PEDS 2007
Piecewise Linear Control Surface for Single
Input Nonlinear P1-Fuzzy Controller
S. M. Ayob, Z. Salam, and N. A. Azli
Department of Energy Conversion, Faculty of Electrical Engineering
Universiti Teknologi Malaysia, 81310 UTM Skudai, Johor, Malaysia
Email: shahrinNfke.utm.my
Abstract--This paper presents an analysis on piecewise
linear control surface for a single-input nonlinear PI Fuzzy
controller. The analysis is carried out to determine the
conditions that should be met in order for the piecewise
linear control surface approximation to be validated. An
equivalent two-input PI-Fuzzy with conventional
fuzzification, inference and deffuzification process is built
for verification purpose and both controllers are then
applied to a single phase inverter system. The dynamic
response of both controllers are examined and compared.
From the simulation results, it is shown that the
approximation is valid.
linear PI controller if the control surface exhibits a linear
surface as depicted in Fig. 1. This type of controller can
also be known as a linear PI-Fuzzy controller while the
nonlinear PI-Fuzzy is referred to a PI-Fuzzy controller
with control surface as shown in Fig. 2. These surfaces
are generated by applying several conditions on the
selection of membership function shape, percentage of
overlap between fuzzy sets, inference method,
fuzzification and defuzzification method. Different
conditions yield different control surfaces.
Index Terms-PI-Fuzzy controller, control surface,
piecewise linear approximation
I. INTRODUCTION
The main feature of a fuzzy controller is that it can
control complex and ill-defined systems by translating
the linguistic control strategy of experts' knowledge into
an automatic control without knowing the detail
mathematical model of the systems. For a UPS inverter
system, its linear model can be obtained via a state-space
averaging method. This linear model has been popularly
used over the years in designing its linear theory based
controllers [1,2]. These controllers basically provide an
excellent steady state performance with low THD
percentage but poor and slower dynamic response with
nonlinear loads or large load disturbances. This is due to
the fact that the model itself is developed based on a
single operating point. Therefore when the operating
point changes, the designed controller is no longer valid.
Moreover, there are several parameters uncertainties that
could not be modeled very well such as switching nonidealities and delays which can affect the overall
controller performance [3]. Therefore, a non-model
based controller such as a PI-Fuzzy controller is
preferable for application purposes. However, since PlFuzzy controllers are not mathematical-based, heuristic
design approaches could not be avoided and popularly
known to be the main drawback of this type of controller.
The controller also lacks on the aspect of stability theory
and therefore impossible to be assessed. However, a lot
of works have been done over the past few years that are
devoted to minimise the approach and as a result, several
design guidelines and stability theories have been
proposed [4,5,6]. Previous researches have shown that
the nonlinearity characteristic of this controller is
strongly dependent on its input-output mapping or known
as control surface. The PI-Fuzzy controller can be a
1-4244-0645-5/07/$20.00©2007 IEEE
<
1
D
-1~~1
X
/
A
v
cerror
error
Fig. 1. Linear control surface
8
6- 4- 2
-
-
a0
-
-2
-4
-6
1
_
-0.5
05
-1
1
1
05
error
Fig. 2. Nonlinear control surface
The single input PI-Fuzzy based on the signeddistance method offers less number of rules, less tuning
parameters and yet capable of providing identical
performance of its equivalent two-input controller [7].
Instead of using two input variables, this controller uses a
single input variable called "distance (d)" and is defined
by (1).
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d = e(k) + 2e(k)
1'[ + A22
(1)
CONTROL SURFACE APPROxiMATION ANALYSIS
Fuzzy controllers are very well known as controllers
with complex algorithm and therefore they always
demand a fast digital processor board in order to run the
program. For power converters control, which associate
with high frequency switching, the real-time
implementation of fuzzy control is quite a challenging
task. The targeted processor should be fast enough to
execute fuzzy algorithm within one switching period.
To alleviate the problem, a signed-distance method has
been proposed [7]. The method has significantly reduced
the complexity of fuzzy algorithm by reducing the
number of rules and inputs. By using this method, the
input is reduced to a single input and the rule table can be
constructed in one-dimension control surface mapping.
To preserve the nonlinearity characteristic of the control
surface and to lessen the on-line computation burden, an
approximation of control surface can be done. For a onedimension mapping, its approximation can be done via a
look-up table. However, in order to do that, several
conditions should be followed.
