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. ([email protected]) 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 /, 2 / 2 , 3 / 3 .., 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
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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,
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