Tuning a Fractional Order PID Controller with Lead Compensator in Frequency Domain
To achieve the desired specifications of gain and
phase margins for plants with time-delay that stabilized with FO-PID
controller a lead compensator is designed. At first the range of
controlled system stability based on stability boundary criteria is
determined. Using stability boundary locus method in frequency
domain the fractional order controller parameters are tuned and then
with drawing bode diagram in frequency domain accessing to desired
gain and phase margin are shown. Numerical examples are given to
illustrate the shapes of the stabilizing region and to show the design
procedure.
[1] I. Podlubny, "Fractional Differential Equations," Academic Press, 1999.
[2] R. Hilfer (ed.2), "Fractional Calculus in Physics," World Scientific,
2000.
[3] D. Xue, Y.Q. Chen. A Comparative Introduction of Four Fractional
Order Controllers. Proceedings of the 4 World Congress on Intelligent
Control and Automation June 10-14, 2002, Shanghai, P.R.China.
[4] S. Manabe, The Non-Integer Integral and its Applications to Control
Systems, ETJ of Japan, vol. 6, no. 3/4, 1961, pp. 83-87.
[5] Monje, V. Feliu, A. J. Calderon, B.M. Vinagre. The Fractional Order
Lead Compensator 0-7803-8588-8/04, 2004, IEEE
[6] M.S. Tavazoei, M. Haeri. Chaos control via a simple fractional-order
controller. Physics Letters A 372 (2008) 798-807.
[7] R.S. Barbosa, J.A.T. Machado, M.F. Silva. Time domain design of
fractional differintegrators using least-squares. Signal Processing 86
(2006) 2567-2581
[8] M. K. Bouafoura, N.B. Braiek. PIλD╬╝ controller design for integer and
fractional plants using piecewise orthogonal functions. Commun
Nonlinear Sci Numer Simulat 15 (2010) 1267-1278.
[9] A. Biswas, S. Das, A. Abraham, S. Dasgupta. Design offractional-order
PIλD╬╝ controllers with an improved differential evolution. Engineering
Applications of Artificial Intelligence 22 (2009) 343-350.
[10] C. A. Monje, B. M. Vinagre, Y.Q. Chen, V. Feliu, P. Lanusse, J.
Sabatier. PROPOSALS FOR FRACTIONAL PIλD╬╝ TUNING.
supported by the Spanish Research Grant 2PR02A024 (Junta de
Extremadura)
[11] C. A. Monje, B.M. Vinagre. Y.Q. Chen. V. Feliu. On Fractional PIλ
Controllers: Some Tuning Rules for Robustness to Plant Uncertainties.
2003 Kluwer Academic Publishers. Printed in the Netherlands.
[12] Y. Luo, H.S. Li, Y.Q. Chen. Fractional Order Proportional and
Derivative Controller Synthesis for A Class of Fractional Order
Systems: Tuning Rule and Hardware-in-the-loop Experiment. Joint 48th
IEEE Conference on Decision and Control and 28th Chinese Control
Conference Shanghai, P.R. China, December 16-18, 2009.
[13] Y. Luo, Y.Q. Chen. Fractional-order [Proportional Derivative]
Controller for Robust Motion Control: Tuning Procedure and
Validation. 2009 American Control Conference Hyatt Regency
Riverfront, St. Louis, MO, USA June 10-12, 2009
[14] H.S. Li, Y. Luo, Y.Q. Chen. A Fractional Order Proportional and
Derivative (FOPD) Motion Controller: Tuning Rule and Experiments.
IEEE Transactions on Control Systems Technology, VOL. 18, NO. 2,
March 2010.
[15] Y. Luo, Y.Q. Chen, C.Y. Wang, Y.G. Pi. Tuning fractional order
proportional integral controllers for fractional order systems. Journal of
Process Control 20 (2010) 823-831
[16] S. E. Hamamci. An Algorithm for Stabilization of Fractional-Order
Time Delay Systems Using Fractional-Order PID Controllers. IEEE
Transactions on Automatic Control, VOL. 52, NO. 10, October 2007.
[17] D.J. Wang. Synthesis of phase-lead/lag compensators with complete
information on gain and phase margins. Automatica 45 (2009)
1026_1031.
