FRACTIONAL ORDER PID CONTROLLER TUNING BASED ON IMC

International Journal of Information Technology, Control and Automation (IJITCA) Vol.2, No.4, October 2012
DOI:10.5121/ijitca.2012.2403 21
FRACTIONAL ORDER PID CONTROLLER TUNING
BASED ON IMC
Mohammad Reza Rahmani Mehdi Abadi1
and Ali Akbar Jalali2
1
Electrical Engineering Department, Iran University of Science and Technology, Tehran,
Iran.
drjalali@iust.ac.ir
2
Electrical Engineering Department, Iran University of Science and Technology,
Tehran, Iran.
mohammadrahmani88@gmail.com
ABSTRACT
In this work, a class of fractional order controller (FOPID) is tuned based on internal model control
(IMC). This tuning rule has been obtained without any approximation of time delay. Moreover to show
usefulness of fractional order controller in comparison with classical integer order controllers, an
industrial PID controller tuned in a similar way, is compared with FOPID and then robust stability of both
controllers is investigated. Robust stability analysis has been done to find maximum delayed time
uncertainty interval which results in a stable closed loop control system. For a typical system, robust
stability has been done to find maximum time constant uncertainty interval of system. Two clarify the
proposed control system design procedure, three examples have been given.
KEYWORDS
Fractional order PID, IMC, Robust Stability
1. INTRODUCTION
Many industrial processes can be modeled by a transfer function in which there is a time delay
element. Time delay in the model of a process appears because of measurement delay, actuator
delay, or approximating high order dynamics of processes by lower order dynamics plus time
delay [1]. In [2] a linear model of active queue management (AQM) router including time delay
has been obtained.
The process identification as a First-Order-Plus-Dead-Time (FOPDT) introduce a model which
represents the process behavior in efficient manner. FOPDT models have been used for
approximating industrial and chemical processes which do not have integral and resonant
characteristics [3, 4]. Although many processes have open-loop stable behavior, in some
engineering fields (such as exothermic chemical processes, batch chemical reactors, biological
reactors, waste treatment processes, etc.), processes have several steady states due to their
nonlinearity. Some of these steady states are unstable. On the other hand some specifications like
maximization of productivity, safety and reduction of economic costs need to model the processes
around an unstable steady state [5, 6]. When a collection of stable open-loop plants are connected,
the resulted open-loop process becomes unstable. Chemical irreversible exothermal reactor is an

22
example of such unstable processes [7].
Cvejn has proposed a method for tuning PI and PID controllers for FOPDT processes which deals
with time delay without approximation [8]. Roy and Iqbal have adopted a Hermite-Biehler
theorem based approach to design PID controllers for stabilization of FOPDT process models [9].
In [10] by employing integral squared time error standard forms, a PI-PD controller has been
designed to control unstable and integrating processes. The design of controllers for stable
processes is mostly based on three criteria, namely, error criteria, time domain and frequency
domain. Out of all these synthesis methods, designs based on desired closed loop specifications
have gained much attention by many researchers. In order to improve the performance of the
process, the model of the system can be incorporated in the design of controller that made huge
success through fabricating internal model control structure in the synthesis of equivalent
controller [11-14] .These equivalent controllers are robust in nature and even they are being used
for higher order systems [15-18] applying direct synthesis approach.
Conventional integer-order differentiation and integration can be extended to allow for orders that
are not necessarily integer. Non-integer differentiation and integration of real functions lead to
fractional differential equations which are dealt with in fractional calculus [19]. These concepts
have been transferred into control engineering as a new methodology of control called fractional
order control [20]. Such controllers are the extended version of conventional integer order
controllers that have some extra parameters which must be tuned more precisely and the control
system design procedure is more complicated than integer order controllers. Previously, fractional
derivative and integral have been used in many engineering fields. Having more degrees of
freedom, fractional order models can approximate processes by fewer parameters. Podlubny has
shown that fractional order PID controllers denoted by PI D
λ µ
, have a better response in
comparison with standard PID controllers, when used for control of fractional-order systems [21].
Fractional order controllers have been applied to FOPDT processes. In [22] a fractional-order
controller has been applied to an FOPDT model. In [23] a method for practical tuning of
Fractional Order Proportional Integral (FOPI) controller in which the system to be controlled has
been modeled by an FOPDT transfer function has been given. In [24-29] recent applications of
fractional-order controllers have been given.
This work gives a FOPID tuning rule for Stable/Unstable- Plus-Dead-Time processes, based on
IMC. Then robust stability of the proposed FOPID has been investigated. A comparison study
between the proposed FOPID and conventional PID has been made to show that FOPID has
better performance than PID. Here, the proposed tuning rule uses delayed time part without any
approximation. However, when controller has a simple pole at origin and system has delayed time
part, this approach will be applicable. Robust stability analysis has been done to find maximum
delayed time uncertainty interval which results in a stable closed loop control system. For a
typical system, which shows a nearly constant phase around phase crossover frequency, robust
stability has been done to find maximum time constant uncertainty interval of the system.
Organization of this paper is as follows: section 2 explains IMC and fractional order controllers.
Section 3 describes tuning rules for a class of fractional order controllers and robust stability is
investigated. In section 4 this tuning rule is applied to three systems. Finally section 5 concludes
and gives some future work suggestions.
2. PRINCIPLES
This section gives preliminaries for next sections, covers IMC approach control design and
fractional order systems.

