Numerical Optimization of Fractional Order PID Controller

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. Ismail1
, I.K. Youssef2
and Tamer M. Rageh3,*
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, Gc(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

(1)
for , is called the Riemann-Liouville fractional integral operator of order
Definition: Let and . The operator defined as
(2)
for , is called the Riemann-Liouville differential operator of order .
Definition: Let and . The operator defined by
(3)
for , is called the Caputo differential operator of order
Definition: Let . The operator defined by
(4)
for , is called the Gr¨unwald-Letnikov fractional derivative of order
From the Riemann-Liouville fractional integral, applying the Laplace transform of the convolution integral,
Equations (1) and (2) will be:
(5)
(6)
III. FRACTIONAL ORDER CONTROLLER [16-19]
Before we introduced the Fractional Order Controller we introduce the fractional-order transfer function
(FOTF) given by the following expression:
(7)
where is an arbitrary real number, , is an
arbitrary constant.

In the time domain, the FOTF corresponds to the n-term fractional –order differential equation (FDE)
(8)
where is caputo’s fractional derivative of order with respect to the variable and with the starting
point at :
The transfer function for conventional PID controller is
(9)
Transfer function for fractional order PID controller is
(10)
FO integro-differential equation
(11)
Where are the parameters of controller to be tuned, and and are the
fractional integral and differential operator respectively, often defined by the Riemann-Liouville definition as
the following:
(12)
(13)
Table 1 Conroller Parameters
Coefficient for the proportional term
Coefficient for the derivative term
Coefficient for the integral term
Fractional order for the derivative term
Fractional order for the integral term
The fractional system is a system which could be better described by fractional order mathematical models, and
its transfer function is at arbitrary real order instead of just integer order.
Podlubny (1999) introduced [1] as a generalization of the classical PID controller, namely the PIλ
Dμ
controller
or FOPID controller with an integrator of order λ and a differentiator of order μ. He also proves the better
response of FOPID controller compered by PID controller special in case of FOS.
Figure 2 : Fractional Order PID Controller
The orders of integration and differentiation (λ, µ) must be positive real numbers, Taking λ =1 and
µ=1, we will have an integer order PID controller. Fig. 2 The classical PID controller has three parameters
( )to be tuned, while the fractional order PID controller has five( ).
The interest of this kind of controller is justified by a better flexibility, since it exhibits fractional
powers (λ and μ) of the integral and derivative parts, respectively. Thus, five parameters can be tuned in this
structure (λ, μ, Kp , Ki and Kd), that is, two more parameters than in the case of a conventional PID controller (λ

= 1 and μ = 1). The fractional orders λ and μ can be used to fulfill additional specifications of design or other
interesting requirements for the controlled system.
Figure 3 types of controllers
From fig. 3 at the corners of square if λ =μ = 1, then it is classical PID controller. If λ = 0 and μ = 1, then it is
classical PD controller. If λ = 1 and μ = 0, then it is classical PI controller. If λ = μ = 0, then it is classical P
controller. But any point inside the square donates a fractional order PID controller.
IV. OPTIMIZATION OF CONTROLLER PARAMETERS
The aims of most interested in FOPID controller is to estimate the controller parameters so many
methods are done for example self - tuning and auto-tuning which introduced by Monje CA at. al [20],rule base
method [21-24] for which FOPID controller based on Ziegler Nichols-type rules, Analytical methods [25-27].
finally numerical treatment for optimization fractional order controllers has been introduced by various authors,
based on the genetic algorithm[28-30],based on particle swarm optimization (PSO) technique[4 and 31-33] has
also been used for estimating the controllers parameters, A multi-objective optimization method was designed
by I. Pan and S. Das [34]
As in the classical root locus method, based on our engineering requirements of the maximum peak
overshoot Mp and rise time trise (or requirements of stability and damping levels)
we find out the damping ratio ζ and the undamped natural frequency ω0. Using the values of ζ and ω0 we then
find out the positions of the dominant poles of the closed loop system,
(14)
Let the closed loop transfer function of the system is:
(15)
Here G(s) = Gc(s).Gp(s) where Gc(s) is the transfer function of the controller to be designed. Gc(s) is of the
form
(16)
Gp(s) is the transfer function of the process we want to control.
If the required closed loop dominant poles are located at s1,2 =p1,2 =-x+ jy,- x –jy , then at s= p1 =-x+jy, we must
have
(17)
we get:
(18)
Assuming H(s) = 1, and Gp(s) being known, (18) can be arranged as:
(19)

