International Journal of Control, Automation, and Systems (2010) 8(5):1009-1017
DOI 10.1007/s12555-010-0510-3
http://www.springer.com/12555
Frequency Response Computation of Fractional Order Interval
Transfer Functions
Celaleddin Yeroğlu, M. Mine Özyetkin, and Nusret Tan
Abstract: The paper present extensions of some results developed in the parametric robust control to
fractional order interval control systems (FOICS). Computation of the Bode and Nyquist envelopes of
FOICS are studied. Using the geometric structure of the value set of fractional order interval polynomials (FOIP), a technique is proposed for computing the Bode and Nyquist envelopes of transfer functions whose numerator and denominator polynomials are fractional order polynomials with interval
uncertainty structure. The results obtained are useful for the analysis and design of FOICS. Numerical
examples are included to illustrate the benefit of the method presented.
Keywords: Bode envelope, fractional order interval transfer functions, nyquist envelope, parametric
uncertainty.
1. INTRODUCTION
A system represented by differential equations where
the orders of derivatives can take any real number, not
necessarily integer number can be considered as a
fractional order system. The significance of fractional
order representation is that fractional order differential
equations are more adequate to describe real world
systems than those of integer order models [1,2].
Therefore, in recent years considerable attention has
been given to the fractional order control systems
(FOCS) due to the better understanding of fractional
calculus [3] and the emergence of a new electrical circuit
called “fractance” [4] which make the implementation of
a fractional order controller feasible [5]. As a result,
some important studies dealing with the applications of
the fractional calculus to the control systems have been
done in [6-10] and references there in. Results related to
the stability analysis and controller design for such
systems can be found in [11,12] and references there in.
For example, synthesis of robust fractional order
controllers using the quantitative feedback theory is
proposed in [13]. This field of research is still new and
there is not much work dealing with robustness analysis
of FOCS with parametric uncertainty [14-16].
The computation of frequency responses of uncertain
transfer functions plays an important role in the
application of frequency domain methods for the analysis
and design of robust control systems. The frequency
__________
Manuscript received September 3, 2009; revised March 12,
2010; accepted April 11, 2010. Recommended by Editorial Board
member Ju Hyun Park under the direction of Editor Hyun Seok
Yang.
Celaleddin Yeroglu is with the School of Computer Engineering, Inonu University, Malatya, Turkey (e-mail: cyeroglu@inonu.
edu.tr).
M. Mine Ozyetkin and Nusret Tan are with the School of Electrical and Electronics Engineering, Inonu University, Malatya,
Turkey (e-mails: {mmozyetkin, ntan}@inonu.edu.tr).
© ICROS, KIEE and Springer 2010
domain analysis of systems is an important topic in
control theory. 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 behaviors of systems. Motivated by
the results especially the Kharitonov and the Edge
theorems [17,18] obtained in the parametric robust
control, there have been several studies [19-23] on the
computation of the frequency responses of control
systems under parametric uncertainty. An interval
analysis algorithm is proposed for automatic synthesis of
fixed structure controllers in quantitative feedback theory
in [24]. However, these results are related to the integer
order control systems with parametric uncertainty.
Therefore, extensions of these results to FOCS with
parametric uncertainty will be very important. Some
preliminary results in this direction have been obtained
in [25].
The purpose of this paper is to present a method for
computation of Bode and Nyquist envelopes of fractional
order interval transfer functions (FOITF). The numerator
and denominator polynomials of a FOITF are a FOIP of
the form
P( s, q ) = q0 sα0 + q1sα1 + q2 sα 2 + q3 sα3 + + qn sα n , (1)
where α 0 < α1 < < α n are generally real numbers,
q = [q0, q1 , q2 ,..., qn ] is the uncertain parameter vector
and the uncertainty box is Q = {q : qi ∈ [qi , qi ], i = 0,
1, 2,..., n}. Here qi and qi are specified lower and
upper bounds of ith perturbation qi, respectively. Thus, a
FOITF can be represented as,
G ( s, a , b ) =
=
N ( s, b)
D ( s, a )
b0 sα 0 + b1sα1 + b2 sα 2 + ... + bm sα m
a0 s β0 + a1 s β1 + a2 s β2 + ... + an s βn
(2)
,
Celaleddin Yeroğlu, M. Mine Özyetkin, and Nusret Tan
1010
where α 0 < α1 < ... < α m
and
β 0 < β1 < .... < β n
are
generally real numbers, a = [a0 , a1 ,..., an ] and b = [b0 ,
b1 ,..., bm ] are uncertain parameter vectors, A = {a : ai
∈ [ai , ai ], i = 0,1, 2,..., n} and B = {b : bi ∈ [bi , bi ], i = 0,
1, 2,..., m} are uncertainty boxes.
