Review, Design, Optimization and Stability Analysis of Fractional-Order PID Controller

Автор: Ammar SOUKKOU, M.C. BELHOUR, Salah LEULMI

Журнал: International Journal of Intelligent Systems and Applications(IJISA) @ijisa

Статья в выпуске: 7 vol.8, 2016 года.

Бесплатный доступ

This paper will establish the importance and significance of studying the fractional-order control of nonlinear dynamical systems. The foundation and the sources related to this research scope is going to be set. Then, the paper incorporates a brief overview on how this study is performed and present the organization of this study. The present work investigates the effectiveness of the physical-fractional and biological-genetic operators to develop an Optimal Form of Fractional-order PID Controller (O2Fo-PIDC). The newly developed Fo-PIDC with optimal structure and parameters can, also, improve the performances required in the modeling and control of modern manufacturing-industrial process (MIP). The synthesis methodology of the proposed O2Fo-PIDC can be viewed as a multi-level design approach. The hierarchical Multiobjective genetic algorithm (MGA), adopted in this work, can be visualized as a combination of structural and parametric genes of a controller orchestrated in a hierarchical fashion. Then, it is applied to select an optimal structure and knowledge base of the developed fractional controller to satisfy the various design specification contradictories (simplicity, accuracy, stability and robustness).

Еще

Optimal fractional-order controllers, BIBO stability analysis, Multiobjective genetic algorithm, CE150 Helicopter model

Короткий адрес: https://sciup.org/15010841

IDR: 15010841

Список литературы Review, Design, Optimization and Stability Analysis of Fractional-Order PID Controller