In this complex equation (19) we have five unknowns, namely {Kp, Ti, Td, λ, µ}. There are an infinite number
of solution sets for s =p1 =-x+ jy . So the equation cannot be unambiguously solved.
Pereto optimization helps us the find the optimal solution set to the complex equation.
Let:
R=real part of the complex expression,
I=imaginary part of the complex expression,
P=phase ( = ).
We define and minimize ‘f’ using the pareto optimization. Our goal is to find out the
optimum solution set for which
The solution space is five-dimensional, the five dimensions being Kp, Ti, Td, and . The personal and
global bests are also five-dimensional. The limits on the position vectors of the particles (i.e. the controller
parameters) are set by us as follows. As a practical assumption, we allow Kp to vary between 1 and 1000, Ti and
Td between 1 and 500, and between 0 and 2.
V. NUMERICAL EXAMPLES
Consider the system of fractional order transfer function which need to be controlled as the following:
(20)
where
and consider the FOPID transfer function as:
(21)
Using mathematica package and apply the pareto optimal algorithm with some constraints on the controler
parameters ( to vary between 1 and 1000, and between 1 and 500, and between 0 and 2) we
estimate the parameters values as ( )
0 1 2 3 4 5 6 7 8 9 10
0
0.5
1
1.5
2
time
output
PIDresponce
FOPIDresponce
openloopresponce
Figure 4: Comparison between PID and FOPID