In this paper, it is first shown that the value set of the
family of polynomial of (1) can be constructed using the
upper and lower values of uncertain parameters. Then,
using the geometric structure of the value set, a
procedure for computing the Bode and Nyquist
envelopes of FOITF represented by (2) is given.
The paper is organized as follows: In Section 2, the
construction of the value set of FOITF is given. An
algorithm is proposed for the computation of the Bode
and Nyquist envelopes of FOITF in Section 3. Numerical
examples are given in Section 4. Section 5 includes
conclusions and remarks.
(a) Uncertainty box in the parameter space.
2. CONSTRUCTION OF THE VALUE SET OF FOIP
The fractional power of jω can be written as
( jω ) µ = ω µ j µ
= ω µ [e
π
j ( + 2 nπ )
2
]µ
= ω µ [e
(3)
π
j ( µ + 2 nµπ )
2
],
(b) Images of exposed edges in the complex plane.
1
2
where n = 0, ± , ± ,... and µ is a rational number.
µ
µ
Thus,
( jω ) µ = ω µ (cos
π
π
µ + j sin µ ).
2
2
(4)
For FOIP of (1), substituting s = jω gives,
P ( jω , q) = q0 (k0 r + jk0i )ω
α0
+ q1 (k1r + jk1i )ω
+ j (q0 k0iω
+ q1k1iω
+ + qn kniω
αn
α1
v2 ( s ) = q0 sα 0 + q1sα1 + q2 sα 2 + + qn sα n
(6)
(5)
= (q0 k0r ω α 0 + q1k1r ω α1 + + qn knr ω α n )
α1
v1 ( s) = q0 sα 0 + q1sα1 + q2 sα 2 + + qn sα n
v3 ( s ) = q0 sα0 + q1sα1 + q2 sα 2 + + qn sα n
+ + qn (knr + jkni )ω α n
α0
Fig. 1. For a polynomial of the form of (1) with 3
uncertain parameters.
),
where klr and kli , l = 1, 2,..., n are constant. From (5),
it is clear that the uncertain parameters appearing both in
the real and imaginary parts are linearly dependent to
each other. The value set of such a polynomial in the
complex plane is a polygon. Thus the corresponding
polytope of a family of (1) in the coefficient space has
2( n+1) vertices and (n + 1)2n exposed edges since the
polynomial family has (n + 1) uncertain parameters.
For example, the uncertainty box in the parameter space
and image of the exposed edges in the complex plane for
a polynomial of the form of (1) with 3 uncertain
parameters are shown in Fig. 1(a) and (b).
Using the upper and lower values of the uncertain
parameters, all 2n+1 the vertex polynomials of P ( s, q)
can be written in the following pattern.
v2( n +1) ( s) = q0 sα0 + q1sα1 + + q2 sα 2 + + qn sα n
From these vertex polynomials, the exposed edges can be
obtained. For example, the vertex polynomial v1(s) and
v2(s) have the same structure except the parameter (q0) is
its lower value (q0 ) in v1(s) and its upper value (q0 ) in
v2(s). Thus, one of the exposed edges can be expressed as,
e(v1 , v2 ) = (1 − λ ) v1 ( s) + λ v2 ( s ),
(7)
where λ ∈ [0,1]. Similarly, the remaining exposed
edges can be constructed. Define the set which contains
all the vertex polynomials as
PV = {v1 , v2 ,....., v2( n +1) }
(8)
and the set of exposed edges as
PE = {e1 , e2 ,......, e
( n +1)2n
}.
(9)
Theorem 1:
∂P ( jω , q) ⊂ PE ( jω ),
(10)
Frequency Response Computation of Fractional Order Interval Transfer Functions
where ∂ denotes the boundary and PE is defined in (9).
