Several research and industrial applications concentrated their
efforts on providing simple and easy control algorithms to cope with the
increasing complexity of the controlled processes/systems (1). The
design method for a controller should enable full flexibility in the
modification of the control surface (2). The systems involved in
practice are, in general, complex and time variant, with delays and
nonlinearities, and often with poorly defined dynamics. Consequently,
conventional control methodologies based on linear system theory have to
simplify/linearize the nonlinear systems before they can be used, but
without any guarantee of providing good performance. To control
nonlinear systems satisfactorily, nonlinear controllers are often
developed. The main difficulty in designing nonlinear controllers is the
lack of a general structure (3). In addition, most linear and nonlinear
control solutions developed during the last three decades have been
based on precise mathematical models of the systems. Most of those
systems are difficult/impossible to be described by conventional
mathematical relations, hence, these model-based design approaches may
not provide satisfactory solutions (4). This motivates the interest in
using FLC; FLCs are based on fuzzy logic theory (5) and employ a mode of
approximate reasoning that resembles the decision making process of
humans. The behavior of a FLC is easily understood by a human expert, as
knowledge is expressed by means of intuitive, linguistic rules.
In contrast with traditional linear and nonlinear control theory, a
FLC is not based on a mathematical model and is widely used to solve
problems under uncertain and vague environments, with high
nonlinearities (6), (7). Since their advent, FLCs have been implemented
successfully in a variety of applications such as insurance and robotics
(8), (9), (10), (11). Fuzzy logic provides a certain level of artificial
intelligence to the conventional PID controllers. Fuzzy PID controllers
have self-tuning ability and on-line adaptation to nonlinear, time
varying, and uncertain systems Fuzzy PID controllers provide a promising
option for industrial applications with many desirable features.
MATERIALS AND METHODS
Fuzzy Controllers includes in their structure the following main
A. Fuzzification: Enabling the input physical signal to use the
rule base, the approach is using membership functions. Four membership
functions are given for the signals e and e in Fig. 1.
B. Programmable Rule Base:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
To implement the FLC on a digital computer according to the
u(t) = u(kT) and u(t+) = u((/k+1)T)
Where, T is the sampling time. The following rule base is applied
u(t) = u(kT) and u(t+) = u((k+ 1)T)
Where, T is the sampling time. The following rule base is applied
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
Where, e (kT) [approximately equal to] l/T [e(k-1)T)], with initial
y(0) = 0, e(-T) = e(0) = r - y(0),
e(0) = 1/T[e(0) - e(-T)] = 0
C. Defuzzification: Select membership functions for the different
control outputs from the rule base
In Figure 2 typical membership functions for u is given. The
overall control signal, u, is generated by a weighted average formula:
u(k + 1)T) = [[[N.summation over (i =
1)][[mu].sub.i][u.sub.i](kT)]/[[N.summation over (i =
1)][[mu].sub.i]]].([[mu].sub.i][greater than or equal
Where control outputs [u.sub.i] (kT), i = 1, N=8 are from the rule
D. Discretization of Conventional PID Controllers:
Digitization of the conventional analog PID controllers by:
S = [2/T][[z-1]/[z+1]]
Where, T > 0 is the sampling time for the PI controller, in Fig.
1 the block diagram for PI digital controller is given:
u(nT) = u(nT - T) + T[DELTA]u(nT) [DELTA]u(nT) = [[~.K].sub.p](nT)
Modeling of the controlling unit: As an example, consider the
voltage raising type-pulse controller. The detailed characteristics of
which are given in (12). The equivalent circuit in view of parasitic
parameters of filtering elements is shown in Fig 4.
Fig.4: Equivalent scheme of boost-converter
Similar structure may be considered as a dynamic system with
external disturbance, in particular, periodic. Using state variables,
the system may be described as
[[dY]/[dt]] = A([S.sub.f])Y + b (1)
[[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
[S.sub.f] is the pulse function which describes a state of the
switch on the specified period of regulation. This function may be
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)
Where, T is the period, [t.sub.k]-the moment of transition of the
switch from one state to another on the specified period of regulation.
As an initial parameters of the model, the range of variation for
the input voltage [U.sub.in] are set with triple overlapping from 20 V
up to 60 V, the range of variation of target resistance [R.sub.0] with
tenfold overlapping from 100 Ohm up to 1000 Ohm and the parasitic
parameters of elements of the filter which define the losses and quality
factor, accordingly, for inductance L = 2 mH; capacitance C = 100
[micro]F; [R.SUB.L] = 0,7 Ohm and [R.sub.c] = 0,2 Ohm.
The block diagram (see Fig.5) of the generalized indistinct
controller consists of four elements (13:
(1.) 1 Fuzzification block, transforming input physical values
[y.sub.i] into corresponding linguistic variables
(2.) Knowledge base, containing rules table for logic output block;
(3.) Logic output block, transforming input linguistic variables
into output with some belonging functions Con;
(4.) Defuzzification block, transforming output linguistic
variables into physical control influence.
Figure 6 shows the structure of P-type a fuzzy controller. In this
case, the error of regulation [epsilon] may be taken as the input
information. The output information is the signal of the relative
duration of conducting state of the switch Con = [t.sub.k]/T-(k-1). The
structure of PI Fuzzy controller is shown in Fig. 7 (13). The input
variables of this controller are, accordingly, the error of regulation
[epsilon] and its derivative [epsilon]. The output is the gain of
relative duration of the switch conducting state [delta] Con. The
membership functions of the input linguistic variables are shown if Fig.
It is expedient to divide a range of values of the normalized input
variables  into five linguistic terms: negative big (NB), negative
small (NS), zero equal (ZE), positive small (PS) and positive big (PB).