23
2.1. IMC
IMC controller has structure shown in fig. 1.
Where ( )
P
G s
% is identified system and ( )
P
G s is actual system. IMC controller defined as
( ) ( ) ( )
C P
C s G s F s
−
= %
(1)
Where ( )
p
G s
−
% expresses minimum phase or invertible part of system ( )
p
G s
% , containing all
stable and unstable poles of system and stable zeros, but not delayed time and unstable zeros of it.
F(s) is a low pass filter designed so that IMC controller can be realizable as well as to reduce
effect of uncertainty of system at high frequency.
Figure 1. IMC control structure
From IMC structure, equivalent and well-known control system structure with unit feedback can
be obtained (fig. 2).
Figure 2. Equivalent IMC control structure
Controller of such a structure is
( )
( )
1 ( ) ( )
C
C P
C s
C s
C s G s
=
− %
(2)

24
2.2. Basic definitions in fractional control
Fractional calculus has been used as a mathematical tool for modelling physical systems and
designing controllers. Fractional calculus is an extension of integer order calculus in which
ordinary differential equations have been replaced by fractional order differential equations. In
fractional order differential equations, derivatives and integrals are not necessarily of integer
order and they span a wider range of differential equations. Fractional calculus deals with
fractional integration and differentiation. Therefore, a generalized differential and integral
operator has been introduced as a single fundamental operator represented by a t
Dλ
where a and t
are the limits and ( )
R
λ λ ∈ the order of the operation. For positiveλ , it denotes derivative and
for negativeλ , it denotes integral action as
( ) 0
1 ( ) 0
( ) 0
( )
a t
t
a
d
Real
dt
D Real
Real
d
λ
λ
λ
λ
λ
λ
λ
τ −

 >


= =

 <



∫
. (3)
Several ways exist to define fractional-order derivatives and integrals. The mostly used
definitions for fractional derivatives are Riemann–Liouville, Grunwald–Letnikov and Caputo
definitions [19]. The Caputo fractional derivative of order λ with respect to the variable t is
defined as







=
<
<
−
−
−
Γ
=
∫ −
+
n
t
f
dt
d
n
n
t
d
f
n
t
f
D
n
n
t
n
n
t
λ
λ
τ
τ
τ
λ λ
λ
),
(
1
,
)
(
)
(
)
(
1
)
(
0 1
0 (4)
where n is the first integer not less than λ , and ( )
Z
Γ is Euler’s Gamma function which is given
by
1
0
( ) z t
Z t e dt
∞
− −
Γ = ∫ . (5)
The Laplace transform of the Caputo fractional derivative is
1
1 ( )
0
0
{ ( )} ( ) (0),
1 .
n
k k
t
k
L D f t s F s s f
n n N
λ λ λ
λ
−
− −
=
= −
− < ≤ ∈
∑ (6)
The main advantage of using the Caputo definition is that, only integer order derivatives of
function ( )
f t at t=0 appear in the Laplace transform of the Caputo fractional derivative [19].
For zero initial conditions in (6), a straightforward result is obtained as
0
{ ( )} ( )
t
L D f t s F s
λ λ
= . (7)

