International Journal of Mathematics and Statistics Invention (IJMSI) E-ISSN: 2321 – 4767 P-ISSN: 2321 - 4759 www.ijmsi.org Volume 5 Issue 2 || February. 2017 || PP-15-20 www.ijmsi.org 15 | Page Numerical Optimization of Fractional Order PID Controller Hassan N.A. Ismail 1 , I.K. Youssef 2 and Tamer M. Rageh 3,* 1,3 Department of Basic Science Engineering, Faculty of Engineering in Benha, Benha University, Benha 13512, Egypt 2 Department of Mathematics, Faculty of Science, Ain Shams University, Cairo 11566, Egypt ABSTRACT: The fractional order PID controller is the generalization of classical PID controller, many Researchers interest in tuning FOPID controller here we use the Pareto Optimum technique to estimate the controller parameter and compare our result with the classical model and with other Researchers result .we used both mathematica package and matlab for tuning and simulation. KEYWORDS: Proportional Integral Derivative (PID) - fractional order PID - Optimization - Pareto Optimum I. INTRODUCTION The fractional order controllers are being the aim of many engineering and scientists in the recent few decay [1-5]. The fractional order Proportional-Integral-Derivative (FOPID) was first introduced by Podlubny [2] and it consider as the generalization case of classical PID controllers. The Proportional-Integral-Derivative (PID) controllers are still the most widely controller in engineering and industrial for process control applications. If the mathematical model of the plant can be derived, then it is possible to apply various design techniques for determining parameters of the controller that will meet the transient and steady state specifications of the closed loop system. In the recent few decay due to the development of fractional calculus(FC) the modeling of engineering system can be appear in fractional order systems(FOS) that require much more than classical PID controller to meet both transient and steady state specifications. There are many methods used to design FOPID, Deepyaman at. al.[4] using Particle Swarm Optimization Technique. Synthesis method which a modified root locus method for fractional-order systems and fractional order controllers was introduced in[8].A state-space design method based on feedback poles placement can be viewed in [10]. The aim of design PID controller is achieve high performance including low percentage overshoot and small settling time. The performance of PID controllers can be further improved by appropriate settings of fractional-I and fractional-D actions. Figure 1 Closed Loop System Consider the simple unity feedback control system shown in fig. 1 where R(s) is an input, G(s) is the transfer function of controlled system, G c (S) is the transfer of the controller, E(s) is an error. U(s) is the controller's output, and C(s) is the system's output. II. FRACTIONAL ORDER CALCULUS [11-15] Fractional calculus (FC) is a generalization of integration and differentiation to non-integer orders. FC provides a more powerful tool for modeling the real live phenomena, and this is actually a natural result of the fact that in FC the integer orders are just special cases. Definition: Let . The operator defined on by