Let us consider the "distance" input has five fuzzy sets
and five fuzzy sets for the output with each input and
output sets labeled as Negative Big (NB), Negative Small
(NS), Zero (Z), Positive Small (PS) and Positive Big
(PB). The rules for this example are as follows:
(b)
For the inference process, the activation operator is set
to be a PRODUCT (*), MAX for the accumulation
operator and Center of Gravity (CoG) as the defuzzifier
operator. For discussion purposes, six different control
surfaces are considered as depicted in Figs. 4(a)-(f). The
red line is the control surface generated from the
inference process while the blue line is a linear unity
slope line which represents a reference.
m°X
(c)
(d)
nE
E
ES-
P.R L11'ILI11
OLZ2EO
(e)
EOS
E
II.
IF d is Negative Big THIEN output is Negative Big
IF d is Negative Small THIEN output is Negative Small
IF d is Zero THEN output is Zero
IF d is Positive Small THEN output is Positive Small
IF d is Positive Big THEN output is Positive Big
(a)
E
E
Where e(k) is the error, e(k) is the change of error and
A is a constant.
It is a well known fact that linear PI controller suffer
from poor large-signal performance, and in order to
improve that, the control surface for the PI-Fuzzy should
be made nonlinear as in Fig. 2. In the case of single input
PI-Fuzzy controller, the nonlinear control surface can be
approximated as a piecewise linear which can be realised
by means of a look up table. This is possible, since the
control surface for the sole input controller is a onedimension mapping. However, several conditions should
be met in order to realise the approximation. Hence, this
paper extensively analyses the conditions that should be
applied for the approximation to be valid.
E -1
-5
0
0
E1
5
05
0
0
(D)
Fig. 4(a)-(f). Six different control surfaces for discussion purposes
From the figures, several conclusions on the generated
control surface approximating the linear line can be made
as follows,
a) The linear line approximation can only be realised
with the singleton output sets.
b) The triangular input set must be used and should
overlap at least 5000 with the adjacent sets.
c) Either Centre of Gravity (CoG) or Centre of Gravity
Singleton (CoGS) defuzzifier can be used if the
output set is a singleton.
By setting fuzzy conditions as above and considering
the generalised output equation of the single-input PlFuzzy as in (2), then the control surface equation of the
single input PI-Fuzzy can be expressed as in (4).
Au =
(2)
l
i
where u, is the membership degree for ith rule and
the singleton membership output.
A
di -d
S, is
(3)
where di and di
are the peak of each fuzzy
membership input sets where measured input is defined.
1
Substituting (3) into (2) gives,
Au =
((Si - Si-' d + (d1S-l - di-,Si) )
(di - di-, )
(di - di-, )
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(4)
Equation (4) demonstrates that the output equation of the
controller is actually a linear equation and can be written
as (5).
C)
(5)
Au =ad+y
The first term in (5), a, represents the ratio of the
peak's location difference between output and input sets
where the exact input value is defined. The equation
actually represents the slope of the line. Therefore,
different slope values can be obtained by changing the
peak locations of input and output sets. Furthermore the
second term in the equation, y, will disappear when all
the input and output sets are equally spaced as in Fig.
4(b). This will result in a straight linear line with a single
slope. To obtain a better dynamic response compared to
linear PI controllers, the surface should be made
nonlinear and in order to obtain that, the input and output
sets are placed unequally as can be depicted in Fig. 4(f).
The nonlinear mapping as in Fig. 4(f) can be
approximated as a piecewise linear mapping which can
be realised using a look-up table. Fig. 5 shows the
comparison of the control surfaces generated using two
different methods. The solid red line is the control
surface generated using the fuzzy conventional process
(the same as in Fig. 4(f)) whiles the dash blue line is
constructed using (4). As can be clearly seen, the dash
blue line can approximate the line generated by the
conventional process very well.
I
+
+
)Au(k)
~~~~d
Fig. 6. Block diagram ofthe single input controller with
approximation control surface.
Inference
Au(k)
e(k)
1
Fig. 7. Block diagram ofthe single input controller with
conventional fuzzy process.
TABLE I
2
0
1-4
-2
-5
-1
R.gi-
/
-0.8
-0.6
-0.4
-0.2
0
input1
0.2
0.4
0.6
0.8
0-
1k__*.
1
Fig. 5 Comparison of control surfaces generated using
conventional process and using (4).
