Enhancement of Fuzzy PID Controller with Fractional Calculus

  • Indranil Pan
  • Saptarshi Das
Part of the Studies in Computational Intelligence book series (SCI, volume 438)


The integer order fuzzy Proportional-Integral-Derivative (PID) controller has been generalized in this chapter to include fractional order derivative and integrals. It works on the closed loop error and its fractional derivative as the input and has a fractional integrator in its output. Other fractional variants of the integer order fuzzy logic controller are also briefly introduced. The fractional order differ-integrations in the proposed fuzzy logic controller (FLC) are kept as design variables along with the input-output scaling factors (SF) and are optimized with Genetic Algorithm (GA) while minimizing several integral error indices along with the control signal as the objective function. Simulations studies are carried out to control a delayed nonlinear process and an open loop unstable process with time delay. The closed loop performances and controller efforts in each case are compared with conventional PID, fuzzy PID and PI λ D μ controller subjected to different integral performance indices. Simulation results show that the proposed fractional order fuzzy PID controller outperforms the others in most cases.


Particle Swarm Optimization Fractional Order Fractional Calculus Fuzzy Logic Controller Load Disturbance 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. Aoun, M., Malti, R., Levron, F., Oustaloup, A.: Numerical simulations of fractional systems: an overview of existing methods and improvements. Nonlinear Dynamics 38, 117–131 (2004)zbMATHCrossRefGoogle Scholar
  2. Arena, P., Caponetto, R., Fortuna, L., Porto, D.: Bifurcation and chaos in noninteger order cellular neural networks. International Journal of Bifurcation and Chaos 8, 1527–1540 (1998)zbMATHCrossRefGoogle Scholar
  3. Arena, P., Fortuna, L., Porto, D.: Chaotic behavior in noninteger-order cellular neural networks. Physical Review E 61, 776 (2000)CrossRefGoogle Scholar
  4. Barbosa, R.S., Jesus, I.S., Silva, M.F.: Fuzzy reasoning in fractional-order PD controllers. New Aspects of Applied Informatics, Biomedical Electronics & Informatics and Communications, 252–257 (2010)Google Scholar
  5. Biswas, A., Das, S., Abraham, A., Dasgupta, S.: Design of fractional-order PIλDμ controllers with an improved differential evolution. Engineering Applications of Artificial Intelligence 22, 343–350 (2009)MathSciNetCrossRefGoogle Scholar
  6. Cai, M., Pan, X., Du, Y.: New Elite Multi-Parent Crossover Evolutionary Optimization Algorithm of Parameters Tuning of Fractional-Order PID Controller and Its Application. In: 2009 Fourth International Conference on Innovative Computing, Information and Control (ICICIC), pp. 64–67 (2009)Google Scholar
  7. Cao, J.Y., Cao, B.G.: Design of fractional order controllers based on particle swarm optimization. In: 2006 1st IEEE Conference on Industrial Electronics and Applications, pp. 1–6 (2006)Google Scholar
  8. Cao, J.Y., Liang, J., Cao, B.G.: Optimization of fractional order PID controllers based on genetic algorithms. In: Proceedings of 2005 International Conference on Machine Learning and Cybernetics, vol. 9, pp. 5686–5689 (2005)Google Scholar
  9. Chang, F.K., Lee, C.H.: Design of fractional PID control via hybrid of electromagnetism-like and genetic algorithms. In: Eighth International Conference on Intelligent Systems Design and Applications, ISDA 2008, vol. 2, pp. 525–530 (2008)Google Scholar
  10. Charef, A.: Analogue realisation of fractional-order integrator, differentiator and fractional PIλDμ controller. IEE Proceedings-Control Theory and Applications 153, 714–720 (2006)CrossRefGoogle Scholar
  11. Das, S.: Functional fractional calculus. Springer (2011)Google Scholar
  12. Das, S., Saha, S., Das, S., Gupta, A.: On the selection of tuning methodology of FOPID controllers for the control of higher order processes. ISA Transactions (2011)Google Scholar
  13. Delavari, H., Ghaderi, R., Ranjbar, A., Momani, S.: Fuzzy fractional order sliding mode controller for nonlinear systems. Communications in Nonlinear Science and Numerical Simulation 15, 963–978 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  14. Dorcak, L., Terpak, J., Papajova, M., et al.: Design of the fractional-order PI λDμ controller based on the optimization with self-organizing migrating algorithm. Acta Montanistica Slovaca 12, 285–293 (2007)Google Scholar
  15. Driankov, D., Hellendoorn, H., Reinfrank, M.: An introduction to fuzzy control. Springer (1993)Google Scholar
