Optimal Controllers with Complex Order Derivatives

  • J. A. Tenreiro Machado


This paper studies the optimization of complex-order algorithms for the discrete-time control of linear and nonlinear systems. The fundamentals of fractional systems and genetic algorithms are introduced. Based on these concepts, complex-order control schemes and their implementation are evaluated in the perspective of evolutionary optimization. The results demonstrate not only that complex-order derivatives constitute a valuable alternative for deriving control algorithms, but also the feasibility of the adopted optimization strategy.


Fractional calculus Optimization Genetic algorithms Control 


  1. 1.
    Oldham, K., Spanier, J.: The Fractional Calculus: Theory and Application of Differentiation and Integration to Arbitrary Order. Academic Press, New York (1974) Google Scholar
  2. 2.
    Miller, K., Ross, B.: An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley, New York (1993) MATHGoogle Scholar
  3. 3.
    Samko, S., Kilbas, A., Marichev, O.: Fractional Integrals and Derivatives: Theory and Applications. Gordon and Breach, London (1993) MATHGoogle Scholar
  4. 4.
    Podlubny, I.: Fractional Differential Equations. Academic Press, San Diego (1999) MATHGoogle Scholar
  5. 5.
    Kilbas, A., Srivastava, H.M., Trujillo, J.: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam (2006) MATHGoogle Scholar
  6. 6.
    Machado, J.T., Kiryakova, V., Mainardi, F.: Recent history of fractional calculus. Commun. Nonlinear Sci. Numer. Simul. 16(3), 1140–1153 (2011) MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Baleanu, D., Diethelm, K., Scalas, E., Trujillo, J.J.: Fractional Calculus Models and Numerical Methods. Series on Complexity, Nonlinearity and Chaos. World Scientific, Singapore (2012) MATHGoogle Scholar
  8. 8.
    Oustaloup, A.: La Commande CRONE: Commande Robuste D’Ordre Non Entier. Hermès, Paris (1991) MATHGoogle Scholar
  9. 9.
    Machado, J.T.: Analysis and design of fractional-order digital control systems. Syst. Anal. Model. Simul. 27(2–3), 107–122 (1997) MATHGoogle Scholar
  10. 10.
    Podlubny, I.: Fractional-order systems and PIλ D μ -controllers. IEEE Trans. Autom. Control 44(1), 208–213 (1999) MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Zaslavsky, G.: Hamiltonian Chaos and Fractional Dynamics. Oxford University Press, New York (2005) MATHGoogle Scholar
  12. 12.
    Magin, R.: Fractional Calculus in Bioengineering. Begell House, Redding (2006) Google Scholar
  13. 13.
    Baleanu, D., Ozlem, D., Agrawal, O.: A central difference numerical scheme for fractional optimal control problems. J. Vib. Control 15(4), 583–597 (2009) MathSciNetCrossRefGoogle Scholar
  14. 14.
    Diethelm, K.: The Analysis of Fractional Differential Equations. Springer, Berlin (2010) MATHCrossRefGoogle Scholar
  15. 15.
    Agrawal, O., Ozlem, D., Baleanu, D.: Fractional optimal control problems with several state and control variables. J. Vib. Control 16(13), 1967–1976 (2010) MathSciNetCrossRefGoogle Scholar
  16. 16.
    Mainardi, F.: Fractional Calculus and Waves in Linear Viscoelasticity: an Introduction to Mathematical Models. Imperial College Press, London (2010) MATHCrossRefGoogle Scholar
  17. 17.
    Monje, C.A., Chen, Y., Vinagre, B.M., Xue, D., Feliu, V.: Fractional-order Systems and Controls: Fundamentals and Applications. Advances in Industrial Control. Springer, Berlin (2010) MATHCrossRefGoogle Scholar
  18. 18.
    Tarasov, V.E.: Fractional Dynamics: Applications of Fractional Calculus to Dynamics of Particles, Fields and Media. Nonlinear Physical Science. Springer, Berlin (2010) MATHGoogle Scholar
  19. 19.
    Petráš, I.: Fractional-Order Nonlinear Systems: Modeling, Analysis and Simulation. Nonlinear Physical Science. Springer, Berlin (2011) MATHCrossRefGoogle Scholar
  20. 20.
    Al-Alaoui, M.A.: Novel digital integrator and differentiator. Electron. Lett. 29(4), 376–378 (1993) CrossRefGoogle Scholar
  21. 21.
    Machado, J.T.: Discrete-time fractional-order controllers. Fract. Calc. Appl. Anal. 4(1), 47–66 (2001) MathSciNetMATHGoogle Scholar
  22. 22.
    Tseng, C.C.: Design of fractional order digital FIR differentiators. IEEE Signal Process. Lett. 8(3), 77–79 (2001) CrossRefGoogle Scholar
  23. 23.
    Chen, Y.Q., Moore, K.L.: Discretization schemes for fractional-order differentiators and integrators. IEEE Trans. Circuits Syst. I, Fundam. Theory Appl. 49(3), 363–367 (2002) MathSciNetCrossRefGoogle Scholar
  24. 24.
    Vinagre, B.M., Chen, Y.Q., Petras, I.: Two direct Tustin discretization methods for fractional-order differentiator/integrator. J. Franklin Inst. 340(5), 349–362 (2003) MathSciNetMATHCrossRefGoogle Scholar
  25. 25.
