths, Exterior Differential Systems, Math. Sci. Res. Inst. Publ. 18, Springer-Verlag, New York, 1991. doi: 10.1007/978-1-4613-9714-4.  Google Scholar [10] D. Casagrande, Ü. Kotta, M. Tőnso and M. Wyrwas, Transfer equivalence and realization of nonlinear input-output delta-differential equations on homogeneous time scales, IEEE Trans. Autom. Contr., 55 (2010), 2601-2606. doi: 10.1109/TAC.2010.2060251.  Google Scholar [11] P. M. Cohn, Free Rings and Their Relations, 2nd edition, London Mathematical Society Monographs, 19, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London, 1985.  Google Scholar [12] R. M. Cohn, Difference Algebra, Interscience Publishers John Wiley & Sons, New York-London-Sydeny, 1965.  Google Scholar [13] G. Conte, C. H. Moog and A. M. Perdon, Algebraic Methods for Nonlinear Control Systems. Theory and Applications, 2nd edition, Communications and Control Engineering Series. Springer-Verlag London, Ltd., London, 2007. doi: 10.1007/978-1-84628-595-0.  Google Scholar [14] Ü. Kotta, Z. Bartosiewicz, S. Nőmm and E. Pawłuszewicz, Linear input-output equivalence and row reducedness of discrete-time nonlinear systems, IEEE Trans. Autom. Contr., 56 (2011), 1421-1426. doi: 10.1109/TAC.2011.2112430.  Google Scholar [15] Ü. Kotta, Z. Bartosiewicz, E. Pawłuszewicz and M. Wyrwas, Irreducibility, reduction and transfer equivalence of nonlinear input-output equations on homogeneous time scales, Systems and Control Letters, 58 (2009), 646-651. doi: 10.1016/j.sysconle.2009.04.006.  Google Scholar [16] Ü. Kotta, B. Rehák and M. Wyrwas, Reduction of MIMO nonlinear systems on homogenous time scales, in 8th IFAC Symposium on Nonlinear Control Systems (NOLCOS), University of Bologna, Bologna, Italy, 2010, 1249-1254. doi: 10.3182/20100901-3-IT-2016.00007.  Google Scholar [17] Ü. Kotta and M. Tőnso, Realization of discrete-time nonlinear input-output equations: Polynomial approach, Automatica, 48 (2012), 255-262. doi: 10.1016/j.automatica.2011.07.010.  Google Scholar [18] Ü. Kotta, M. Tőnso and Y. Kawano, Polynomial accessibility condition for the multi-input multi-output nonlinear control system, Proc. Estonian Acad. Sci., 63 (2014), 136-150. doi: 10.3176/proc.2014.2.04.  Google Scholar [19] J. C. McConnell and J. C. Robson, Noncommutative Noetherian Rings, Graduate Studies in Mathematics, 30. American Mathematical Society, Providence, RI, 2001. doi: 10.1090/gsm/030.  Google Scholar [20] M. Ondera, Computer-Aided Design of Nonlinear Systems and their Generalized Transfer Functions, PhD thesis, Slovak University of Technology in Bratislava, 2008. Google Scholar [21] O. Ore, Theory of non-commutative polynomials, Annals of Mathematics, 34 (1933), 480-508. doi: 10.2307/1968173.  Google Scholar [22] J.-F. Pommaret, Partial Differential Control Theory. Vol. I. Mathematical Tools; Vol. II Control Systems, Mathematics and Its Applications 530, Kluwer Academic Publishers, Dordrecht, 2001. doi: 10.1007/978-94-010-0854-9.  Google Scholar [23] V. M. Popov, Some properties of the control systems with irreducible matrix-transfer functions, Differential Equations and Dynamical Systems, II (Univ. Maryland, College Park, Md., 1969), 169-180. Lecture Notes in Math., 144, Springer, Berlin, 1970.  Google Scholar [24] A. J. van der Schaft, On realization of nonlinear systems described by higher-order differential equations, Mathematical Systems Theory, 19 (1987), 239-275. doi: 10.1007/BF01704916.  Google Scholar [25] J. C. Willems, The behavioral approach to open and interconnected systems, IEEE Control Systems Magazine, 27 (2007), 46-99. doi: 10.1109/MCS.2007.906923.  Google Scholar
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