Author Topic: Tuning of FOPID Controller Using Taylor Series Expansion  (Read 2518 times)

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Tuning of FOPID Controller Using Taylor Series Expansion
« on: August 17, 2011, 06:43:14 am »
Author : Ali Akbar Jalali, Shabnam Khosravi
International Journal of Scientific & Engineering Research, Volume 2, Issue 5, May-2011
ISSN 2229-5518
Download Full Paper : PDF

Abstractó In this paper, a direct synthesis approach to fractional order controller design  Is investigated. The proposed algorithm makes use of Taylor series of both desired closed-loop and actual closed-loop transfer function which is truncated to the first five terms. FOPID Controller parameters are synthesized in order to match the closed-loop response of the plant to the desired closed-loop response. The standard and stable second-order model is considered for both plant and the desired closed-loop transfer functions. Therefore for a given plant with damping ratio  and natural frequency . The tuned FOPID controller results in the desired closed-loop response with damping ratio and natural frequency .  An example is presented that indicates the designed FOPID results in actual closed-loop response very close to desired response rather than PID controller. It is shown that the proposed method performs better than Genetic Algorithm in obtaining the desired response.
Index Termsó FOPID controller, Taylor series expansion, second order model. 

1   INTRODUCTION                                                                     
For many decades, proportional-integral-derivative (PID) controllers have been very popular in industries for process control applications. The popularity and widespread use of PID controllers are attributed primarily to their simplicity and performance characteristics. Owing to the paramount importance of PID controllers, continuous efforts are being made to improve their quality and robustness [1], [2].
An elegant way of enhancing the performance of PID controllers is to use fractional order controllers where the integral and derivative operators have non-integer orders. Podlubny proposed the concept of fractional order control in 1999 [3]. In FOPID controller, despite of the proportional, integral and derivative constants, there are two more adjustable parameters: the power of s in integral and derivative operators,   respectively. Therefore this type of controllers is generalizations of PIDs and consequently has a wider scope of design, while retaining the advantages of classical ones.
Several methods have been reported for FOPID de-sign. Vinagre, Podlubny, Dorcak, Feliu [4] proposed a frequency domain approach based on expected crossover frequency and phase margin. Petras came up with a method based on the pole distribution of the characteristic equation in the complex plane [5]. In recent years evolutionary algorithms are used for FOPID tuning. YICAO, LIANG, CAO [6], presented optimization of FOPID controller parameters based on Genetic Algorithm. A method based on Particle Swarm Optimization was proposed [7]. In this paper a tuning method for FOPID controller is proposed. Suppose a standard and stable second order plant such that desired response is not available. Tuning FOPID con-troller by the proposed method results in desired closed-loop response. The standard second order is considered for desired response. It is shown that the proposed me-thod performs better than Genetic Algorithm in obtaining the desired response. The rest of the paper is organized as follows: In section 2 the tuning method for FOPID controller is described. An example is investigated in section 3 and finally Section 4 draws some conclusions.
 Consider the block diagram of feedback control system in fig. 1. The objective is design a FOPID controller,   such that for a given plant,   with standard second order model, the actual closed-loop response results in desired closed-loop response. Desired closed-loop response denoted by  and described by standard second order model as follows

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