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American Journal of Computational Mathematics, 2013, 3, 7-12
doi:10.4236/ajcm.2013.33B002 Published Online September 2013 (http://www.scirp.org/journal/ajcm)
Simplified Optimization Routine for Tuning Robust
Fractional Order Controllers
Cristina I. Muresan
Department of Automation, Faculty of Automation and Computer Science, Technical, University of Cluj-Napoca, Romania
Email: [email protected]
Received June, 2013
ABSTRACT
Fractional order controllers have been used intensively over the last decades in controlling different types of processes.
The main methods for tuning such controllers are based on a frequency domain approach followed by optimization rou-
tine, generally in the form of the Matlab fminsearch, but also evolving to more complex routines, such as the genetic
algorithms. An alternative to these time consuming optimization routines, a simple graphical method has been proposed.
However, these graphical methods are not suitable for all combinations of the imposed performance specifications. To
preserve their simplicity, but also to make these graphical methods generally applicable, a modified graphical method
using a very straightforward and simple optimization routine is proposed within the paper. Two case studies are pre-
sented, for tuning fractional order PI and PD controllers.
Keywords: Fractional Order Controllers; Graphical Tuning; Simplified Optimization Routine
1. Introduction
Fractional order PIDs (FO-PIDs) have been employed in
various engineering fields ranging applications in a wide
variety of domains. The fractional order PID controller is
in fact a generalization of the classical integer order PID.
In the fractional order PID control algorithm, the error
signal is integrated and differentiated to any order, rather
than to an integer order as with the traditional PIDs. The
fractional order PIDs have two supplementary parame-
ters compared to the traditional PIDs. It is for this reason,
that the fractional order PIDs have the potential to meet
more design specifications than the traditional PIDs and
hence to increase the performance and robustness of
closed loop systems [1-4]. A couple of interesting meth-
ods have been proposed for tuning such FO-PIDs with
the great majority centered upon Matlab’s fminsearch or
graphical approaches [1, 5-7]. The current trend nowa-
days is directed to the latter methods, since they require
less computational and time resources. Nevertheless, if
no exact solution exists, the current graphical methods
fail at the tuning of the FO-PID controller.
The purpose of this paper is to design an improved
graphical method for tuning FO-PI and FO-PD control-
lers, based upon an optimization routine that selects the
best possible tuning option even in the case of no exact
solution. For exemplification, two case studies are con-
sidered. The first case study implies the design of FO-PI
control for a simple first order process. The second case
study consists in the design of a FO-PD controller for a
second order process with integrator effect. Simulation
results in both case studies show that the fractional order
controllers tuned using the proposed algorithm can meet
all performance specifications. To exemplify the opti-
mized graphical methods for tuning fractional order con-
trollers, the first case study has no exact solution, while
the second case study has an exact solution.
The paper is organized as follows. Section 2 contains
the main contribution of the present paper, with a de-
scription of the fractional order PI and PD optimized
graphical tuning algorithms, while Section 3 presents the
two case studies. The final section contains the conclud-
ing remarks.
2. Optimization Routine for Tuning
Fractional Order Controllers
The transfer function of the fractional order PI (FO-PI)
controller is given by:
()1 i
FO PIp
k
Hsk
s


(1)
while the transfer function for the fractional order PD
(FO-PD) controller is given by:
() 1
FO PDpd
H
sk ks
 (2)
where kp, ki and kd are the proportional, integral and de-
rivative gains and
and
 are the fractional
Copyright © 2013 SciRes. AJCM
C. I. MURESAN
8
order. If 1
, then the FO-PI controller in (1) is re-
duced to a traditional PI controller:
()1 i
FO PIp
k
Hsk
s