III. SIMULATIONS
Fig. 6 and Fig. 7 show the structure of the single-input
PI-Fuzzy controller with the approximation and with
conventional process, respectively. The I is a piecewise
linear mapping which represents the nonlinear control
surface for the single input PI-Fuzzy controller. The
variable, i, has been defined in order for the controller to
exhibit a similar linear PI's performance for input jdj<20
units. The approximated control surface used in this work
has unity gain for input range of Jdj<20 and 3.1875 unit
gain for Idl beyond that. The surface is set to saturate at
input over 100 units. Table I shows the rule table used
for the system in Fig. 7, while Fig. 8 shows the input and
output sets used for the same system.
Fig. 8. Input and output fuzzy sets used for system in Fig. 7
Both controllers are built using MATLAB-Simulink.
MIATLAB's Fuzzy Toolbox is used to build the controller
with the conventionl fuzzy process. Both controllers are
then applied to control a typical single phase inverter
system. To examine the excellent performance of the
controllers, a small load (partial load disturbance) and
large load disturbance (no load to full load) are imposed
on the system. Both controllers reveal fast dynamic
response that is identical to each other as shown in Fig. 9
and 10 which verifies the approximation.
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90
[3]
--l- X
80
70S - -
-
t
-II - -T
t
- Sr
-
- - -X;- -- - - _4 _[4]
A. Kawamura, T. Haneyoshi and R.G. Hoft, "Deadbeat Controlled
PWM Inverter with Parameter Estimation Using Only Voltage
Sensor", IEEE Transaction on Power Electronics, Vol. 3 (2),
ppl 18 125, April 1988.
"Fuzzy Control: Synthesis and Analysis", Edited by S. S.
Farinwata, D. Filev and R. Langari, John Wiley & Sons, 2000.
60
50
I
[5]
A. G. Perry, Y.-F. Liu and P. C. Sen, "A New Design Method for
PI-like Fuzzy Logic Controllers for DC-to-DC Converters", The
35th Annual IEEE Power Electronics Specialists Conference
(PESC 2004), Vol.5, pp3751-3757, June 2004.
[6]
A. Kandel,Y. Luo and Y.-Q. Zhang, "Stability Analysis of Fuzzy
40 L - - - -/2 - - - - S - - - - X - - - - -\- - - - _
0.02 0.021 0.022 0.023 0.024 0.025 0.026 0.027 0.028 0.029
0.03
Fig. 9. Voltage response comparison between system in Fig. 6 and
Fig. 7 for small load disturbance.
82
Control Systems", Fuzzy Sets and Systems, Vol.105, Issue 1,
pp33-48, July 1999.
[7]
B. J Choi, S.W. Kwak and B. K.Kim, " Design and Stability
Analysis of Single-Input Fuzzy Logic Controller", IEEE
Transaction on Systems, Man and Cybernetics-Part B:
Cybernetics, Vol.30, No. 2, pp303-309, April 2000.
-
80
7876
F - -F
74
72 -
-
70
-
0-
-
-
0
-
-
i-
--
-
-
-
-
-
-
-
-
-
t-
-
-
-
-
-
68
66
0.0235
0.024
0.0245
0.025
0.0255
0.026
0.0265
0.027
Fig. 10. Voltage response comparison between system in Fig. 6
and Fig. 7 for large load disturbance.
IV. CONCLUSION
In this paper, an analysis on a piecewise linear
approximation control surface for a single input PI-Fuzzy
controller has been presented. The approximation can be
made by applying several soft conditions to the fuzzy
as discussed in detail earlier. For verification
purposes, the single input PI-Fuzzy with conventional
inference process and the single-input controller with the
control surface approximation have been developed using
MlATLAB-Simulink. Both controllers have been applied
to control the output voltage of a single phase inverter.
The dynamic response of both controllers are found to be
identical hence verifying the approximation.
parameters
REFERENCES
[1]
M. J. Ryan, W. E. Brumsickle and R..D. Lorenz, "Control
Topology Options for Single-Phase UPS Inverters IEEE
Transaction on Industry Applications, Vol. 33, No. 2, pp 4 -5 0 1,
March/April 1997.
[2] H. Deng, R.Oruganti and D. Srinivasan, "Modelling and Control
of Single-Phase UPS Invter: A Survey", The 6th International
Conference on Power Electronics and Drive Systems (PEDS
2005), pp848-853, December 2005.
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