The 33rd Annual Conference of the IEEE Industrial Electronics Society (IECON)
Nov. 5-8, 2007, Taipei, Taiwan
A Discrete Single Input P1 Fuzzy Controller for
Inverter Applications
S. M. Ayob, Student member, IEEE, Z. Salam, Member, IEEE and N. A. Azli, Member IEEE
Department of Energy Conversion, Faculty of Electrical Engineering
Universiti Teknologi Malaysia, 81310 UTM Skudai, Johor
E-mail: shahringke.utm.my
In order to generate a pure sinusoidal output waveform,
immune to any disturbances and robust operation, a non-model
based controller is preferable. Fuzzy and Sliding Mode
controller are the best candidates as both are non-model based
controllers. Sliding mode controller is inherently robust and
insensitive to parameter changes. However the difficulty of
obtaining the Lyapunov function and the complex mathematics
associated with its design is limitin its applications. Fuzzy
controller on the other hand requires much simpler
mathematics and if tuned appropriately, it can approximate
Sliding mode controller very close [8].
However, fuzzy controller implementation requires a very
fast processor to compute the output since the accuracy of the
controller is dependent on the number of rules and its fuzzy
I. INTRODUCTION
sets. For fuzzy controller with more inputs, more rules and a
faster
processor is required.
Inverters have found widespread applications in AC power
This
paper proposed a single input PI Fuzzy controller for
conditioning systems, such as automatic voltage regulators,
inverter
applications. The reduction in the number of inputs
programmable AC power sources and uninterruptable power
reduced
the number of rules, tuning parameters and thus
supply (UPS). In these applications, the inverter is required to
the overall fuzzy design. In addition, for single
simplified
synthesize a pure sinusoidal waveform with specified
input
fuzzy
control, the input-output mapping or the control
frequency and amplitude at its output regardless of load type.
surface
can
be
presented as a simple one-dimensional control
Since the inverter plays a major role in converting DC voltage
surface.
It
can
be
easily realised using a look-up table. This in
to AC voltage, the performance of the corresponding system
turn
contributes
to
faster control output calculation and allows
will be highly dependent upon its controller [1]. Several
the
controller
to
be
easily realised using inexpensive digital
controllers have been applied to inverter system namely,
An
in the design of the proposed controller
processor.
example
Sliding Mode [2], Deadbeat [3] and Fuzzy [4].
for
a
inverter is presented. To analyse
controlling
single-phase
The inverter is a nonlinear system by nature due to the
the
of
the
performance
proposed
controller, simulations have
existence of the switching devices.
State space and
been
conducted
MATLAB-Simulink.
Simulation results
using
linearisation techniques can be applied to model the switching
have
shown
that
the
controller
exhibits
proposed
equivalent
converters as linear systems provided that the modulation
with
the
controller
and
PI
performance
typical
Fuzzy
capable
frequency is very low compared to the switching frequency.
of
and
nonlinear
load
well.
handling
cyclic
very
With such approach, the switching converter can be modeled
as a linear gain, Kp,, while the dynamics of the system
majorly comes from the LC filter. However, even though the
II. THE PROPOSED CONTROLLER
model is very well established, some uncertainties still exists
A. Rules and Inputs Reduction
and not be very well modeled. They are listed as follows [1]:
The number of inputs can be reduced by using a method
1) The real values of capacitance and inductance must be proposed
in [5]. By taking advantage of Toeplitz or near
measured at their operating points, but it is nearly impossible Toeplitz structure [6] of the rule table, the number of inputs
in practice.
can be reduced into a single input without degrading the
2) The load of the inverter is not fixed. Unlike DC-DC original performance. Consequently, the number of rules is
converter, there are many types of loads may be connected to also reduced. Therefore fast processor requirement for fuzzy
the output port of the inverter system.
control can be softened. Table I below shows the rule table
3) The characteristics of the switching devices are not ideal that exhibits a Toeplitz structure. The structure exhibits the
in practice.
same output membership association in diagonal direction.
Abstract - In this paper, a discrete single input PI Fuzzy controller
for inverter applications is presented. In contrary to the typical
PI Fuzzy which requires minimum of two inputs variables, the
proposed controller requires only one input. However the
performance of the controller is not degraded. The reduction in
the number of inputs minimized the number of fuzzy rules, tuning
parameters and thus simplifies the overall fuzzy controller design.
