7th WSEAS Int. Conf. on ARTIFICIAL INTELLIGENCE, KNOWLEDGE ENGINEERING and DATA BASES (AIKED'08),
University of Cambridge, UK, Feb 20-22, 2008
COMPARATIVE ANALYSIS OF CLASSICAL AND FUZZY PI
ALGORITHMS
JENICA ILEANA CORCAU
Division Avionics
University of Craiova, Faculty of Electrotechnics
Blv. Decebal, nr. 107, Craiova, Dolj
ROMANIA
ELEONOR STOENESCU
University of Craiova, Faculty of Electrotechnics
Blv. Decebal, nr. 107, Craiova, Dolj
ROMANIA
MIHAI LUNGU
Division Avionics
University of Craiova, Faculty of Electrotechnics
Blv. Decebal, nr. 107, Craiova, Dolj
Abstract: - In this paper is presented a fuzzy PI controller. Fuzzy PI controller and principles their performing is
considered in large applications. Some investigations deal with the comparison of PI and Fuzzy PI algorithms for
application. The settings of classical controllers are based on deep common physical background. Fuzzy controller can
employ better behavior comparing with classical PI controller because of its nonlinear characteristics.
Key-Words: - Fuzzy Logic Control, linear control, nonlinear control, discrete, continuous.
1 Introduction
Fuzzy Logic Control (FLC) is one of most successful
application of fuzzy Set Theory, introduced by Zadeh in
1965. Its major features are the use of linguistic
variables rather than numerical variables. Linguistic
variables, defined as variable, whose values are
sentences in a natural language, may be represented by
fuzzy sets.
The principles of fuzzy controllers design are
investigated in a number of applications [1].
The main aims of fuzzy controller application are:
- improving of dynamic
behavior of the system (in
comparison with classical
algorithm);
- adaptation to change of a
system parameters.
Analysis shows a fuzzy PI controller is a fuzzy controller
with two crisp input signals, which represent: error e(t ),
integral of the error and one analog crisp output signal
u (t ).
ISSN: 1790-5109
The Fuzzy PI a discrete digital controller: the initial
information is represented in digital form by values
e[(k − 2)T ]
all necessary
e[kT ],
e[(k − 1)T ] ,
computations are digital.
Obviously, sampling time must be greater than the
computation time. The quantization of signal in time
reduces the stability margin of the system.
The Fuzzy PI controller can differ by:
- basis of control rule;
- number of terms for each
input variable;
- shapes by fuzzy sets;
- fuzzification methods;
- deffuzzification methods;
- structure controller.
Page 189
ISBN: 978-960-6766-41-1
7th WSEAS Int. Conf. on ARTIFICIAL INTELLIGENCE, KNOWLEDGE ENGINEERING and DATA BASES (AIKED'08),
University of Cambridge, UK, Feb 20-22, 2008
2 Fuzzy PI controller design
A classical PI controller is described by equation (1)
[1],
t
⎛
⎞
1
u (t ) = K ⎜⎜ e(t ) + ∫ e(τ )dτ ⎟⎟,
(1)
Ti 0
⎝
⎠
Where K is the gain of PI controller, Ti is an integral
constant, e(t ) is an error signal, e(t ) = w(t ) − y (t ), y (t )
is the output from process and u (t ) is the output from
controller (the action value).
Deriving equation (1) we get
⎛
⎞
1
u& (t ) = K ⎜⎜ e&(t ) + e(t ) ⎟⎟.
(2)
Ti
⎝
⎠
For a local extreme location we put
1
u& (t ) = e&(t ) + e(t ) = 0
(3)
Ti
The solution of equation (3), is
1
e&(t ) = − e(t ),
(4)
Ti
because the PI controller gain K > 0.
The equation (4) depends only on the PI controller
integral time constant Ti .
By comparison we find out that the deviation of action
values depends on the derivation of error magnitude.
Fuzzy sets must be defined for each input and output
variable. Seven fuzzy subsets PB (Positive Big), PM
(Positive Medium), PS (Positive Small), ZO (Zero), NB
(Negative Big), NM (Negative Medium), NS (Negative
Small) have been chosen for input variables and output
variable. In table 1 is presented map the rule base to the
discrete state space. We use the zero elements on the
diagonal for mapping of equation (4).
Table 1. The rule base of fuzzy PI controller mapped to
the discrete state space
Δe(k )
ZO PS PM PB PB PB PB
NS ZO PS PM PB PB PB
NM NS ZO PS PM PB PB
NB NM NS ZO PS PM PB
NB NB NM NS ZO PS PM
NB NB NB NM NS ZO PS
NB NB NB NB NM NS ZO
⎛
⎞
1
Δu ( k ) = K ⎜⎜ Δe(k ) + e(k ) ⎟⎟,
(5)
Ti
⎝
⎠
u (k ) − u (k − 1)
e(k ) − e(k − 1)
, Δe(k ) =
,
Where Δu ( k ) =
T
T
T is the sampling period, k is the step. The time
constant Ti determine relation to the change in
error. For a fuzzy PI controller [2]
K
(Ti Δe(k ) + e(k )).