25
3. TUNING BASED ON IMC
In eq. (2), it can be seen that for a delayed time system, C(s) has a simple pole at s=0.
Considering this, if C(s) is rewritten as
( )
( )
f s
C s
s
= (8)
For tuning a specified controller, the controller must have a simple pole at origin, consequently.
Here it is assumed that ( ) ( )
P P
G s G s
=
% .
Controllers that are used to tune are a well-known PID, and a class of fractional order controller,
as formulated below
1
1
( ) 1
1
d
i p
i d
k s
C s k
k s ak s
 
  +
= +  
 
+
  
(9)
( )
( ) 1 1
m
i
f p d
k
C s k k s
s
 
= + +
 
 
(10)
Where m may be any real number. Now tuning rules are as following
3.1 Integer order PID tuning rule
As it is seen from eq.(9), ( )
i
C s can be written in form of
( ) (1 ) ( )
( ) ,
(1 ) (1 )
i d
g s s f s
C s ak
s s s s
β
β
β β
+
= = =
+ +
(11)
From equations (2, 8), it is obvious that f(s) is only a function of system parameters. To determine
a value for β , As a result of previous work [13], one can guess that 0.25max( , )
d p
k l t
= ,
where l and p
t denote delayed time and slow pole of system respectively, and a=0.1 is chosen.
Thus in eq.(11), ( )
1 ( )
s f s
β
+ is known. On the other hand, from eq.(9) it is seen that equivalent
term to ( )
1 ( )
s f s
β
+ is ( )( )
1 1
p
i d
i
k
k s k s
k
+ + . To find out values of three unknown
parameters , ,
p i d
k k k ,Taylor series of g(s) at point s=0 is used. Comparing coefficients of s
terms of this Taylor series and eq.(9) gives values of unknowns. Results have been stated below
[13].

26
(0) (0)
(0) (0) (0)
(0) (0) 2 (0)
2! 2
p
i
c
c d
k
g f
k
k g f f
g f f
k k
β
β
= =
′ ′
= = +
′′ ′′ ′
+
= =
(12)
This PID tuning rule has some disadvantages, that are as follows. First of all, it is required to
guess the value of β , in turn value of d
k , and then in tunig rules eq.(12) find the value of
d
k again. Secondly, it is needed to define low pass filter F(s) in eq.(1) in a complex form, that is
not a easy design of filter. Finally and worst of all is that there is one and only one set of solution
for eq.(12); this, makes achievement to a more robust stability solution hard. With fractional
definition, as given below, these disadvantages will be eliminated.
3.2 Fractional order controller tuning rule
Considering eq.(10), there are four unknowns that must be determined. To find these parameters,
( )
F
C s will be rewritten as
( )
( )
F
g s
C s
s
= (13)
Where
( )( )
( ) 1
m
p i d
g s k s k k s
= + + (14)
On the other side, from eqs. (2, 8) ( )
F
C s can be defined as function of system parameters. From
relation ( ) ( )
f s g s
= , where f(s) and g(s) are defined in eq.(8) and eq.(14) respectively, it is
possible to determine unknowns , , ,
p i d
k k k m . This tuning can be done by using Taylor series
as done below.
2 3 2 3
(0) (0) (0) (0)
( ) (0) (0) (0) (0)
2! 3! 2! 3!
g s g s f s f s
g s g g s f f s
′′ ′′′ ′′ ′′′
′ ′
= + + + + = + + + +
L L (15)
Because of four unknowns, one way to tune is that first four Taylor series terms of g(s) and f(s)
are used. This results in following relation.
2 2 2
2 2 2 3 3 2 3 3
2( ( )) ( )( )
(0) 3( ( ))
(0)
(0) ( )( )
(0)
( )( 3 2 )
p d p i d d
p d d p i d d d
p i
p p i d
g k k
g k k k
k mk k k m k mk
g k m k mk k k m k m k
m
g
mk
k
+ −
=
′ = +
′′ =
′′′ = − + − +
(16)
To find solutions of this set of nonlinear equations with 4 equations and 4 unknowns, procedure
below is suggested.

