Journal of Central South University

, Volume 23, Issue 10, pp 2616–2629 | Cite as

Proportional integral derivative controller design using Legendre orthogonal functions

Mechanical Engineering, Control Science and Information Engineering
First Online:
Received:
Accepted:
  • 33 Downloads

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.

Key words

PID controllers orthogonal functions legendre functions moment matching PI controller PD controller 
This is a preview of subscription content, log in to check access.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [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.MathSciNetCrossRefMATHGoogle Scholar
  2. [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.MathSciNetCrossRefMATHGoogle Scholar
  3. [3]
    JUNG S M. Approximation of analytic functions by Legendre functions [J]. Nonlinear Analysis: Theory, Methods & Applications, 2009, 71(12): e103–e108.MathSciNetCrossRefMATHGoogle Scholar
  4. [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.CrossRefGoogle Scholar
  5. [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.CrossRefGoogle Scholar
  6. [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.MathSciNetCrossRefMATHGoogle Scholar
  7. [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.MathSciNetCrossRefMATHGoogle Scholar
  8. [8]
    BENNETT S. The past of PID controllers [J]. Annual Reviews in Control, 2001, 25: 43–53.CrossRefGoogle Scholar
  9. [9]
    ÅSTRÖM K J, HÄGGLUND T. PID Controllers: Theory, design and Tuning [M]. USA: Instrumentation Society of America, 1995: 120–273.Google Scholar
  10. [10]
    DATTA A, HO M T, BHATTACHARYYA S P. Structure and synthesis of PID controllers [M]. London, UK: Springer-Verlag, 2000: 15–235.MATHGoogle Scholar
  11. [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.CrossRefGoogle Scholar
  12. [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.CrossRefGoogle Scholar
  13. [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.CrossRefGoogle Scholar
  14. [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.CrossRefGoogle Scholar
  15. [15]
    VILANOVA R. IMC based robust PID design: Tuning guidelines and automatic tuning [J]. Journal of Process Control, 2008, 18(1): 61–70.CrossRefGoogle Scholar
  16. [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.CrossRefGoogle Scholar
  17. [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.CrossRefGoogle Scholar
  18. [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.MathSciNetCrossRefMATHGoogle Scholar
  19. [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.MathSciNetCrossRefMATHGoogle Scholar
  20. [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. [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.MathSciNetCrossRefMATHGoogle Scholar
  22. [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.CrossRefGoogle Scholar
  23. [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

Copyright information

© Central South University Press and Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Department of Electrical Engineering, Khomeinishahr BranchIslamic Azad UniversityIsfahanIran

Personalised recommendations