Integer Order Approximation of Uncertain Fractional Order Differentiations and Integrations M. Mine Özyetkin, Nusret Tan
Abstract—This paper deals with the robust stability of the fractional order system having interval order uncertainty. For this aim, integer order equivalences of fractional order system having interval order uncertainty are obtained. Kharitonov stability criterion is applied to check the stability of the system. The proposed idea is supported through numerical examples.
I. INT RODUCT ION
T
HE notion of non-integer order (or fractional order)
differentiation/integration emerged in 1695 [1]. Firstly, mathematicians studied it only as a theoretical subject due to its complexity. Thus, other science disciplines could not use it effectively owing to the absence of exact solution methods of non-integer order differential equations [2]. Today, fractional order differentiation/integration is gaining more and more attentions and importance for control systems especially robust control performance since fractional order calculation is closer to real world systems , which they have most likely fractional structure [2]. Therefore, it describes them more accurately than those of integer order models [25]. In other words, Fractional Order Systems (FOS) can be seen as effective tools to describe the real world systems [1]. Fractional calculus has been extensively employed in many science disciplines such as engineering, chemistry, physical, mechanical and other sciences. For instance, cardiac behavior, dielectric polarizations, electrochemical processes, long lines, viscoelastic materials, chaos, turbulence, signal processing, anomalous diffusion etc. [1, 6-8]. In recent years, important studies dealing with fractional calculus and its applications have been done in [9-14]. Fractional order control systems (FOCS) have transfer functions with fractional derivatives and/or fractional integrals where [15]. A control system may have both fractional order dynamics and be controlled by fractional order controller, namely, controller, controller, controller, TID (Tilted Proportional and Integral) controller, CRONE (Controle Robuste d’Ordre Non Entier) controller, fractional lag-lead compensator etc. [2]. However, in most cases, studies mainly utilize fractional order controller instead of fractional order plant models [2] since fractional order controllers may be more convenient for fractional order models to succeed best performance [3]. M. M. Özyetkin is with the Department of Electrical and Electronics Engineering, İnönü University, Malatya, 44280 T urkey (
[email protected]). N. T an is with the Department of Electrical and Electronics Engineering, İnönü University, Malatya, 44280 T urkey (n
[email protected]).
It is not easy to calculate fractional order transfer functions by using available software programs since they have been prepared to compute integer order system dynamics [15]. There are some studies based on MATLAB tools to model the fractional order dynamic system and control [3, 12, 1620]. In the literature, many studies dealing with fractional order control systems utilize integer order approximation methods, in other words, an integer order transfer function which almost have the same behaviors with real transfer function can be used instead of fractional order transfer function as it is much easier to tackle [15]. Besides, the power of for a fractional order control system having fractional derivatives and/or fractional integrals may vary between and (interval order uncertainty) where , . In practice, systems may have parametric uncertainty, in other words , the model of control system may not be known explicitly [21]. Thus, parametric uncertainty is an inevitable truth and the fractional order power varying in the specified interval named as interval fractional order polynomials is a more convenient representation for real world systems. The existing stability checking methods used for classical control systems cannot be utilized to the FOCS directly [22]. If FOCS have parametric uncertainty, the problem will become more complex. There are some studies about FOS with parametric uncertainty. In [22], Linear Time Invariant (LTI) FOS with parametric and interval order uncertainties are studied. In [9, 23-25], robust stability is studied for LTI FOS with interval uncertainties. Robust stability is investigated for interval plant with time delay using controller in [26]. Robust controllability problem for interval fractional order LTI systems is studied in [27]. However, the studies mainly are focused on interval uncertainties (coefficient uncertainties). But, a FOCS may have both interval uncertainty and interval order uncertainty. Besides, it may have only interval order uncertainty or interval uncertainty. There is not much work in the literature dealing with this problem. The aim of this paper, which the main idea is based on [22], is to compute integer order approximations of . Using interval arithmetic rules, integer order equivalences of are obtained in terms of first, second, third and fourth order integer approximations dependent on . And, it is shown that integer order equivalences of have interval transfer function, in other words, coefficients of equivalence transfer function have interval structure. Thus, the results obtained from practical robust control field can be employed
for analysis of fractional order systems having interval order uncertainty. II. REVIEW OF INT ERVAL CONTROL SYST EMS In this section, a short review of interval control systems is given. A control system with parametric uncertainty can be described using interval plant model whose numerator and denominator polynomials have interval structure. The results obtained in the area of parametric robust control are based on the Kharitonov theorem dealing with the stability analysis of interval polynomials. It can be said that the Kharitonov theorem is the extended version of the Routh stability criterion for interval polynomials [28]. According to this theorem, an interval polynomial is Hurwitz stable if and only if four Kharitonov polynomials, which can be obtained using upper and lower values of the unknown parameters, are Hurwitz stable [14, 28]. A. Kharitonov Theorem Consider an integer order interval polynomial as ( )
(1) {
),
where (
[
]
},
and are specified lower and upper bound of the th perturbation, respectively [21]. Four Kharitonov polynomials can be denoted as follows
{
( ) where
( )
( )|
( ) ( )
}
(4)
. III. INTEGER ORDER APPROXIMATION
There are several methods to obtain integer order equivalence of fractional order systems. For example Carlson’s method, Matsuda’s method, Oustaloup’s method, the Grünwald-Letnikoff approximation, Maclaurin series based approximations, time response based approximations etc. [13]. One of the most important approximations for fractional order systems is the CFE method. The CFE method is employed to obtain the realization of ( ) in this paper. This method can be expressed in the following form [29] (1 x)
1 x (1 ) x (1 ) x (2 ) x (2 ) x 1 1 2 3 2 5 ...