  • D. A. Benson, M. M. Meerschaert and J. Revielle, ‘‘Fractional calculus in hydrologic modeling: A numerical perspective,’’ Advances in Water Resources, vol. 51, 2013, pp. 479-497.
  • V. Feliu-Batlle, R. Rivas-Perez, F.J. Castillo-Garcia, L. Sanchez-Rodriguez, A. Linarez-Saez, ‘‘Robust fractional order controller for irrigation main canal pools with time-varying dynamical parameters,’’ Computers and Electronics in Agriculture, vol. 76, no. 2, May 2011, pp. 205-217.
  • Concepción A. Monje, Blas M. Vinagre, Vicente Feliu, YangQuan Chen, ‘‘Tuning and auto-tuning of fractional order controllers for industry applications,’’ Control Engineering Practice, vol. 16, 2008, pp. 798-812.
  • Indranil Pan, Saptarshi Das, Amitava Gupta, ‘‘Handling packet dropouts and random delays for unstable delayed processes in NCS by optimal tuning of controllers with evolutionary algorithms,’’ ISA Transactions, vol. 50, 2011, pp. 557-572.
  • Chia-Hung Lin, Cong-Hui Huang, Yi-Chun Du, Jian-Liung Chen, ‘‘Maximum photovoltaic power tracking for the PV array using the fractional-order incremental conductance method,’’ Applied Energy, vol. 88, 2011, pp. 4840-4847.
  • Shahab Ghasemi, Ahmadreza Tabesh and Javad Askari-Marnani, ‘‘Application of Fractional Calculus Theory to Robust Controller Design for Wind Turbine Generators,’’ IEEE Transactions on Energy Conversion, vol. 29, no. 3, September 2014, pp. 780-787.
  • Zhihuan Chen, Xiaohui Yuan, Bin Ji, Pengtao Wang, Hao Tian, ‘‘Design of a fractional order PID controller for hydraulic turbine regulating system using chaotic non-dominated sorting genetic algorithm II,’’ Energy Conversion and Management, vol. 84, pp. 390-404, 2014.
  • Haiyang Chao, YingLuo, LongDi, YangQuanChen, ‘‘Roll-channel fractional order controller design for a small fixed-wing unmanned aerial vehicle,’’ Control Engineering Practice, vol. 18, pp. 761-772, 2010.
  • Ivo Petráš, Fractional-order nonlinear systems: Modeling, Analysis and Simulation, Higher Education Press, Beijing and Springer-Verlag Berlin Heidelberg 2011.
  • Sezgin Kaçar, Akgül Akgül, A. Turan Ergüzel, Muhammed M. Öztürk, Abdullah Sevin,"Design of a Web Interface for Fractional Chaotic Systems", IJCNIS, vol.7, no.1, pp.46-53, 2015. DOI: 10.5815/ijcnis.2015.01.07
  • Asim Kumar Das, Tapan Kumar Roy, “Fractional Order EOQ Model with Linear Trend of Time-Dependent Demand,” IJISA, vol.7, no.3, pp.44-53, 2015. DOI: 10.5815/ijisa.2015.03.06.
  • Anguluri Rajasekhar, Ravi Kumar Jatoth, Ajith Abraham, ‘‘Design of intelligent PID/PIλDμ speed controller for chopper fed DC motor drive using opposition based artificial bee colony algorithm,’’ Engineering Applications of Artificial Intelligence, vol. 29, pp. 23–32, 2014.
  • Ramiro S. Barbosa, J.A. Tenreiro Machado, Isabel S. Jesus, ‘‘Effect of fractional orders in the velocity control of a servo system,’’ Computers and Mathematics with Applications, vol. 59, 2010, pp. 1679-1686.
  • B. Maundy, A.S.Elwakil, T.J.Freeborn, ‘‘On the practical realization of higher-order filters with fractional stepping,’’ Signal Processing, vol. 91, pp. 484-491, 2011.
  • Richa Sharma, K.P.S. Rana, Vineet Kumar, ‘‘Performance analysis of fractional order fuzzy PID controllers applied to a robotic manipulator,’’ Expert Systems with Applications, vol. 41, 2014, pp. 4274-4289.
  • A.C. Sparavigna and P. Milligan, ‘‘Using fractional differentiation in astronomy,’’ Instrumentation and Methods for Astrophysics (astro-ph.IM), arXiv:0910.4243, 2009.
  • A. Oustaloup, La Commande CRONE: Commande Robuste d'Ordre Non Entier, Editions Hermès, Paris, 1991.
  • A. Oustaloup, La Dérivation non entière, Hermès, Paris 1991.
  • I. Podlubny, Fractional differential équations, San Diego: Academic Press, 1999.
  • I. Podlubny, “Fractional-order systems and controllers,’’ IEEE Trans. Automatic Control, vol. 44, no.1, 1999, pp. 208-214.
  • Ivo Petráš, ‘‘Tuning and implementation methods for fractional-order controllers,’’ Fractional Calculus & Applied Analysis : An International Journal of Theory and Applications, vol. 15, no. 2, 2012, pp. 282-303.
