1. JCPEE
  2. Всі випуски
  3. Випуск 3, Номер 1, 2013
  4. Додатні дробові та конічні дробові лінійні системи із затримкою

Додатні дробові та конічні дробові лінійні системи із затримкою

JCPEE.
2013;
Випуск 3, Номер 1
: с. 41-52
Автори:
  1. Тадеуш Качорек
1
Білостоцький політехнічний університет

У статті розглянуто додатні та конічні дробові неперервні та дискретні в часі лінійні системи. Наведено достатні умови для досяжності таких систем. Встановлено необхідні та достатні умови для додатності та асимптотичної стабільності неперервних у часі лінійних систем із затримкою. Сформульовано та розв’язано проблему реалізації додатних дробових неперервних у часі систем. Встановлено необхідні та достатні умови для додатності та практичної стабільності дробових дискретних у часі лінійних систем Застосовано підхід лінійної матричної нерівності (ЛМН) для перевірки асимптотичної стабільності додатних дробових дискретних у часі лінійних систем. Встановлено достатні умови для існування та запропоновано процедури для розрахунку додатних та конічних реалізацій дискретних у часі лінійних систем.

fractional
Positive
continuous-time
discrete-time
linear
система
Reachability
controllability
realization problem
LMI approach.
  1. M.Busłowicz, “Robust stability of positive of discrete-time linear systems with multiple delays with unity rank uncertainty structure or non-negative perturbation matrices,” Bull. Pol. Acad. Sci. Techn., vol. 55, no. 1,  pp. 347-350, 2007.
  2. M.Busłowicz, “Simple stability conditions for linear positive discrete-time systems with delays,” Bull. Pol. Acad. Sci. Techn., vol. 56, no. 4, pp. 325-328, 2008.
  3. M.Busłowicz, “Stability of linear continuous-time fractional order systems with delays of the retarded type,” Bull. Pol. Acad. Sci. Techn., vol. 56, no. 4, pp. 318-324, 2008. 
  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., vol. 52, no. 2,  pp. 99-102, 2004.
  5. M.Busłowicz and T. Kaczorek, “Robust stability of positive discrete-time interval systems with time delays,” Bull. Pol. Acad. Sci. Tech., vol. 55, no. 1, pp. 1-5, 2007.
  6. M.Busłowicz and 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, vol. 1, pp. 105-108, Międzyzdroje, Poland. 2004.
  7. M.Busłowicz and 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, Paper 0169 (on CD-ROM), Międzyzdroje, Poland, 2005.
  8. L.Farina and S.Rinaldi, Positive Linear Systems; Theory and Applications, New York, USA: J. Wiley, 2000.
  9. E.Fornasini and G.Marchesini, “State-space realiza­tion theory of two-dimensional filters,” IEEE Trans, Autom. Contr., vol. AC-21, pp. 481-491, 1976.
  10. E.Fornasini and G.Marchesini, “Double indexed dynamical systems,” Math. Sys. Theory, vol. 12, pp. 59-72, 1978.
  11. D.Hinrihsen, N.K.Sin, and P.H.A.Hgoc, “Stability radii of positive higher order difference systems,” System&Control Letters, vol. 49, pp. 377-388, 2003.
  12. A.Hmamedd, A.Benzaouia, M.Ait Rami, and F.Tadeo, “Positive stabilization of discrete-time systems with unknown delays and bounded control,” in Proc. European Control Conference, Kos, Greece, pp. 5616-5622 (paper ThD07.3) , 2007.
  13. T.Kaczorek, Positive 1D and 2D Systems, London, UK: Springer-Verlag, 2002.
  14. T.Kaczorek, “Stability of positive discrete-time systems with time-delays,” in Proc. 8th World Multiconference on Systemics, Cybernetics and Informatics, pp. 321-324, Orlando, Florida, USA, 2004.
  15. T.Kaczorek, “Choice of the forms of Lyapunov function for positive 2D Roesser model,” Int. J. Applied Math. Comp. Sci., vol. 17, no. 4, pp. 471-475, 2007.