(5)
In this formulation used for the computation of . Using (5) first, second, third and fourth order integer approximations which are dependent on are obtained as follows [30]: First order approximation is (1 ) s (1 ) (1 ) s (1 )
(6)
Second order approximation is
( )
( 2 3 2) s 2 (2 2 8) s ( 2 3 2) ( 2 3 2) s 2 (2 2 8) s ( 2 3 2)
( ) (2)
( )
Third order approximation is ( 3 6 2 11 6) s 3 (3 3 6 2 27 54) s 2 (3 3 6 2 27 54) s ( 3 6 2 11 6) ( 3 6 2 11 6) s 3 (3 3 6 2 27 54) s 2
( )
(7)
(8)
(3 3 6 2 27 54) s ( 3 6 2 11 6)
Kharitonov indicated that the stability of the interval polynomial family of Eq. (1) could be found by applying the Routh criterion to these four polynomials [14]. B. Sixteen Kharitonov Plant Family
Fourth order approximation is ( 4 10 3 35 2 50 24) s 4 (4 4 20 3 40 2 320 384) s 3 (6 4 150 2 864) s 2 (4 4 20 3 40 2 320 384) s ( 4 10 3 35 2 50 24) ( 4 10 3 35 2 50 24) s 4 (4 4 20 3 40 2 320 384) s 3
An interval system can be described as
(9)
(6 4 150 2 864) s 2 (4 4 20 3 40 2 320 384) s
( ) where
( )
(3)
( )
[
]
and
[
]
. In Eq. (3), numerator and denominator polynomials ( ) and ( ) are interval polynomials described as Eq. (1). So, it can be seen that each polynomials (numerator and denominator polynomials ) have four Kharitonov polynomials. Therefore, if all combinations of these four polynomials for numerator and denominator are taken, sixteen Kharitonov transfer functions are obtained. And, they can be defined as follows [21]
( 4 10 3 35 2 50 24)
IV. INTEGER ORDER EQUIVALENCES OF In this section, it is shown that derivative can be represented by an interval integer order transfer function. As can be seen from eq. (5), each boundary can be shown by an integer order approximation (transfer function). Thus, turns into from interval order uncertainty to interval integer order uncertainty. For the robust stability test of this system, interval order system stability test procedure can be applied. Let consider fractional order derivative , where and ( ) are lower and upper boundaries of
uncertain order . Using approximations dependent on terms of first, second, third, approximations. For example First order integer approximation
Eq. (5) integer order of can be obtained in fourth, etc. integer order
Example 1:
of
Consider a control system with fractional order transfer function having interval order uncertainty
is
V. NUMERICAL EXAMPLES
( )
(15)
(10) where [
The integer order interval transfer function equivalences for this system using first, second, third and fourth order integer approximations of can be obtained as follows:
] (11)
Second order integer approximation of
The integer order equivalence can be obtained as in (16) using first order integer approximation of :
is
(16)
(12) where ( [( (
Using second order integer approximation of
)]
)( ) ( )(
)]
:
(13)
(17)
)
Third order integer approximation of
Using third order integer approximation of
is
:
(14)
(18)
where [( [( [( (
)(
)] )(
)]
)( )(
(15)
where ), )] ), )] )(
)] ), )] ), )]
(19)
is (13)
[( [( ( [( (
:
)] )
Fourth order integer approximation of
[( ( [( (
Using fourth order integer approximation of
(14)
Bode plots for lower limit ( ) and upper limit ( ) of the original system, and second order approximations of ( ) and ( ) is given in Fig 1. It can be seen that Bode plots of original system and second order approximations are very close each other from the Fig 1. Bode plots for lower limit and upper limit of original system and different values within given intervals is given in Fig. 2. And Bode plots of sixteen Kharitonov transfer functions of second order approximate interval model with original system can be seen in Fig. 3. From Fig. 3, it can be seen that sixteen Kharitonov transfer functions contain lower and upper bounds of original system.