Proof: From (5), it can be seen that the uncertain
parameters in the real and imaginary parts are related
with each other linearly. This type of polynomial family
is called polytopic family [22]. Therefore, a linear map
of the uncertainty box Q from the parameter space to the
complex plane is a polygon whose vertices and exposed
edges can be obtained by mapping the vertices and
exposed edges of Q as shown in Figs. 1(a) and (b). As a
result, the boundary of the value set of P ( s, q) at
*
s = jω can be obtained from the images of exposed
edges. Therefore, ∂P ( jω , q) ⊂ PE ( jω ) for all real ω
[25].
vertices of the polygon.
Consider the transfer function given in (2), and let
n1 , n2 ,..., n2m +1 and d1 , d 2 ,..., d 2n +1 be the vertex
polynomials of N ( s, b) and D( s, a) polynomials,
respectively. Define the sets NV and NE which contains
the vertices and edges of N ( s, b) as
NV = {n1 , n2 , n3 , n4 ,..., n2m +1 },
N E = {ne1 , ne2 , ne3 , ne4 ,..., ne
3.1. Bode envelopes of FOITF
The numerator and denominator polynomials of
FOITF of (2) are in the form of P ( s, q) of (1).
Therefore, the results given in the previous section can
be used to obtain the Bode envelopes of FOITF.
Theorem 2: The magnitude extremums of P ( s, q) at
}
(14)
similarly define DV and DE for the D( s, a) as
DV = {d1 , d 2 , d3 , d 4 ..., d 2n +1 },
( n +1)2n
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.
However, in order to apply this classical design method
to fractional order uncertain control systems, it is
necessary to compute the Bode and Nyquist envelopes of
a given FOITF.
(13)
( m +1)2m
DE = {de1 , de2 , de3 , de4 ..., de
3. FREQUENCY RESPONSES OF FOITF
1011
(15)
}.
(16)
Then, the magnitude and phase extremums of G ( s, a, b)
at s = jω * can be calculated from the following
theorem.
Theorem 3:
max G ( jω * , a, b)
=
max N ( jω * , b)
min D( jω * , a)
=
max nk ∈NV nk
min del ∈DE del
(17)
,
min G ( jω * , a, b)
=
min N ( jω * , b)
max D( jω * , a)
=
min ne p ∈N E ne p
max dr ∈DV d r
,
max arg[G ( jω * , a, b)]
= max arg[ N ( jω * , b)] − min arg[ D( jω * , a)]
= max arg nk ∈NV [nk ] − min arg dr ∈DV [d r ],
ma x P ( jω * , q ) = ma xvk ∈PV vk ,
(11)
min P ( jω * , q ) = minel ∈PE el
*
and the phase extremums of P( jω , q ) are
min arg[ P( jω * , q)] = min arg vk ∈PV [vk ],
ma x arg[ P ( jω * , q)] = max arg vk ∈PV [vk ],
(12)
where k = 1, 2,..., 2n +1 and l = 1, 2,...., ( n + 1)2n.
Proof: The proof of this theorem can be done by using
the geometric structure of the value set of P ( s, q) at
s = jω * . Since the value set of P ( jω * , q ) is contained
in a polygon as shown in Fig. 1, the maximum distance
from the origin to the edges of the polygon is achieved at
the corner of the polygon and minimum distance comes
from the edges. The phase extremums will be on the
(19)
min arg[G ( jω * , a, b)]
= min arg[ N ( jω * , b)] − max arg[ D( jω * , a)]
= min arg nk ∈NV [nk ] − max arg dr ∈DV [d r ],
s = jω * can be written as
(18)
where k = 1, 2,..., 2m +1 ,
p = 1, 2,..., ( m + 1)2m ,
(20)
r = 1, 2,
..., 2n+1 and l = 1, 2,..., (n + 1)2n.
Proof: The proof of this theorem follows from the
results of Theorem 2. Since at each frequency, the value
set of N ( s, b) and D( s, a) are similar to the value set
of P( s, q), the magnitude and phase extremums of
G ( s, a, b) can be found from the magnitude and phase
extremums of polygons corresponding to N ( s, b) and
D( s, a).
3.2. Nyquist envelope of FOITF
The numerator and denominator polynomials of
FOITF of (2) are in the form of P( s, q) of (1).
