Upload
doliem
View
229
Download
0
Embed Size (px)
Citation preview
1
Extensions of Classical Automatic Control Methods to Fractional Order Systems: An
Educational Perspective
Nusret Tan
Inonu University, Engineering Faculty, Dept. of Electrical and Electronics Engineering,
44280, Malatya, Turkey.
Abstract
The rapid development of control technology has an impact on all fields of control theory.
This development forced researchers to produce new mathematical control theory, new
controllers and design methods, actuators, sensors, new industrial processes, computer
methods, new applications, new philosophies and solution techniques for new challenges. One
of such new theory is fractional order control system which is based on fractional order
calculus. In recent years there has been considerable interest in the study of feedback systems
containing processes whose dynamics are best described by fractional order derivatives. It
seems that the developments in the area of fractional order control systems can give many
opportunities for new advancements in the control and automation field which is an important
part of science and technology. Therefore, it is important to teach developed results based on
fractional order calculus to see the effects of fractional order integrator and derivative on
control system performance. This can be done by using new and high-quality educational
methods such as advanced computer software programs and interactive tools. The purpose of
this paper is to show how fractional order control methods can be introduced into a first
course on classical control using interactive tools such as Matlab, Simulink and LabView.
Key words: Education, Control Theory, Fractional Order Systems, Interactivity, LabVIEW
1. Introduction
In recent years, fractional calculus has been an important tool to be used in engineering,
chemistry, physics, mechanical, bioengineering and other sciences. It can be seen in the
literature that there have been many publications in this field (Das, 2008; Xue at al., 2007;
Monje et al, 2010). The reason of this amount of interest to this subject is that the future
scientific developments will be based on fractional calculus and the attempts to find solutions
of unsolved complex problems. It is known that the differential operators which we
encountered in mathematics, science and engineering are in the form of 𝑑𝑓/𝑑𝑡, 𝑑2𝑓/𝑑𝑡2,
𝑑3𝑓/𝑑𝑡3.., or in other words, the systems are described by integer order differential
equations, however, is it necessary that the orders have to be integer? Why can it not be a
rational, fractional or complex number? The correspondence between Leibniz and L’Hospital
drawn attention to this subject at the beginning of differential and integral calculus era and
this subject is now called fractional calculus. We have entered the era of fractional
mathematics especially due to the computational facilities provided by technological
developments. Therefore, hereafter developed theories, innovations and applications in
science and technology will be based on fractional calculus. Automatic control expressed as
“hidden technology” by Karl Astrom will be a field which uses fractional calculus the most.
Fractional order control systems are needed for better modelling and performance of
dynamical systems. Therefore, in order to obtain effective solutions to the problems in science
2
and technology, it is necessary to do work on this subject which has a great capacity of
original value.
On the other hand, the subjects being taught to students on automatic control, despite
advances in the field of control theory, seem to have changed little over last many years.
However, developments in computer technology now allow us to use new and high-quality
educational methods such as interactive tools, virtual and remote laboratories to make use of
World Wide Web etc. (Dormido, 2002; Tan et al, 2016a). Today’s computer software
programs such as Matlab and LabVIEW can be used effectively to teach some advanced
subjects to the students without going into details of mathematical derivations. Although the
fundamental control theory concepts known as classical control theory are necessary for
control theory education and because of its many advantages we think this must continue in
the future, it will be important that any developments in control theory which can be linked to
classical control are good candidates for consideration in basic feedback control education by
using interactive tools. The interactive tools are extremely useful and they enable students to
explore changes in system performance as parameters are varied, and to do so looking at
several diagrams simultaneously. This can be done without any programming by just using a
mouse to adjust any parameters and the effects can be immediately seen. Matlab, LabVIEW
and Simulink provide an excellent environment for such studies to teach additional topics
besides subjects of classical control theory. One such topic is the recent development in
methods to analyse systems using fractional order calculus. As summarized above this is an
important topic since most real physical system can be modeled more adequately by fractional
order differential equations. The applications of fractional order differential equations to the
problems in the control theory have been increased in recent years and promising results have
been obtained. It has been shown that there is a strong link between classical control methods
and fractional order approaches. Therefore, from educational point of view, it is important to
teach developed results based on fractional order calculus to see the effects of fractional order
integrator and derivative on control system performance. The purpose of this paper is to show
how the results based on fractional order concepts can be introduced into a basic classical
feedback control theory course given for undergraduate students and the teaching can be
supported by software tools. This will accelerate future developments in the control and
automation field which is an important part of science and technology.
2. Formulization of Fractional Order Derivative and Integrator
Fractional order derivative and integrator can be considered as an extension of integer order
derivative and integrator operators to the case of non-integer orders and it is defined in
general form as the following (Chen et al., 2009),
( )
0
1 0
( ) 0
a t
t
a
d
dt
D
d
(1)
where, a tD represents fundamental non-integer order operator of fractional calculus.