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
0.2
0.4
0.6
0.8
1
1.2
1.4
time
O/Presponce
FOPID using PSO
FOPID
Figure 5: Comparison between Our FOPID and POS Method
Fig (4) show comparison between the output response of open loop transfer function (red line) and the
classical PID controller (green line) and the FOPID controller using pareto optimization (blue line) and it is
clearly how worest the open loop system with long time response and large peak over shot, but using PID
controller all system requirements improved but still need more improvement, after using pareto optimal to
estimate the controller parameter which make the system response be better with less peak over shot ( we can
claim that no peak over shot) and very small time response.
Fig (5) show comparison between the output response of closed loop transfer function and fractional
order PID by using Particle Swarm Optimization Technique [4] (red line) and our method by using pareto
optimization (blue line)
VI. RESULTS AND CONCLUSION
Her we used pareto method for numerical optimization of the FOPID which give an estimation of the controller
parameter to meet the engineering specification needs, our result compared by classical PID and POS method
REFERENCES
[1]. Podlubny I., Fractional-order systems and PIλDμ –controllers, IEEE Transactions on Automatic Control. 44(1) (1999) 208-214.
[2]. Podlubny I., Fractional-order systems and fractional-order controllers, UEF-03-94, Slovak Academy of Sciences, Kosice, 1994.
[3]. Dorčák Ľ., Valsa J., Gonzalez E, Terpák J, Petráš I, Pivka L, Analogue Realization of Fractional-Order Dynamical Systems,
Entropy. 15 (2013) 4199-4214.
[4]. Maiti D., Biswas S., Konar A., Design of a Fractional Order PID Controller Using Particle Swarm Optimization Technique, 2nd
National Conference on Recent Trends in Information Systems (ReTIS-08)
[5]. Shah P., Agashe S., Review of fractional PID controller, Mechatronics. 38 (2016) 29-41.
[6]. Petras I., The fractional order controllers: Methods for their synthesis and application, Journal of Electrical Enginnering, 50(9-
10)(1999) 284-288.
[7]. Podlubny, I., 1999. Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to
methods of their solution and some of their applications. Academic Press, San Diego.
[8]. Miller, K. S., and Ross, B., An introduction to the fractional calculus and fractional differential equations. John Wiley and Sons,
New York. 1993
[9]. Oldham, K.B.; Spanier J. The Fractional Calculus; Academic Press: New York, NY, USA, 1974.
[10]. Caponetto R., Dongola G., Fortuna L., Petráš I., Fractional Order Systems: Modeling and Control Applications, World Scientific,
Singapore.2010
[11]. Herrmann R., Fractional Calculus an Introduction For Physicists, World Scientific, Singapore.2011
[12]. Xue D., Zhao C., Chen Y., Fractional order PID control of a dc-motor with elastic shaft a case study, Proceedings of American
control conference, 7 (2006)3182–3187
[13]. Monje C.A., Vinagre B.M., Feliu V., Chen Y.Q., Tuning and auto-tuning of fractional order controllers for industry applications,
Control Engineering Practice. 16 (2008) 798–812.
[14]. Valerio D., da Costa J.S., A review of tuning methods for fractional PIDs, 4th IFAC workshop on fractional differentiation and its
applications, FDA, 10; 2010 .
[15]. Sabatier J., Agrawal O.P., Tenreiro Machado J.A., Advances in fractional calculus, (2007).
[16]. Barbosa R.S., Tenreiro Machado J.T., Jesus I.S., Effect of fractional orders in the velocity control of a servo system, Computers &
Mathematics with Applications. 59(5)(2010) 1679–1686 .
[17]. Valerio D., da Costa J.S., Tuning of fractional PID controllers with ziegler–nichols- type rules, Signal Processing.86(10)( 2006)
2771–2784 .
[18]. Zhao C., Xue D., Chen Y.Q., A fractional order PID tuning algorithm for a class of fractional order plants, IEEE International
Conference Mechatronics and Automation. 1(2005) 216-221.
[19]. Bouafoura M.K., Braiek N.B., PIλ
Dμ
controller design for integer and fractional plants using piecewise orthogonal functions,
Communications in Nonlinear Science and Numerical Simulation.15(5)( 2010)1267–1278 .

[20]. Das S., Saha S., Das S., Gupta A., On the selection of tuning methodology of FOPID controllers for the control of higher order
processes, ISA Transactions. 50(2011) 376–388 .
[21]. Chang L-Y, Chen H-C, Tuning of fractional PID controllers using adaptive genetic algorithm for active magnetic bearing system,
WSEAS transactions on systems. 8(1)( 2009) 158–167.
[22]. Cao J-Y, Liang J., Cao B-G, Optimization of fractional order PID controllers based on genetic algorithms, Machine learning and
cybernetics. 9(2005) 5686–9 .
[23]. Das S., Pan I., Das S., Gupta A . Improved model reduction and tuning of fractional- order PIλ
Dμ
controllers for analytical rule
extraction with genetic programming. ISA Transactions 51(2)(2012)237–261 .
[24]. Cao J.Y., Cao B.G., Design of fractional order controllers based on particle swarm optimization, 1ST
IEEE conference on Industrial
electronics and applications. (2006) 1–6.
[25]. Zamani M., Karimi-Ghartemani M., Sadati N., Parniani M., Design of a fractional order PID controller for an AVR using particle
swarm optimization. Control Engineering Practice. 17 (2009) 1380–1387.
[26]. Karimi-Ghartemani M., Zamani M., Sadati N., Parniani M., An optimal fractional order controller for an AVR system using
particle swarm optimization algorithm, Power engineering. (2007) 244–249 .
[27]. Pan I., Das S., Chaotic multi-objective optimization based design of fractional order PIλDμ controller in AVR system,
International Journal of Electrical Power & Energy Systems.43(1)( 2012) 393–407 .

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