With the application of indistinct logic, the logic choice for a P-type
controller can be obtained on the basis of table -1 (the definition
rules of the normalized error of regulation). The specified table is
filled on the basis of the following logic expression:
[epsilon] is Ai, then [Con.sub.k] is [C.sub.j], (3)
Where, [A.sub.i], B-terms of indistinct variables, [C.sub.j] - the
centre of j- accessory function.
Calculation of output signal Con of P-type controller is carried
out according to the following equation:
[Con.sub.k] = [[[n.summation over (j =
1)][[mu].sub.j](k[U.sub.in][epsilon])[C.sub.j]]/[[n.summation over (j =
Where, k[U.sub.in] is the weighting factor which normalizes the
input error [epsilon] to the unit.
The logic choice for the PI controllers with the application of
indistinct logic can be lead on the basis of table-2 (the definition
rules for the normalized error of regulation). The specified table is
filled on the basis of following logic expression:
[epsilon] is [A.sub.i] and )[EPSILON] is [B.sub.i], then
[Con.sub.k] is [Cj. (5)
Calculation for the target signal Con is carried out according to
the following equation:
[Con.sub.k]= [Con.sub.K-1] + 0[delta] [Con.sub.K]. (6)
Where, * * is a weighting factor which normalizes the target value
Con to unity.
[delta]Co[n.sub.k] = [[[n.summation over (j =
1)][[mu].sub.j](y)[C.sub.j]]/[[n.summation over (j -
Where, y - The input linguistic variable. The next values (0.1,
0.2, 0.3) of 0-coefficient were used when indistinct PI-regulator was
Comparison for quality parameters of P and PI controllers: The
following values were taken for comparison: [U.Sub.ref] = 3; [BETA]
=0,04, k[U.sub.in]: 0,25; 0,5; 1,0; 2,0; 4,0; 0: 0,1; 0,2; 0,3. The
Simulation of the structure of fig. 4 allows defining the value of the
static regulation error > and the values of overcorrection 8. For
that, it was necessary to vary the parameters of an input voltage in the
above-mentioned range and the factor of error scaling k[U.sub.in] The
results given in tables 3, 4 are obtained at a value of loading
resistance [R.sub.0] = 300 Ohm. It is found that with the increasing of
error scaling factor k[U.sub.in], the static error is decreased and the
overregulation is increased. The value of static error was defined for
the input voltage [U.sub.in] = 60 V only, quasiperiodic oscillations
were observed for other values of the input voltage. The estimation of
the specified parameters of the controller structure of Fig. 7
isn't given, as it is practically static (>[approximately equal
to] 0,1 %) with a periodic transient.
Two-parametrical diagrams of synchronous mode existence areas are
given for the structures of controllers on Fig. 6 and Fig. 7 accordingly
in Fig. 9 and Fig. 10 for two values of k[U.sub.in] and 0. The area of
existence of a synchronous mode is shaded. Time-domain diagrams of a
current [i.sub.1] flowing in the coil and voltage across the capacitor
[u.sub.c], are presented on Fig. 11 and Fig. 12, respectively. For a
fuzzy P-type controller a value of k[U.sub.in]=1 is chosen, and for
PI-type a value of * * = 0.1 is chosen.
Fuzzy logic provides a certain level of artificial intelligence to
the conventional controllers, leading to the effective fuzzy
controllers. Process loops that can benefit from a non-linear control
response are excellent candidates for fuzzy control. Since fuzzy logic
provides fast response times with virtually no overshoot. Loops with
noisy process signals have better stability and tighter control when
fuzzy logic control is applied.
P Fuzzy controller has smaller sensitivity to the change in the
input voltage, however, more sensitivity is observed to load changes.
PI- Fuzzy controller has less sensitivity to load changes, where, higher
sensitivity to the change of the input voltage is observed.
Analysis of transient and static error of regulation has shown
advantage of an indistinct PI- controller for the output voltage over
the P-type fuzzy controller.
P Fuzzy controller has faster transient as compared to PI
controller, while, transient for PI Fuzzy controller is almost periodic.
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Electrical Engineering Department. Faculty of Engineering Mutah
Abdullah I. Al-Odienat, Department of Electrical Engineering,
Faculty of Engineering, Mutah University
Table 1: The definition rules of [epsilon] for P controller
NB NS ZE PS PB
[C.sub.j] 0 0.225 0.45 0.675 0.9
Table 2: The definition rules of [EPSILON] for controller
NB NS ZE PS PB
* [epsilon] PB -0.3 -0.35 -0.45 -0.65 -1.0
PS 0.0 -0.1 -0.2 -0.35 -0.5
ZE 0.2 0.1 0.0 -0.1 -0.2
NS 0.5 0.35 0.2 0.1 0.0
NB 1.0 0.65 0.45 0.35 0.3
Table 3. The Static error of regulation >, %
V 0.25 0.5 1.0 2.0 4.0
20 37.30 27.5 17.0 9.6 *
30 18.10 13.3 8.5 4.8 *
40 1.80 1.3 0.8 0.5 *
50 -12.70 -9.2 -6.0 -3.5 *
60 -25.90 -18.7 -12,3.0 -7.3 -4.0
* - a quasiperiodic mode.
Table 4. An overcorrection 8, %
Uin, V 0.25 0.5 1.0 2.0 4.0
20 0.30 29.0 41.0 47.0 50.0
30 38.00 73.0 88.0 95.0 98.0
40 71.00 111.0 127.0 139.0 143.0
50 102.00 148.0 171.0 180.0 185.0
60 129.00 183.0 208.0 220.0 225.0