(3)
and the FO-PD is reduced to the classical PD controller
by setting 1
in (2):
() 1
FO PDpd
sk ks
 (4)
A proper tuning of the FO-PI and FO-PD controllers in
(1) and (3), as well as of the PI and PD controllers in (3)
and (4), respectively, implies determining the values for
the parameters, three in the case of the FO-PI and FO-PD
controllers and two in the case of the traditional PI and
PD controllers. For tuning FO-PI and FO-PD controllers,
in order to uniquely determine the three parameters -
,
kp and ki in the case of the FO-PI and
, kp and kd in the
case of the FO-PD– three equations are used that de-
scribe the performance of the closed loop system. The
general approach regarding the tuning of fractional order
controllers is based on frequency domain performance
specifications [8-10], which refer to imposing a gain
crossover frequency, a phase margin and robustness to
open loop gain variations.
For a general process transfer function Hp(s), the open
loop system when
s
j
is written as:
()() ()
oopFOPIPopen l
H
jH jHj

(5)
where
is the frequency.
In order for the open loop system to attain an imposed
gain crossover frequency
g
c
, then the following rela-
tion must hold:
()
open loopgc
Hj
1 (6)
where denotes the modulus of the complex function.
The open loop phase margin, m
, is also computed at
the gain crossover frequency as:
()
open loopgcm
Hj
 (7)
where denotes the phase of the complex function.
Finally, the last performance specification, robustness
to gain variations, implies that the phase of the open loop
system at the gain crossover frequency should be flat:

()
0
open loopgc
gc
dH j
d
(8)
2.1. Optimization Routine for Tuning Fractional
Order PI Controllers
The transfer function of the FO-PI controller, in the fre-
quency domain, may be written as:
()
PI
Hj 1cos sin
22
FOp i
kk j




 




(9)
in which

11 cos sin
22
j
sj


 


(10)
Equations (6), (7) and (8) imply a certain behavior of
the closed loop system, according to the specified values
for the gain crossover frequency and the phase margin,
and may further be used to determine all three values for
the kp, ki and
parameters of the FO-PI controller:
1cossin1(
22 gc
pigc p
kkj Hj
 







)
(11)

sin 2()
cos 2
i
mpgc
gc i
k
tgH j
k

 








(12)
1
22
sin ()
2
12 cos
2
igc
p
gc
gc
igci gc
kdH j
d
kk









(13)
where Hp(s) is the process transfer function.
Using solely equations (12) and (13), ki and
may
be determined uniquely, while (11) may be then used to
determine kp. The simplest method for computing the
FO-PI parameter values is based on a graphical approach
[1, 5-7], which implies that ki is computed and plotted as
a function of
using equations (12) and (13). The in-
tersection point of the resulting two curves yields the
final values for ki and
. Consequently, kp is deter-
mined using (11) and the previously graphically selected
values for ki and
. Such an example is given in Fig-
ure 1.
Although this method proves to be highly efficient and
simple, the graphical approach is based upon the inter-
section of the curves resulting from (12) and (12). Such
0.4 0.5 0.6 0.7 0.80.91
-10
0
10
20
30
40
50
60
k
i
k
i
as a funct ion of using (12)
k
i
as a funct ion of using (13)
Figure 1. Selection of ki and
according to the intersection
point of the curves.
Copyright © 2013 SciRes. AJCM
C. I. MURESAN 9
an intersection point depends upon the imposed criteria
for the gain crossover frequency and the phase margin.
For a specified set of gain crossover frequency and phase
margin, such an intersection point might not exist. Thus,
the existing graphical methods cannot be used to com-
pute the parameters and optimization algorithms need to
be used instead.
In order to facilitate the use of the simplicity of the
graphical methods in tuning the FO-PI controllers and to
avoid complex optimization algorithms, a simple approach
is proposed that combines the graphical methods with a
very simple optimization routine. The idea behind the
optimization routine consist in plotting the two curves for
ki as a function of
and selecting the values that
minimize the distance between the two plotted curves.
The proposed tuning algorithm is given below:
for 0:1
compute ki using (12)
store result in vector ki1
compute ki using (13)
store result in vector ki2
end
plot ki1 as a function of
plot ki2 as a function of
compute absolute value of distance = ki1-ki2
determine optim = min(distance)
return optim
corresponding to optim
compute ki using (13) and optim
compute kp using (11)
The algorithm for computing PI controllers is based
upon setting 1
and computing ki using either (12) or
(13) and kp using (11). Since, the PI controller in (3) has
only two design parameters, the tuning of the PI control-
ler may be done using any combination of two perform-
ance criteria in (11), (12) or (13). Thus, imposing (11)
and (12) means that (13) will not necessarily be ensured,
which is the main drawback of traditional PI controllers
as compared to FO-PI controllers.
2.2. Optimization Routine for Tuning Fractional
Order PD Controllers
The tuning of the FO-PD controller is achieved in a
similar manner to the FO-PI. The transfer function of the
FO-PD controller, in the frequency domain, may be
written as:
() 1cossin
22
FO PIpd
Hjkk j


