1. JCPEE
  2. All Volumes and Issues
  3. Volume 2, Number 2, 2012
  4. Positive fractional and cone fractional linear systems

Positive fractional and cone fractional linear systems

JCPEE.
2012;
Volume 2, Number 2
: pp. 31-40
Authors:
  1. Tadeusz Kaczorek
1
Białystok University of Technology

The positive fractional and cone fractional continuous-time and discrete-time linear systems are addressed. Sufficient conditions for the reachability of positive and cone fractional continuous-time linear systems are given. Necessary and sufficient conditions for the positivity and asymptotic stability of the continuous-time linear systems are established. The realization problem for positive fractional continuous-time systems is formulated and solved.

cone
continuous-time
fractional
Positive
realization problem
  1. M. Busłowicz, Robust Stability of Positive Discrete-time Linear Systems with Multiple Delays with Unity Rank Uncertainty Structure or Non-negative Perturbation Matrices // Bull. Pol. Acad. Sci., Techn. Sci. – Warsaw, Poland: Publishing house of Polish Academy of Sciences. – Vol. 55, №. 1. – 2007. – P. 347 - 350.
  2. M. Busłowicz, Simple Stability Conditions for Linear Positive Discrete-time Systems with Delays // Bull. Pol. Acad. Sci., Techn. Sci. – Warsaw, Poland: Publishing house of Polish Academy of Sciences. - Vol. 56, No. 4. - 2008 - P. 325-328.
  3. M. Busłowicz, Stability of Linear Continuous-Time fractional Order Systems with Delays of the Retarded Type // Bull. Pol. Acad. Sci., Techn. Sci. – Warsaw, Poland: Publishing house of Polish Academy of Sciences. – Vol. 56, №. 4. – 2008. – P. 318-324.
  4. M. Busłowicz, Robust Stability of Positive Discrete-time Linear Systems with Multiple Delays with Linear Unit Rank Uncertainty Structure or Non-negative Perturbation Matrices // Bull. Pol. Acad. Sci., Techn. Sci. – Warsaw, Poland: Publishing house of Polish Academy of Sciences. – Vol. 52, №. 2. – 2004. – P. 99-102.
  5. M. Busłowicz and T. Kaczorek, Robust Stability of Positive Discrete-time Interval Systems with Time Delays // Bull. Pol. Acad. Sci. Tech. – Vol. 55, №. 1. – 2007. – P. 1-5.
  6. M. Busłowicz, T. Kaczorek, Stability and Robust Stability of Positive Discrete-time Systems with Pure Delays// In Proc. 10th IEEE Int. Conf. On Methods and Models in Automation and Robotics. – Międzyzdroje, Poland. – 2004. – P. 105-108.
  7. M. Busłowicz, T. Kaczorek, Robust Stability of Positive Discrete-time Systems with Pure Delays with Linear Unit Rank Uncertainty Structure // In Proc. 11th IEEE Int. Conf. On Methods and Models in Automation and Robotics. – Międzyzdroje, Poland. – 2005.
  8. L. Farina, S. Rinaldi, Positive Linear Systems; Theory and Applications. - New York, USA: Wiley. – 2000.
  9. T. Kaczorek, Positive 1D and 2D Systems, London, UK: Springer-Verlag. – 2002.
  10. T. Kaczorek, Computation of Realizations of Discrete-time Cone Systems // Bull. Pol. Acad. Sci., Techn. Sci. – Warsaw, Poland: Publishing house of Polish Academy of Sciences. – Vol. 54, №. 3. – 2006. – P. 347-350.
  11. T. Kaczorek, Reachability and Controllability to Zero Tests for Standard and Positive Fractional Discrete-time Systems // JESA Journal. – Vol. 42, №. 6-9. – 2008. – P. 770-787.