27
first and second equations of eq.(16) leads to relations below
1 ( ) (0)
(0)
( )
(0)
p i
p
a d
C k k f
f k
C mk
f
= =
′ −
= =
(17)
By replacing p i
k k and d
mk with their equivalent term from eq.(17), in two other equations in
eq.(16), It can be seen that relations (18) will be obtained.
3 2 2
1
1
C (2k +C (-k +C ))=f (
3 (
0),
C C C C C
) ( 3 2 ) (0)
C
p d d d
a p d a
a a a a a
k k k k f ′′′
+ =
′
+
′
− + −
(18)
Now, from the set of 2 equation-2 unknown in (18), ,
d p
k k can be obtained. From first equation
of eq.(18) d
k is defined, then putting it in second relation of eq.(18), it can be seen that a fourth
order equation of p
k will be achieved. Thus there will be four choices to define p
k . These
choices make a more powerful robust stability design.
3.3 Defining low pass filter F(s)
Low pass filter for all stable pole systems is defined as
1
( )
( 1)n
F s
s
λ
=
+
(19)
And for unstable systems defined as
1
( )
( 1)n
s
F s
s
γ
λ
+
=
+
(20)
Where γ is chosen so that the following relation is satisfied.
( ) 1
1/
P
p
F s G
s t
+
=
=
(21)
Where ( )
P
G s
+
is noninvertible part of system and n is an integer number that is selected so that
IMC controller becomes realizable (proper). To select λ tradeoffs between speed of response and
stability of the closed loop must be considered. Here, following rules explained will be used [13].
FOPDT system: max(0.2 ,1.7 )
SOPDT system: max(0.2 ,0.25 )
IPDT system: 10
p
p
for t l
for t l
for l
λ
λ
λ
=
=
=
(22)

28
3.4 Robust stability investigation
Robust stability will be investigated from two aspects, from phase margin and gain margin. Using
phase margin, makes it possible to investigate uncertainty occurred in phase of closed loop. On
the other side, using gain margin, one can investigate uncertainty occurred in gain of system.
These two robust stability methodologies will be clarified below.
3.4.1 Robust stability from phase margin point of view
Using phase margin, it is necessary for magnitude of open loop bode diagram not to change by
uncertainty. To provide this condition, it is supposed that there exists uncertainty in delayed time
of system. For stability of closed loop control system with open loop stable system, it is
necessary that phase of open loop transfer function at gain crossover frequency, not to reach π
−
radian. According to this, it is possible to find maximum uncertainty interval, at which closed
loop is stable. Thus to find maximum uncertainty resulting stable closed loop response, procedure
is as follows. At first, gain crossover frequency is determined from magnitude relation of open
loop transfer function. Then from phase relation of loop gain at this frequency, delayed time
uncertainty will be achieved.
3.4.2 Robust stability from gain margin point of view
When using gain margin, it is necessary for phase of open loop bode diagram not to change by
uncertainty. For some typical cases it is reasonable that with uncertainty in time constant of
system, this condition approximately is provided. For stability of closed loop control system with
open loop stable transfer function, it is necessary that magnitude of open loop transfer function at
phase crossover frequency, is less than 1. According to this, it is possible to find maximum
uncertainty interval for time constant of system, at which closed loop is stable.
4. SOME EXAMPLES
4.1 Example 1
Consider a FOPDT system ( ) 10exp( 0.1 ) / (5 1)
P
G s s s
= − + . Here for 5, 0.1, 10
p
t l k
= = = ,
from eq.(22) 1
λ = for first order low pass filter ( ) 1/ ( 1)
F s s
λ
= + and in eq.(11) 0.125
β = is
chosen. According to tuning rules for integer order PID,
k = .46632,
p = 5.12952, k = 0.12636
i d
k . For fractional order controller, there will be three
reasonable selections (that result in positive p
k ), 0.40371e-3, 0.45333, 0.45455, which
provide desired response to step input and load disturbance and very similar to integer order one.
But difference of each selection will become important, when stability robustness gets critical.
This point has been discussed below.
4.1.1 Delayed time Uncertainty interval for integer order PID
To find uncertainty interval, from open loop magnitude equation, gain crossover frequency 0
ω
obtained.

