# 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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##### References:
 [1] E. Aranda-Bricaire, Ü. Kotta and C. Moog, Linearization of discrete-time systems,, SIAM J. Contr. Optim., 34 (1996), 1999. doi: 10.1137/S0363012994267315. Google Scholar [2] E. Artin, Geometric Algebra,, Interscience Publishers, (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. 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. 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. 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, (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. 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, (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. doi: 10.1109/TAC.2010.2060251. Google Scholar [11] P. M. Cohn, Free Rings and Their Relations,, 2nd edition, (1985). Google Scholar [12] R. M. Cohn, Difference Algebra,, Interscience Publishers John Wiley & Sons, (1965). Google Scholar [13] G. Conte, C. H. Moog and A. M. Perdon, Algebraic Methods for Nonlinear Control Systems. Theory and Applications,, 2nd edition, (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. 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. 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), (2010), 1249. 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. 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. doi: 10.3176/proc.2014.2.04. Google Scholar [19] J. C. McConnell and J. C. Robson, Noncommutative Noetherian Rings,, Graduate Studies in Mathematics, (2001). doi: 10.1090/gsm/030. Google Scholar [20] M. Ondera, Computer-Aided Design of Nonlinear Systems and their Generalized Transfer Functions,, PhD thesis, (2008). Google Scholar [21] O. Ore, Theory of non-commutative polynomials,, Annals of Mathematics, 34 (1933), 480. 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, 530 (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, 144 (1969), 169. 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. 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. doi: 10.1109/MCS.2007.906923. Google Scholar
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