# Fractional positive continuous-time linear systems and their reachability

International Journal of Applied Mathematics and Computer Science (2008)

- Volume: 18, Issue: 2, page 223-228
- ISSN: 1641-876X

## Access Full Article

top## Abstract

top## How to cite

topTadeusz Kaczorek. "Fractional positive continuous-time linear systems and their reachability." International Journal of Applied Mathematics and Computer Science 18.2 (2008): 223-228. <http://eudml.org/doc/207879>.

@article{TadeuszKaczorek2008,

abstract = {A new class of fractional linear continuous-time linear systems described by state equations is introduced. The solution to the state equations is derived using the Laplace transform. Necessary and sufficient conditions are established for the internal and external positivity of fractional systems. Sufficient conditions are given for the reachability of fractional positive systems.},

author = {Tadeusz Kaczorek},

journal = {International Journal of Applied Mathematics and Computer Science},

keywords = {fractional systems; positive systems; reachability},

language = {eng},

number = {2},

pages = {223-228},

title = {Fractional positive continuous-time linear systems and their reachability},

url = {http://eudml.org/doc/207879},

volume = {18},

year = {2008},

}

TY - JOUR

AU - Tadeusz Kaczorek

TI - Fractional positive continuous-time linear systems and their reachability

JO - International Journal of Applied Mathematics and Computer Science

PY - 2008

VL - 18

IS - 2

SP - 223

EP - 228

AB - A new class of fractional linear continuous-time linear systems described by state equations is introduced. The solution to the state equations is derived using the Laplace transform. Necessary and sufficient conditions are established for the internal and external positivity of fractional systems. Sufficient conditions are given for the reachability of fractional positive systems.