29
( )
( )( )
2 2
0
2 2 2 2 2 2
0 0 0
1
1
1 1
1 1
d
p
i p
k
kk
k t
ω
ω ω β ω
+
 
+ =
 
+ +
 
(23)
From above equation 0 0.93176
ω = . Now equality of phase equation of open loop with π
− , will
give maximum uncertainty in delayed time that provide stable response.
0 0 0 0 0
0
arctan( ) arctan(1/(k ))+arctan(k )-arctan( )+
p i d
l t
l
ω ω ω ω βω π
ω
− − −
∆ = (24)
Uncertainty interval for delayed time 1.59275
l
∆ = has been calculated.
4.1.2 Delayed time Uncertainty interval for Fractional order controller
Procedure is as same as for integer order case, but equations are changed. Magnitude equation is.
( )
( )
2 2
2
0
2 2 2
0 0
1
1 1
1
m
d
i
p
p
k
k
kk
t
ω
ω ω
+
 
+ =
 
+
 
(25)
Whereas said before, there are three choices for , , ,
p i d
k k k m . For all of these choices,
0 0.90911
ω = that is approximately as same as integer order case. But, for these three selections,
uncertainty interval changes. Uncertainty interval is obtained from relation below.
0 0 0 0
0
arctan( ) arctan(k / )+marctan(k )+
p i d
l t
l
ω ω ω ω π
ω
− − −
∆ = (26)
For selected 0.45333
p
k = , from eq.(16) 7.47426
d
k = , 0.20054
i
k = , m=0.00240 are
calculated. Thus 1.62745
l
∆ = is obtained. If k =0.00040
p is selected, thus 225.18636
i
k = ,
4.99995
d
k = , 1.00003
m = , and 1.62880
l
∆ = is determined. It is seen that if 0.40e-3
p
k =
is chosen, maximum uncertainty in delayed time is obtained, but for all of choices uncertainty
intervals are approximately the same. Besides, remembering, integer order uncertainty interval, it
becomes obvious that fractional order is a more robust than integer order for this typical system.
4.1.3 Time constant Uncertainty interval for integer order PID
To find uncertainty interval, at first from open loop phase equation, phase crossover frequency
p
ω obtained.
arctan( ) arctan(1/(k ))+arctan(k ) arctan( )=
p p p i p d p p
l t
ω ω ω ω βω π
− − − − − (27)
Because of existence of function arctan(.) in eq.(27), it is not possible to find an analytical
solution for p
ω from above equation. But it is possible to find p
ω graphically, or use following
approximation.

30
, 1
arctan( ) 1
, 1
2
x x
x
x
x
π
<


= 
− >


(28)
Considering values of , , , ,
p i d p
k k k l t , and knowing that 0 1
p
ω ω
> , eq.(27) reformed to
1 0.1949490474 1 1
0.1
2 5 2 0.1263590714 2 0.125
p
p p p p
π π π
ω π
ω ω ω ω
       
− − − − + − − − =−
       
       
       
(29)
In above equation, it is seen that with 5
p
t = , arctan( )
2
p p
t
π
ω → , ensuring that phase equation
will not be altered by variable p
t . From above relation, =15.76574
p
ω is calculated.
Now equality of magnitude equation of loop gain with 1, will give maximum uncertainty in time
constant of system that provide stable response.
( )
( )( )
2 2
2 2 2 2 2 2
1
1
1 1
1 ( ) 1
d p
p
i p p p p p
k
kk
k t t
ω
ω ω β ω
+
 
