Design of a Fractional Order PID Controller Using Particle Swarm Optimization Technique

Design of a Fractional Order PID Controller Using Particle Swarm Optimization Technique #Deepyaman Maiti, Sagnik Biswas, Amit Konar
Department of Electronics and Telecommunication Engineering, Jadavpur University Kolkata - 700 032 deepyamanmaiti@gmail.com, sagnik_agp@rediffmail.com, konaramit@yahoo.co.in

Abstract

Particle Swarm Optimization technique offers optimal or sub- optimal solution to multidimensional rough objective functions. In this paper, this optimization technique is used for designing fractional order PID controllers that give better performance than their integer order counterparts. Controller synthesis is based on required peak overshoot and rise time specifications. The characteristic equation is minimized to obtain an optimum set of controller parameters. Results show that this design method can effectively tune the parameters of the fractional order controller.

1. INTRODUCTION

Proportional-Integral-Derivative (PID) controllers are widely being used in industries for process control applications. The merit of using PID controllers lie in its simplicity of design and good performance including low percentage overshoot and small settling time for slow industrial processes. The performance of PID controllers can be further improved by appropriate settings of fractional-I and fractional-D actions. This paper attempts to study the behavior of fractional PID controllers over integer order PID controllers.
In a fractional PID controller, the I- and D-actions being fractional have wider scope of design. Naturally, besides setting the proportional, derivative and integral constants i T

and

d T , p K respectively, we have two more parameters: the power of s in integral and derivative actions- λ and δ respectively. Finding δ]

λ, , T , T , [K i d p as an optimal solution to a given process thus calls for optimization on the five-dimensional space. Classical optimization techniques cannot be used here because of the roughness of the objective function surface. We, therefore, use a derivative-free optimization, guided by the collective behavior of social swarm and determine optimal settings of p K , Td, Ti, λ and δ. The performance of the optimal fractional PID controller is better than its integer counterpart. Thus the proposed design will find extensive applications in real industrial processes. Traces of work on fractional PID are available in the current literature [1] – [9] on control engineering. A frequency domain approach based on the expected crossover frequency and phase margin is mentioned in [2]. A method based on pole distribution of the characteristic equation in the complex plane was proposed in [5]. A state-space design method based on feedback poles placement can be viewed in [6]. The fractional controller can also be designed by cascading a proper fractional unit to an integer-order controller. Our design focuses on positioning closed loop dominant poles, and the constraints thus obtained on the characteristic equation are optimally satisfied by particle swarm optimization algorithm. The work is thus original and may open up new avenues for the next generation fractional order controller design. It is necessary to understand the theory of fractional calculus in order to realize the significance of fractional order integration and derivation. Fractional calculus is the branch of calculus that generalizes the derivative or integral of a function to non-integer order, allowing calculations such as deriving a function to 1/2 order. Since sα indicates deriving to the order α, knowledge in the subject of fractional calculus is essential to design fractional order controllers. Of the several definitions of fractional derivatives, the Grunwald-Letnikov and Riemann-Liouville definitions are the most used. These definitions are required for the realization of discrete control algorithms.

2. THE INTEGER AND FRACTIONAL ORDER PID CONTROLLERS

The integer order PID controller has the following transfer function: s T s T K d 1 i p + + − . Here, the orders of integration and derivation are both unity.

             Fig. 1: Generic Closed Loop System 

The real objects or processes that we want to control are generally fractional (for example, the voltage-current relation of a semi-infinite lossy RC line). However, for many of them the fractionality is very low. In general, the integer-order approximation of the fractional systems can cause significant differences between 2nd National Conference on Recent Trends in Information Systems (ReTIS-08)

mathematical model and real system. The main reason for using integer-order models was the absence of solution methods for fractional-order differential equations. PID controllers belong to dominating industrial controllers and therefore are o

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