Parameters a and t are the lower and upper bounds of integration, and R denotes the
fractional-order. Two definitions used for the general fractional derivative and integrator are
the Riemann-Liouville definition and the Caputo definition. The Riemann–Liouville
definition for the fractional-order derivative of order R has the following form
3
1
1 ( )( )
( ) ( )
tn
a t n n
a
d f dD f t
n dt t
(2)
where (.) is Euler’s gamma function and 1n n , n N . An alternative definition for the
fractional-order derivative was given by Caputo as
1
1 ( )
( ) ( )
t n
a t n
a
fD d
n t
(3)
where 1n n , n N . The Laplace transform of the Caputo fractional order derivative has
the following result that is particularly significant for fractional order system modeling 1
1 ( )
0
00
( ) ( ) (0)n
st k k
t
k
e D f t dt s F s s f
(4)
When (1) (2) (3) ( 1)(0) (0) (0) (0) ,.., (0) 0nf f f f f is considered, a basic property
facilitating for design and analysis of fractional order systems is expressed for Laplace
transform of fractional order derivative as ( ) ( )L D f t s F s .
In general, fractional order LTI systems were described by the following fractional order
differential equation form as (Vinagre et al, 2000) 1 0 1 0
1 0 1 0( ) ( ) ( ) ( ) ( ) ( )n n m m
n n m ma D y t a D y t a D y t b D r t b D r t b D r t
(5)
By applying ( ) ( )L D f t s F s , a general form of transfer function of fractional order LTI
systems were expressed as, 1 0
1 0
1 0
1 0
( )( )
( )
m m
n n
m m
n n
b s b s b sY sG s
R s a s a s a s
(6)
where ia , jb ( 0,1,2,...,i n and 0,1,2,...,j m ) are real parameters and i ,
j are real
positive numbers with 0 1 n and 0 1 m .
3. Why Fractional Order Control?
Many real systems are known to display fractional order dynamics. For example, it is known
that the semi-infinite lossy (RC) transmission line demonstrates fractional behaviour since the
current into the line is equal to the half derivative of the applied voltage that is
( ) (1/ ) ( )V s s I s . Thus, the significance of fractional order representation is that fractional
order differential equations are more adequate to describe some real world systems than those
of integer order models. Many physical systems such as viscoelastic materials,
electromechanical processes , long transmission lines, dielectric polarisations , coloured noise,
cardiac behaviour, problems in bioengineering, and chaos can be described using fractional
order differential equations. Thus, fractional calculus has been an important tool to be used in
engineering, chemistry, physical, mechanical and other sciences (Magin, 2006; Ppoudlubny,
1999a; Harley et al., 1995; Koeller, 1984; Perdikaris and Karniadakis, 2014; Oldham and
Spanier; 1974).
In feedback control theory the proportional, derivative, and integral control actions affect the
performance of closed loop control system. The closed loop behaviours such as speed of the
response, elimination of steady-state error, relative stability and sensitivity to noise can be
changed by proportional, derivative and integral actions (Monje et al., 2010). By introducing
more general control actions of the form s or 1/ s , R , we can achieve more
4
satisfactory results from closed loop control system. Clearly s represents fractional order
derivative and 1/ s represents fractional order integral.
For modelling dynamic systems, frequency domain experiments are usually performed in
order to obtain equivalent electrical circuits which represent real dynamical systems.
Generally, a frequency domain behaviour of the form / ( )k j , R and in the Laplace
domain which is /k s is required for accurate modelling. /k s is known as Bode’s ideal
transfer function (Monje et al. 2010).
4. Frequency Domain Analysis
The computation of frequency responses of transfer functions plays an important role in the
application of frequency domain methods for the analysis and design of control systems.
There are some powerful graphical tools in classical control, such as the Nyquist plot, Bode
plots and Nichols charts, which are widely used to evaluate the frequency domain behaviours
of the systems. The Bode and Nyquist envelopes of a transfer function are important in
classical control theory for the analysis and design. For example, the frequency domain
specifications such as gain and phase margins can be obtained using the Bode and Nyquist
envelopes of a transfer function. The Bode plot of a control system provides a clear indication
of how the Bode plot should be modified to meet given specifications. Therefore, controller
design based on the Bode plot is simple and straightforward.
A transfer function including fractional powered s terms can be called a fractional order
transfer function, FOTF. The frequency response computation of FOTF can be obtained
similar to integer order transfer functions (Tan et al., 2009). For example, with the FOTF 2.3 0.8( ) 1/ ( 4 1)G s s s replacing s by j and using ( ) (cos / 2 sin / 2)j j ,
one obtains
0.8 2.3 0.8 2.3
1( )
(1 1.236 0.891 ) (3.8044 0.4540 )G j
j
(7)
Bode and Nyquist diagrams of this equation can then be obtained as shown in Figs. 1 (a) and
(b).
a) b)
Figure 1: a) Bode plot b) Nyquist plot
10-2
10-1
100
101
102
-100
-50
0Bode diagram
frequency(rad/sec)
gain
(dB
)
10-2
10-1
100
101
102
-300
-200
-100
0
frequency(rad/sec)
phase(d
egre
e)
-0.2 0 0.2 0.4 0.6 0.8 1 1.2-0.4
-0.35
-0.3
-0.25
-0.2
-0.15
-0.1
-0.05
0
0.05Fractional Nyquist diagram
Real
Imagin
ary
5
5. Time Domain Analysis
For time domain computation, there is not a general analytical method for determining the
output of a control system with an FOTF. There have been many studies over the years some
of them are based on integer approximation models and others based on numerical
approximation of the non-integer order operator. The methods developing integer order
approximations can be used for time domain analysis of fractional order control systems
similar to classical control approaches. Some of the well known methods for evaluating
rational approximations are the Continued Fractional Expansion (CFE) method, Outaloup’s
method, Carlson’s method, Matsuda’s method, Chareff’s method, least square methods and
others (Oustaloup et al., 2000; Matsuda and Fujii, 1993; Vinagre et al., 2000).
