PEDS 2007
Analysis of Double Loops Discrete Single
Input
P1
Fuzzy for Single phase
Inverter
S.M. Ayob, Z. Salam, and N.A. Azli
Department of Energy Conversion, Faculty of Electrical Engineering
University Teknologi Malaysia, 81310 UTM Skudai, Malaysia
Email:shahrinNfke.utm.my
Abstract--This paper presents an analysis of double loop
PI-Fuzzy controller for a single phase inverter. The scheme
comprises two feedback control loops arranged in a
cascaded manner. The current in the filter inductor and
voltage across the filter capacitor are sensed and fed back to
the system to form the inner loop and the outer loop,
respectively. Both loops are controlled by a discrete single
input PI-Fuzzy controller. It is derived using the "signdistance" method, which simplifies fuzzy set into a single
input single output system. The work analyses the effect of
fuzzy control surface for both loops in improving largesignal performance of the controller. Based on the analysis,
the replacement of single input PI Fuzzy for voltage loop
with a simple discrete linear PI controller will be justified.
Index Terms--PI Fuzzy, UPS inverter, Double loops
controller
I. INTRODUCTION
The PI-Fuzzy and PD-Fuzzy are the general types of
fuzzy logic control that have been used to control power
electronic converters. Although PD type provides faster
rising time and smaller overshoot compared to the PI
type, the former suffered from significantly large steady
state error. The existence of the error is due to the lack of
integration operation in the controller itself. As a result,
in many cases PD-Fuzzy controller seems to be an
unpopular choice. To overcome the drawback of the PDFuzzy, PID-Fuzzy controller has been proposed.
Theoretically such controller exhibits good dynamic
performance i.e. small overshoot, fast settling time and
fast rising time. However, due to the three-input based
operation of fuzzy system, its 3-dimension rule table is
difficult to implement. Therefore many fuzzy based
controlled systems prefer to use PI type rather than PD or
PID type.
Over the years, significant research has been carried
out to study the fundamental mathematics of PI-Fuzzy.
Compared to other nonlinear controllers, fuzzy
controllers exhibit simpler mathematics and offer a
higher degree of freedom in tuning its control parameters.
Depending on the tuning method used, the PI-Fuzzy can
be classified as a linear PI-Fuzzy and a nonlinear PlThe linear PI-Fuzzy yields
Fuzzy controller.
performance identically similar to its linear PI
counterpart and the stability condition can be simply
This project is supported by the ScienceFund grant from the
Ministry of Science, Technology and Innovation, Malaysia (MOSTI).
1-4244-0645-5/07/$20.00©2007 IEEE
assessed using conventional frequency analysis. In
contrary to linear PI-Fuzzy, the nonlinear PI-Fuzzy is not
supported by adequate control theory mathematics [1].
Its performance is quite unpredictable. In addition it is
somewhat impossible to assess its stability due to the lack
of stability theory. Most of the PI-Fuzzy designs were
conducted by trial-and-error method.
However, by applying certain tuning methods, one can
make the nonlinear PI-Fuzzy approximates other
nonlinear controllers, such as the layered Sliding Mode
controller (SMC). Previous works have shown that the
nonlinear PI-Fuzzy controller is capable of performing
similarly to the original SMC [2,3,4]. Furthermore, since
similar mathematical principle of SMC is applied to
design this nonlinear PI-Fuzzy controller, the stability
assessment can simply be done using the well known
Lypunov's method. This type of nonlinear PI-Fuzzy
controller is also known as Fuzzy Sliding Mode
controller (FSMC) [4].
Fundamentally, fuzzy control is a non-mathematical
based controller. It is more of a human logic way of
thinking based controller. Therefore the heuristic fuzzy
design approach will always be faced and could not be
totally eliminated. However, the approach can be made
minimal by reducing the number of tunable parameters in
the controller. A single input PI-Fuzzy employed in this
paper, is a simple controller which yields an identical
performance of the typical two-input PI-Fuzzy controller.
The reduction in the number of inputs to a single input
has significantly reduced the number of tuning
parameters, thus simplifies the overall design of PI-Fuzzy
control. Furthermore the input-output mapping or the
control surface (P) of the single input fuzzy controller
can be easily constructed as a piecewise linear
approximation. This provides a better insight on how
exactly fuzzy controller works and makes the analysis of
fuzzy controller less complex.
In this paper, an analysis on a double loop feedback
single input PI-Fuzzy for inverter control is presented.
For inverter systems, a single loop control structure, also
referred as Voltage Mode Control (VMC) is not enough
to damp the undamped poles introduced by the LC filter
[10]. Hence, in order to obtain better dynamic response,
a double loop consisting of current and voltage loop is
preferable. Analysis of double loop with sliding mode
control, deadbeat and linear PI controller for the control
of single phase inverters can be found in [9,10,11],
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respectively. However, no such concrete analysis is found
for a fuzzy controller. This may be due to the complexity
of analysing the behavior of Fuzzy control even for the
case of the VMC.