 




(14)
in which

cos sin
22
sj j




 
(15)
Similar to the FO-PI situation, equations (6), (7) and (8)
may be used to determine the three parameters of the
FO-PD controller, kp, kd and
:
1
1cossin
22 ()
gc
pdgc
p
kk jHj
 







(16)

sin 2()
cos 2
d
mpgc
gc d
k
tgH j
k

 








(17)
1
22
sin ()
2
12 cos
2
dgc
p
gc
gc
dgc dgc
kdH j
d
kk







(18)
Then, (17) and (18) may be employed to determine
using the optimized graphical algorithm the parameters
kd and
, and then, kp may be computed directly using
(16), as described below:
for 1:0
compute kd using (17)
store result in vector kd1
compute kd using (18)
store result in vector kd2
end
plot kd1 as a function of
plot kd2 as a function of
compute absolute value of distance = kd1-kd2
determine optim = min(distance)
return optim
corresponding to optim
compute kd using (18) and optim
compute kp using (16)
The algorithm for computing PD controllers is based
upon setting 1
and computing kd using either (17)
or (18) and kp using (16). Since, the PD controller in (4)
has only two design parameters; the tuning of the PD
controller may be done using any combination of two
performance criteria in (16), (17) or (18). Thus, imposing
(16) and (17) means that (18) will not necessarily be en-
sured, which is the main drawback of traditional PD con-
trollers as compared to FO-PD controllers.
3. Case Studies
3.1. Tuning an FO-PI Controller for a First
Order Process
The process transfer function is given by:
27.5
() 0.26 1
p
Hs
s
(19)
For a gain crossover frequency of cg
=15 rad/s and
a phase margin of m
=70o, the curves in Figure 1 are
obtained. Thus, the existing graphical methods may be
used to determine the final values for the FO-PI control-
Copyright © 2013 SciRes. AJCM
C. I. MURESAN
10
ler parameters. Imposing slightly different performance
criteria, such as cg
=30 rad/s, m
=70o and robustness
to gain uncertainties, the two curves in Figure 2 are ob-
tained.
For these particular performance criteria, the two plots
for ki do not intersect. Nevertheless, using the algorithm
proposed in Section 2, the minimum distance between
the two curves is computed, yielding 0.55
and ki =
5.69. Finally, using (11) the remaining FO-PI parameter
is computed as kp=0.1677.
The resulting (FO-PI) is:
0.55
5.69
(
PI
Hs
) 0.1677
FO s