In the recent papers by the authors (Atherton et al., 2015), two new computational methods to
obtain solutions were given. One was based on the Fourier series of a square wave and is
called the Fourier Series Method(FSM) and the other is based on the Inverse Fourier
Transform Method (IFTM). Following this study, the method has been used to study the time
response of closed loop fractional order systems with time delay and fractional order
controller in (Tan et al., 2016b; Tan et al., 2016c). The results obtained from these methods
are exact, to numerical accuracy in summing infinite series, since they use frequency response
information for an FOTF which can be computed exactly by using the relationship
( ) [cos( / 2) sin( / 2)]j j .
The block diagram of a closed loop fractional order control system with time delay is shown
in Fig. 2
Figure 2: A fractional order closed loop control system with time delay
Here, ( ) ( ) ( )pL s C s G s is the open loop transfer function which is in the form of Eq. (6) such
as 1 0
1 0
1 0
1 0
( ) ( ) ( ) ( ) ( )m m
n n
s sm mp
n n
b s b s b sL s C s G s C s G s e e
a s a s a s
(8)
Then the closed loop transfer function can be written as 1 0
1 0 1 0
1 0
1 0 1 0
( )( ) ( )( )
( ) 1 ( ) ( )
m m
n n m m
s
m m
s
n n m m
b s b s b s eY s L sP s
R s L s a s a s a s b s b s b s e
(9)
Letting
( ) ( )( ) ( )( )
( ) 1 ( ) 1 ( ) ( )
p
p
C s G sY s L sP s
R s L s C s G s
(10)
and replacing s by j in Eq. (10), one obtains
6
( ) ( )( )( )
( ) 1 ( ) ( )
(Re[ ( )] Im[ ( )])(Re[ ( )] Im[ ( )] ( ) ( )
1 (Re[ ( )] Im[ ( )])(Re[ ( )] Im[ ( )] ( ) ( )
p
p
p p
p p
C j G jYP j
R j C j G j
C j j C j G j j G j U jV
C j j C j G j j G j Z jQ
(11)
where
( ) Re[ ( )]Re[ ( )] Im[ ( )]Im[ ( )]p pU C j G j C j G j (12)
( ) Re[ ( )]Im[ ( )] Im[ ( )]Re[ ( )]p pV C j G j C j G j (13)
( ) 1 Re[ ( )]Re[ ( )] Im[ ( )]Im[ ( )]p pZ C j G j C j G j (14)
( ) Re[ ( )]Im[ ( )] Im[ ( )]Re[ ( )]p pQ C j G j C j G j (15)
and
Re[ ( )] Re[ ( )]cos( ) Im[ ( )]sin( )pG j G j G j (16)
Im[ ( )] Im[ ( )]cos( ) Re[ ( )]sin( )pG j G j G j . (17)
Thus, ( )P j can be written as
2 2
[ ( ) ( ) ( ) ( )] [ ( ) ( ) ( ) ( )]( )
( ) ( )
U Z V Q j V Z U QP j
Z Q
(18)
so that the real part and imaginary parts of the closed loop transfer function ( )P j are
2 2
[ ( ) ( ) ( ) ( )]Re[ ( )]
( ) ( )
U Z V QP j
Z Q
(19)
2 2
[ ( ) ( ) ( ) ( )]Im[ ( )]
( ) ( )
V Z U QP j
Z Q
(20)
The Fourier series for a square wave of -1 to 1 with frequency 2 /s T can be written as
1(2)
4 1( ) sin( )s
k
r t k tk
(21)
where T is the period of the square wave. If ( )r t passes through the transfer function ( )P s
then the output, which is the unit step response if T is sufficiently large, can be written as
1(2)
2 21(2)
4 1( ) Re ( ) sin( )
[ ( ) ( ) ( ) ( )]4 1sin( )
[ ( ) ( ) ]
s s s
k
s s s ss
k s s
y t P jk k tk
U k Z k V k Q kk t
k Z k Q k
(22)
Similarly, the impulse response, which is the derivative of the step response is given by
1(2)
2 21(2)
( ) 4( ) Re ( ) cos( )
[ ( ) ( ) ( ) ( )]4cos( )
[ ( ) ( ) ]
si s s s
k
s s s ss s
k s s
dy ty t P jk k t
dt
U k Z k V k Q kk t
Z k Q k
(23)
This method is called FSM.