Furthermore, with single input, the input-output mapping can be
presented as a simple control surface. In this case, it is mapped as
a piecewise linear and can be realized by using a simple look-up
table. As a result, a faster fuzzy calculation can be achieved using
an inexpensive digital processor. A design example of the
proposed controller for a single phase inverter is outlined. Using
Matlab simulations, it can be shown that the proposed controller
exhibits equivalent performance with the typical PI fuzzy
controller.
1-4244-0783-4/07/$20.00 C 2007 IEEE
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Diagonal line consisted of ZERO output membership is
defined as the main diagonal line.
Au(k)
TABLE I
RULE TABLE WITH TOEPLITZ STRUCTURE
z
NS
NB
NB
NB
-H-
-H-
PS
z
NS
NB
NB
-H-
-H-
PB
PS
z
NS
NB
-H-H-
PB
PB
PS
z
NS
-H-H-
Fig. 2(a). Simplified PI Fuzzy structure
PB
PB
PB
e(k)
PS
z
By using this method, the state variables, (i.e. error ancI the
change of error) can be represented by a single variable, which
is a distance (d). The distance is defined as the absc)lute
distance of any state points perpendicular to any points orithe
main diagonal line. The distance can be defined as in (1). The
X in (1) is the slope of the main diagonal line. Then the new
rule table, with distance variable can be constructed as in T able
II.
e+Ae
1j + A22
(1)
TABLE II
REDUCED RULE TABLE
RB
'B
Rule table with this type of structure can be found wiidely
used in fuzzy control for power electronic inverter cointrol
system and can be considered as the popular rule ttable
structure [4, 9]. Therefore the proposed method in [5] caiIn 1De
applied to control the inverter system.
e(k)
Fig. 2(b). Discrete linear PI controller structure
(-n)
u(k) = (m + n)[e(k) +( ) e(k)]
The second term of (3) represents the addition of error and
change of error with a gain. The block diagram of Fig. 2(b)
has been drawn using (3). From Fig. 2(a) and Fig. 2(b), the
similarity of the simplified PI Fuzzy with discrete linear PI
controller can be seen. However, it should be noted that the
proposed controller has a control surface that can be tuned to
achieve superior dynamic performance for large disturbance
over its linear counterpart.
To obtain a good performance for small disturbance similar
to its linear controller counterpart, the parameters of the
proposed controller obtained upon replacing the control surface
to unity may be matched with those of the linear PI.
The distance function, d, of the system in Fig. 2(a) can be
rewritten as:
d=d=e+>e
+
Controller Description
Fig. 2(a) and 2(b) show the block diagram of the prop osed
controller and discrete linear PI controller, respectively. The
proposed controller is equivalent to a discrete linear PI vvhen
the control surface, xv, is replaced by unity gain. To show this
let us consider a discrete PI controller transfer function griven
by (2):
(3)
(m + n)
e
A2
~j
+
+
Le
(4)
B.
C(z)
Where
m=
K1
mz+n
-+-] andn
The transfer function of C (z)
form [7] as in (3):
1
-n
1+A2.
m+n
Then, the term
(2)
z -1
By comparing the term in (4) and the second term of (3), X can
be obtained as:
T, as follows:
=
can
K1
m+n
-n
-
be expressed in differential
-n
,
m+n2
(5)
-n
can be expressed in terms of Kp, Ki and
T
-2K
KjTK
2Ki
2K1T
(6)
KjT -2Kp
From (6), since K1 is always larger than Kp in many
inverter systems, then m + n > n which means
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m+n
-n
>>1 will
always hold. Then the parameter of the proposed controller
that exhibits similar performance to its linear PI with the
control surface is set unity can be found as in (7):
A-
m+n
m
nand r = m+n
-n
C,(s) =2300 0.00002174s+1]
(7)
C,(s) = 126 0.00025s+1
s
(8a)
I
(8b)
For large disturbance, the gain of the control surface of the
Then their equivalent discrete control forms are obtained
proposed controller is set higher than unity. This will result in
A bilinear
via
Zero Order Hold (ZOH) emulation.
faster response and superior performance over its linear
Laplace
form
transform
the
transformation
has
been
used
to
counterpart for large disturbance. Thus, while the proposed
as:
can
be
found
Z-form
and
they
into
controller offers the performance similar to its linear PI for
small-signal performance, it also offers a superior performance
165z + 0.165
over its linear PI controller for large disturbance.
C () 0.
