, Ülle Kotta , Małgorzata WyrwasThe observability property of the nonlinear system, defined on a homogeneous time scale, is studied. The observability condition is provided through the notion of the observable space. Moreover, the observability filtration and observability indices are defined and the decomposition of the system into observable/unobservable subsystems is considered.
1. Albertini, F. and D’Alessandro, D. Remarks on the observability of nonlinear discrete time systems. In Proceedings of the 17th IFIP TC7 Conference on System Modelling and Optimization, Prague, Czech Republic, July 10–14, 1995 (Doležal, J. and Fidler, J., eds). Chapman & Hall, London, 1996, 155–162.
2. Albertini, F. and D’Alessandro, D. Observability and forward-backward observability of discrete-time nonlinear systems. Math. Contr. Sign. Syst., 2002, 15(4), 275–290.
http://dx.doi.org/10.1007/s004980200011
3. Aranda-Bricaire, E., Kotta, Ü., and Moog, C. H. Linearization of discrete-time systems. SIAM J. Contr. Optim., 1996, 34(6), 1999–2023.
http://dx.doi.org/10.1137/S0363012994267315
4. Bartosiewicz, Z., Kotta, Ü., Pawłuszewicz, E., and Wyrwas, M. Algebraic formalism of differential one-forms for nonlinear control systems on time scales. Proc. Estonian Acad. Sci. Phys. Math., 2007, 56, 264–282.
5. Bartosiewicz, Z. and Pawłuszewicz, E. Realizations of linear control systems on time scales. Control Cybern., 2006, 35(4), 769–786.
6. Bohner, M. and Peterson, A. Dynamic Equations on Time Scales. Birkhäuser, Boston, 2001.
http://dx.doi.org/10.1007/978-1-4612-0201-1
7. Casagrande, D., Kotta, Ü., Tõnso, M., and Wyrwas, M. Transfer equivalence and realization of nonlinear input-output delta-differential equations on homogeneous time scales. IEEE Trans. Autom. Contr., 2010, 55(11), 2601–2606.
http://dx.doi.org/10.1109/TAC.2010.2060251
8. Choquet-Bruhat, Y., DeWitt-Morette, C., and Dillard-Bleick, M. Analysis, Manifolds and Physics, Part I: Basics. North-Holland, Amsterdam, 2004.
9. Cohn, R. M. Difference Algebra. Wiley-Interscience, New York, USA, 1965.
10. Conte, G., Moog, C. H., and Perdon, A. M. Algebraic Methods for Nonlinear Control Systems. Theory and Applications. 2nd edition, Springer-Verlag, London, UK, 2007.
11. Davis, J. M., Gravagne, I. A., Jackson, B. J., and Marks II, R. J. Controllability, observability, realizability, and stability of dynamic linear systems. Electronic J. Differential Equations, 2009, 2009(37), 1–32.
12. Goodwin, G. C., Graebe, S. F., and Salgado, M. E. Control System Design. Prentice Hall, Upper Saddle River, New Jersey, 2001.
13. Grizzle, J. W. A linear algebraic framework for the analysis of discrete-time nonlinear systems. SIAM J. Contr. Optim., 1993, 31(4), 1026–1044.
http://dx.doi.org/10.1137/0331046
14. Hermann, R. and Krener, A. J. Nonlinear controllability and observability. IEEE Trans. Autom. Contr., 1977, 22(5), 728–740.
http://dx.doi.org/10.1109/TAC.1977.1101601
15. Isidori, A. Nonlinear Control Systems. Springer, Berlin, 1995.
http://dx.doi.org/10.1007/978-1-84628-615-5
16. Kotta, Ü. Decomposition of discrete-time nonlinear control systems. Proc. Estonian Acad. Sci. Phys. Math., 2005, 54, 154–161.
17. Kotta, Ü., Bartosiewicz, Z., Pawłuszewicz, E., and Wyrwas, M. Irreducibility, reduction and transfer equivalence of nonlinear input-output equations on homogeneous time scales. Syst. & Contr. Lett., 2009, 58(9), 646–651.
http://dx.doi.org/10.1016/j.sysconle.2009.04.006
18. Kotta, Ü., Rehák, B., and Wyrwas, M. Reduction of MIMO nonlinear systems on homogeneous time scales. In 8th IFAC Symposium on Nonlinear Control Systems, University of Bologna, Italy, September 01–03, 2010 (Marconi, L., ed.). International Federation of Automatic Control, 2010, 1249–1254.
19. Kotta, Ü. and Schlacher, K. Possible non-integrability of observable space for discrete-time nonlinear control systems. In Proceedings of the 17th IFAC World Congress, Seoul, South Korea, July 6–11, 2008 (Chung, M. J., Misra, P., and Shim, H., eds). Seoul, 2008, 9852–9856.
20. Middleton, R. H. and Goodwin, G. C. Digital Control and Estimation: A Unified Approach. Prentice Hall, Englewood Cliffs, New Jersey, 1990.
21. Nijmeijer, H. and van der Schaft, A. J. Nonlinear Dynamical Control Systems. Springer, 1990.
http://dx.doi.org/10.1007/978-1-4757-2101-0
22. Pawłuszewicz, E. Observability of nonlinear control systems on time scales. Int. J. Syst. Sci., 2012, 43(12), 2268–2274.
http://dx.doi.org/10.1080/00207721.2011.569771
23. Wang, Y. and Sontag, E. D. Orders of input/output differential equations and state-space dimensions. SIAM J. Contr. Optim., 1995, 33(4), 1102–1126.
http://dx.doi.org/10.1137/S0363012993246828
