Design of fractional-order PIlambdaDµ controllers with an improved differential evolution

Differential evolution (DE) has recently emerged as a simple yet very powerful technique for real parameter optimization. This article describes an application of DE to the design of fractional-order proportional-integral-derivative (FOPID) controllers involving fractional-order integrator and fractional-order differentiator. FOPID controllers' parameters are composed of the proportionality constant, integral constant, derivative constant, derivative order and integral order, and its design is more complex than that of conventional integer-order proportional-integral-derivative (PID) controller. Here the controller synthesis is based on user-specified peak overshoot and rise time and has been formulated as a single objective optimization problem. In order to digitally realize the fractional-order closed-loop transfer function of the designed plant, Tustin operator-based continuous fraction expansion (CFE) scheme was used in this work. Several simulation examples as well as comparisons of DE with two other state-of-the-art optimization techniques (Particle Swarm Optimization and binary Genetic Algorithm) over the same problems demonstrate the superiority of the proposed approach especially for actuating fractional-order plants. The proposed technique may serve as an efficient alternative for the design of next-generation fractional-order controllers.

[1]  Igor Podlubny,et al.  Fractional-order systems and PI/sup /spl lambda//D/sup /spl mu//-controllers , 1999 .

[2]  C. Lubich Discretized fractional calculus , 1986 .

[3]  I. Kostial,et al.  State-Space Controller Design for the Fractional-Order Regulated System , 2002 .

[4]  Riccardo Poli,et al.  New ideas in optimization , 1999 .

[5]  Saman K. Halgamuge,et al.  Self-organizing hierarchical particle swarm optimizer with time-varying acceleration coefficients , 2004, IEEE Transactions on Evolutionary Computation.

[6]  Karl Johan Åström,et al.  PID Controllers: Theory, Design, and Tuning , 1995 .

[7]  John H. Holland,et al.  Adaptation in Natural and Artificial Systems: An Introductory Analysis with Applications to Biology, Control, and Artificial Intelligence , 1992 .

[8]  Alain Oustaloup,et al.  Fractional order sinusoidal oscillators: Optimization and their use in highly linear FM modulation , 1981 .

[9]  R. Storn,et al.  Differential Evolution: A Practical Approach to Global Optimization (Natural Computing Series) , 2005 .

[10]  Amit Konar,et al.  Two improved differential evolution schemes for faster global search , 2005, GECCO '05.

[11]  René Thomsen,et al.  A comparative study of differential evolution, particle swarm optimization, and evolutionary algorithms on numerical benchmark problems , 2004, Proceedings of the 2004 Congress on Evolutionary Computation (IEEE Cat. No.04TH8753).

[12]  Bing-Gang Cao,et al.  Optimization of fractional order PID controllers based on genetic algorithms , 2005, 2005 International Conference on Machine Learning and Cybernetics.

[13]  K. Miller,et al.  An Introduction to the Fractional Calculus and Fractional Differential Equations , 1993 .

[14]  Kenneth V. Price,et al.  An introduction to differential evolution , 1999 .

[15]  M. Nakagawa,et al.  Basic Characteristics of a Fractance Device , 1992 .

[16]  Robert F. Stengel,et al.  Optimal Control and Estimation , 1994 .

[17]  I. Podlubny Fractional-order systems and PIλDμ-controllers , 1999, IEEE Trans. Autom. Control..

[18]  Yangquan Chen,et al.  Fractional Calculus and Biomimetic Control , 2004, 2004 IEEE International Conference on Robotics and Biomimetics.

[19]  Yangquan Chen,et al.  Robust controllability of interval fractional order linear time invariant systems , 2006, Signal Process..

[20]  M. Caputo Linear Models of Dissipation whose Q is almost Frequency Independent-II , 1967 .

[21]  I. Podlubny Fractional differential equations , 1998 .

[22]  Vicente Feliu,et al.  On Fractional PID Controllers: A Frequency Domain Approach , 2000 .

[23]  Rainer Storn,et al.  Differential Evolution – A Simple and Efficient Heuristic for global Optimization over Continuous Spaces , 1997, J. Glob. Optim..

[24]  I. Petras,et al.  The fractional - order controllers: Methods for their synthesis and application , 2000, math/0004064.