  16. T.Kaczorek, “Asymptotic stability of positive 1D and 2D linear systems,” Recent Advances in Control and Automation, Acad. Publ. House EXIT, pp. 41-52, 2008.
  17. T.Kaczorek, “Practical stability of positive fractional discrete-time systems,” Bull. Pol. Acad. Sci. Techn., vol. 56, no. 4, 313-318, 2008.
  18. T.Kaczorek, “Computation of realizations of discrete-time cone systems,” Bull. Pol. Acad. Sci. Techn., vol. 54, no. 3, pp. 347-350, 2006.
  19. T. Kaczorek, “Reachability and controllability to zero tests for standard and positive fractional discrete-time systems,” JESA Journal, vol. 42, no. 6-9, pp. 770-787, 2008.
  20. T.Kaczorek, “Positive minimal realizations for singular discrete-time systems with delays in state and delays in control,” Bull. Pol. Acad. Sci. Techn., vol. 53, no. 3, pp. 293-298, 2005.
  21. T.Kaczorek, “Asymptotic stability of positive 2D linear systems with delays,” Bull. Pol. Acad. Sci. Techn., vol. 52, no. 2, pp.133-138, 2009.
  22. T.Kaczorek, “Realization problem for singular positive continuous-time systems with delays,” Control and Cybernetics, vol. 36, no. 1, pp. 2-11, 2007.
  23. T.Kaczorek, “Realization problem for positive continuous-time systems with delays,” Int. J. Comp. Intellig. And Appl., vol. 6, no. 2, pp. 289-298, 2006.
  24. T.Kaczorek, “Realization problem for positive fractional hybrid 2D linear systems,” Fractional Calculus and Applied Analysis, vol. 11, no. 3, pp. 1-16, 2008.
  25. T.Kaczorek, “Reachability and controllability to zero of positive fractional discrete-time systems,” Machine Intelligence and Robotic Control, vol. 6, no. 4, pp. 139-143, 2007.
  26. T.Kaczorek, “Reachability and controllability to zero of cone fractional linear systems,” Archives of Control Scienes, vol. 17, no. 3, pp. 357-367, 2007.
  27. T.Kaczorek, “Fractional positive continuous-time linear systems and their reachability,” Int. J. Appl. Math. Sci., vol. 18, no. 2, pp. 223-228, 2008.
  28. T.Kaczorek, “Positive 2D hybrid linear systems,” Bull. Pol. Acad. Techn. Sci., vol. 55, no. 4, pp. 351-358, 2007.
  29. T.Kaczorek, “Realization problem for positive 2D hybrid systems,” COMPEL, vol. 27, no. 3, pp. 613-623, 2008.
  30. T.Kaczorek, “Positivity and stabilization of fractional 2D Roesser model by state feedbacks, LMI approach,” Archives of Control Sciences, vol. 19, no. 2, pp. 165, 2009.
  31. T.Kaczorek, “Realization problem for positive fractional discrete-time linear systems,” Int. J. Fact. Autom. Robot. Soft Comput., vol. 2, pp. 76-88, July 2008.
  32. T.Kaczorek, “Cone-realizations for multivariable continuous-time systems with delays,” in Proc. 5th Workshop of the IIGSS, Wuham, China, 14-17 June 2007.
  33. T.Kaczorek, “LMI approach to stability of 2D positive systems with delays,” Multidim. Syst. Sign  Process., vol. 20, pp. 39-54, 2009.
  34. T.Kaczorek, “Reachcability of cone fractional continuous-time linear systems,” Int. J. Appl. Math. Comput. Sci., vol. 19, no. 1, pp.89-93, , 2009.
  35. T.Kaczorek, “Stability of positive continuous-time linear systems with delays,” Bull. Pol. Acad. Sci. Techn., vol. 57, no. 4, pp. 395-398, 2009.
  36. T.Kaczorek, “Independence of the asymptotic stability of positive 2D linear systems with delays of their delays” Int. J. Appl. Math. Comput. Sci., vol. 19, no. 2, pp. 255-261, 2009.