Therefore, the results given in the Section 2 can be used
to obtain the Nyquist envelope of FOITF.
Celaleddin Yeroğlu, M. Mine Özyetkin, and Nusret Tan
1012
Define the extremal system as
GE ( s ) =
NV ( s ) N E ( s )
,
∪
DE ( s ) DV ( s )
(21)
K * = infG ( s )∈GE ( s ) K G , θ * = infG ( s )∈GE ( s ) θG ,
where NV, NE, DV and DE are defined in (13)-(16).
Theorem 4: At s = jω ,
∂G ( jω , a, b) ⊂ GE ( jω ) =
NV
N
∪ E,
DE DV
(22)
where GE is defined in (21) and ∂ denotes the boundary
[25].
Proof: Let A1 and A2 be the two complex plane
polygons with vertex sets VA1 , VA 2 and edge sets E A1 ,
EA 2
respectively. Then, from the complex plane
geometry, the following is known [22].
 A   VA1
∂ 1  ⊂ 
 A2   E A 2
  E A1
∪
  VA 2
 




Since the value set of the numerator and denominator
of (2) are independent polygons, one can write
(24)
∂G ( jω , a, b) ⊂ GE ( jω ) =
NV
N
∪ E.
DE DV
(25)
3.3. Robust Gain and Phase Margins:
Gain and phase margins are two important frequency
domain specifications. For systems with a nominal
transfer function, these margins are computed from the
Nyquist or Bode plots of the open-loop transfer function.
However, in the case of systems with parametric
uncertainties, the computation of the gain and phase
margins becomes very complex. In this section, we deal
with the calculation of the robust gain and phase margins
for systems with an uncertain transfer function of the
form of (2).
Suppose that a closed-loop system with an uncertain
plant of the form of (2) is stable. The robust gain margin
is then the largest value of the gain K greater than 1 for
which the stability of KG ( s, a, b) is preserved, and the
robust phase margin is the largest value of phase θ for
which the uncertain system with e− jθ G ( s, a, b) is
robustly stable. Thus, the worst case gain margin K *
and phase margin θ * can be stated as
*
E A1 and E A 2 , respectively. From the complex plane
geometry, the following is then known:
∂( A1 + A2 ) ⊂ ( E A1 + VA 2 ) ∪ (VA1 + E A 2 ).
*
K = infG ( s )∈G ( s , a,b) KG , θ = infG ( s )∈G ( s, a,b ) θG , (26)
where KG is gain margin of G ( s ) and θG is phase
margin of G ( s ). Using Theorem 5, values of K * and
θ * can be computed from the extremal system GE ( s).
(28)
Now, for the calculation of gain margin, we need to find
the maximum value of K greater than 1 for which
(29)
is stable. The multiplication of a convex polygon with a
fixed K is still a convex polygon. Thus, for a fixed
value of K , one can write
VA1 = KNV , VA 2 = DV , E A1 = KN E , E A 2 = DE . (30)
From (28), following equation can be written:
∆( jω ) ⊂ ∆ E ( jω ) = ( KN E + DV ) ∪ ( KNV + DE ).
Thus
(27)
where GE (s) defined in (21).
Proof: Let A1 and A2 be the two complex plane
polygons with vertex sets VA1 and VA 2 , and edge sets
∆ ( s) = KN ( s, b) + D( s, a)
(23)
VA1 = NV , VA 2 = DV , E A1 = N E , E A 2 = DE .
Theorem 5: Suppose a unity feedback system with
G ( s, a, b) is stable. The robust gain and phase margins
are then
(31)
Therefore, stability of ∆ E ( s) implies stability of ∆ ( s ).
For the phase margin calculation, the gain K will be a
complex gain and the same proof will be valid [25].
Following procedure can be used for computation
of the Bode and Nyquist envelopes of a given FOITF:
• Construct the vertices and edges of N ( s, b) and
D( s, a) using (8) and (9).
• From (13)-(16), obtain the vertex and edge sets.
• For each s = jω , using (17)-(20), obtain the
magnitude and phase extremums, and construct the
Bode envelopes.
• For each s = jω , using (22), obtain the Nyquist
envelope.
• From the Bode and Nyquist envelopes and using
Theorem 5, compute the robust gain and phase
margins.