For IFTM, the impulse response, ( )p t , corresponding to the transfer function ( )P s of Eq. (9)
is given by 1( ) ( )p t L P s where 1L denotes the inverse Laplace transform. The impulse
response can be computed by
7
2 2
0 0
2 2 [ ( ) ( ) ( ) ( )]( ) Re[ ( )]cos( ) ( ) cos( ) ( )
[ ( ) ( ) ]
U Z V Qp t P j t d t d
Z Q
(24)
or alternatively by
2 2
0 0
2 2 [ ( ) ( ) ( ) ( )]( ) Im[ ( )]sin( ) ( ) sin( ) ( )
[ ( ) ( ) ]
V Z U Qp t P j t d t d
Z Q
(25)
Example 1: In this example ( )L s of Fig. 2 was taken as 1.15
5.3 3.65 2.82 1.7 1.42
0.5 0.4( ) ( ) ( )
3 7 4 1.4p
sL s C s G s
s s s s s
(26)
Thus, the closed loop transfer function is 1.15
5.3 3.65 2.82 1.7 1.42 1.15
( ) 0.5 0.4( )
1 ( ) 3 7 4 1.4 0.5 0.4
L s sP s
L s s s s s s s
(27)
The step responses of ( )P s using the FSM and the impulse responses of 1
( )P ss
which are the
step responses of ( )P s using the IFTM for the selected range of frequencies are plotted in
Figs. 3 (a) and (b). Step responses obtained from FSM and IFTM are given in Fig. 4.
a) b)
Figure 3: a) Step responses obtained from FSM for different frequency ranges b) Step
responses obtained from IFTM for different frequency ranges
8
Figure 4: Step responses obtained from FSM and IFTM
6. Fractional Order Control and Implementation
Fractional order functions are infinite dimensional functions and hence it is very difficult to
implement them practically or simulate them numerically (Vinagre et al. ,2000). Therefore,
several integer order approximation methods were proposed for realization of fractional order
systems by replacing them with integer order approximations. Oustaloup presented an
approximation method based on recursive distribution of poles and zeros in a limited
frequency (Oustaloup et al, 2000). Another approximation method using the gain of the
fractional order transfer functions at certain frequencies was suggested by Matsuda (Matsuda
et al., 1993). Recently, SBL fitting approximation method has been presented for fractional
order derivative and integrator operators (Deniz et al., 2016). The integer order
approximations model obtained by SBL fitting method is given in Table 1. Rational
approximations for 0.5( ) 1/G s s obtained using different methods are shown in Table 2.
The exact Bode plots of 0.5( ) 1/G s s and its approximations are shown in Fig. 5 where one
can see that the approximations are good for a range of frequency but fail to give an exact
matching. It is known that the exact step response of 0.5( ) 1/G s s is equal to
0.5
( ) 2 /sy t t and its impulse response equals ( ) 1/iy t t . The exact step response and
the approximations by different methods are shown in Fig. 6 (a), where it can be seen that the
approximations are worse as time increases. Fig. 6 (b) shows the errors in the approximations.
0 10 20 30 40 50 60 70 80 90 1000
0.5
1
1.5
Time(sec)
Outp
ut(
ste
p r
esponse)
12 13 14 15 16 17
1.35
1.36
1.37
1.38
1.39
1.4
FSM
IFTM
9
Table 1. Lists of fractional order derivative operator approximations determined using
proposed method for 0.01,1.2 .
s Fractional Order Derivative Approximations by SBL
fitting 0.1s
4 4 3 2
4 4 3 2
5532 1.338 10 4147 191.5 1
4392 1.353 10 5063 284.5 1.894
s s s s
s s s s
0.2s
4 4 3 2
4 4 3 2
7718 1.714 10 4974 214.2 1
4810 1.743 10 7372 470.2 3.587
s s s s
s s s s
0.3s
4 4 4 3 2
4 4 3 4 2
1.087 10 2.227 10 6052 242.6 1
5224 2.263 10 1.083 10 783.7 6.829
s s s s
s s s s
0.4s
4 4 4 3 2
4 4 3 4 2
1.553 10 2.949 10 7507 279.8 1
5622 2.972 10 1.615 10 1324 13.16
s s s s
s s s s
0.5s
4 4 4 3 2
4 4 3 4 2
2.271 10 4.012 10 9566 330.7 1
5987 3.978 10 2.46 10 2283 25.87
s s s s
s s s s
0.6s
4 4 4 3 4 2
4 4 3 4 2
3.447 10 5.677 10 1.268 10 405.6 1
6300 5.493 10 3.88 10 4074 52.66
s s s s
s s s s
0.7s
4 4 4 3 4 2
4 4 3 4 2
5.564 10 8.562 10 1.791 10 528.2 1
6537 8.007 10 6.487 10 7704 113.8
s s s s
s s s s
0.8s
5 4 5 3 4 2
4 5 3 5 2 4
1.008 10 1.452 10 2.844 10 770.1 1
6666 1.298 10 1.213 10 1.628 10 275.5
s s s s
s s s s
0.9s
5 4 5 3 4 2
4 5 3 5 2 4
2.428 10 3.28 10 6.012 10 1488 1
6648 2.773 10 3.006 10 4.561 10 887.3
s s s s
s s s s
Table 2: Integer approximations of 0.5( ) 1/G s s using different methods