[18] Q.G. Wang, Z. Ye, C.C. Hang. Tuning of phase-lead compensators for
exact gain and phase margins. Automatica 42 (2006) 349 - 352.
[19] A.P. Loh, X. Cai, W.W. Tan. Auto-tuning of phase lead/lag
compensators. Automatica 40 (2004) 423 - 429.
[20] K. S. Yeung, K. O. Chaid, T. X. Dinh, (1995). Bode design chart for
continuous-time and discrete-time compensators. IEEE Transactions on
Education, 38(3), 252_257.
[21] F.Y. Wang (2003). The exact and unique solution for phase-lead and
phase-lag compensation. IEEE Transactions on Education, 46(2),
258_262.
[22] Q. G. Wang, Z. Ye, C. C. Hang. (2006). Tuning of phase-lead
compensators for exact gain and phase margins. Automatica, 42,
349_352.
[23] I. Podlubny, "Fractional-order systems and PIλD╬╝ controllers," IEEE
Trans. Autom. Control, vol. 44, no. 1, pp. 208-214, 1999.
[24] R.C. Dorf, R.H. Bishop. Modern Control Systems (10th Edition).
Pearson Prentice Hall (2010).
[25] C. Hwang and Y.-C. Cheng, "A numerical algorithm for stability testing
of fractional delay systems," Automatica, vol. 42, no. 5, pp. 825-831,
2006.
[26] J. I. Neimark, "D-decomposition of the space of quasi-polynomials (on
the stability of linearized distributive systems)," in Amer. Math. Soc.
(AMS) Transl., ser. 2. Providence, R. I.: AMS Soc., 1973, vol. 102, Ten
papers in analysis, pp. 95-131.
[27] Y.C. Cheng , C. Hwang, "Stabilization of unstable first-order time delay
systems using fractional-order PD controllers," J. Chinese Inst. Eng.,
vol. 29, no. 2, pp. 241-249, 2006.
[1] I. Podlubny, "Fractional Differential Equations," Academic Press, 1999.
[2] R. Hilfer (ed.2), "Fractional Calculus in Physics," World Scientific,
2000.
[3] D. Xue, Y.Q. Chen. A Comparative Introduction of Four Fractional
Order Controllers. Proceedings of the 4 World Congress on Intelligent
Control and Automation June 10-14, 2002, Shanghai, P.R.China.
[4] S. Manabe, The Non-Integer Integral and its Applications to Control
Systems, ETJ of Japan, vol. 6, no. 3/4, 1961, pp. 83-87.
[5] Monje, V. Feliu, A. J. Calderon, B.M. Vinagre. The Fractional Order
Lead Compensator 0-7803-8588-8/04, 2004, IEEE
[6] M.S. Tavazoei, M. Haeri. Chaos control via a simple fractional-order
controller. Physics Letters A 372 (2008) 798-807.
[7] R.S. Barbosa, J.A.T. Machado, M.F. Silva. Time domain design of
fractional differintegrators using least-squares. Signal Processing 86
(2006) 2567-2581
[8] M. K. Bouafoura, N.B. Braiek. PIλD╬╝ controller design for integer and
fractional plants using piecewise orthogonal functions. Commun
Nonlinear Sci Numer Simulat 15 (2010) 1267-1278.
[9] A. Biswas, S. Das, A. Abraham, S. Dasgupta. Design offractional-order
PIλD╬╝ controllers with an improved differential evolution. Engineering
Applications of Artificial Intelligence 22 (2009) 343-350.
[10] C. A. Monje, B. M. Vinagre, Y.Q. Chen, V. Feliu, P. Lanusse, J.
Sabatier. PROPOSALS FOR FRACTIONAL PIλD╬╝ TUNING.
supported by the Spanish Research Grant 2PR02A024 (Junta de
Extremadura)
[11] C. A. Monje, B.M. Vinagre. Y.Q. Chen. V. Feliu. On Fractional PIλ
Controllers: Some Tuning Rules for Robustness to Plant Uncertainties.
