Accessibility conditions of MIMO nonlinear control systems on homogeneous time scales

Previous Article
Partial null controllability of parabolic linear systems
MCRF Home
This Issue
Next Article
Determination of time dependent factors of coefficients in fractional diffusion equations

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

Abstract
Related Papers

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 & 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. 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

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. 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

[1]	Chunjiang Qian, Wei Lin, Wenting Zha. Generalized homogeneous systems with applications to nonlinear control: A survey. Mathematical Control & Related Fields, 2015, 5 (3) : 585-611. doi: 10.3934/mcrf.2015.5.585
[2]	Ugo Boscain, Yacine Chitour. On the minimum time problem for driftless left-invariant control systems on SO(3). Communications on Pure & Applied Analysis, 2002, 1 (3) : 285-312. doi: 10.3934/cpaa.2002.1.285
[3]	B. Kaymakcalan, R. Mert, A. Zafer. Asymptotic equivalence of dynamic systems on time scales. Conference Publications, 2007, 2007 (Special) : 558-567. doi: 10.3934/proc.2007.2007.558
[4]	M. Motta, C. Sartori. Exit time problems for nonlinear unbounded control systems. Discrete & Continuous Dynamical Systems - A, 1999, 5 (1) : 137-156. doi: 10.3934/dcds.1999.5.137
[5]	J. Gwinner. On differential variational inequalities and projected dynamical systems - equivalence and a stability result. Conference Publications, 2007, 2007 (Special) : 467-476. doi: 10.3934/proc.2007.2007.467
[6]	Yilei Tang, Long Wang, Xiang Zhang. Center of planar quintic quasi--homogeneous polynomial differential systems. Discrete & Continuous Dynamical Systems - A, 2015, 35 (5) : 2177-2191. doi: 10.3934/dcds.2015.35.2177
[7]	Hebai Chen, Jaume Llibre, Yilei Tang. Centers of discontinuous piecewise smooth quasi–homogeneous polynomial differential systems. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 1-15. doi: 10.3934/dcdsb.2019150
[8]	P.E. Kloeden. Pitchfork and transcritical bifurcations in systems with homogeneous nonlinearities and an almost periodic time coefficient. Communications on Pure & Applied Analysis, 2004, 3 (2) : 161-173. doi: 10.3934/cpaa.2004.3.161
[9]	Ta T.H. Trang, Vu N. Phat, Adly Samir. Finite-time stabilization and $H_\infty$ control of nonlinear delay systems via output feedback. Journal of Industrial & Management Optimization, 2016, 12 (1) : 303-315. doi: 10.3934/jimo.2016.12.303
[10]	Alex Bombrun, Jean-Baptiste Pomet. Asymptotic behavior of time optimal orbital transfer for low thrust 2-body control system. Conference Publications, 2007, 2007 (Special) : 122-129. doi: 10.3934/proc.2007.2007.122
[11]	Xiaowei Tang, Xilin Fu. New comparison principle with Razumikhin condition for impulsive infinite delay differential systems. Conference Publications, 2009, 2009 (Special) : 739-743. doi: 10.3934/proc.2009.2009.739
[12]	Pham Huu Anh Ngoc. Stability of nonlinear differential systems with delay. Evolution Equations & Control Theory, 2015, 4 (4) : 493-505. doi: 10.3934/eect.2015.4.493
[13]	Zhiming Guo, Zhi-Chun Yang, Xingfu Zou. Existence and uniqueness of positive solution to a non-local differential equation with homogeneous Dirichlet boundary condition---A non-monotone case. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1825-1838. doi: 10.3934/cpaa.2012.11.1825
[14]	Raffaela Capitanelli. Robin boundary condition on scale irregular fractals. Communications on Pure & Applied Analysis, 2010, 9 (5) : 1221-1234. doi: 10.3934/cpaa.2010.9.1221
[15]	Mike Crampin, David Saunders. Homogeneity and projective equivalence of differential equation fields. Journal of Geometric Mechanics, 2012, 4 (1) : 27-47. doi: 10.3934/jgm.2012.4.27
[16]	Tayel Dabbous. Adaptive control of nonlinear systems using fuzzy systems. Journal of Industrial & Management Optimization, 2010, 6 (4) : 861-880. doi: 10.3934/jimo.2010.6.861
[17]	José Miguel Pasini, Tuhin Sahai. Polynomial chaos based uncertainty quantification in Hamiltonian, multi-time scale, and chaotic systems. Journal of Computational Dynamics, 2014, 1 (2) : 357-375. doi: 10.3934/jcd.2014.1.357
[18]	Robert E. Miller. Homogenization of time-dependent systems with Kelvin-Voigt damping by two-scale convergence. Discrete & Continuous Dynamical Systems - A, 1995, 1 (4) : 485-502. doi: 10.3934/dcds.1995.1.485
[19]	Piermarco Cannarsa, Carlo Sinestrari. On a class of nonlinear time optimal control problems. Discrete & Continuous Dynamical Systems - A, 1995, 1 (2) : 285-300. doi: 10.3934/dcds.1995.1.285
[20]	Alexander Mielke, Sina Reichelt, Marita Thomas. Two-scale homogenization of nonlinear reaction-diffusion systems with slow diffusion. Networks & Heterogeneous Media, 2014, 9 (2) : 353-382. doi: 10.3934/nhm.2014.9.353

2018 Impact Factor: 1.292

American Institute of Mathematical Sciences