    Chen, Y.Q., Vinagre, B.M.: A new IIR-type digital fractional order differentiator. Signal Process. 83(11), 2359–2365 (2003) MATHCrossRefGoogle Scholar
  26. 26.
    Barbosa, R.S., Machado, J.T., Silva, M.: Time domain design of fractional differintegrators using least squares approximations. Signal Process. 86(10), 2567–2581 (2006) MATHCrossRefGoogle Scholar
  27. 27.
    Holland, J.H.: Adaptation in Natural and Artificial Systems. University of Michigan Press, Ann Arbor (1975) Google Scholar
  28. 28.
    Goldenberg, D.E.: Genetic Algorithms in Search Optimization, and Machine Learning. Addison-Wesley, Reading (1989) Google Scholar
  29. 29.
    Cao, J.-Y., Cao, B.-G.: Design of fractional order controller based on particle swarm optimization. Int. J. Control. Autom. Syst. 4(6), 775–781 (2006) Google Scholar
  30. 30.
    Valério, D., da Costa J, S.: Tuning of fractional controllers minimising H 2 and H norms. Acta Polytech. Hung. 3(4), 55–70 (2006) Google Scholar
  31. 31.
    Maiti, D., Acharya, A., Chakraborty, M., Konar, A., Janarthanan, R.: Tuning PID and PIλ D δ controllers using the integral time absolute error criterion. In: 2008 IEEE Forth International Conference on Information and Automation for Sustainability, Colombo, Sri Lanka (2008) Google Scholar
  32. 32.
    Biswas, A., Das, S., Abraham, A., Dasgupta, S.: Design of fractional-order PIλ D μ controllers with an improved differential evolution. Eng. Appl. Artif. Intell. 22, 343–350 (2009) MathSciNetCrossRefGoogle Scholar
  33. 33.
    Machado, J.T., Galhano, A., Oliveira, A.M., Tar, J.K.: Optimal approximation of fractional derivatives through discrete-time fractions using genetic algorithms. Commun. Nonlinear Sci. Numer. Simul. 15(3), 482–490 (2010) CrossRefGoogle Scholar
  34. 34.
    Machado, J.T.: Optimal tuning of fractional controllers using genetic algorithms. Nonlinear Dyn. 62(1–2), 447–452 (2010) MathSciNetMATHGoogle Scholar
  35. 35.
    Padhee, S., Gautam, A., Singh, Y., Kaur, G.: A novel evolutionary tuning method for fractional order PID controller. Int. J. Soft Comput. Eng. 1(3), 1–9 (2011) Google Scholar
  36. 36.
    Gao, Z., Liao, X.: Rational approximation for fractional-order system by particle swarm optimization. Nonlinear Dyn. 67(2), 1387–1395 (2012) MathSciNetMATHCrossRefGoogle Scholar
  37. 37.
    Hartley, T.T., Adams, J.L., Lorenzo, C.F.: Complex-order distributions. In: ASME International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, Long Beach, CA, DETC2005-84952 (2005) Google Scholar
  38. 38.
    Hartley, T.T., Lorenzo, C.F., Adams, J.L.: Conjugated-order differintegrals. In: ASME International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, Long Beach, CA, DETC2005-84951 (2005) Google Scholar
  39. 39.
    Silva, M.F., Machado, J.T., Barbosa, R.S.: Complex-order dynamics in hexapod locomotion. Signal Process. 86(10), 2785–2793 (2006) MATHCrossRefGoogle Scholar
  40. 40.
    Adams, J.L., Hartley, T.T., Lorenzo, C.F.: Complex-order distributions using conjugated-order differintegrals. In: Sabatier, J., Agrawal, O.P., Tenreiro Machado, J.A. (eds.) Theoretical Developments and Applications in Physics and Engineering, pp. 347–360. Springer, Berlin (2007) Google Scholar
  41. 41.
    Barbosa, R.S., Machado, J.T., Silva, M.F.: Discretization of complex-order algorithms for control applications. J. Vib. Control 14(9–10), 1349–1361 (2008) MathSciNetMATHCrossRefGoogle Scholar
  42. 42.
    Adams, J.L., Hartley, T.T., Lorenzo, C.F.: Identification of complex order-distributions. J. Vib. Control 14(9–10), 1375–1388 (2008) MathSciNetMATHCrossRefGoogle Scholar
  43. 43.
    Pinto, C.M., Machado, J.T.: Complex-order Van der Pol oscillator. Nonlinear Dyn. 65(3), 247–254 (2011) CrossRefGoogle Scholar
  44. 44.
    Pinto, C.M., Machado, J.T.: Complex order biped rhythms. Int. J. Bifurc. Chaos Appl. Sci. Eng. 21(10), 3053–3061 (2011) CrossRefGoogle Scholar
  45. 45.
    Machado, J.T., Galhano, A., Oliveira, A.M., Tar, J.K.: Approximating fractional derivatives through the generalized mean. Commun. Nonlinear Sci. Numer. Simul. 14(11), 3723–3730 (2009) MATHCrossRefGoogle Scholar
  46. 46.
    Machado, J.T.: Multidimensional scaling analysis of fractional systems. Comput. Math. Appl. (2012). doi: 10.1016/j.camwa.2012.02.069 Google Scholar
  47. 47.
    Barbosa, R.S., Machado, J.T.: Describing function analysis of systems with impacts and backlash. Nonlinear Dyn. 29(1–4), 235–250 (2002) MATHCrossRefGoogle Scholar
  48. 48.
    Duarte, F., Machado, J.T.: Describing function of two masses with backlash. Nonlinear Dyn. 56(4), 409–413 (2009) MATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.Dept. Electrical EngineeringInstitute of Engineering of the Polytechnic of PortoPortoPortugal

Personalised recommendations