2003 Kluwer Academic Publishers. Printed in the Netherlands.
[12] Y. Luo, H.S. Li, Y.Q. Chen. Fractional Order Proportional and
Derivative Controller Synthesis for A Class of Fractional Order
Systems: Tuning Rule and Hardware-in-the-loop Experiment. Joint 48th
IEEE Conference on Decision and Control and 28th Chinese Control
Conference Shanghai, P.R. China, December 16-18, 2009.
[13] Y. Luo, Y.Q. Chen. Fractional-order [Proportional Derivative]
Controller for Robust Motion Control: Tuning Procedure and
Validation. 2009 American Control Conference Hyatt Regency
Riverfront, St. Louis, MO, USA June 10-12, 2009
[14] H.S. Li, Y. Luo, Y.Q. Chen. A Fractional Order Proportional and
Derivative (FOPD) Motion Controller: Tuning Rule and Experiments.
IEEE Transactions on Control Systems Technology, VOL. 18, NO. 2,
March 2010.
[15] Y. Luo, Y.Q. Chen, C.Y. Wang, Y.G. Pi. Tuning fractional order
proportional integral controllers for fractional order systems. Journal of
Process Control 20 (2010) 823-831
[16] S. E. Hamamci. An Algorithm for Stabilization of Fractional-Order
Time Delay Systems Using Fractional-Order PID Controllers. IEEE
Transactions on Automatic Control, VOL. 52, NO. 10, October 2007.
[17] D.J. Wang. Synthesis of phase-lead/lag compensators with complete
information on gain and phase margins. Automatica 45 (2009)
1026_1031.
[18] Q.G. Wang, Z. Ye, C.C. Hang. Tuning of phase-lead compensators for
exact gain and phase margins. Automatica 42 (2006) 349 - 352.
[19] A.P. Loh, X. Cai, W.W. Tan. Auto-tuning of phase lead/lag
compensators. Automatica 40 (2004) 423 - 429.
[20] K. S. Yeung, K. O. Chaid, T. X. Dinh, (1995). Bode design chart for
continuous-time and discrete-time compensators. IEEE Transactions on
Education, 38(3), 252_257.
[21] F.Y. Wang (2003). The exact and unique solution for phase-lead and
phase-lag compensation. IEEE Transactions on Education, 46(2),
258_262.
[22] Q. G. Wang, Z. Ye, C. C. Hang. (2006). Tuning of phase-lead
compensators for exact gain and phase margins. Automatica, 42,
349_352.
[23] I. Podlubny, "Fractional-order systems and PIλD╬╝ controllers," IEEE
Trans. Autom. Control, vol. 44, no. 1, pp. 208-214, 1999.
[24] R.C. Dorf, R.H. Bishop. Modern Control Systems (10th Edition).
Pearson Prentice Hall (2010).
[25] C. Hwang and Y.-C. Cheng, "A numerical algorithm for stability testing
of fractional delay systems," Automatica, vol. 42, no. 5, pp. 825-831,
2006.
[26] J. I. Neimark, "D-decomposition of the space of quasi-polynomials (on
the stability of linearized distributive systems)," in Amer. Math. Soc.
(AMS) Transl., ser. 2. Providence, R. I.: AMS Soc., 1973, vol. 102, Ten
papers in analysis, pp. 95-131.
[27] Y.C. Cheng , C. Hwang, "Stabilization of unstable first-order time delay
systems using fractional-order PD controllers," J. Chinese Inst. Eng.,
vol. 29, no. 2, pp. 241-249, 2006.
@article{"International Journal of Electrical, Electronic and Communication Sciences:60649", author = "Tahmine. V. Moghaddam and N. Bigdeli and K. Afshar", title = "Tuning a Fractional Order PID Controller with Lead Compensator in Frequency Domain", abstract = "To achieve the desired specifications of gain and
phase margins for plants with time-delay that stabilized with FO-PID
controller a lead compensator is designed. At first the range of
controlled system stability based on stability boundary criteria is
determined. Using stability boundary locus method in frequency
domain the fractional order controller parameters are tuned and then
with drawing bode diagram in frequency domain accessing to desired
gain and phase margin are shown. Numerical examples are given to
illustrate the shapes of the stabilizing region and to show the design
procedure.", keywords = "Fractional controller, Lead compensator, Stabilityregions, Stability boundary locus", volume = "5", number = "3", pages = "445-6", }