4. NUMERICAL EXAMPLES
Example 1: Consider a negative unity feedback
control system with the following FOITF
G ( s , a, b) =
N ( s , b)
1
=
,
D( s, a) a0 + a1s 0.9 + a2 s 2.2
(32)
where a0 ∈ [0.8,1.2], a1 ∈ [0.5, 0.9] and a2 ∈ [0.6,1].
The objective is to compute the Bode and Nyquist
envelopes and investigate robust stability of the system.
It can be seen that N ( s, b) = 1 which is constant and
Frequency Response Computation of Fractional Order Interval Transfer Functions
1013
D( s, a) has 3 uncertain parameters. Therefore, there are
23 = 8 vertex polynomials and 3 x 22 = 12 exposed
edges for D( s, a). The vertex polynomials of D( s, a)
are
d1 ( s) = 0.8 + 0.5s 0.9 + 0.6 s 2.2 ,
d 2 ( s) = 1.2 + 0.5s 0.9 + 0.6s 2.2 ,
d3 ( s) = 0.8 + 0.9s 0.9 + 0.6 s 2.2 ,
d 4 ( s) = 1.2 + 0.9s 0.9 + 0.6s 2.2 ,
d5 ( s) = 0.8 + 0.5s 0.9 + s 2.2 ,
(33)
d6 ( s) = 1.2 + 0.5s 0.9 + s 2.2 ,
d7 ( s) = 0.8 + 0.9s 0.9 + s 2.2 ,
d8 ( s) = 1.2 + 0.9s
0.9
+s
2.2
Fig. 3. Value sets of D(s,a) for 0 ≤ ω ≤ 3.
,
thus
DV = {d1 , d 2 , d3 , d 4 , d5 , d6 d7 , d8 }
(34)
and the set of exposed edges of D( s, a) is
DE = {de1 , de2 ,...., de12 }
e(d1 , d 2 ), e( d1 , d3 ), e( d1 , d5 ), e( d 2 , d 4 ), 


= e(d 2 , d 6 ), e( d3 , d 4 ), e( d3 , d7 ), e( d 4 , d8 ),  .
e(d , d ), e( d , d ), e( d , d ), e( d , d ) 
5 7
6 8
7 8 
 5 6
(35)
The value set of D( s, a) is bounded by the images of
the exposed edges which are given in (35). The value
sets of D( s, a) for ω = 1rad / sec. and for 0 ≤ ω ≤ 3
are shown in Figs. 2 and 3, respectively.
The 8 vertex transfer functions of G ( s, a, b) can be
written as
Gi ( s) =
1
,
di ( s )
Fig. 4. Bode plots of 8 vertex transfer functions.
(36)
where di ( s ), i = 1, 2,...,8 are given in (33). The Bode
plots of these 8 transfer functions are shown in Fig. 4. On
the other hand, the Bode plots of 1000 transfer functions,
Fig. 5. Bode plots of 1000 transfer functions.
Fig. 2. Value set of D(s,a) at ω = 1rad / sec.
which are obtained by taking 10 points within the range
of each uncertain parameter, of G(s,a,b) are shown in Fig.
5. When Fig. 4 and Fig. 5 are compared, it can be seen
that the boundary of the phase envelopes and the
minimum boundary of the gain come from the Bode
plots of Gi(s). However, the maximum boundary of the
gain is mostly but not completely covered by Bode plot
of Gi(s). Using the results obtained in this paper, the
magnitude and phase envelopes of G(s,a,b) are obtained
1014
Celaleddin Yeroğlu, M. Mine Özyetkin, and Nusret Tan
Fig. 6. Bode envelopes of G(s,a,b) with Bode plot of
nominal (---) transfer function.
Fig. 9. Nyquist plots of 8 vertex transfer functions.
Fig. 7. Nyquist template at ω = 1rad / sec.
Fig. 10. Nyquist plots of 1000 transfer functions.
Fig. 8. Nyquist envelope.
Fig. 11. Bode envelopes of C ( s) G ( s, a, b).
as shown in Fig. 6. Looking at Fig. 6, one can say that
the given FOICS is not robust bounded input bounded
output (BIBO) stable.