  12. T. Kaczorek, Positive Minimal Realizations for Singular Discrete-time Systems with Delays in State and Delays in Control // Bull. Pol. Acad. Sci., Techn. Sci. – Warsaw, Poland: Publishing house of Polish Academy of Sciences. – Vol. 53, № 3. – 2005. – P. 293-298.
  13. T. Kaczorek, Asymptotic Stability of Positive 2D Linear Systems with Delays // Bull. Pol. Acad. Sci., Techn. Sci. – Warsaw, Poland: Publishing house of Polish Academy of Sciences. – Vol. 52, №. 2. – 2009. – P.133-138.
  14. T. Kaczorek, Realization Problem for Singular Positive Continuous-time Systems with Delays // Control and Cybernetics. – Vol. 36, №. 1. – 2007. P. 2-11.
  15. T. Kaczorek, Realization Problem for Positive Continuous-time Systems with Delays // Int. J. Comp. Intellig. And Appl. – Vol. 6, №. 2. – 2006. P. 289-298.
  16. T. Kaczorek, Reachability and Controllability to Zero of Positive Fractional Discrete-time Systems // Machine Intelligence and Robotic Control. – Vol. 6, №. 4. – 2007. – P.139-143.
  17. T. Kaczorek, Reachability and Controllability to Zero of Cone Fractional Linear Systems // Archives of Control Scienes. – Vol. 17, №. 3. – 2007. – P. 357-367.
  18. T. Kaczorek, Fractional Positive Continuous-time Linear Systems and Their Reachability // Int. J. Appl. Math. Sci. – Vol. 18, №. 2. – 2008. – P. 223-228.
  19. T. Kaczorek, Realization Problem for Positi­ve Fractional Discrete-time Linear Systems // Int. J. Fact. Autom. Robot. Soft Comput. – Vol.2. – 2008. – P.76-88.
  20. T. Kaczorek, Cone-realizations for Multivariable Continuous-time Systems with Delays // In Proc. 5th Workshop of the IIGSS. – Wuham, China. – 2007.
  21. T. Kaczorek, Reachability of Cone Fractional Continuous-time Linear Systems // Int. J. Appl. Math. Comput. SCI. – Vol. 19, №. 1. – 2009.
  22. T. Kaczorek, Realization Problem for Fractional Continuous-time Systems // Archives of Control Sciences. – Vol. 18, №. 1. – 2008. – P. 43-58.
  23. T. Kaczorek, Selected Problems in Fractional System Theory. – London, UK: Springer-Verlag. – 2011.
  24. T. Kaczorek, M. Busłowicz, Reachability and Minimum Energy Control of Positive Linear Discrete-time Systems with Multiple Delays in State and Control // Pomiary, Automatyka, Kontrola. – Warsaw, Poland: PAK. – №. 10. – 2007. – P. 40-44.
  25. K. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations. – New York, USA: Wiley. – 1993.
  26. K. Nishimoto, Fractional Calculus. Koriama, Japan: – Decartess Press. – 1984.
  27. K. Oldham., and J. Spanier, The Fractional Calculus. - New York, USA: Academic Press. – 1974.
  28. P. Ostalczyk, The Non-integer Difference of the Discrete-time Function and its Application to the Control System Synthesis // Int. J. Syst. Sci. – Vol. 31, №. 12. – 2000. P. 1551–1561.
  29. A. Oustaloup, Fractional Derivation. – Paris, France: Hermés. – 1995. (French)
  30. I. Podlubny, Fractional Differential Equations. – San Diego, USA: Academic Press. – 1999.
  31. B. Vinagre, C. Monje, A. Calderon, Fractional Order Systems and Fractional Order Control Actions //  41st IEEE Conf. Decision and Control. - TW#2: Fractional Calculus Applications in Autiomatic Control and Robotics. – Las Vegas, USA. – 2002
  32. B. Vinagre, V. Feliu, Modelling and Control of Dynamic System using Fractional Calculus: Application to Electrochemical Processes and Flexibles Structures // In Proc. 41st IEEE Conf. Decision and Control. – Las Vegas, USA. – 2002. – P. 214-613.