  • B. J. Lurie, ‘‘Three-parameter tunable Tilt-integral-derivative (TID) controller,’’ United States Patent, vol. 5, 1994, pp. 371-670.
  • C. A. Monje, Y. Q. Chen, B. M. Vinagre, D. Xue, V. Feliu, Fractional-order systems and control: Fundamentals and applications, Advanced Industrial Control Series, Springer-Verlag, London, 2010.
  • Saptarshi Das, Suman Saha, Shantanu Das, Amitava Gupta, ‘‘On the selection of tuning methodology of FOPID controllers for the control of higher order processes,’’ ISA Transactions, vol. 50, 2011, pp. 376-388.
  • Cao Junyi and Cao Binggang, ‘‘Fractional-order control of pneumatic position servo systems,’’ Mathematical Problems in Engineering, vol. 2011, pp. 1-14.
  • Ying Luo, YangQuan Chen, ‘‘Stabilizing and robust fractional order PI controller synthesis for first order plus time delay systems,’’ Automatica, vol. 48, 2012, p. 2159-2167.
  • Ivo Petráš, ‘‘The fractional-order controllers: methods for their synthesis and application,’’ Journal of Electrical Engineering, vol. 50, 1999, pp. 284-288.
  • C. Yeroglu, MM. Özyetkin, N. Tan, ‘‘Frequency response computation of fractional order interval transfer functions,’’ International Journal of Control, Automation, and Systems, vol. 8, no. 5, 2010, pp. 09-17.
  • D. Valério, J. S. da Costa, ‘‘Tuning of fractional PID controllers with Ziegler-Nichols type rules,’’ Signal Process, vol. 86, no. 10, 2006, pp. 2771-2784.
  • Saptarshi Das, Indranil Pan, Shantanu Das, Amitava Gupta, ‘‘A novel fractional order fuzzy PID controller and its optimal time domain tuning based on integral performance indices,’’ Engineering Applications of Artificial Intelligence, vol. 25, 2012, pp. 430-442.
  • Prateek Puri, Subhojit Ghosh, ‘‘A hybrid optimization approach for PI controller tuning based on gain and phase margin specifications,’’ Swarm and Evolutionary Computation, vol. 8, 2013, pp. 69-78.
  • Muhammad Asif Zahoor Raja, Junaid Ali Khan, I M Qureshi, ‘‘Heuristic computational approach using swarm intelligence in solving fractional differential equations,’’ Genetic and Evolutionary Computation Conference GECCO 2010, Proceedings, Portland, Oregon, USA, July 7-11, 2010, Companion Material. ACM 2010, pp. 2023-2026.
  • Debarati Kundu, Kaushik Suresh, Sayan Ghosh, and Swagatam Das, ‘‘Designing fractional-order controller using a modified invasive Weed optimization algorithm,’’ World Congress on Nature & Biologically Inspired Computing, NABIC 2009, 2009, pp. 1315-1320.
  • Sanjoy Debbarma, Lalit Chandra Saikia, Nidul Sinha, ‘‘Automatic generation control using two degree of freedom fractional order PID controller,’’ Electrical Power and Energy Systems, vol. 58, 2014, pp. 120-129.
  • Celaleddin Yeroglu, Nusret Tan, ‘‘Classical controller design techniques for fractional order case,’’ ISA Transactions, vol. 50, 2011, pp. 461-472.
  • D. Valério, J. S. da Costa, ‘‘Tuning of fractional controllers minimizing H2 and H∞ norms,’’ Acta Polytech. Hung, Vol. 3, No. 4, 2006, pp. 55-70.
  • Shantanu Das, Functional fractional calculus for system identification and controls, Berlin: Springer; 2008.
  • Arijit Biswas, Swagatam Das, Ajith Abraham, Sambarta Dasgupta, ‘‘Design of fractional-order controllers with an improved differential evolution,’’ Engineering Applications of Artificial Intelligence, vol. 22, 2009, pp. 343-350.
  • Dingyü Xue, YangQuan Chen, Derek P. Atherton, Linear feedback control analysis and design with MATLAB, SIAM Publisher, Philadelphia, 2007.
  • Aleksei Tepljakov, Eduard Petlenkov and Juri Belikov, ‘‘FOMCON: Fractional-order modeling and control toolbox for MATLAB,’’ MIXDES 2011, 18th International Conference "Mixed Design of Integrated Circuits and Systems, June 16-18, 2011, Gliwice, Poland, 2011, pp. 684-689.
  • YangQuan Chen, Ivo Petráš and Dingyü Xue, ‘‘Fractional order control - A tutorial,’’ 2009 American Control Conference, Hyatt Regency Riverfront, St. Louis, MO, USA, June 10-12, 2009, pp. 1397-1411.
  • R. Gorenflo, Fractional calculus: Some numerical methods, In Fractals and Fractional Calculus in Continuum Mechanics, Springer Verlag, Vienna-New York, 1997.