The Nyquist template at ω = 1rad / sec. and the
Nyquist envelope of G(s,a,b) are shown in Figs. 7 and 8,
respectively. From Fig. 8, one can also say that the given
FOICS is not robust BIBO stable. The 8 vertex transfer
functions of G(s,a,b) can be written as in (36). The
Nyquist plots of 8 transfer functions are shown in Fig. 9.
The Nyquist plots of 1000 transfer functions, which are
obtained by taking 10 points within the range of each
uncertain parameter of the G(s,a,b) are shown in Fig. 10.
When Fig. 9 and Fig. 10 are compared, it can be seen
that the boundary of the Nyquist envelope for this
example come from the Nyquist plots of Gi(s). However,
in general, the boundary of the Nyquist envelope is
Frequency Response Computation of Fractional Order Interval Transfer Functions
1015
d1 ( s ) = 0.4 + 2 s 0.8 + 3s 2.2 + s 3.1 ,
d 2 ( s) = 0.8 + 2s 0.8 + 3s 2.2 + s3.1 ,
d3 ( s) = 0.4 + 3s 0.8 + 3s 2.2 + s 3.1 ,
d 4 ( s) = 0.8 + 3s 0.8 + 3s 2.2 + s3.1 ,
d5 ( s) = 0.4 + 2s 0.8 + 6 s 2.2 + s 3.1 ,
(42)
d6 ( s) = 0.8 + 2 s 0.8 + 6s 2.2 + s 3.1 ,
d7 ( s) = 0.4 + 3s 0.8 + 6 s 2.2 + s3.1 ,
d8 ( s) = 0.8 + 3s 0.8 + 6 s 2.2 + s3.1 ,
thus
DV = {d1 , d 2 , d3 , d 4 , d5 , d6 d7 , d8 }
Fig. 12. Nyquist envelope of C ( s) G ( s, a, b).
(43)
and the set of exposed edges of D( s, a ) is
(37)
In this case, the Bode envelopes of C ( s) G ( s, a, b) are
shown in Fig. 11 and the Nyquist envelope of
C ( s) G ( s, a, b) are shown in Fig. 12. By using Fig. 11
and 12, it can be calculated that the robust gain margin is
equal to ∞ and the robust phase margin is about 30o.
Therefore, the given system is now robust stable.
Example 2: The Bode and Nyquist envelopes of a
fractional order control system with the following
transfer function is studied in this example
G ( s, a, b) =
b0 + b1s 0.9
N ( s, b)
, (38)
=
D ( s, a ) a0 + a1s 0.8 + a2 s 2.2 + a3 s 3.1
⎧e(d1 , d 2 ), e(d1 , d3 ), e(d1 , d5 ), e(d 2 , d 4 ), ⎫
⎪
⎪
= ⎨e(d 2 , d 6 ), e(d3 , d 4 ), e(d3 , d7 ), e(d 4 , d8 ), ⎬ .
⎪
⎪
⎩e(d5 , d 6 ), e(d5 , d 7 ), e(d 6 , d8 ), e(d 7 , d8 ) ⎭
(44)
Bode plots of 625 transfer functions, which are obtained
by taking 5 points within the range of each uncertain
Bode diagram
20
0
gain / dB
C ( s ) = K p + K d s μ = 20 + 3s1.15 .
DE = {de1 , de2 ,...., de12 }
-20
-40
-60
-1
10
0
10
frequency(rad/s)
1
10
0
-50
phase / º
mostly but not completely covered by Nyquist plots of
Gi ( s ).
Now, consider that the following fractional order
PD μ controller is connected in the forward path of the
given control system.
-100
-150
21 = 2 vertex polynomials and 1x 20 = 1 exposed edge.
D( s, a ) have 3 uncertain parameters. Therefore, it have
23 = 8 vertex polynomials and 3 x 22 = 12 exposed
edges. The vertex polynomials of N ( s, b) are
n1 ( s ) = 1 + s
0.9
, n2 ( s ) = 1.2 + s
0.9
,
-200
-1
10
0
The vertex polynomials of D( s, a ) are
-20
-40
-60
-1
10
(39)
0
10
frequency(rad/sec)
1
10
0
(41)
Phase deg
(40)
and the set of exposed edges of N ( s, b) is
N E = {ne1} = {e(n1 , n2 )}.