(9a)
C System Description and Control Design Example
The single-phase inverter system and the controller are
designed in MATLAB-Simulink. The inverter system is
comprised of a full-bridge inverter and an L-C filter. Fig. 3
shows the complete system structure with the controller. The
control structure uses a cascaded double feedback loop with
the inductor current as the inner loop and the voltage output as
the outer loop. Both loops are controlled using fuzzy
controllers. Table III summarises the parameters of the system
used in the simulation.
FLC1
F PM
3.
z-1
C - 0.03465z - 0.02835
C (z)
(9b)
z-1
The parameters of the proposed controller that ensures
identical performance with its linear PI counterpart is derived
using (7) and given as follows:
Al =-3.538 and ri=0.23
A, = 0.222 and t = 0.063
(lOa)
(lOb)
In single input fuzzy control, the input/output mapping or
the control surface can be formed in a 1-dimension surface.
The surface can be linear or nonlinear surface, depending on
the location of the input and output peak of the membership.
Fig. 4 shows the control surface used in this paper.
INVERTER
Fig. 3. Control structure
5
4
Parameter
VDC
Lfilter
Cfilter
Rated load
Reference Voltage
Output Power
Switching frequency, f,
-X
Rg
Region 2
3
TABLE III
SYSTEM'S PARAMETERS
R
g,on
2
Value
1 O0V
250pfH
33 iF
20 Q
80V
0.32KW
20 KHz
50us
100us
:5
o
--------
-1
-2
-3
-4
.//
-1 V
-1
-0.8
-0.6
-0.4
-0.2
00.2
input1
0.4
0.6
0.8
Fig. 4. Fuzzy control surface used to control the inverter
For design simplicity, both loops are set to have identical
fuzzy control surface. The control surface used in the
proposed controller is divided into two linear regions. Region
1 comprised of a linear line with unity slope while Region 2 is
A discrete linear PI controller that ensures good a line with a slope higher than unity. Region 1 is considered as
performance for small load disturbance has been designed first the range of input where the small-signal performance for the
using frequency analysis in continuous time mode. The proposed controller is similar to its linear PI controller.
transfer function of the controller for both loops can be
obtained as:
Inner loop's sampling time
Outer loop's sampling time
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A-br. -_T
The range of the input for Region 1 is decided by
conducting simulations. In the simulation, small load
disturbance is given to both inverter system controlled by the
proposed controller and linear PI controller. Matching smallsignal performances with both controllers were obtained when
the control surface is set to have a unity gain for ldl<20 units.
For Region 2, the gain has been tuned through simulations in
order to obtain good performance against large load
disturbance. The control surface is set to saturate at Id1.100.
D.
To demonstrate the approximation of the proposed
controller with the typical PI Fuzzy, simulations are conducted
on the system for large load disturbance (from no load to full
load). Fig. 7 shows the voltage response for both controllers.
It can be seen, that the response for both controllers are very
much identical and indistinguishable. Thus, verifies the
approximation of the proposed controller.
Verification of the Proposed Controller Approximuting the
Typical PI Fuzzy Controller
E
Z
NS
-100
-50
50
100
TABLE IV
RULE TABLE FOR TYPICAL PI FUZZY CONTROLLER
-
-
-
-H-H-
275
20
0
-20
-275
-
-
275
83.75
_
_ 20
0
_
-211.25
_
- - ---
-
I- I- -I-I
I
-
-
-
a)
ol
'o0
-
-
-
I-
I
,MIS0 PI FozzyI
I
/1
80
L
L
-L-
L
lb
-
J
-
IbSO11
a)
-o
.F. 70
D-
E
< 65
60
55
50
i
2.5
3
3.5
4
4.5
5
5.5
6
6.5
7
7.5
x 10-,
Time(s)
Fig. 7. Comparison of voltage output response for typical PI Fuzzy and the
proposed controller when a large load disturbance is given to the system
III. SIMULATION RESULTS
Fig. 5. Input membership function for typical PI Fuzzy
controller for both loops
0 _-_ 211.25
0
-211.25
-83.75 _-_ -20
-275
-83.75
-275
-275
I
PB
Ps
0.5-
0
-
90
85
To verify the approximation ofthe proposed controller to the
typical PI Fuzzy controller, a Sugeno-Type two-input PI Fuzzy
equivalent to the proposed controller has been designed. The
inputs of the typical PI Fuzzy are the error (with the scaling
factor of unity), and the change of error (with the scaling factor
of -n/(m+n)) for both loops). The input range of d which is
from -100 to 100 units is divided into five fuzzy sets as shown
in Fig. 5 for both inputs. Table IV is the generated rule table
and Fig. 6 shows the control surface for both loops.