1
(20)
Figure 3 shows that the Bode plot of the open loop
system with a FO-PI controller. It can be seen that the
phase margin is slightly increased from 70o as imposed
to74o. This is due to the optimization algorithm, in which
the final value for ki is chosen in order to meet the ro-
bustness criteria, rather than the phase margin criteria.
However, thanks to the optimal choice for the fractional
order
, the phase margin criteria obtained does not
vary significantly from the one imposed in the design
phase. The Bode plot also indicates that the modulus
crosses the zero axes at 30rad/s, as imposed in the design
specifications. Most importantly, it can be seen, that
changing the open loop gain will not reduce the phase
margin, but rather increase it, which means that the
overshoot of the closed loop system will not vary sig-
nificantly from the nominal value. Hence, the closed loop
system should behave robustly despite uncertainties in
the gain variations.
The closed loop results considering ±50% gain uncer-
tainty are given in Figure 4. It can be seen that the FO-PI
controller maintains the overshoot below 5%, while the
settling time varies slightly between 0.15-0.25 seconds.
0.4 0.45 0.50.55 0.6 0.65 0.70.75 0.8
0
2
4
6
8
10
12
k
i
k
i
as a f unct ion of using ( 12)
k
i
as a f unct ion of using ( 13)
Figure 2. Plots of ki as a function of μ using (12) and (13) for
ωcg = 25 rad/s and φm = 70o
-100
-50
0
50
100
Magnitud e (dB)
Bode Diagram
F requency (rad/sec)
10
-4
10
-3
10
-2
10
-1
10
0
10
1
10
2
10
3
10
4
-120
-90
-60
-30
Phase (deg)
cg
=30
m
=74
o
Figure 3. Open loop Bode diagram using FO-PI controller.
00.1 0.2 0.3 0.4 0.50.6
0
0.2
0.4
0.6
0.8
1
1.2
1.4
Time (s)
Output
+50% ga in unce r t ain t y
-50% gain u ncertainty
nominal case
Figure 4. Closed loop results with FO-PI controller consid-
ering ±50% process gain variation.
3.2. Tuning an FO-PD Controller for a Second
Order Process
The process transfer function is given as:
1
() (0.5)
p
Hs ss
(21)
Taking ωcg = 15 and m = 50O and using the algorithm
described in Section 2, the plots of kd as a function of μ
are derived as given in Figure 5. In this case, the algo-
rithm presented in Section 2 yields the same result as any
of the existing graphical methods, since the two curves
intersect. Figure 5 finally yields a fractional order λ=
0.573 and kd =2.59.
Using (16), the following value is obtained for the kp
parameter, as a function of the previously determined μ
and kd values: kp =17.5.
Copyright © 2013 SciRes. AJCM
C. I. MURESAN 11
The resulting (FO-PD) is:
0.573
()17.5 12.59
FO PD
Hs s
 (22)
The Bode diagram of the open loop system using the
previously determined FO-PD controller is given in Fig-
ure 6, while the closed loop system considering ±50%
gain uncertainty is given in Figure 7.
The Bode diagram in Figure 6 shows that variations of
the open loop gain will not have a negative effect on the
overshoot of the closed loop system, but only on the set-
tling time, which demonstrates that the designed frac-
tional order PD controller ensures the robustness of the
closed loop system despite gain variations. As compared
to the fractional order PI controller, the solution of the
PD controller at the intersection of the two curves im-
plies that all performance specifications are met: the gain
crossover frequency is exactly 15 rad/sec, as specified,
the phase margin is exactly 50o and the phase plot is flat
around the gain crossover frequency.
0.56 0.580.60.62 0.64 0.66
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
5.5
k
d
k
d
as a func tion of using ( 17)
k
d
as a func tion of using ( 18)
Figure 5. Selection of the fractional order μ and kd parameter.
-100
-50
0
50
100
150
Magnitude (dB)
Bode Diagram
F requency (rad/sec)
10
-4
10
-3
10
-2
10
-1
10
0
10
1
10
2
10
3
10
4
-150
-120
-90
-60
Phase (deg)
gc =15
m=50o
Figure 6. Bode diagram of the open loop system with FO-PD
controller.
00.2 0.4 0.6 0.81
0
0.2
0.4
0.6
0.8
1
1.2
1.4
Time (s)
Out put
+50% gain uncer t ainty
-50% gain unc er t ain ty
nominal case
Figure 7. Closed loop responses considering ±50% gain
uncertainty with a FO-PD controller.
As expected from the Bode plot, the overshoot is
maintained in all three case scenarios at the value of 25%,
while the settling time varies between 0.3-0.6 seconds.
4. Conclusions
The purpose of the present paper was to present a simple
and efficient optimization algorithm for tuning fractional
order PI and PD controllers. For specific performance
criteria, the existing graphical methods may not yield an
exact solution. Thus, optimization routines need to be
used in order to tune the fractional order controllers. The
paper shows that even in the case of no exact solution,
the graphical methods may still be employed with a
slight modification that implies computing and selecting
the minimum distance between the possible solutions. It
is shown through simulations that the fractional order
controllers designed using the proposed method yield
satisfactorily results in terms of closed loop performance
and robustness.
5. Acknowledgements
This work was supported by a grant of the Romanian
National Authority for Scientific Research, CNCS – UE-
FISCDI, project number PN-II-RU-TE-2012-3-0307.
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