  • C. H. Lubich, ‘‘Discretized fractional calculus,’’ SIAM J. Math. Anal., vol. 17, no. 3, 1986, pp. 704-719.
  • Ramon Vilanova, Antonio Visioli, PID control in the third millennium: Lessons learned and new approaches, Advances in Industrial Control, Springer-Verlag London Limited, 2012.
  • B. M. Vinagre, I. Podlubny, A. Hernandez, V. Feliu, ‘‘Some approximations of fractional order operators used in control theory and applications,’’ Fract. Calculus Appl. Anal., vol. 3, no. 3, 2000b, pp. 231-248.
  • I. Podlubny, I. Petráš, B. Vinagre, P. O’Leary, L. Dorčak, ‘‘Analogue realizations of fractional-order controllers,’’ Nonlinear Dynamics, vol. 29, 2002, pp. 281-296.
  • A. Charef, H. H. Sun, Y. Y. Tsao, and B. onaral, ‘‘Fractal system as represented by singularity function,’’ IEEE Transactions on Automatic Control, vol. 37, no. 9, 1992, pp. 1465-1470.
  • G.E. Carlson, C.A. Halijak, ‘‘Approximation of fractional capacitors by a regular Newton process,’’ IRE Transactions on Circuit Theory, CT-11 No. 2, 1964, pp. 210-213.
  • K. Matsuda, H. Fujii, ‘‘ optimized wave absorbing control: Analytical and experimental results,’’ J. Guidance, Control and Dynamics, vol. 16, no. 6, 1993, pp. 1146-1153.
  • M. Amairi, M. Aoun, S. Najar, M. N. Abdelkrim, ‘‘Guaranteed frequency-domain identification of fractional order systems: application to a real system,’’ International Journal of Modeling, Identification and Control, Vol. 17, No. 1, 2012, pp. 32-42.
  • Duarte Valéerio, Manuel Duarte Ortigueira, and José Sá da Costa, ‘‘Identifying a transfer function from a frequency response,’’ Transactions of the ASME -Journal of Computational and Nonlinear dynamics, Vol. 3, No. 2, April 2008, pp. 21-35.
  • Saptarshi Das, Indranil Pan, Fractional order signal processing, Verlag/Jahr: SPRINGER, BERLIN 2011.
  • B. M. Vinagre, I. Podlubny, A. Hernandez, V. Feliu, ‘‘Some approximations of fractional order operator used in control theory and applications,’’ Fractional Canculus and Applied Analysis, vol. 3, no. 3, 2000, pp. 231-248.
  • Ramiro S. Barbosa, J. A. T. Machado, and M. F. Silva, ‘‘Time domain design of fractional differintegrators using least-squares,’’ Signal Processing, vol. 86, 2006, pp. 2567-2581.
  • J. A. T. Machado, ‘‘Analysis and design of fractional-order digital control systems,’’ J. Systems Anal.-Model.-Simulation, vol. 27, 1997, pp. 107–122
  • Y. Q. Chen, K. L. Moore, ‘‘Discretization schemes for fractional-order differentiators and integrators,’’ IEEE Trans. Circuits Systems-I: Fundam. Theory Appl., vol. 49, no. 3, 2002, pp. 363-367.
  • Hu Sheng, YangQuan Chen, TianShuang Qiu, Fractional processes and fractional-order signal processing: Techniques and applications, Springer London Dordrecht Heidelberg New York, 2012.
  • Shantanu Das, Functional fractional calculul, Second Edition, Springer-Verlag Berlin Heidelberg; 2011.
  • Filipe Silva, Vitor Santos, ‘‘Towards an Autonomous small-size Humanoid Robot: Design Issues and control strategies,’’ Proceedings 2005 IEEE International Symposium on Computational Intelligence in Robotics and Automation, June 27-30, 2005, Espoo, Finland, pp. 87-92.
  • Ying Luo, YangQuan Chen, ‘‘Fractional order [proportional derivative] controller for a class of fractional order systems,’’ Automatica, vol. 45, 2009, pp. 2446-2450
  • Fabrizio Padula, Antonio Visioli, ‘‘Tuning rules for optimal PID and fractional-order PID controllers,’’ Journal of Process Control, vol. 21, 2011, pp. 69-81
  • R. Isermann, Digital Control Systems, Springer-Verlag, 1989
  • Wei Zhao, Byung Hwa kim, Amy C. Larson and Richard M. Voyles, “FPGA implementation of closed loop control system for small scale robot,’’ In Proceedings, 12th International conference on advanced robotics-ICAR 05, 2005, pp. 70-77
  • Ammar SOUKKOU, Salah LEULMI, “Controlling and Synchronizing of Fractional Order Chaotic Systems via Simple and Optimal Fractional-order Feedback Controller,’’ IJISA, Accepted paper 2016.