(6)
Ti
In the next step it is necessary to map the rule base to the
discrete state space Δe(k ), e(k ).
It is introduced the scaling factor M for the universe
range, M > 0. This scale factor sets the universe range
for the error and its first difference (see to table 1).
We extend the equation (6) obtains
KM ⎛ Ti
1
⎞
Δu ( k ) =
e(k ) ⎟.
(7)
⎜ Δe ( k ) +
Ti ⎝ M
M
⎠
Δu ( k ) =
We apply fuzzification to input variables and after
deffuzzification we get the equation
KM ⎧ ⎧ Ti
1
⎫⎫
D ⎨ F ⎨ Δe(k ) +
e(k )⎬⎬.
(8)
Ti
M
⎭⎭
⎩ ⎩M
Where F is an operation for fuzzyfication and D for
deffuzzification.
Δu (k )
we can written
For action value
u (k ) − u (k − 1)
Δu ( k ) =
=
T
1
KM ⎧ ⎧ Ti
⎫⎫
(9)
=
D ⎨ F ⎨ Δe ( k ) +
e(k )⎬⎬.
Ti
M
⎭⎭
⎩ ⎩M
Δu ( k ) =
The control signal generated by fuzzy PI controller
in step k is
⎧ T
⎫
u (k ) = KM D ⎨ F ⎧⎨ i Δe(k ) + 1 e(k )⎫⎬⎬ + u (k − 1).
Ti
⎩ ⎩M
M
⎭⎭
(10)
In figure 1 is presented the fuzzy PI controller realized
using the toolbox of Simulink (anfisedit). As shown in
figure 1, triangular and trapezoidal shapes have been
adopted for the membership functions; the values of
each input and output variable are [-1, 1].
e(k )
If we translate the equation (3) to a discrete form, we get
the equation for action value change of the discrete PI
controller
ISSN: 1790-5109
Page 190
ISBN: 978-960-6766-41-1
7th WSEAS Int. Conf. on ARTIFICIAL INTELLIGENCE, KNOWLEDGE ENGINEERING and DATA BASES (AIKED'08),
University of Cambridge, UK, Feb 20-22, 2008
Figure 3. The structure system using
Fuzzy PI controller
A physical meaning of the parameters for fuzzy PI
controller remains the same like for the PI controller (the
gain controller K and time integral constant Ti ).
For exemplifies let us assume plant described by
following transfer function to illustrate and to compare
behavior of fuzzy PI controller with classical continuous
PI controller
1
H (s) =
.
(11)
(10 s + 1)( s + 1) 2
The parameters Ti = 5,8; M = 10 and sample period was
set to T = 0.1 In figure 2 is presented the block scheme
realized in Simulink of system using the PI classic, and
in figure 3 is presented the structure of system using the
fuzzy PI controller.
PID
Step
PID Controller
1
The saturation limits present in figure 3, limits of the
action values realizes antiwindup. The inference method
min-max and deffuzzification COG method was utilized
in simulation.
output
Figure 1. The fuzzy PI controller
Scope
Figure 4. The simulation results using
classical and fuzzy PI controller
den(s)
Transfer Fcn
yl
To Workspace
The simulation results obtained using classical PI
controller and fuzzy PI is shown in figure 4. Simulation
results of controllers confirm the validity of the proposed
control technique.
Scope1
Scope2
3 Conclusions
Figure 2. The structure system using the classical PI
ISSN: 1790-5109
In this paper is presented comparative analysis of
classical and fuzzy PI algorithms. Is realized a fuzzy PI
controller using same like parameters for PI controller.
Comparing the results obtained using classical PI and
fuzzy PI controllers following discussion can take place.
The output of the system has very small overshoot when
it is controlled with fuzzy regulator. The disturbance
Page 191
ISBN: 978-960-6766-41-1
7th WSEAS Int. Conf. on ARTIFICIAL INTELLIGENCE, KNOWLEDGE ENGINEERING and DATA BASES (AIKED'08),
University of Cambridge, UK, Feb 20-22, 2008
rejection using fuzzy controller is comparable with
disturbance rejection of classical PI controller.
References
[1]. Pivonka P. Analysis and design of fuzzy PID
controller based on classical PID controllers approach,
Physica-Verlag, 2000.
[2]. Edik Arakeljan, Mark Panko, Vasili Usenko.
Comparative analysis of classical and fuzzy PID
algorithms, Physica-Verlag, 2000.
[3]. Sofron, E., Bizon, N., Ioniţă, S., Răducu, R. Sisteme
de control fuzzy. Modelare şi proiectare asistate de
calculator. Editura ALL EDUCAŢIONAL, Bucureşti,
1998.
ISSN: 1790-5109
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ISBN: 978-960-6766-41-1