  16. Efe, M.O.: Fractional fuzzy adaptive sliding-mode control of a 2-DOF direct-drive robot arm. IEEE Transactions on Systems, Man, and Cybernetics, Part B: Cybernetics 38, 1561–1570 (2008)CrossRefGoogle Scholar
  17. Er, M.J., Sun, Y.L.: Hybrid fuzzy proportional-integral plus conventional derivative control of linear and nonlinear systems. IEEE Transactions on Industrial Electronics 48, 1109–1117 (2001)CrossRefGoogle Scholar
  18. Golob, M.: Decomposed fuzzy proportional-integral-derivative controllers. Applied Soft Computing 1, 201–214 (2001)CrossRefGoogle Scholar
  19. Hu, B., Mann, G.K.I., Gosine, R.G.: New methodology for analytical and optimal design of fuzzy PID controllers. IEEE Transactions on Fuzzy Systems 7, 521–539 (1999)CrossRefGoogle Scholar
  20. Kadiyala, V.K., Jatoth, R.K., Pothalaiah, S.: Design and implementation of Fractional Order PID controller for aerofin control system. In: World Congress on Nature & Biologically Inspired Computing, NaBIC 2009, pp. 696–701 (2009)Google Scholar
  21. Karimi-Ghartemani, M., Zamani, M., Sadati, N., Parniani, M.: An optimal fractional order controller for an AVR system using particle swarm optimization algorithm. In: 2007 Large Engineering Systems Conference on Power Engineering, pp. 244–249 (2007)Google Scholar
  22. Krishna, B.: Studies on fractional order differentiators and integrators: A survey. Signal Processing 91, 386–426 (2011)zbMATHCrossRefGoogle Scholar
  23. Kundu, D., Suresh, K., Ghosh, S., Das, S.: Designing Fractional-order PIλDμ controller using a modified invasive Weed Optimization algortihm. In: World Congress on Nature & Biologically Inspired Computing, NaBIC 2009, pp. 1315–1320 (2009)Google Scholar
  24. Lee, C.H., Chang, F.K.: Fractional-order PID controller optimization via improved electromagnetism-like algorithm. Expert Systems with Applications 37, 8871–8878 (2010)CrossRefGoogle Scholar
  25. Li, T.H.S., Shieh, M.Y.: Design of a GA-based fuzzy PID controller for non-minimum phase systems. Fuzzy Sets and Systems 111, 183–197 (2000)MathSciNetCrossRefGoogle Scholar
  26. Li, W.: Design of a hybrid fuzzy logic proportional plus conventional integral-derivative controller. IEEE Transactions on Fuzzy Systems 6, 449–463 (1998)CrossRefGoogle Scholar
  27. Li, W., Chang, X.: Application of hybrid fuzzy logic proportional plus conventional integral-derivative controller to combustion control of stoker-fired boilers. Fuzzy Sets and Systems 111, 267–284 (2000)MathSciNetCrossRefGoogle Scholar
  28. Li, W., Chang, X., Farrell, J., Wahl, F.M.: Design of an enhanced hybrid fuzzy P+ ID controller for a mechanical manipulator. IEEE Transactions on Systems, Man, and Cybernetics, Part B: Cybernetics 31, 938–945 (2001a)CrossRefGoogle Scholar
  29. Li, W., Chang, X., Wahl, F., Farrell, J.: Tracking control of a manipulator under uncertainty by FUZZY P+ ID controller. Fuzzy Sets and Systems 122, 125–137 (2001b)MathSciNetzbMATHCrossRefGoogle Scholar
  30. Luo, Y., Li, J.: The controlling parameters tuning and its application of fractional order PID bacterial foraging-based oriented by particle swarm optimization. In: IEEE International Conference on Intelligent Computing and Intelligent Systems, ICIS 2009, vol. 1, pp. 4–7 (2009)Google Scholar
  31. Maiti, D., Acharya, A., Chakraborty, M., et al.: Tuning PID and PIλDδ Controllers using the Integral Time Absolute Error Criterion. In: 4th International Conference on Information and Automation for Sustainability, ICIAFS 2008, pp. 457–462 (2008a)Google Scholar
  32. Maiti, D., Chakraborty, M., Acharya, A., Konar, A.: Design of a fractional-order self-tuning regulator using optimization algorithms. In: 11th International Conference on Computer and Information Technology, ICCIT 2008, pp. 470–475 (2008b)Google Scholar
  33. Majid, Z., Masoud, K., Nasser, S.: Design of an H∞-optimal FOPID controller using particle swarm optimization. In: Control Conference, CCC 2007, Chinese, pp. 435–440 (2007)Google Scholar
  34. Malki, H.A., Misir, D., Feigenspan, D., Chen, G.: Fuzzy PID control of a flexible-joint robot arm with uncertainties from time-varying loads. IEEE Transactions on Control Systems Technology 5, 371–378 (1997)CrossRefGoogle Scholar
  35. MathworksInc. Global Optimisation Toolbox, User’s Guide. 2010 (2010)Google Scholar
  36. Meng, L., Xue, D.: Design of an optimal fractional-order PID controller using multi-objective GA optimization. In: Control and Decision Conference, CCDC 2009, Chinese, pp. 3849–3853 (2009)Google Scholar