  • Fujio Ikeda, ‘‘A numerical algorithm of discrete fractional calculus by using Inhomogeneous sampling data,’’ Transactions of the Society of Instrument and Control Engineers, vol. E-6, no. 1, 207, pp. 1-8
  • Brian P. Sprouse, Christopher L. MacDonald, Gabriel A. Silva, ‘‘Computational efficiency of fractional diffusion using adaptive time step memory,’’ 4th IFAC Workshop on Fractional Differentiation and Its Applications, 2010, pp. 1-6
  • Zhe Gao, Xiaozhong Liao, ‘‘Discretization algorithm for fractional order integral by Haar wavelet approximation,’’ Applied Mathematics and Computation, vol. 218, no. 5, 2011, pp. 1917-1926
  • D. Matignon, ‘‘Stability result on fractional differential equations with applications to control processing,’’ IMACS-SMC Proceedings, Lille, France, July, 1996, pp. 963-968.
  • Y. Li, Y. Q. Chen, I. Podlubny and Y. Cao, ‘‘Mittag-Leffler stability of fractional order nonlinear dynamic system,’’ Proc. of the 3rd IFAC Workshop on Fractional Differentiation and its Applications, November 05–07, Ankara, Turkey, 2008.
  • Saeed Balochian, Ali Khaki Sedigh, ‘‘Sufficient condition for stabilization of linear time invariant fractional order switched systems and variable structure control stabilizers,’’ ISA Transactions, vol. 51, no. 1, January 2012, pp. 65-73
  • Yan Li, YangQuan Chen, Igor Podlubny, ‘‘Stability of fractional-order nonlinear dynamic systems: Lyapunov direct method and generalized Mittag-Leffler stability,’’ Computers and Mathematics with Applications, vol. 59, 2010, pp. 1810-1821.
  • Ibrahima N’Doye, Holger Voos and Mohamed Darouach, ‘‘Chaos in a fractional-order Cancer system,’’ 2014 European Control Conference (ECC), June 24-27, 2014. Strasbourg, France, pp. 171-176.
  • Mohammad Reza Faieghi, Suwat Kuntanapreeda, Hadi Delavari, and Dumitru Baleanu, ‘‘Robust stabilization of fractional-order chaotic systems with linear controllers: LMI-based sufficient conditions,’’ Journal of Vibration and Control, vol. 20, no. 7, 2014, pp. 1042-1051.
  • Kamran Akbari Moornani and Mohammad Haeri, ‘‘Robust stability check for fractional PID-based control systems,’’ Transactions of the Institute of Measurement and Control, vol. 35, no. 2, 2013, pp. 236-246.
  • Shankar Sastry, Nonlinear Systems: Analysis, Stability and Control, Springer-Verlag New York, Inc, 1999.
  • M. Vidyasagar, Nonlinear systems analysis, 2nd ed.., Prentice-Hall, Engle Wood, 1993.
  • Kit Po Wong and Zhaoyang Dong, ‘‘Differential evolution, an alternative approach to evolutionary algorithm’’, in ‘‘Modern heuristic optimization techniques : Theory and applications to power systems’’, IEEE Press Editorial Board Kwang Y. Lee, Mohamed A. EL-Sharkawi, Chapter 9, pp. 171-187, 2008.
  • Rolf Isermann, Marco Münchhof, Identification of dynamic systems: An introduction with applications, Springer-Verlag Berlin Heidelberg, 2011.
  • Shaminder Singh, Jasmeen Gill,"Temporal Weather Prediction using Back Propagation based Genetic Algorithm Technique", IJISA, vol.6, no.12, pp.55-61, 2014. DOI: 10.5815/ijisa.2014.12.08
  • Osama I. Hassanein,Ayman A. Aly,Ahmed A. Abo-Ismail,"Parameter Tuning via Genetic Algorithm of Fuzzy Controller for Fire Tube Boiler", IJISA, vol.4, no.4, pp.9-18, 2012.
  • Molly Mehra, M.L. Jayalal, A. John Arul, S. Rajeswari, K. K. Kuriakose, S.A.V. Satya Murty,"Study on Different Crossover Mechanisms of Genetic Algorithm for Test Interval Optimization for Nuclear Power Plants", IJISA, vol.6, no.1, pp.20-28, 2014. DOI: 10.5815/ijisa.2014.01.03
  • A. Soukkou, A. Khellaf, S. Leulmi, M. Grimes, “Control of dynamical systems: An intelligent approach,’’ International Journal of Control, Automation, and Systems, vol. 6, no. 4, August 2008, pp. 583-595
  • Humusoft, CE 150 helicopter model: User's manual, Humusoft, Prague 2002.
  • H. Boubertakh, M. Tadjine, Pierre-Yves Glorennec, Salim Labiod, ‘‘Tuning fuzzy PD and PI controllers using reinforcement learning,’’ ISA Transactions, vol. 49, 2010, pp. 543-551.
  • A. Soukkou, S. Leulmi, A. Khellaf, ‘‘Intelligent nonlinear optimal controller of a biotechnological process,’’ Archives of Control Sciences, vol. 19, no. 2, 2009, pp. 217-240.
Еще
Статья научная