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Proportional integral derivative controller design using Legendre orthogonal functions

  • Mechanical Engineering, Control Science and Information Engineering
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Abstract

The Legendre orthogonal functions are employed to design the family of PID controllers for a variety of plants. In the proposed method, the PID controller and the plant model are represented with their corresponding Legendre series. Matching the first three terms of the Legendre series of the loop gain with the desired one gives the PID controller parameters. The closed loop system stability conditions in terms of the Legendre basis function pole (λ) for a wide range of systems including the first order, second order, double integrator, first order plus dead time, and first order unstable plants are obtained. For first order and double integrator plants, the closed loop system stability is preserved for all values of λ and for the other plants, an appropriate range in terms of λ is obtained. The optimum value of λ to attain a minimum integral square error performance index in the presence of the control signal constraints is achieved. The numerical simulations demonstrate the benefits of the Legendre based PID controller.

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References

  1. HWANG C, GUO T Y. Parameter identification of a class of time-varying systems via orthogonal shifted Legendre polynomials [J]. Journal of the Franklin Institute, 1984, 318(1): 56–69.

    Article  MathSciNet  MATH  Google Scholar 

  2. PARASKEVOPOULOS P N. Legendre series approach to identification and analysis of linear systems [J]. IEEE Transactions on Automatic Control, 1985, 30(6): 585–589.

    Article  MathSciNet  MATH  Google Scholar 

  3. JUNG S M. Approximation of analytic functions by Legendre functions [J]. Nonlinear Analysis: Theory, Methods & Applications, 2009, 71(12): e103–e108.

    Article  MathSciNet  MATH  Google Scholar 

  4. WANG H, YU S. Tracking control of robot manipulators based on orthogonal neural network [J]. International Journal of Modelling, Identification and Control, 2010, 11(1/2): 130–135.

    Article  Google Scholar 

  5. MOHAN B M, KAR S K. Optimal control of singular systems via orthogonal functions [J]. International Journal of Control, Automation and Systems, 2011, 9(1): 145–150.

    Article  Google Scholar 

  6. MARZBAN H R, RAZZAGHI M. Optimal control of linear delay systems via hybrid of block-pulse and Legendre polynomials [J]. Journal of the Franklin Institute, 2004, 341(3): 279–293.

    Article  MathSciNet  MATH  Google Scholar 

  7. RAZZAGHI M, YOUSEFI S. Legendre Wavelet method for constrained optimal control problems [J]. Mathematical Methods in the Applied Sciences, 2002, 25(7): 529–539.

    Article  MathSciNet  MATH  Google Scholar 

  8. BENNETT S. The past of PID controllers [J]. Annual Reviews in Control, 2001, 25: 43–53.

    Article  Google Scholar 

  9. ÅSTRÖM K J, HÄGGLUND T. PID Controllers: Theory, design and Tuning [M]. USA: Instrumentation Society of America, 1995: 120–273.

    Google Scholar 

  10. DATTA A, HO M T, BHATTACHARYYA S P. Structure and synthesis of PID controllers [M]. London, UK: Springer-Verlag, 2000: 15–235.

    MATH  Google Scholar 

  11. ZIEGLER J G, NICHOLS N B. Optimum settings for automatic controller [J]. Journal of Dynamic Systems, Measurement, and Control, 1993, 115(2B): 220–222.

    Article  Google Scholar 

  12. WANG J S. Optimal design of PI/PD controllers for non-minimum phase system [J]. Transactions of the Institute of Measurement and Control, 2006, 28(1): 27–35.

    Article  Google Scholar 

  13. HERREROS A, BAEYENS E, PERAN J R. Design of PID-type controllers using multiobjective genetic algorithms [J]. ISA Transactions, 2002, 41(4): 457–472.

    Article  Google Scholar 

  14. MIKHALEVICH S S, BAYDALI S A, MANENTI F. Development of a tunable method for PID controllers to achieve the desired phase margin [J]. Journal of Process Control, 2015, 25: 28–34.

    Article  Google Scholar 

  15. VILANOVA R. IMC based robust PID design: Tuning guidelines and automatic tuning [J]. Journal of Process Control, 2008, 18(1): 61–70.

    Article  Google Scholar 

  16. PANDA R C, YU C C, HUANG H P. PID tuning rules for SOPDT systems: review and some new results [J]. ISA Transactions, 2004, 43(2): 283–295.

    Article  Google Scholar 

  17. RAMASAMY M, SUNDARAMOORTHY S. PID controller tuning for desired closed loop responses for SISO systems using impulse response [J]. Computers & Chemical Engineering, 2008, 32(8): 1773–1788.

    Article  Google Scholar 

  18. XU L. A proportional differential control method for a time-delay system using the Taylor expansion approximation [J]. Applied Mathematics and Computation, 2014, 236(1): 391–399.

    Article  MathSciNet  MATH  Google Scholar 

  19. HORNG I R, CHOU J H. Digital PID controller design via general discrete orthogonal polynomials [J]. International Journal of Control, 1988, 47(1): 188–192.

    Article  MathSciNet  MATH  Google Scholar 

  20. AYADI B, BRAIEK N B. MIMO PID controllers synthesis using orthogonal functions [C]// Proceedings of the 16th IFAC World Congress. Prague, Czech Republic: IFAC, 2005: 508–513.

    Google Scholar 

  21. BOUAFOURA M K, BRAIEK N B. PI?Dµ controller design for integer and fractional plants using piecewise orthogonal functions [J]. Communications in Nonlinear Science and Numerical Simulation, 2010, 15(5): 1267–1278.

    Article  MathSciNet  MATH  Google Scholar 

  22. HEUBERGER P S C, van den HOF P M J, WAHLBERG B. Modelling and identification with rational orthogonal basis functions [M]. London, UK: Springer-Verlag, 2005: 41–59.

    Book  Google Scholar 

  23. COHEN G H, COON G A. Theoretical consideration of retarded control [J]. Transactions of American Society of Mechanical Engineers, ASME, 1953, 75(1): 827–834.

    Google Scholar 

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Authors and Affiliations

  1. Department of Electrical Engineering, Khomeinishahr Branch, Islamic Azad University, Isfahan, Iran

    Reza Moradi & Mohammad Tabatabaei

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  1. Reza Moradi
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  2. Mohammad Tabatabaei
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Correspondence to Mohammad Tabatabaei.

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Moradi, R., Tabatabaei, M. Proportional integral derivative controller design using Legendre orthogonal functions. J. Cent. South Univ. 23, 2616–2629 (2016). https://doi.org/10.1007/s11771-016-3323-6

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  • DOI: https://doi.org/10.1007/s11771-016-3323-6

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