  37. Misir, D., Malki, H.A., Chen, G.: Design and analysis of a fuzzy proportional-integral-derivative controller. Fuzzy Sets and Systems 79, 297–314 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  38. Monje, C.A., Vinagre, B.M., Feliu, V., Chen, Y.Q.: Tuning and auto-tuning of fractional order controllers for industry applications. Control Engineering Practice 16, 798–812 (2008)CrossRefGoogle Scholar
  39. Mudi, R.K., Pal, N.R.: A robust self-tuning scheme for PI-and PD-type fuzzy controllers. IEEE Transactions on Fuzzy Systems 7, 2–16 (1999)CrossRefGoogle Scholar
  40. O’Dwyer, A.: Handbook of PI and PID controller tuning rules. Imperial College Pr. (2006)Google Scholar
  41. Ou, B., Song, L., Chang, C.: Tuning of fractional PID controllers by using radial basis function neural networks. In: 2010 8th IEEE International Conference on Control and Automation (ICCA), pp. 1239–1244 (2010)Google Scholar
  42. Oustaloup, A., Levron, F., Mathieu, B., Nanot, F.M.: Frequency-band complex noninteger differentiator: characterization and synthesis. IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications 47, 25–39 (2000)CrossRefGoogle Scholar
  43. Pan, I., Das, S., Gupta, A.: Tuning of an optimal fuzzy PID controller with stochastic algorithms for networked control systems with random time delay. ISA Transactions 50, 28–36 (2011a)CrossRefGoogle Scholar
  44. Pan, I., Das, S., Gupta, A.: Handling packet dropouts and random delays for unstable delayed processes in NCS by optimal tuning of PIλDμ controllers with evolutionary algorithms. ISA Transactions 50, 557–572 (2011b)CrossRefGoogle Scholar
  45. Podlubny, I.: Fractional-order systems and PIλDμ controllers. IEEE Transactions on Automatic Control 44, 208–214 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  46. Podlubny, I., Petravs, I., Vinagre, B.M., et al.: Analogue realizations of fractional-order controllers. Nonlinear Dynamics 29, 281–296 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  47. Sadati, N., Ghaffarkhah, A., Ostadabbas, S.: A new neural network based FOPID controller. In: IEEE International Conference on Networking, Sensing and Control, ICNSC 2008, pp. 762–767 (2008)Google Scholar
  48. Sadati, N., Zamani, M., Mohajerin, P.: Optimum design of fractional order PID for MIMO and SISO systems using particle swarm optimization techniques. In: 4th IEEE International Conference on Mechatronics, ICM2007, pp. 1–6 (2007)Google Scholar
  49. Saha, S., Das, S., Ghosh, R., et al.: Design of a Fractional Order Phase Shaper for Iso-Damped Control of a PHWR Under Step-Back Condition. IEEE Transactions on Nuclear Science 57, 1602–1612 (2010)CrossRefGoogle Scholar
  50. Skoczowski, S., Domek, S., Pietrusewicz, K., Broel-Plater, B.: A method for improving the robustness of PID control. IEEE Transactions on Industrial Electronics 52, 1669–1676 (2005)CrossRefGoogle Scholar
  51. Tang, K., Man, K.F., Chen, G., Kwong, S.: An optimal fuzzy PID controller. IEEE Transactions on Industrial Electronics 48, 757–765 (2001)CrossRefGoogle Scholar
  52. Tavazoei, M.S.: Notes on integral performance indices in fractional-order control systems. Journal of Process Control 20, 285–291 (2010)CrossRefGoogle Scholar
  53. Valério, D., Sá da Costa, J.: Variable-order fractional derivatives and their numerical approximations. Signal Processing 91, 470–483 (2011)zbMATHCrossRefGoogle Scholar
  54. Visioli, A.: Optimal tuning of PID controllers for integral and unstable processes. IEE Proceedings-Control Theory and Applications 148, 180–184 (2001)CrossRefGoogle Scholar
  55. Woo, Z.W., Chung, H.Y., Lin, J.J.: A PID type fuzzy controller with self-tuning scaling factors. Fuzzy Sets and Systems 115, 321–326 (2000)zbMATHCrossRefGoogle Scholar
  56. Yesil, E., Güzelkaya, M., Eksin, I.: Self tuning fuzzy PID type load and frequency controller. Energy Conversion and Management 45, 377–390 (2004)CrossRefGoogle Scholar
  57. Zamani, M., Karimi-Ghartemani, M., Sadati, N., Parniani, M.: Design of a fractional order PID controller for an AVR using particle swarm optimization. Control Engineering Practice 17, 1380–1387 (2009)CrossRefGoogle Scholar
  58. Zhao, Y., Gao, Y., Hu, Z., et al.: Damping inter area oscillations of power systems by a fractional order PID controller. In: International Conference on Energy and Environment Technology, ICEET 2009, vol. 2, pp. 103–106 (2009)Google Scholar
  59. Zhuang, M., Atherton, D.: Automatic tuning of optimum PID controllers. IEE Proceedings Control Theory and Applications D 140, 216–224 (1993)zbMATHCrossRefGoogle Scholar

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© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Department of Power EngineeringJadavpur UniversityKolkataIndia

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