1
10
20
thus
NV = {n1 , n2 }
0
10
frequency(rad/sec)
Fig. 13. Bode plots of 625 transfer functions.
Gain db
where a0 ∈ [0.4, 0.8], a1 ∈ [2,3], a2 ∈ [3, 6], a3 = 1,
b0 ∈ [1,1.2] and b1 = 1. Thus, the transfer function has
four uncertain parameters. It can be seen that N ( s, b)
has one uncertain parameter. Thus, N ( s, b) have
-50
-100
-150
-200
-1
10
0
10
frequency(rad/sec)
Fig. 14. Bode envelopes of G ( s, a, b).
1
10
Celaleddin Yeroğlu, M. Mine Özyetkin, and Nusret Tan
1016
obtained from, ∆ ( s, a, b) = 1 + G ( s, a, b) = 0 which gives,
0
-0.2
∆( s, a, b) = b0 + a0 + a1s 0.8 + b1s 0.9 + a2 s 2.2 + a3 s 3.1. (45)
-0.4
The value sets of ∆( s, a, b) for 0 ≤ ω ≤ 2 are shown
in Fig. 17. Looking at Fig. 17, one can see that “0” is not
included in the value sets. Therefore, from zero
exclusion principle [22], one can conclude that the
system is robust stable [15].
yr -0.6
an
ig
a
m
I -0.8
-1
5. CONCLUSIONS
-1.2
-1.4
-1.1
-1
-0.9
-0.8
-0.7
-0.6
-0.5
-0.4
-0.3
-0.2
-0.1
Real
Fig. 15. Nyquist template at ω = 1 rad / sec.
0
-0.5
-1
yr
an
ig
a
m
I -1.5
-2
-2.5
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
3
Real
Fig. 16. Nyquist envelopes of G(s,a,b).
In this paper, a method has been presented for the
computation of the Bode and Nyquist envelopes of
fractional order control systems with interval uncertainty
structure. The results obtained are basically extensions of
some results developed in the parametric robust control
to the FOICS. The given method is based on the
computation of the value set of FOIP. It has been shown
that the value set of a FOIP family can be constructed
using the exposed edges. Since the value set of a FOIP is
a convex polygon, the phase extremums and the value of
the maximum magnitude can be calculated from the
vertices of the value set, however, the value of the
minimum magnitude can be calculated from the exposed
edges. Thus, using the geometric structure of the value
set of a FOIP, an effective algorithm has been proposed
for the computation of the Bode and Nyquist envelopes
of a FOITF whose numerator and denominator
polynomials are FOIP. Numerical examples have been
given to illustrate the application of the method
presented. The results obtained will be very important for
the robust stability analysis and design of FOCS with
parametric uncertainty.
4
2
[1]
0
-2
[2]
-4
[3]
-6
[4]
-8
-10
-12
-25
-20
-15
-10
-5
0
5
Fig. 17. Value sets of ∆ ( s, a, b) for 0 ≤ ω ≤ 2.
[5]
[6]
parameter, of G ( s, a, b) are shown in Fig. 13. Using
the algorithm provided in this paper, boundary of the
Bode envelopes of G ( s, a, b) are obtained as in Fig. 14.
The Nyquist template at ω = 1rad / sec. and the Nyquist
envelope of G ( s, a, b) are shown in Figs. 15 and 16,
respectively.
The characteristic equation of the G ( s, a, b) can be
[7]
[8]
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Celaleddin Yeroğlu received his B.Sc.
degree in Electrical and Electronics Engineering from Hacettepe University in
1990. He received his Ph.D. degree in
Computer Engineering from Trakya University in 2000. His research interests
include fractional order control systems,
robust control, nonlinear control, modeling and simulation.
M. Mine Özyetkin received her B.Sc.
degree in Electrical and Electronics Engineering from İnönü University in 2003.
Her research interests include robust
analysis and design of fractional order
control systems and nonlinear control.
Nusret Tan received his B.Sc. degree in
Electrical and Electronics Engineering
from Hacettepe University in 1994. He
received his Ph.D. degree in control engineering from University of Sussex,
Brighton, U.K., in 2000. He is currently
working as a professor in the department
of electrical and electronics engineering
at Inonu University. His primary research
interest lies in the area of systems and control.