  37. T.Kaczorek, “Realization problem for fractional continuous-time systems,” Archives of Control Sciences, vol. 18, no. 1, pp. 43-58, 2008.
  38. T.Kaczorek, “LMI approaches to practical stability of positive fractional discrete-time linear systems,” Int. J. Appl. Math. Comput. Sci., vol. 19, no. 4. 2009 (in Press).
  39. T.Kaczorek, “Asymptotic stability of positive 2D linear systems,” Proc. of 13th Scientific Conf. Computer Applications in Electrical Engineering, April 14-16, Poznan, Poland, pp. 1-5.
  40. T.Kaczorek, “Practical stability of positive fractional discrete-time linear systems,” Bull. Pol. Acad. Techn. Sci., vol. 56, no. 4, pp. 313-324, 2008.
  41. T.Kaczorek, “Reachability and controllability of non-negative 2D Roesser type models,” Bull. Pol. Acad. Sci. Techn., vol. 44, no. 4, pp. 405-410, 1966.
  42. T.Kaczorek, “Positive different orders fractional 2D linear systems,” Acta Mechanica et Automatica, vol. 2, no. 2, pp. 1-8, 2008.
  43. T.Kaczorek, “Fractional 2D linear systems,” Journal of Automation, Mobile Robotics and Intelligent Systems, vol. 2, no. 2, pp. 5-9, 2008.
  44. T.Kaczorek, Selected Problems in Fractional System Theory, London, UK: Springer–Verlag, 2011.
  45. T.Kaczorek, “New stability tests of positive standard and fractional linear systems,” Circuits and Systems, vol. 2, pp. 261-268, 2011.
  46. T.Kaczorek, “Positive switched 2D linear systems described by the Roesser Models,” European Journal of Control, vol. 3, pp. 1-8, 2012.
  47. 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, no. 10, pp. 40-44, 2007.
  48. T.Kaczorek and K.Rogowski, “Positivity and stabi­lization of fractional 2D linear systems described by the Roesser model,” in Proc. MMAR Conference, Międzyzdroje, Poland, 2009.
  49. J.Kurek, “The general state-space model for a two-dimensional filters,” IEEE Trans. Autom. Contr., AC-30, pp. 600-602, 1985.
  50. K.S.Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, New York, USA: Willey, 1993.
  51. K.Nishimoto, Fractional Calculus, Decartess Press, Koriama, 1984.
  52. K.B.Oldham and J.Spanier, The Fractional Calculus, New York, USA: Academic Press, 1974.
  53. 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, no. 12, pp. 1551-1561, 2000.
  54. A.Oustaloup, Fractional Derivative, Paris, France: Hermés, 1995. (French)
  55. Podlubny, Fractional Differential Equations, San Diego, USA: Academic Press, 1999.
  56. P.Przyborowski and T. Kaczorek, “Positive 2D discrete-time linear Lyapunov systems,” Int. J. Appl. Math. Comput. Sci., vol.19, no.1, pp. 95-1005, 2009.
  57. R.P.Roesser, “A discrete state-space model for linear image processing,” IEEE  Trans. Autom. Contr., AC-20, vol. 1, pp. 1-10, 1975.
  58. M.Twardy, “An LMI approach to checking stability of 2D positive systems,” Bull. Pol. Acad. Sci. Techn., vol. 55, no.4, pp. 386-395, 2007. 
  59. M.E. Valcher, “On the initial stability and asymptotic behavior of 2D positive systems,” IEEE Trans. on Circuits and Systems – I, vol. 44, no. 7, pp. 602-613, 1997.
  60. B.E.Vinagre, C.A.Monje, and A.J.Calderon, “Frac­tional order systems and fractional order control actions,” Lecture 3 IEEE CDC’02 TW#2: Fractional Calculus Applications in Autiomatic Control and Robotics.
  61. M.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, pp. 214-613, Las Vegas, USA, 2002.