Method Integer approximations of
0.5( ) 1/G s s
CFE high frequency
Method
4 3 2
4 3 2
0.3513 1.405 0.8433 0.1574 0.008995( )
1.333 0.478 0.064 0.002844cfe
s s s sH s
s s s s
Carlson’s Method 4 3 2
4 3 2
36 126 84 9( )
9 84 126 36 1car
s s s sH s
s s s s
Matsuda’s Method 4 3 2
4 3 2
0.08549 4.877 20.84 12.995 1( )
13 20.84 4.876 0.08551mat
s s s sH s
s s s s
Least-Squares
Method
4 3 2
4 3 2
0.1002 4.011 11.26 5.076 0.3694( )
8.654 9.364 1.771 0.03744ls
s s s sH s
s s s s
Charref’s Method 4 3 2
5 4 3 2
6.3 74.84 121.1 29.79 0.9986( )
29.85 121.8 76.85 7.497 0.1cha
s s s sH s
s s s s s
Oustaloups’s Method 5 4 3 2
5 4 3 2
74.97 768.5 1218 298.5 10( )
10 298.5 1218 768.5 74.97 1ous
s s s s sH s
s s s s s
10
Fig. 5: Exact Bode plots of 0.5( ) 1/G s s and its approximations
a) b)
Fig.6: a) Step responses of 0.5( ) 1/G s s b) Errors in the approximations
An example is provided below relating to SBL fitting method. For this case, the values in
Table 1 are used to obtain integer order approximation model of fractional order derivative
operator. Also, these values can be used as 1/ s for integer order approximation model of
fractional order integrator operator.
Example 2: Consider the fractional order transfer function
1.2
1( )
1G s
s
(28)
For the fractional order derivative 0.2s , SBL fitting method, Matsuda’s method and
Oustaloup’s method provide the integer order approximation model in Table 3. Also,
equivalent integer order transfer function ( )TG s of fractional order transfer function ( )G s can
10-4
10-2
100
102
104
-50
0
50
Frequency(rad/sec)
Magnitude(d
B)
10-4
10-2
100
102
104
-60
-40
-20
0
Frequency(rad/sec)
Phase(d
eg)
Hcfe
Hcar
Hmat
Hls
Hcha
Hous
Exact
0 20 40 60 80 100 120 140 160 180 2000
2
4
6
8
10
12
14
16
Time(sec)
Outp
ut(
Ste
p R
esponse)
Analy tical Function
CFE Method
Carlson's Method
Matsuda's Method
Least-Squares Method
Oustaloups's Method
Charref 's Method
0 20 40 60 80 100 120 140 160 180 200-2
0
2
4
6
8
10
12
14
Time(sec)
Err
or
CFE Method
Carlson's Method
Matsuda's Method
Least-Squares Method
Oustaloups's Method
Charref 's Method
11
be seen in Table 3. Fig. 7 shows a comparison of the amplitude and phase responses. Here, the
exact solution for 1.2( ) 1/ ( 1)G s s was obtained by calculating phase and amplitude of
1.21/ [( ) 1]j .
Figure 7: Amplitude and phase responses of ( )G s and approximate integer order models,
SBL fitting method, Matsuda’s method and Oustaloup’s method
Table 3: Equivalent integer order transfer function ( )TG s of fractional order transfer function
( )G s
Met
hod
Integer order approximation models for 1.2
1( )
1G s
s
SB
L
4 4 3 20.2
4 4 3 2
7718 1.714 10 4974 214.2 1
4810 1.743 10 7372 470.2 3.587
s s s ss
s s s s
4 4 3 2
5 4 4 4 3 2
4810 1.743 10 7372 470.2 3.587( )
7718 2.195 10 2.241 10 7586 471.2 3.587T
s s s sG s
s s s s s
Mat
suda
4 3 20.2
4 3 2
3.357 161 453.9 95 1
95 453.9 161 3.357
s s s ss
s s s s
4 3 2
5 4 3 2
95 453.9 161 3.357( )
3.357 162 548.9 548.9 162 3.357T
s s s sG s
s s s s s
Oust
aloup
5 4 3 20.2
5 4 3 2
2.512 98.83 531.7 442.3 56.87 1
56.87 442.3 531.7 98.83 2.512
s s s s ss
s s s s s
5 4 3 2
6 5 4 3 2
56.87 442.3 531.7 98.83 2.512( )
2.512 99.83 588.6 884.5 588.6 99.83 2.512T
s s s s sG s
s s s s s s
Fractional order PID controller (PIλDμ) which has five tuning controller parameters including
an integrator of order λ and an differentiator of order μ provides a better response than the
10-3
10-2
10-1
100
101
102
103
-80
-60
-40
-20
0
20Bode diagram
frequency(rad/sec)
gain
(dB
)
10-3
10-2
10-1
100
101
102
103
-120
-100
-80
-60
-40
-20
0
frequency(rad/sec)
phase(d
egre
e)
exact
SBL
Matsuda
Oustaloup
exact
SBL
Matsuda
Oustaloup
12
integer order PID controller when used both for the integer-order systems and fractional-order
systems. PIλDμ is described in time domain in (Podlubny, 1999b) as follows;
( ) ( ) ( ) ( )p i du t k e t k D e t k D e t
(29)
where pk is the proportional gain, ik is the integral gain, dk is the derivative gain; and
are positive real numbers. The frequency domain formula is given as
( )( )
( )
ip d
kU sC s k k s
E s s
(30)
In Equation (30), classical PID controller can be obtained for 1 and 1 .