+ =
 
  + + ∆ +
 
(30)
Uncertainty interval for time constant -4.70845
p
t
∆ = is calculated. This means that up to about
30% faster system can be stable in closed loop.
4.1.4 Time constant Uncertainty interval for Fractional order controller
Procedure is as same as for integer order case, but equations are changed. Phase equation is
arctan( ) arctan(k / )+marctan(k )=
p p p i p d p
l t
ω ω ω ω π
− − − − (31)
Whereas said before, there are three choices for , , ,
p i d
k k k m . Almost, for all of these choices,
15.74512
p
ω = that is approximately as same as integer order case. But, for these three
selections, uncertainty interval changes very little and is bigger than integer order. Uncertainty
interval is obtained from relation below:
( )
( )
2 2
2
2 2 2
1
1 1
1 ( )
m
d p
i
p
p p p p
k
k
kk
t t
ω
ω ω
+
 
+ =
 
  + + ∆
 
(32)
For selected 0.45333
p
k = , 7.47426
d
k = , 0.20054
i
k = , m=0.00240 and -4.71575
p
t
∆ =
are obtained. But if k =0.00040
p is selected, 225.18636
i
k = , 4.99995
d
k = , 1.00003
m = ,
and -4.73209
p
t
∆ = . It is seen that if 0.40e-3
p
k = is selected, maximum uncertainty in time
constant will be obtained, but for all of choices uncertainty intervals are approximately the same.
Besides, remembering, integer order uncertainty interval, it becomes obvious that fractional order
is a more robust than integer order for this typical system.

31
Figure 3, shows step response of two controllers. IAE measure for integer order is 0.1 but for
fractional order is 0.0. Load disturbance rejection for disturbance value 0.1 applied in time
interval [4,5] seconds, is also shown with IAE=0.28 for I.O., and 0.29 for F.O.
0 2 4 6 8 10
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
step response
t
output
0 5 10 15 20 25 30
-0.02
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
load distorbance with value=.1 response
t
output
desired response
integer order
fractional
integer order
fractional
Figure 3. step response and load distorbance rejection of exa. 1
4.2 Example 2
Considering system ( ) 10(0.1 1)exp( 0.1 ) / (5 1)
P
G s s s s
= + − + , thus 5, 0.1
p
t l
= = are defined.
From eq.(22) 1
λ = and low pass filter ( ) 1/ ( 1)
F s s
λ
= + is selected.
Now for case of integer order, calculating k = .45723, = 5.02955, t = 0.28871e-1
p i d
k , to
find uncertainty interval, from
( )( )
( )( )
2 2 2 2
1 0 0
1
2 2 2 2 2 2
0 0 0
1 1
1
1 1, 0.1, 5
1 1
d
p p
i p
t k
kk t t
k t
ω ω
ω ω β ω
+ +
 
+ = = =
 
+ +
 
(33)
gain crossover frequency 0 .91239
ω = is determined. From Phase equation at 0
ω , uncertainty
0 0 0 0 0 1 0
0
arctan( ) arctan(1/(k ))+arctan(k )-arctan( )+arctan(t )+
p i d
l t
l
ω ω ω ω βω ω π
ω
− − −
∆ = (34)
1.62710
l
∆ = is calculated. Now considering fractional order controller, there are two
reasonable choices 0.45455, 0.47992
p
k = . Selected .45455
p
k = , gives