~B
95
4
275
275
275
211.25
0
In this section, simulations are carried out on the inverter
system controlled by the proposed controller and discrete linear
PI controller. The output voltage response for both controllers
for different types of loads is compared to analyse the
effectiveness of the proposed controller and its superiority in
terms of performance over linear PI controller.
To analyse the small-signal performance of the proposed
controller, a small load disturbance (change from its full load,
20Q to 25Q) is given to the inverter system. For this type of
load disturbance, the proposed controller should exhibit similar
performance to the linear PI controller. Fig. 8 shows the
voltage output response for both controllers. From the
simulation, it is found that the response for both controllers is
identical. The result shows that the proposed controller can
perform very well and can approximate its linear PI's response
for small load disturbances.
-
-1- -
-1 ----1__S:-_
-_-1:___2-
80 _
70
s\\\
60
I
bscrete
I-near PI
ol
'2 50
0
_
i-
1-
-
\_
__
SSISO
1--I
PI Fuzzy
340
E 30 7
-100- - 2<
lll
20
-3
00
3OoOos~~
~~~~5
--50
-
-
10 _
_
10
o
CError
Error
Fig. 6. Fuzzy control surface for typical PI Fuzzy used
for both loops
-
r - T
- -
2
-
13
4
-
'I-
I-
5
6
Time(s)
7
8
9
x
10-3
Fig. 8. Comparison of voltage output response for the proposed controller
and linear PI controller when a small load disturbance is given to the system
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To perform large-signal analysis, a sudden load change
from no load to full load is given to the system. Fig. 9 shows
the voltage output response. The result yields that the
proposed controller result a superior performance compared to
its linear counterpart. The proposed controller offers a faster
rising time with smaller overshoot and faster settling time
compared to the linear PI controller.
-2 -
0.02
-I
-
0.025
0.035
0.03
--
0.04
--
0.045
0.05
--
0.055
0.06
1:
90
SISO
.l Fuzzy
Discrete .inear PI
L
-I-
80
J1,11,
o5 70
Fig. 11. Response of output voltage for inductive load. R = 20 ohm
and L = 100mH. Power factor =0.535 lagging
-o
E E60
50
10F
3
5
4
Time(s)
6
7
8
x
lo-,
Fig. 9. Comparison of voltage output response for the proposed controller
and linear PI controller when a large load disturbance is given to the system
-10
0.01
The proposed controller is also has been designed such that
it is capable of handling cyclic load and nonlinear load, which
are the type of loads typically handled by inverters. Fig. 10
and Fig. 11 show the voltage response for cyclic load and
inductive load with p.f of 0.535 lagging, respectively. While
Fig. 12 shows the voltage output response of the inverter when
connected to a rectifier load. The THD ofthe output waveform
for the rectifier load is calculated as 3.18% over 1000
harmonics.
0
Ql
0002 0004 0006 0008
l0
001
Time(s
0012 0014 0016 0018
002
0.015
0.02
100
0.025
0.03
0.035
0.04
0.025
0.03
0.035
0.04
1I
50
50
-1000.01
,/
-
0.015
-
0.02
Fig. 12. Output voltage response for rectifier load. R = 100 ohm and C
470uF.
closely approximate the typical PI Fuzzy. The proposed
controller exhibits small-signal performance similar to its
linear PI controller and superior over the linear PI controller
for large load disturbance. The simulation results also have
shown that the proposed controller is capable of handling
cyclic and nonlinear load very well.
100
4
-50
-
4l
4
REFERENCES
50
E
100
0
0.002 0.004 0.006 0.008
0.01
Time(s)
0.012 0.014 0.016 0.018
0.02
Fig. 10. Output voltage response for cyclic load
IV. CONCLUSION
This paper proposed a single input PI Fuzzy controller
which can exhibits similar performance of the typical twoinput PI Fuzzy controller. By taking advantage of Toeplitz
structure of the rule table, a new single input variable for fuzzy
control is derived and applied to control a single-phase inverter
system. The reduction of inputs reduced the number of rules,
which inherently contributes to a faster control output
calculation and allows the controller to be easily realised using
inexpensive digital processor. Through simulations conducted
using MATLAB-Simulink, the proposed controller is shown to
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