Thus, this motivates the writers to present an analysis
on the behavior of a double loop single-input PI Fuzzy
controller. The controller is derived based on the signeddistance method [6]. The employment of a single-input
PI-Fuzzy has made the analysis of the fuzzy control less
complex as well as made the analysis of double loop
possible. In the analysis, the control surface for each loop
is varied and the dynamic response for the respective
control surface is recorded and analysed. Then based on
the simulation results conducted using MATLAB,
justification on replacing the single input PI-Fuzzy for the
voltage loop with a simple discrete linear PI controller is
made.
II. DISCRETE SINGLE INPUT PI-Fuzzy CONTROLLER
The design of the discrete single input PI Fuzzy
controller is generally based on a previous work proposed
by Viswanathan et. al. [7]. The conventional two input
variables of PI-Fuzzy, namely the error (e) and the
change of error (ce) have been replaced by a new single
input variable which is the distance (d). In this method,
the distance is defined as the absolute distance of any
state points perpendicular to any points in the main
diagonal line where by the main diagonal line is a line
consisted of ZERO output membership in the rule table.
The method, however can only be applied if the rule table
exhibits a Toeplits structure or near Toeplits structure [7]
as shown in Table I.
TABLE I
T-TTTT 't--%U]T)T
L
PS
NS
NB
NB
NB
NS
NB
NB
z
I
PB3
PS
z
NS
NB
PB
PB
PS
z
NS
PB
PB
PB
PS
z
The controller developed in [7] is an analog-based
controller and has an identical small-signal performance
of its linear PI and exhibits a superior performance over
its linear controller for large load disturbance. The
superiority can be achieved by varying the control surface
(v) higher than unity gain. For digital implementation
purpose, the output equation of the discrete single input
PI-Fuzzy has been derived in [8]. To obtain a similar
small-signal performance of its discrete linear PI
controller when V =1, the controller parameters have
been derived as follows,
Au = (m + n)y!t
)+e]
where
I.r T
M=,K. -+-,K. 2
n
=
T
K.-2 Kp A
K.
m+n
,r= m+n
-n
11
-
The block diagram representation of (1) can be
depicted as in Fig. 1.
Au(k)
Fig. 1. Block diagram representation of (1).
In [8], analysis was conducted with both fuzzy
controllers set to have identical control surfaces. This is
done for simplification purposes. The control surface
used in [8] and applied in this paper is as shown in Fig. 2
2
Regiont
T
le
i
~~~~~~~~~-1--4/--------Fig2.Conro surac use fo th analysis
1
-2
5-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
nt.
0.6
0.8
Fig. 2. Control surface
withi th inu,ag
f used
2 for the analysis
Thi isdn,s
The control surface for the single input PI-Fuzzy
controller is constructed based on two n linear lineswith
different slopes. The first linear line is fixed at unity gain
within the input range of ldl<520 units. This is done, so
that the controller can exhibit similar small-signal
performance of its linear PI controller. For input range
beyond Id1>20 units, the gain of the second linear line is
made higher than unity so that a better performance can
be obtained for large load disturbance. The control
surface is set to saturate for Id1>100 units.
III. SYSTEM DESIGN EXAMPLE
A complete single phase inverter system has been
developed using MATLAB-Simulink. The system
consisted of a full-bridge inverter with four switches, an
L-C filter and a load. Table II summarises the parameters
used in the simulations while Fig. 3 shows the block
diagram of the control system. The inductor current and
output voltage are sensed and fed back to the controller.
Both loops are controlled by single input PI-Fuzzy
controller.
(1)
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To demonstrate that, a large load disturbance (no load to
full load) is imposed on the system. To start with, let the
control surface's slope for the voltage loop varies
according to the values of Kj= 2.25, K2 =3.1875 and K3 =
8.1875. In the mean time, the control surface for the
current loop is fixed to unity for the whole input range.
Fig. 4 shows the result.
From Fig. 4, it can be clearly seen that there is no
performance change even though the surface is varied
from K, to K3. This is probably because the maximum
value of the distance input driven by the disturbance has
not yet exceed the breakpoint, set at d=20. Applying the
same procedure as above, different responses are
obtained for different surface slopes when the breakpoint
is set at Id1=10 as can be shown in Figure 5.
TABLE II: SYSTEM PARAMETERS
Value
Parameter
VDC
1OOV
Lfilter
250 jil
Cfilter
Rated load
Reference Voltage
Output Power
Switching frequency, f,
33 piE
20 Q
80V
0.32KW
20 KHz
Vouu
Vll
+
+
IVE
E
Fig. 3. Double feedback loop consisted of inductor current and
output voltage
95
In a cascaded control scheme, the output voltage
usually has slower response compared to the inductor
current. Based on this, it is beneficial that the sampling
time, T, for the outer loop is set to have slower sampling
rates. For the inner loop the sampling time is chosen to
be similar to the switching frequency, f, Since, f, = 20
KHz, then the sampling time is set to be equal to 50jts.
Usually for the outer loop, the sampling time is set to be
twice slower than the inner. Therefore the sampling time
for the outer loop is set as 100[ts.