Example 3: In this example, SBL fitting method is used to simulate the closed loop control
system with PIλ controller as shown in Fig. 8 in MATLAB. Consider the fractional order PI
controller
( ) ip
kC s k
s (31)
To use fractional order controller ( )C s for simulation of the closed loop control system in
MATLAB, an integer order approximation method has to be applied for fractional operator
s . When an equivalent integer order approximation model of ( )C s is determined using an
integer order approximation method, the closed loop control system as shown in Fig. 8 can be
simulated to obtain the closed loop response. For this, fractional order operator is replaced
with its equivalent integer order model in closed loop control system.
An illustrative example which contains SBL fitting approximation method is given to clarify
this strategy. Assuming that the plant transfer function is given by 2
1( )
3 1G s
s s
and PI
controller is formed as 0.9
( ) ip
kC s k
s in Figure 2, one can found stability region of the
closed loop control system using SBL analysis (Hamamci, 2008). Then, the controller
parameters pk and ik can be selected in stability region to obtain the step response of
configuration in Fig. 8.
Figure 8: The closed loop control system with PIλ simulated in MATLAB
Fig. 9 shows different step responses obtained for different controller parameters ( , )p ik k with
0.9 selected from stability region as (0.2604, 2.2281), (0.6843, 4.7544) and (0.1129,
( )G sue
ik
s
1
1s
Integer approximation
model by SBL fitting
pk
PI
yr
13
1.2807). Better closed loop step responses can be obtained for different controller parameters
( , , )p ik k selected from stability region.
Figure 9: Step responses of the closed loop control system with PIλ for different values pk
and ik
7. Teaching Fractional Order Control Systems
MATLAB is a high-performance language for technical computing. It integrates computation,
visualization, and programming in an easy-to-use environment where problems and solutions
are expressed in familiar mathematical notation. Simulink can easy analyze model and
simulate control systems (MathWorks, http://www.mathworks.com/). Simulink can work with
the MathWorks Real-Time Workshop. Real-Time Workshop generates and executes stand-
alone C code for developing and testing algorithms modeled in Simulink. The resulting code
can be used for many real-time and non-real-time applications, including simulation
acceleration, rapid prototyping, and hardware-in-the-loop testing (Rodriguez, et al. 2005). So
Simulink is also widely used in simulation of all kind of systems. However, Simulink lack the
imitation of physical instruments or equipment in appearance and operation. That’s why, it is
a good idea to combine LabVIEW and MATLAB in simulation of control system (Xuejun et
al, 2007).
The LabVIEW ( Laboratory Virtual Instrument Engineering Workbench) software is a
graphical programming language and used to develop a virtual instrument (vi) that includes a
front panel and a functional block diagram. User enters input from the front panel of the vi.
LabVIEW has become a vital tool for engineers and scientists in research throughout
academia, industry, and government labs. LabVIEW is taught at many universities in Europe
and USA. LabVlEW has been used in the classroom for teaching of difficult subjects. For
example, the graphical programming approach of LabVlEW has been used to teach computer
science concepts. The graphics make the concepts more intuitive and easier to understand.
LabVlEW is used to teach control systems. The graphical programming approach is based on
dataflow theory which is an ideal platform for learning how signals flow from one function to
another such as from an acquisition function through a filter to a spectrum analysis and finally
0 5 10 15 20 25 300
0.5
1
1.5
t (sec)
y(t
)
kp=0.2604, k
i=2.2281
kp=0.6843, k
i=4.7544
kp=0.1129, k
i=1.2807
14
to a graph. Each function is an icon that is wired to other icons as in the example given in
Figure 10. The wires are the signals flowing from one icon to the next. Research and industry
use LabVlEW for automated test (ATE), medicine, chemistry, process control, simulation,
calibration, and general-purpose data acquisition and analysis (Vento, 1988).
Figure 10: The front and block diagram panel image of LabVIEW
The temperature control system given in Figure 10 performs on LabVIEW environment. The
application includes two windows. The first window is front panel and second is block
diagram panel of the program. The front panel is developed as a graphical interface and it is
performed interactively. The block diagram panel is that provided wire connection and data
flow.
Figure 11: The front panel image of frequency response application of FOTF using LabVIEW
The application given in Figure 11 has been developed in the LabVIEW environment for the
frequency domain analysis of fractional order control systems. The Bode, Nyquist and
Nichols diagrams for any fractional order transfer function can be plotted by using the
program. The program with this properties helps to design suitable controller for a given
control system. The program is especially suitable for use in the field of education since it has
interactive features.
