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A Sliding Mode Controller in Single Phase Voltage
Source Inverters
H.
Pmheiro,
A.S. Martins,
J.R. plnheiro
Universidade Federal de Santa Maria, CT-DELC-NUPEDEE-DESP, 97119-900, Santa Maria, RS, Brazil
Fax:O5
226
1
975, e-mail
:
Abstract- This
paper
deals with a
sliding
mode controller
for single phase inverter used in UPS applications. The
proposed system provides
overload
and short Circuit
protection. It c n
operate in
constant
or
variable
frequency. The
use
of areduced order observerelhinates
the requirement of the load current measurement
and
improves the noise immunity Experimental results
obtained
in
laboratoryarepresented and they confirm the
simulation
and theoretical
analysis.
JNTRODUCIION
Static converters are intrinsically variable structure
systems, so the slidmg mode control
SMC)
approach is a
strong candidate method for the inverter controller design.
Many papers have been published in the field of
slidmg mode control[2]-[10]. It is claimed th the
use
of the
sliding mode control result in order reduction, disturbance
rejection, insensitivity to parameter variations ancl simple
implementation.
Implementation of sliding mode control implies hgh
frequency discontinuous signals. This is not a problem static
converters, because d~scontinuous ignals is always present.
Furthermore, SMC can provide a possibility to solve the
closed loop system design and to create the switching signals
(PWM pattem) in the same framework.
Some paper have been published about
shdmg
mode
control applied to DC/AC inverters. In [8] a three level
PWM
inverter with fixed frequency is presented, the proposed
method is complex, because it needs
two
frequency closed
loops and a midpoint control. The proposed current limiter
degrades
the
transients response during large perhx-bations. In
[9] is presented a two level PWM inverter with fixed
switching frequency and current limiter. The overall
perFormance is good, but it needs two current measurement,
namely, the load and filter indqctor current. In
[lo] is
presented a comparison of three and
two
levelsP W M mverter.
In the
above mentioned works the output voltage posses
steady-state error.
l b s paper is concemed on a
slidmg
mode controller
implementation for
U P S
inverters. The suggested system
posses the following features: first, it does not need
measurement of the load current; second, it is possible to
obtain zero steady state error to step input; thnd, it can operate
with constant frequency; fourth, it provides overload protection
for the power semiconductors.
SYSTEM DESCRIPTION
Figure 1 shows the basic circuit of single phase
inverter with a LC output filter whch is often used in single
phase U P S .
T
,L
L
P P
Fig.1: Single phase voltage source inverter.
The control goal is to make the output voltage v, be
equal to a reference input v4 (sinusoidal) while
the
inductor
current absolute value
ir
is kept lower
than
a maximum
value
ik.
SLIDINGMODE
ROLLER
T&ng into account
that
the output voltage
v,
must
track the reference voltage
vmf
and the inductor current must
be limited, the sliding mode control
in
the voltage and current
error coordinates is formulated. The dynamic behaviour of the
inverter, shown in Fig.1, is govemed by the
state
space
equation 1).
h 2
=
- X I
-
ref
t L L L d t
Where x , is the voltage error
(
vref-vc
,
x, is the current
error (i&) and L and C re inductance and capacitance of
0-7803-1328 -3/94 03.000 1994 IEEE 394
8/12/2019 A Sliding Mode Controller in Single Phase Voltage Source Inverters
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the output filter respectively.
defined as:
The sliding surface and the control action
U
are
s=
C g x , czx,
o
and
U=E sign(s)
Average Model
The equation of the motion along
the
surface
s=O
is
given by
In order to achieve the invariance property in slidmg
mode the inductor reference current
is
made to
be:
Substituting
(3)
into
2)
yelds:
(3)
4)
So,
once the system is
in
slidmg mode it behaves
as
a stable first order system and the voltage error decays
exponentially to zero
C,IC,>O).
Existence Co ndition
The state space equation l),when the i s given by
(3),
is:
U
dt L L
- - + g ( t )
or
Where
and
g( t )
is given
by:
g ( t )
=
v4 lL
diJdt +
C&vJ&.
If s
and its time denvatwe have opposite signs the
state trajectory reaches the slidmg
surface s=O
after finite time
interval. This constraint can
be
achieved
if
the following
inequality is true:
possible to construct a reduced dimension observer to obtain
io. The state space equation of a Luenberger reduced-order
observer, which has
been
implemented, is:
l
c z
+Ck0vc
,
where w s the natural frequency of the output filter and z
is
an
auxiliary variable.
The observed load current is given by:
Q ? .
1 =1 I
o c l
9)
By proper choice of the gain
k, the
desired
convergence rate of
I
to ic and, consequently, to &may
be
provided.