32
0.20000, k 0.10179, m= 0.93774
i d
k = = − , and then for such a tuning 0 0.91239
ω = and
1.62617
l
∆ = are calculated. If .47992
p
k = is chosen, calculated
0.18943, k 7.68554, m=-0.48739e-1
i d
k = = , result in 1.65716
l
∆ = at 0 .912393
ω = . It is
seen that for this typical system, robust stability is the same for both integer and fractional order.
But difference in their step responses is obvious. Figure 4 shows this.
0 5 10
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
step response
t
o
u
tpu
t
0 10 20 30
-0.02
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
t
o
u
tpu
t
desired response
integer order
fractional
integer order
fractional
Figure 4. step response and load distorbance rejection of exa. 2
As it is seen, transient response of integer order is not as good as fractional ones. For I.O. step
response IAE=0.1, but for F.O.=0.0 And in load disturbance rejection for disturbance value of 0.1
in time interval [4,5] seconds, for I.O. IAE=0.28 and for F.O. IAE=0.29 are achieved.
4.3 Example 3
Considering system ( ) exp( 0.5 ) / (2 1)(0.5 1)
P
G s s s s
= − − + , thus 2, 0.5
p
t l
= = are defined.
From eq.(22) 0.4
λ = and low pass filter 2
( ) ( 1) / ( 1)
F s s s
γ λ
= + + is selected. Reason of
selecting a second order filter it to make IMC controller realizable. On the other side, to obtain
positive unknowns, it is needed to choose γ from below eq.(21). For this system with
( ) exp( 0.5 )
P
G s s
+
= − , 1.6918
γ = has been calculated.
Now for the case of integer order, by calculating k = 2.12342, = 11.64156, t =0.63135
p i d
k ,
to find uncertainty interval, gain crossover frequency 0 1
ω = is obtained, then from phase
relation, maximum 1
l
∆ = − is calculated.
Now considering fractional order controller, there are three reasonable choices for
0.11566, 1.68415, 2.00265
p
k = . 2.00265
p
k = gives 0.91079e-1, k 0.44968
i d
k = =
, m=1.36119 , thus for such a tuning, 0 1
ω = and 0.6279
l
∆ = is determined. If

33
0.11566
p
k = is selected, 1.57701, k 10.98993, m=0.99704
i d
k = = will be calculated, and
results in 0.6135
l
∆ = at same 0
ω . It is seen that for this typical system, robust stability for
different fractional order are similar and is greater than for integer order controller. However,
their step responses are not as much different as their robust stability. Figure 5 shows this.
Figure5 step response and load distorbance rejection of exa. 3
Table 1 shows IAE measure for different tunings.
TABLE 1.IAE FOR EXA. 3.
0 2 4 6 8 10 12 14 16 18 20
0
0.5
1
1.5
2
2.5
3
step response
t
output
0 5 10 15 20 25 30
-0.04
-0.03
-0.02
-0.01
0
0.01
0.02
0.03
0.04
0.05
0.06
t
output
desired response
integer order
fractional with kp=0.11566
integer order

34
5. CONCLUSION
In this work a class of fractional order controller has been tuned. Results show that without any
robust condition there is at least one set of solution for fractional order controller that is more
robust than conventional PID controller. Besides, it is seen that in desired response tracking,
when fractional order has robust stability near to integer order, F.O. with smaller IAE is better
than conventional PID. Moreover, owning to its more degree of freedom, it is possible to add
robust conditions in fractional order controllers tunings. Latter can be future work.
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Authors
Mohammad Reza Rahmani Mehdi Abadi was born in Yazd, Iran, in 1987. He received
his B.S. degree in electronics engineering from Yazd University in 2008, and the M.S.
degree in control engineering from Iran University of science and technology, in 2011. His
main area of interest includes robust control.
Ali Akbar Jalali was born in Damghan, Iran, in 1954. He received his B.Sc. degree in
Electronics Engineering from Khajeh Nasiredin Toosi University of Technology, Tehran,
Iran, May 1985. He obtained his M.Sc. degree in Electrical Engineering from Oklahama
University, Norman, US, in 1988. Then, he earned his Ph.D. and Post Doctoral in
Electrical Engineering, from West Virginia University, Morgantown, US, in 1993 and
1994 respectively.Dr. Jalali became a member of the Lane Department of Computer
Science and Electrical Engineering, Collage of Engineering and Mineral Resources, West Virginia
University, as an adjunct Professor in 2002. Currently, he is working in the Department of Electrical
Engineering, Iran University of Science and Technology (IUST) where he has been since 1994 as an
associate professor. His research field interests include mainly Extended Kalman Filtering, Robust
Control, H-infinity and Fractional order control. Furthermore, study of Information Technology and its
applications like: Virtual Learning, Virtual Reality, Internet City, Rural ICT developments and Designing
ICT Strategic Plan are his other research interests.

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