9.8 Vibration Damping System
203
Fig. 9.8 Spring mass
viscodamped dynamics
x0
m
F1
k1q1
F2
k2q2
xm 1
xm 2
k
k3q3
xi
q1
F1 = k10 dt (x 0 x m1 )
F1 = k(x m1 x i )
q2
F2 = k 0 dt (x 0 x m2 )
q3
F2 = k30 dt (x m2 x i ), and
F1 + F2 = m 0 dt2 x 0
where ks are
8
1 Introduction to Fractional Calculus
1.4.1.2 GrunwaldLetnikov: (Differintegrals)
" #
[ t a
h ]
1 !
f (t) = lim
(1) j
f (t j h)
j
h0 h
j =0
%
$
t a
I NT EGE R
h
a Dt
1.4.1.3 M. Caputo (1967)
C
a Dt
1
f (t) =
(n )
&t
a
f (n) ( )
d, (n 1) < n,
(t )+1n
9.6 Feedback Control System
199
0 1 0
0
%
&
A = 0 0 1 , B = 0 , C = 1 0 0 , D = [0]
6 11 6
1
100
C
The normal observability matrix is C A = 0 1 0 and has full rank. There001
C A2
fore, the system is observable. The characteristic polynomial is
(w) = (
9.6 Feedback Control System
187
polynomial, the realization with CFE can have R, C, L components, combination
of them, or even negative impedances, in the ladder form.
The fundamentals of circuit synthesis are applicable for this fractance realization,
of
8.2 Electronics Operational Amplifier Circuits
!
1
k
vo (t) =
C
"
1
c Dt vi (t)
161
1
where (vi , 1, a, c, t) =
k
#t
1
1 ( )d + C vo (c)
k
c
Selecting the coefficient values and circuit constants as unity we have
vo (t) = c Dt1 vi (t). This is the same e
1.5 About Fractional Integration Derivatives and Differintegration
Integration
f (3)
f (2)
f (1)
(i)
11
Differentiation
f (0)
0.7
f (1)
f (2)
(m ) = 0.7
f (3)
2.3
m=3
(ii)
Fig. 1.4 Fractional differentiation of 2.3 times in LHD
In LHD and RHD the integer
8.2 Electronics Operational Amplifier Circuits
167
8.2.11 Cascaded Semi-integrators
Figure 8.11 represents two semi-integrator circuits in series, which gives integral
operation of order one.
The circuit of
Fig. 8.11 has the following expressions:
r1 1
1/
8.5 Fractional Order State Vector Representation in Circuit Theory
179
The time domain representation of the transfer function (without initialization) is
thus the following:
3/2
0 dt v0 (t)
+ v0 (t) = vi (t).
Here initialization is zero,
(v0 , 3/2, a, 0,
8.5 Fractional Order State Vector Representation in Circuit Theory
177
8.5 Fractional Order State Vector Representation
in Circuit Theory
This example, in this chapter, is chosen as a working model for vector space representation. The vector initializatio
9.6 Feedback Control System
193
The transfer characteristic equation in the state-space format is discretized using
the definition of GrunwaldLetnikov (GL) differintegral. For the state variables,
()
()
()
= Dt
x 2 (t), GL expansion is
fractional derivati
x
Acknowledgments
learnt the subject from several presentations and works of Dr. Alain Oustaloup,
CNRS-University Bordeaux, Dr. Francesco Mainardi, University Bologna Italy,
Dr. Stefan G Samko University do Algarve Portugal, Dr. Katsuyuki Nishimoto,
Insti
10.11 Ultra-damped System Response
229
10.10 Frequency Domain Response of Sinusoidal Inputs
for Fractional Order Operator
In earlier section, it was mentioned that replacement of s q ( j )q gives the
steady-state response of the transfer function. However
8.3 Battery Dynamics
171
we get the charging equation as:
!
"
$ %&'
$ %
#
t
t
B t
B t
R ( B )2 t
2
+ 2B
e R er f c
1+
C
B
R
R
(
*
*+'
)
!