LA - eng

KW - fractional systems; positive systems; reachability

UR - http://eudml.org/doc/207879

ER -

## References

top- Engheta N. (1997). On the role of fractional calculus in electromagnetic theory, IEEE Transactions on Atennas and Propagation 39(4): 35-46.
- Farina L.and Rinaldi S.(2000). Positive Linear Systems, Theory and Applications, J. Wiley, New York. Zbl0988.93002
- Ferreira N.M.F. and Machado J.A.T. (2003). Fractional-order hybrid control of robotic manipulators, Proceedings of the 11th International Conference on Advanced Robotics ICAR'2003, Coimbra, Portugal, pp. 393-398.
- Gałkowski K. and Kummert A. (2005). Fractional polynomials and nD systems, Proceedings of the IEEE International Symposium on Circuits and Systems, ISCAS'2005, Kobe, Japan, CD-ROM.
- Kaczorek T. (2002). Positive 1D and 2D Systems, Springer-Verlag, London Zbl1005.68175
- Kaczorek T. (2006). Computation of realizations of discretetime cone systems, Bulletin of the Polish Academy of Sciences: Technical Sciences 54(3): 347-350. Zbl1194.93129
- Kaczorek T. (2007a). Reachability and controllability to zero of positive fractional discrete-time systems, Machine Intelligence and Robotic Control 6(4), (in press). Zbl1152.93393
- Kaczorek T.(2007b). Reachability and controllability to zero of cone fractional linear systems, Archives of Control Sciences 17(3): 357-367. Zbl1152.93393
- Klamka J. (2002). Positive controllability of positive dynamical systems, Proceedings of American Control Conference, ACC-2002, Anchorage, AL, CD-ROM. Zbl1017.93014
- Klamka J. (2005). Approximate constrained controllability of mechanical systems, Journal of Theoretical and Applied Mechanics 43(3): 539-554.
- Miller K.S. and B. Ross (1993). An Introduction to the Fractional Calculus and Fractional Differenctial Equations, Willey, New York. Zbl0789.26002
- Moshrefi-Torbati M. and K. Hammond (1998). Physical and geometrical interpretation of fractional operators, Journal of the Franklin Institute 335B(6): 1077-1086. Zbl0989.26004
- Nishimoto K. (1984). Fractional Calculus, Koriyama: Decartes Press. Zbl0605.26006
- Oldham K.B. and J. Spanier (1974). The Fractional Calculus, New York: Academic Press. Zbl0292.26011
- Ortigueira M.D. (1997). Fractional discrete-time linear systems, Procedings of the IEEE International Conference on Acoustics, Speech and Signal Processing, Munich, Germany, Vol. 3, pp. 2241-2244.
- Ostalczyk P. (2000). The non-integer difference of the discretetime function and its application to the control system synthesis, International Journal of Systems Science 31(12): 1551-1561. Zbl1080.93592
- Ostalczyk P. (2004a). Fractional-order backward difference equivalent forms Part I Horner's form, IFAC Workshop on Fractional Differentation and its Applications, FDA'04, Bordeaux, France, pp. 342-347.
- Ostalczyk P. (2004b), Fractional-order backward difference equivalent forms Part II Polynomial Form. Proceedings the 1st IFAC Workshop Fractional Differentation and its Applications, FDA'04, Bordeaux, France, pp. 348-353.
- Oustalup A. (1993). Commande CRONE, Paris, Hermès.
- Oustalup A. (1995). La dérivation non entiére, Paris: Hermès.
- Podlubny I. (1999). Fractional Differential Equations, San Diego: Academic Press. Zbl0924.34008
- Podlubny I. (2002). Geometric and physical interpretation of fractional integration and fractional differentation, Fractional Calculs and Applied Analysis 5(4): 367-386. Zbl1042.26003
- Podlubny I., L. Dorcak and I. Kostial (1997). On fractional derivatives, fractional order systems and $P{I}^{\lambda}{D}^{\mu}$-controllers, Procedings of the 36th IEEE Conference on Decision and Control, San Diego, CA, USA, pp. 4985-4990.
- Reyes-Melo M.E., J.J. Martinez-Vega, C.A. Guerrero-Salazar and U. Ortiz-Mendez (2004). Modelling and relaxation phenomena in organic dielectric materials. Application of differential and integral operators of fractional order, Journal of Optoelectronics and Advanced Materials 6(3): 1037-1043.
- Riu D., N. Retiére and M. Ivanes (2001). Turbine generator modeling by non-integer order systems, Proceedings of the IEEE International Conference on Electric Machines and Drives, IEMDC 2001, Cambridge, MA, USA, pp. 185-187.
- Samko S. G., A.A. Kilbas and O.I. Marichev (1993). Fractional Integrals and Derivatives. Theory and Applications. London: Gordon and Breach. Zbl0818.26003
- Sierociuk D. and D. Dzieliński (2006). Fractional Kalman filter algorithm for the states, parameters and order of fractional system estimation, International Journal of Applied Mathematics and Computer Science 16(1): 129-140. Zbl1334.93172
- Sjöberg M. and L. Kari (2002). Non-linear behavior of a rubber isolator system using fractional derivatives, Vehicle System Dynamics 37(3): 217-236.
- Vinagre M., C. A. Monje and A.J. Calderon (2002). Fractional order systems and fractional order control actions. Lecture 3 IEEE CDC'02 TW#2: Fractional Calculus Applications in Automatic Control and Robotics.
- Vinagre M. and V. Feliu (2002) Modeling and control of dynamic systems using fractional calculus: Application to electrochemical processes and flexible structures, Proceedings of the 41st IEEE Conference Decision and Control, Las Vegas, NV, USA, pp. 214-239.
- Zaborowsky V. and R. Meylaov (2001). Informational network traffic model based on fractional calculus, Proceedings of the International Conference Info-tech and Info-net, ICII 2001, Beijing, China, Vol. 1, pp. 58-63.

## Citations in EuDML Documents

top- Bartosz Bandrowski, Anna Karczewska, Piotr Rozmej, Numerical solutions to integral equations equivalent to differential equations with fractional time
- Mikołaj Busłowicz, Tadeusz Kaczorek, Simple conditions for practical stability of positive fractional discrete-time linear systems
- Tadeusz Kaczorek, Reachability of cone fractional continuous-time linear systems
- Tadeusz Kaczorek, Descriptor fractional linear systems with regular pencils
- Tadeusz Kaczorek, Singular fractional linear systems and electrical circuits
- Tadeusz Kaczorek, Reduced-order fractional descriptor observers for a class of fractional descriptor continuous-time nonlinear systems

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.