To find the values of variables m and n, the linear PI
controller transfer functions, C(s) for both loops are
obtained using conventional frequency analysis. The
controller transfer functions for the voltage loop and
current loop are defined by equations (2a) and (2b),
Using the Zero Order Hold (ZOH)
respectively.
emulation and Bilinear Transformation to transform C(s)
to C(z), the parameters of m and n can be obtained for
both loops as in (3a) and (3b).
C, (s) 2300 0.00002174s±+
=
C,(s) = 126L0.00025s+1]
M
n=
K,
90
K 2425-
85
K3=8.1l875
K2=3.i1875
80
>
75 -
0
65____8
-
70 -
-I
- - -
-
- I- - -
- - - -
3
5
Time(s)
4
IV. SIMULATION AND ANALYSIS
In this section, detail analysis on double-loop single
input PI-Fuzzy controller with different control surfaces
is presented. It should be noted that the analysis is
conducted on large-signal performance of the controller.
6
7
8
x
10-3
90
2 85
o
0)
80
>
*Q
75
L
1
70
---
65
<
--------
--- --- T
--I
--- --bu
.
3
3.5
-,
-I-
4
,I-
4.5
.
5
Time(s)
(3b)
-
- - -
Fig. 4. Voltage output with the control surface for voltage
loop is varied
(2b)
_KI
- -
55
(2a)
(3a)
- - -1
- -
65
-a
+
-
E
.
55.5
-,I-
6
,-I-
6.5
-.-
7
X
10-3
Fig. 5. Response for different slope with the breakpoint of
d= lOunits
As can be observed in the figure, a sharp overshoot
along with oscillation can be seen for a steep slope. Less
overshoot and oscillation with slower rising time can be
seen when the slope is set more moderate.
To analyse the effect of PI-Fuzzy for the current loop,
the voltage loop control surface is set to unity for the
whole range of input. Then the control surface for input
Jdj>20 is varied. The result for voltage response is as
shown in Fig. 6. As can be observed, each slope value
portrays distinct performance. Large and sharp overshoot
can be examined for high gain value. A lower overshoot
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height is expected when the gain is reduced. Besides, an
acceptable overshoot height with slower rising time can
be obtained when the gain is set to unity. A better
performance can be achieved by a slight increase above
unity gain. A lower overshoot and faster rising time is
obtained when the gain is set at 3.5. A fine-tuning
process is then conducted around that value to obtain a
better response. The best performance is obtained when
the gain is set equal to 3.1875. As can be seen from the
figure, the value gives similar rising time performance as
for the case of K=3.5, but offers lower overshoot and
faster settling time.
95
K = .0
K2=i
K3=3.2
90
85
K4-f.
_s
a 80
° 75
-a
. 70
E
< 65
60
55
50
2.5
3
3.5
4
4.5
5
5.5
Time(s)
6
6.5
7
7.5
x
10-3
Fig. 7. Performance when both loops is PI-Fuzzy
95I
gain is unity, which actually reflects the performance of
its linear PI controller. Thus, it is sufficiently justified to
replace the single input PI-Fuzzy controller with a simple
linear PI controller.
90
85 t
3=3.5
'- 1 875-
80
75
V. CONCLUSIONS
< 70
65
60
55 L _
3
3.5
4
4.5
L _
5
lime(s)
5.5
6
6.5
7
x
10-3
Fig. 6. Different responses are obtained when fuzzy control
surface of current loop is varied
A further analysis is carried out with both loops
controlled by a single input PI-Fuzzy controller with each
loop having its own control surface. The analysis is
carried out as follows,
*
*
The control surface for the voltage loop has a
breakpoint at Id1=10 and the gain for Id1>10 is
varied.
The control surface for the current loop has a
breakpoint at Jdj=20 and for Jdj>20, the slope is set
In this paper, an analysis on double loop PI-Fuzzy for
controlling a single phase inverter has been presented. A
cascaded control structure is employed with the inductor
current as the inner loop and the output voltage as the
outer loop. Both loops are controlled by PI-Fuzzy
controllers with each loop having a different control
surface. By varying the control surface, an analysis is
conducted to see the effect of each loop on the system's
large-signal response. From the observations and
discussions, it has been shown that fuzzy control for the
voltage loop has minimal impact in improving the largesignal performance The best performance is achieved
when the gain is set to unity, which is equivalent to a
linear PI controller. Thus, the replacement with a linear
PI controller is justified.
REFERENCES
as 3.1875.
The result is as shown in Fig. 7. From the figure, it
can be clearly seen that the performance is not affected at
all even though the control surface for the voltage loop is
varied.
For voltage loop, the rising time can be considered
unaffected by the gain values. However, the overshoot
and settling time can be controlled by varying the gain.
The best performance, i.e. small overshoot and fast
settling time is obtained when the gain is set equal to
unity. For current loop with single input PI-Fuzzy, the
rising time, overshoot and settling time is largely affected
by the gain variation. Different gain values provide
different performance. The best gain value that offers an
excellent large-signal performance is K = 3.1875.
From the discussion, current loop seems to have a
dominant impact in improving the large-signal dynamic
response. The results have also shown that voltage loop
has minimal effect in improving the system performance.
Moreover the best performance is only achieved when the
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