15
Figure 12: The front panel image of PI controller design application for the fractional order
plant using LabVIEW
The application given in Figure 12 runs the stability boundary locus (SBL) method to find all
stabilizing PI controllers in a stability region for closed loop control systems with fractional
order plant transfer function using LabVIEW application. Effect of the parameters of
controllers selected from the stability region can be immediately observed and step response
of the system can be immediately plotted. Thus, the controller which gives the best results can
be designed based on proposed interactive approach.
Example 4: Consider the transfer function as below,
2.2 0.9
5( )
7 2G s
s s
(32)
Figure 12: Transfer function input panel for frequency response application
16
Figure 13: Bode Diagram
a) b)
Figure 14: a) Nyquist diagram b) Nichols diagram
Figure 15: Gain and phase margin panel
In this example, frequency response application of fractional order transfer functions is
examined. Transfer function given in Eq. (32) is entered to panel as shown in Fig. 12. Bode,
Nyquist Nichols diagrams are simultaneously plotted as shown in Fig. 13 and Fig. 14. Also,
the application computes gain and phase margin values which are shown in Fig. 15. The
application has interactive features, thus user can try different values, while the application is
performing.
Example 5: Consider the transfer function with time delay as below,
2
2.4 1.2 0.4
2( )
2.5
sG s es s s
(33)
Figure 16: Transfer function input panel for PI controller design application
17
a) b)
Figure 17: a) Selected point in stability region b) Stable step response for closed loop system
In this example, PI controller design for fractional order transfer functions is examined.
Transfer function given in Eq. (33) is entered to panel as shown in Fig. 16. SBL graph is first
plotted when the program is run. Then, user can select any points in the controller parameter
plane which is shown in Fig. 17 (a) by moving mouse on the opened SBL plot window. As
seen in Fig. 17 (a), selected point is 0.262545pK , 0.06iK and this point is in stability
region. Thus, the closed loop system is stable and step response of the system is plotted for
the selected point and the result is given in Fig. 17 (b). According to selected point, the PI
controller is designed as shown in Fig. 18.
Figure 18: Designed PI controller according to selected point
a) b)
Figure 19: a) Selected point on stability curve b) Critical step response for closed loop
system
Now, consider another point as seen in Fig. 19 (a), where selected point is 0.546182pK ,
0.21iK and this point is exactly on SBL curve. Thus, the closed loop system is critically
stable and step response of the system is plotted for the selected point and the result is given
in Figure 19 (b). According to selected point, the PI controller is designed as shown in Fig.
20.
18
Figure 20: Designed PI controller according to selected point
Figure 21: Selected point outside stability region
Figure 22: Stability status panel
Consider another point as seen in Fig. 21, where selected point is outside the SBL curve.
Thus, the closed loop system is unstable and step response is not plotted, because the
apllication uses Inverse Fourier Transform Method (IFTM) for plotting the time response and
IFTM method is defined in stable condition (Atherton et al, 2015). Thus, the application
writes ‘SYSTEM IS UNSTABLE’ on stability status panel as shown in Fig. 22.
8. Stability Analysis
Stability analysis of fractional order system has been an interesting area of research in that,
the theorems proposed for classical systems are usually inadequate for fractional order
system. There can be found numerous stability analysis methods for classical systems in the
literature. For instance, Routh Hurwitz method can be used for stability analysis of
polynomials with integer orders, but does not work for fractional order polynomials. On the
other hand, it is well known that the frequency domain methods of classical control theory are
applicable to the fractional order control systems. Therefore, frequency domain based stability
criterion can be applied to the fractional order polynomials.
The stability test of fractional-order systems is different from the integer-order systems. The
only known stability test method is for commensurate-order systems. For a commensurate-
order polynomial, if the absolute values of the angles of all the poles are larger than απ/2, the
system is stable. The stable region of commensurate-order systems is shown in Fig. 23. The
BIBO(Bounded Input Bounded Output) stability analysis of fractional order control system
can be done by using time domain simulation.
19
Figure 23: Stability region for commensurate-order systems.
9. Conclusions
The objective of this paper has been to draw attention to the new developments in the field of
control systems with fractional order derivative and integrator and to show how these ideas
can be introduced into a first course on classical control theory. The methods are based on
fractional order calculus and allow students to think more practically in terms of
representation of real systems with fractional order differential equations. It has also been
shown that with suitable software, the theoretical results obtained in the field of fractional
order control can be used for the analysis and design of systems. Mathematical concepts of
fractional order calculus, the advantages of fractional order control, time and frequency
domain analysis of non-integer order control systems, teaching and stability issue have been
summarized.
Acknowledgment
This work is supported by the Scientific and Research Council of Turkey(TÜBİTAK) under
Grant no. EEEAG-115E388.
References
Atherton, D. P., Tan, N., Yüce, A. (2015). Methods for Computing the Time Response of
Fractional Order Systems. IET Control Theory & Application, 9(6), 817-830.
Chen, Y. Q., Petras, I., & Xue, D. (2009). Fractional order control-a tutorial. In 2009
American Control Conference pp. 1397–1411.
