Abstract
The relationships between the reachability of positive standard and fractional discrete-time and continuous-time linear systems are addressed. It is shown that: 1) The fractional positive discrete-time and linear systems are reachable in one step if and only if the corresponding positive standard system is reachable in one step; 2) If the positive standard discrete-time linear system with single input is unreachable, then the corresponding fractional positive system is also unreachable; 3) The fractional positive continuous-time linear system is reachable if and only if the corresponding continuous-time positive standard system is reachable.
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References
1. Kalman R.: Mathematical description of linear systems, SIAM J. Control, vol. 1, no. 2, pp. 152-192, 1963.
2. Kalman R.: On the general theory of control systems, Prof. First Intern. Congress on Automatic Control, Butterworth, London, pp. 481-493, 1960.
3. Antsaklis P., Michel A.: Linear Systems, Birkhauser, Boston, 2006.
4. Kaczorek T.: Linear Control Systems, Vol. 1, J. Wiley, New York, 1999.
5. Kaczorek T.: Vectors and Matrices in Automation and Electrotechnics, WNT, Warsaw, 1998 (in Polish).
6. Kailath T.: Linear Systems, Prentice Hall, Englewood Cliffs, New Yok, 1980.
7. Klamka J.: Controllability of Dynamical Systems, Kluwer, Academic Press, Dordrecht, 1991.
8. Rosenbrock H.: State-space and multivariable theory, J. Wiley, New York, 1970.
9. Wolovich W.: Linear multivariable systems, Springer-Verlag, New York, 1974.
10. Żak S.H.: Systems and Control, Oxford University Press, New York, 2003.
11. Farina L., Rinaldi S.: Positive Linear Systems: Theory and Applications, J. Wiley & Sons, New York, 2000.
12. Kaczorek T.: Constructability and observability of standard and positive electrical circuits, Electrical Review, vol. 89, no. 7, pp. 132-136, 2013.
13. Kaczorek T.: Controllability and observability of linear electrical circuits, Electrical Review, vol. 87, no. 9a, pp. 248-254, 2011.
14. Kaczorek T.: Positive 1D and 2D systems, Springer-Verlag, London, 2002.
15. Kaczorek T.: Positive linear systems consisting of n subsystems with different fractional orders, IEEE Trans. Circuits and Systems, vol. 58, no. 6, pp. 1203-1210, 2011.
16. Kaczorek T.: Reachability and controllability to zero tests for standard and positive fractional discrete-time systems, Journal Européen des Systemes Automatisés, JESA, vol. 42, no. 6-8, pp. 769-787, 2008.
17. Kaczorek T.: Reachability and observability of fractional positive electrical circuits, Computational Problems of Electrical Engineering, vol. 23, no. 2, pp. 28-36, 2013.
18. Kaczorek T.: Relationship between the observability of standard and fractional linear systems, 2016.
19. Kaczorek T.: Selected Problems of Fractional Systems Theory, Springer-Verlag, Berlin, 2011.
20. Oldham K., Spanier J.: The Fractional Calculus: Integrations and Differentiations of Arbitrary Order, Academic Press, New York, 1974.
21. Ostalczyk P.: Epitome of the Fractional Calculus, Theory and its Applications in Automatics, Technical University of Lodz Press, Lodz, 2008 (in Polish).
22. Podlubny I.: Fractional Differential Equations, Academic Press, San Diego, 1999.
23. Klamka J.: Relationship between controllability of standard and fractional linear systems, Submitted to KKA 2017.
24. Gantmacher F.R.: The Theory of Matrices, Chelsea Pub. Comp., London, 1959.
Acknowledgement
This work was supported by National Science Centre in Poland under work No. 2014/13/B/ST7/03467.
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Kaczorek, T. (2017). Relationships between the reachability of positive standard and fractional discrete-time and continuous-time linear systems. In: Mitkowski, W., Kacprzyk, J., Oprzędkiewicz, K., Skruch, P. (eds) Trends in Advanced Intelligent Control, Optimization and Automation. KKA 2017. Advances in Intelligent Systems and Computing, vol 577. Springer, Cham. https://doi.org/10.1007/978-3-319-60699-6_39
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DOI: https://doi.org/10.1007/978-3-319-60699-6_39
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