)
"
2
t b R ( B )2 (tb)
B t b
t b
B t b
R
e
1+
er f c
Ic u(t b)
+2B
C
B
R
R
v1 (t) v3 (t) = Ic
Figure 8.14 gives
1.4 Historical Development of Fractional Calculus
1
f (x) =
2
!+
5
!+
f (z)dz
cos( px pz)d p .
He made a remark as
d n f (x)
1
=
n
dx
2
!+
!+
f (z)dz
cos( px pz + n )d p,
2
and this relationship could serve as a definition of nth order derivative for noni
8.2 Electronics Operational Amplifier Circuits
Fig. 8.8 Semi-infinite lossy
line (half-order element)
165
i(t)
v(t)
2
The diffusion equation corresponding to this lossy line is v(x,t)
= v(x,t)
. In
t
x2
Fig. 8.8, v(t) and i (t) are the voltage and current
10.6 Variable Order System
223
The variable order structure differintegration allows the introduction of a new
transfer function concept. Refer Fig. 10.4; the conventional transfer function relates
the Laplace transform of the output to the transform of t
Chapter 10
System Order Identification and Control
10.1 Introduction
For unknown systems, system identification has become the standard tool of the
control engineer and scientists. Identifying a given system from data becomes more
difficult, however when
9.3 ElectrodeElectrolyte Interface Impedance
183
boundary x = L, so the system will behave as semi-infinite media. Anomalous
diffusion is characterized by a mean square displacement of the diffusing particles that does not follow the ordinary linear law <
Bibliography
235
46. A. Oustaloup, From fractality to non-integer order derivation through recursivity, a property common to these two concepts: a fundamental idea for a new process control strategy, Proceedings of the 12th IMACS World Congress Paris July
8.2 Electronics Operational Amplifier Circuits
163
Putting i i (t) = i f (t) and v (t) = 0, we have the differentiator expression:
!
$
#
d "
1
vi (t) vi() (c) + (i f , 0, a, c, t) or
vo (t) = C
k
dt
$
!
1 d
1
vi (t) + (vi , 1, a, c, t)
vo (t) = C
k dt
C
I
9.5 Fractance Circuit
185
9.5 Fractance Circuit
Electrical circuit related to fractional calculus is fractance, an electrical circuit
behaving in between capacitance and resistance. An example of fractance is tree
fractance shown in Fig. 9.2, a self-simil
8.2 Electronics Operational Amplifier Circuits
1
vo (t) =
RC
!t
c
vi ( )d + vo (c) =
159
1
1
c D vi (t)
RC t
with (vi , 1, a, c, t) = RCvo (c) as initializing function.
Then with RC = 1, this integrator is represented as vo (t) = c Dt1 vi (t). The
initi
10.3 Continuous Order Distribution
207
For integer order systems once the maximum order of the system to be identified
is chosen, the parameters of the model can be optimized directly. For fractional
order systems, the identification requires (a) the choi
9.6 Feedback Control System
197
0 1 0
0
%
&
Consider A = 0 0 1 , B = 0 , C = 1 2 3 , D = [0], the system
2 4 6
1
q
y
is 0 Dt x(t) = A x +B u and (t) = C x(t) with controller as u(t) = K x(t).
The control parameter (gains) will be chosen such that the po
8.3 Battery Dynamics
173
From charging analysis of equation V1 (s) V3 (s) and with
Ic (s) = I RL (s) = cfw_V3 (s) V1 (s) /R L
and taking a = 0, c = 1
V1 (s) V3 (s) =
!
" 1/2
#
R L s s B + R1
"
#
3/2
R
1
R L s B + 1+ RL s+ C1B s 1/2 + RC
$!
v12 (1)
s
W (v2
Chapter 8
Application of Generalized Fractional Calculus
in Electrical Circuit Analysis
8.1 Introduction
The fractional calculus is widely popular, especially in the field of viscoelasticity.
In this chapter, a variety of applications are discussed. This
10.4 Determination of Order Distribution from Frequency Domain Experimental Data
209
orders are point property, the question is if at all these are point quantities? If so
then should we have mass assigned to a point? But in reality, the mass is assigned