Das, S. (2008). Functional Fractional Calculus for System Identification and Control.
Springer-Verlag, Berlin Heidelberg, New York.
Deniz, F. N., Alagöz, B. B., Tan, N., Atherton, D. P. (2016). An integer order approximation
method based on stability boundary locus for fractional order derivative/integrator operators.
20
ISA Transactions, Vol. 62, pp. 154-163, doi:10.1016/j.isatra.2016.01.020,
http://dx.doi.org/10.1016/j.isatra.2016.01.020.
Hamamci, S. E. (2008). Stabilization using fractional-order PI and PID controllers. Nonlinear
Dynamics, 51, 329–343.
Dormido, S., (2002). Control learning: present and future. 15th IFAC World Congress on
Automatic Control, Barcelona, Spain.
Hartley, T. T., Lorenzo, C. F., Qammer, H. K. (1995). Chaos in Fractional Order Chua’s
System‖, IEEE Transactions on Circuits and Systems-I: Fundamental Theory and
Applications, Vol. 42, No. 8, 485-490.
Koeller, R. C. (1984). Amlication of Fractional Calculus to The Theory of Viscoelasticity‖, J.
Appl. Mech., Vol. 51, 299-307.
Magin, R. L. (2006). Fractional Calculus in Bioengineering. USA: Begell House Publishers.
Matsuda, K., Fujii, H. (1993). H (infinity) optimized wave-absorbing control-Analytical and
experimental results. Journal of Guidance, Control, and Dynamics, 16(6), 1146–1153.
MathWorks, Inc. Using Simulink. Retrieved from http://www.mathworks.com/.
Monje C. A., Chen, Y. Q., Vinagre, B. M., Xue D., Feliu, V. (2010). Fractional-order Systems
and Controls-“Fundamentals and Applications”. London: Springer-Verlag.
Oldham, K. B., Spanier, J. (1974). Fractional Calculus. San Diego: Academic Press.
Oustaloup A., Levron F., Mathieu B., and Nanot F.M. (2000). Frequency- band complex
noninteger differentiator: characterization and synthesis. 1410 IEEE Trans. on Circuit and
Systems - I: Fundamental Theory and Application, vol. 47, no. 1, pp. 25–3.
Perdikaris, P. and Karniadakis, G.E. 2014. ―Fractional-order viscoelasticity in one-
dimensional blood flow models‖, Ann Biomed Eng., 42(5):1012-23. doi: 10.1007/s10439-
014-0970-3.
Podlubny,I., (1999a). Fractional Differential Equation. Academic Press, San Diego.
Podlubny, I. (1999b). Fractional-order systems and PIλDµ -controllers. IEEE Transactions on
Automatic Control, vol. 44, no. 1, pp. 208– 214.
Rodriguez, A. A., Metzger, R. P., Cifdaloz, O., Dhirasakdanon, T. (2005). Description of a
modeling, simulation, animation, and real-time control (MoSART) environment for a class of
electromechanical systems. IEEE Transactions on Education, 48(3), 359-374.
Tan, N., Yüce, A, Deniz, F. N. (2016a). Teaching fractional order control systems using
interactive tools. International Conference on Education in Mathematics, Science &
Technology (ICEMST 2016), Bodrum, Turkey.
Tan, N., Özyetkin, M. M., Yeroglu, C. (2009). Nyquist envelope of fractional order transfer
functions with parametric uncertainty, New Trends in Nanotechnology and Fractional
21
Calculus Applications (Baleanu, Dumitru; Güvenç, Ziya Burhanettin; Machado, J.A. Tenreiro
(Eds.), Springer.
Tan, N., Atherton, D. P., Yüce, A., Deniz, F. N. (2016b). Exact time response computation of
control systems with fractional order lag and lead compensators. International Journal of
Circuits, Systems and Signal Processing, Vol: 10, pp. 260-268, ISSN: 1998-4464.
Tan, N., Atherton, D. P., Yüce, A. (2016c). Computing Step and Impulse Responses of
Closed Loop Fractional Order Time Delay Control Systems using Frequency Response Data.
International Journal of Dynamics and Control, DOI 10.1007/s40435-016-0237-y.
Xue, D., Chen, Y. Q., & Atherton, D. P., (2007). Linear Feedback Control - Analysis and
Design with Matlab. SIAM Press, ISBN: 978-0-898716-38-2. (348 pages) Chapter-8:
Fractional-order Controller - An Introduction.
Xuejun, X., Ping, X., Sheng, Y., & Ping, L. (2007). Real-time Digital Simulation of Control
System with LabVIEW Simulation Interface Toolkit. Proceedings of the 26th Chinese Control
Conference, Zhangjiajie, Hunan, China, 318-322.
Vinagre, B. M., Podlubny, I., Hernandez A., Feliu, V. (2000). Some Approximations of
Fractional Order Operators used in Control Theory and Applications. Fractional Calculus &
Applied Analysis, 3(3), 231–248.
Vento, J. A. (1988). Application of Labview In Higher Education Laboratories. 1988
Frontiers In Education Conference Proceedings, 444-447.