Example 2:
Bode Diagram for 1/(s [1.2,1.4]+1)
Magnitude(dB)
10
Consider a fractional order control system as follows
0 -10 -20 -30 -1 10
original system for s
1.2
( )
original system for s 1.4
(20)
second order approximation for s 1.4 second order approximation for s 1.2 0
1
10 Frequency(rad/sec)
10
fourth order integer approximation can be obtained as
Phase(deg)
0
(21)
-50 -100 -150 -1 10
where 0
1
10 Frequency(rad/sec)
10
Fig. 1. Bode plots for lower limit ( ) and upper limit ( ) of original system, and second order approximations of ( ) and ( ) Bode Diagram for 1/(s
(22)
[1.2,1.4]
+1)
Magnitude (dB)
10 0 -10
and
-20 -30 -1 10
0
10 Frequency (rad/sec)
1
10
Phase (deg)
0
-50
(23) -100
-150 -1 10
0
10 Frequency (rad/sec)
1
10
Fig. 2. Bode plots for lower limit (*) and upper limit (*) of the original system and different values within given intervals (black)
Since the given system is unstable, consider a fractional order controller which is given by
Magnitude (dB)
10 0
( )
-10 -20 -30 -1 10
0
10 Frequency (rad/sec)
1
10
Phase (deg)
50 0 -50 -100 -150 -1 10
0
10 Frequency (rad/sec)
1
10
Fig. 3. Bode plots for lower limit(*) and upper limit (*) of the original system and sixteen Kharitonov transfer functions of second order approximate interval model (black)
(24)
Step responses for sixteen Kharitonov transfer functions and fourth order approximations of the original system are shown in Fig. 4.
Characteristic equation of the closed loop system with unity feedback can be obtained as
Step response
( )
1.2
1
( )
(29)
( )
(30)
y(t)
0.8
(
0.6
)
(
)
(
)
(
)
(31) 0.4
(32)
0.2
0
0
0.5
1
1.5
2
2.5 time(sec)
3
3.5
4
4.5
5
And it can be written as follows
Fig. 4. Step responses for sixteen Kharitonov transfer functions and second order approximations of real system (lower limit, solid red, upper limit, solid blue, Kharitonov, solid black)
As can be seen from the Eq. (20) Kharitonov stability criterion cannot be applied to investigate system stability since the given system has interval order uncertainty. However, Kharitonov stability criterion can be utilized to test the stability of system as obtained integer order approximations for the system have coefficient uncertainty. As seen from the Fig. 4, sixteen Kharitonov transfer functions step responses enclose original system lower and upper bounds. So, it can be infer from the Fig. 4 that the stability test of four Kharitonov polynomials is enough for the system stability test. Example 3:
( )
(33)
Four Kharitonov polynomials are ( )
( )
( )
( )
Consider a control system having interval order uncertainty as follows
( )
(25)
Fourth order integer approximation is (26) where
(27)
and
(28)
For the integer order LTI system, if all roots of characteristic polynomial are located on the left of imaginary axis, in other words if they have negative real parts or are negative, it can be said that the system is stable. So, the given system is stable since all roots of the four Kharitonov polynomials have negative real parts (they are located in the left half complex plane). The value sets of interval polynomial given in Eq. (33) are shown in Fig. 5 where it can be seen that the zero is not included in the value sets. Therefore, from the zero exclusion principle it can be said that the polynomial family is Hurwitz stable which agree with the Kharitonov theorem.
Value Set 500
0
Im
-500
-1000
-1500
-2000
-2500 -2500
-2000
-1500
-1000 Re
-500
0
500
Fig. 5 Value set of interval polynomial of Eq (32) for
VI. CONCLUSION In this paper, the robust stability of fractional order systems having interval order uncertainty has been studied. To utilize classical control stability methods, integer order equivalences (first, second, third or fourth order etc.) of fractional order system having interval order uncertainty are obtained. As these equivalences have coefficient uncertainties, stability test procedure for classical interval order systems can be employed to fractional order systems with interval order uncertainty. For the next studies, it can be investigate fractional order systems with both interval order uncertainty and coefficient uncertainty. REFERENCES [1]
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