American Institute of Mathematical Sciences

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

Accessibility conditions of MIMO nonlinear control systems on homogeneous time scales

Zbigniew Bartosiewicz 1, , Ülle Kotta 2, , Maris Tőnso 2, and  Małgorzata Wyrwas 3,

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.
Keywords: transfer equivalence., Nonlinear control systems, homogeneous time scale, accessibility condition, left differential polynomials.
Mathematics Subject Classification: Primary: 34N05, 93B25; Secondary: 93C1.
Citation: Zbigniew Bartosiewicz, Ülle Kotta, Maris Tőnso, Małgorzata Wyrwas. Accessibility conditions of MIMO nonlinear control systems on homogeneous time scales. Mathematical Control and Related Fields, 2016, 6 (2) : 217-250. doi: 10.3934/mcrf.2016002
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.

[2]

E. Artin, Geometric Algebra, Interscience Publishers, Inc., New York-London, 1957. doi: 10.1002/9781118164518.

[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.

[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.

[5]

J. Belikov, V. Kaparin, Ü. Kotta and M. Tőnso, NLControl website,, 2014. Available from: , (). 

[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.

[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.

[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.

[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.

[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.

[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.

[12]

R. M. Cohn, Difference Algebra, Interscience Publishers John Wiley & Sons, New York-London-Sydeny, 1965.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[20]

M. Ondera, Computer-Aided Design of Nonlinear Systems and their Generalized Transfer Functions, PhD thesis, Slovak University of Technology in Bratislava, 2008.

[21]

O. Ore, Theory of non-commutative polynomials, Annals of Mathematics, 34 (1933), 480-508. doi: 10.2307/1968173.

[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.

[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.

[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.

[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.

show all references

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.

[2]

E. Artin, Geometric Algebra, Interscience Publishers, Inc., New York-London, 1957. doi: 10.1002/9781118164518.

[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.

[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.

[5]

J. Belikov, V. Kaparin, Ü. Kotta and M. Tőnso, NLControl website,, 2014. Available from: , (). 

[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.

[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.

[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.

[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.

[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.

[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.

[12]

R. M. Cohn, Difference Algebra, Interscience Publishers John Wiley & Sons, New York-London-Sydeny, 1965.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[20]

M. Ondera, Computer-Aided Design of Nonlinear Systems and their Generalized Transfer Functions, PhD thesis, Slovak University of Technology in Bratislava, 2008.

[21]

O. Ore, Theory of non-commutative polynomials, Annals of Mathematics, 34 (1933), 480-508. doi: 10.2307/1968173.

[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.

[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.

[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.

[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.

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