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Calculation of all stabilizing fractional-order PD controllers for integrating time delay systems. (English) Zbl 1189.93125
Summary: A simple and effective stabilization method for integrating time delay systems using fractional order PD controllers $C(s)=kp+kds\mu $ is proposed. The presented method is based on finding the stability regions according to the fractional orders of the derivative element in the range of (0, 2). These regions are computed by using three stability boundaries: Real Root Boundary (RRB), Complex Root Boundary (CRB) and Infinite Root Boundary (IRB). The method gives the explicit formulae corresponding to these boundaries in terms of fractional order PD controller ($PD^\mu$ controller) parameters. Thus, the complete set of stabilizing controllers for an arbitrary integrating time delay system can be obtained. In order to demonstrate the effectiveness in solution accuracy and the simplicity of this method, two simulation studies are given. The simulation results indicate that the $PD^\mu$ controllers can provide larger stability regions than the integer order PD controllers.

93D15Stabilization of systems by feedback
93B51Design techniques in systems theory
Full Text: DOI
[1] Wang, L.; Cluett, W. R.: Tuning PID controllers for integrating processes, IEE proc. Control theory appl. 144, 385-388 (1997) · Zbl 0900.93089 · doi:10.1049/ip-cta:19971435
[2] Chidambaram, M.; Sree, R. Padma: A simple method of tuning PID controller for integrator/dead-time processes, Comput. chem. Eng. 27, 211-215 (2003)
[3] M. Shamsuzzoha, M. Lee, J. Park, Robust PID controller design of time delay processes with/without zero, in: Proc. of the ICIT 2006 IEEE Int. Conf. on Industrial Technology, Mumbai, India, Dec. 15--17, 2006
[4] Kaya, I.; Atherton, D. P.: Use of Smith predictor in the outer loop for cascaded control of unstable and integrating processes, Ind. eng. Chem. res. 47, 1981-1987 (2008)
[5] Zhang, W.; Xu, X.; Sun, Y.: Quantitative performance design for integrating processes with time delay, Automatica 35, 719-723 (1999) · Zbl 0925.93246 · doi:10.1016/S0005-1098(98)00207-6
[6] Aström, K. J.; Hägglund, T.: Advanced PID control, (2005)
[7] Ou, L.; Zhang, W.; Gu, D.: Sets of stabilising PID controllers for second-order integrating processes with time delay, IEE proc. Control theory appl. 153, 607-614 (2006)
[8] H. Taguchi, M. Kokawa, M. Araki, Optimal tuning of two-degree-of-freedom PD controllers, in: Proc. of the 4th Asian Control Conference, Singapore, September 25--27, 2002
[9] Leu, J. -F.; Tsay, S. -Y.; Hwang, C.: Design of optimal fractional-order PID controllers, J. chin. Inst. chem. Engrs. 33, 193-202 (2002)
[10] Agrawal, O. P.: A general formulation and solution scheme for fractional optimal control problems, Nonlinear dynam. 38, 323-337 (2004) · Zbl 1121.70019 · doi:10.1007/s11071-004-3764-6
[11] Y.Q. Chen, H. Dou, B.M. Vinagre, C.A. Monje, A robust tuning method for fractional order PI controllers, in: Proc. of the 2nd IFAC Workshop on Fractional Differentiation and its Applications, Porto, July 19--21, 2006
[12] Podlubny, I.: Fractional-order systems and $PI{\lambda}$D${\mu}$-controllers, IEEE trans. Automat. control 44, 208-214 (1999) · Zbl 1056.93542 · doi:10.1109/9.739144
[13] Barbosa, R. S.; Machado, J. A. T.; Ferreira, I. M.: Tuning of PID controllers based on bode’s ideal transfer function, Nonlinear dynam. 38, 305-321 (2004) · Zbl 1134.93334 · doi:10.1007/s11071-004-3763-7
[14] C.A. Monje, B.M. Vinagre, V. Feliu, Y.Q. Chen, On auto-tuning of fractional order PID controllers, in: Proc. of the 2nd IFAC Workshop on Fractional Differentiation and its Applications, Porto, July 19--21, 2006
[15] L. Ou, Y. Tang, D. Gu, W. Zhang, Stability analysis of PID controllers for integral processes with time delay, in: Proc. of the American Control Conference, Portland, June 08--10, 2005
[16] G.J. Silva, A. Datta, S.P. Bhattacharyya, Stabilization of first-order systems with time delay using the PID controller, in: Proc. of the American Control Conference, Arlington, June 25--27, 2001 · Zbl 1023.93052
[17] N. Hohenbichler, J. Ackermann, Synthesis of robust PID controllers for time delay systems, in: Proc. of the European Control Conference, Cambridge, 2003
[18] Tan, N.: Computation of stabilizing PI and PID controllers for processes with time delay, ISA trans. 44, 213-223 (2005)
[19] S. Jung, R.C. Dorf, Analytic PIDA controller design technique for a third order system, in: Proc. of the Decision and Control Conference, Davis, CA, USA, 1996
[20] Hamamci, S. E.: An algorithm for stabilization of fractional-order time delay systems using fractional-order PID controllers, IEEE trans. Automat. control 52, 1964-1969 (2007)
[21] Cheng, Y. -C.; Hwang, C.: Stabilization of unstable first-order time-delay systems using fractional-order PD controllers, J. chin. Inst. eng. 29, 241-249 (2006)
[22] Hamamci, S. E.; Tan, N.: Design of PI controllers for achieving time and frequency domain specifications simultaneously, ISA trans. 45, 529-543 (2006)
[23] Hamamci, S. E.: Stabilization using fractional-order PI and PID controllers, Nonlinear dynam. 51, 329-343 (2008) · Zbl 1170.93023 · doi:10.1007/s11071-007-9214-5
[24] Majhi, S.; Atherton, D. P.: Autotuning and controller design for processes with small time delays, IEE proc. Control theory appl. 146, 415-425 (1999)
[25] Hwang, C.; Cheng, Y. -C.: A numerical algorithm for stability testing of fractional delay systems, Automatica 42, 825-831 (2006) · Zbl 1137.93375 · doi:10.1016/j.automatica.2006.01.008
[26] Hwang, C.; Leu, J. -F.; Tsay, S. -Y.: A note on time-domain simulation of feedback fractional-ordersystems, IEEE trans. Automat. control 47, 625-631 (2002)
[27] Vinagre, B. M.; Chen, Y. Q.; Petras, I.: Two direct tustin discretization methods for fractional-order differentiator/integrator, J. franklin inst. Appl. math. 340, 349-362 (2003) · Zbl 1051.93031 · doi:10.1016/j.jfranklin.2003.08.001
[28] I. Petras, A Matlab Script for unit step characteristics of FOC systems. http://ivopetras.tripod.com/foc_t_ch.zip (2003)
[29] Chen, Y. Q.; Vinagre, B. M.; Podlubny, I.: Continued fraction expansion approaches to discretizing fractional order derivatives--an expository review, Nonlinear dynam. 38, 155-170 (2004) · Zbl 1134.93300 · doi:10.1007/s11071-004-3752-x