Current Limiter
A mod~fied
eference current may be obtained by
rearranging the sliding surface equation
as
shown
in l0).
L
s c , x , c , x ,
=C, ( r , +x , )
C,
L
s=c,( X,
rt,- l
=C, i
Lf i t )
C
L
It is possible to limit the inverter output current
through the limitation i , assuring, in
this
way, overload and
short-circuit protection, as shown in Fig.2.
0 a b i 8.h B B3 s)
-Me
Fig.2
:
Output voltage and inductor current when a
temporary short-circuit in the load occurs.
Elimination of the Steady State Error
This inequality also
detennines
the
minimum
inverter
DC
voltage needed for enforcing
the
sliding mode.
The Use of Reduced-Order Observer
As
the state variables
v,
and
il
are available, it is
Whenever the steady-state specification
is
not
satisfied, due to the use of hysteresis comparator leadmg to
finite
switching frequency, it is possible
to
use extra state
variables in order to cancel the voltage errors. For instance,
the steady-state error to a constant reference input is
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eliminated by introducing an extra state variable, as shown
below.
= x ,
r
dt
=
- - - - + g ( t )
I U
dt
L L
t B i
The slidmg surface is gven by:
Fig.4: State trajectories in the vicinity of slidmg surface
when the hysteresis comparator
is
used.
s=
CdE, +c ,x ,
Q2 =o (12)
~=f x8,g t>);f=f x,-E,g t)).
The average dynamic behaviour is govemed by:
The maximum switchmg frequency is obtained setting
v , ~ ~o zero, which results in the following equation:
(13)
'x,
C,
dx
CO
- - - r l = O
d t2 C2C t C,C
Freq- =-CP 15)
4AL
Figure 3 shows the output voltage error for a step
Constant Frequency O peration
nput.
It is easily
seen
from the equation (14), that the
switchmg frequency depends on the reference and
DC
nput
voltages. It can result in a very wide switching frequency
range. This is a disadvantage from the point of view of the
output filter design and implementation There
are
basically
two methods to achieve constant frequency operation:
a) Variable Hysteresis Band with feed fo mad or feed
back techmque. In t h ~ s ase some points must
be
carefully
investigatedsuch
as:
stability problems, transient
perform nce
limitations, complexity of implementation[121 131.
b) Disturbance Method:
it is possible to obtain
1.t-3
2.t-3
3 t - 3
4.1 3
constant frequency operation adding an
adequate
constant
Fig.3: Output voltage error for a step input.
SWITCHING
FREQUENCY
Variable Frequency O peration
The hysteresis is responsible for the fact th t once the
state trajectories hit the dscontinuity surface s=U, the
describing points do not move exactly along
the
surface but
wll oscillate in its vicinity with width equal to
2A
(Fig.4).
Under the assumption th t the states trajectories are constant
near
the
surface s=U the switchmg frequency is given by:
frequency signal to
s
variable, whch is comparable to the
approach proposed in
[9].
Figure
5
shows athree evel disturbance signal, whch
with an apropriate design provides constant switching
frequency operation. 6t determines the maximum duty-cycle
at steady state. The amplitude D must be greater
than
maximum s value. Disturbance
signal
frequency
Fd
must obey
the folowing inequality:
d t
Fig5 Three level dlsturbance signal.
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THREE LEVEL PWM
The three level P W M method proposed in [8]
presents a good performance, but it is complex. A simpler way
to implement a three levelP W M s presented in [lo]. The later
treats the three level
P W M
s a double two level
PWM,
ne
of them is active when
ueq
s positive and the other when
ueq
is negative
(ueq
s defined in [l]). However, it was used vref
instead of ueq.
This
practice resulted in degradation of the
transient response and &stortion around the zero
(ueq
may have
opposite sign of
vref
, as illustrated in Fig.6.
Fig.6:
Three
Level P W M nverter with 0.7 inductive load.
INSULATINGRANSFORMER
Usually it is required galvanic insulation of the load.
There are basically two ways of implementing the galvanic
insulation: the first one is by using a low frequency
transformer and the second one by using hgh frequency
transformer.
Low Frequency Insulating Tramformer
To improve the performance of the system in low
frequency it is possible to substitute the derivative of the
reference voltage by the integral of the output voltage in the
equation (3) [81, it because the reference signal is sinusoidal
in
U P S
applications.
High Frequency Insulating Transformer
Several works have been published on inverters using
hgh
frequency transformer link [11,14,15].
The
DC/AC
converter proposed in [ l l ] is shown in Fig.7. The phase
control used there can be repalced by the sliding mode
controller previously presented here.
Fig.7: DC/AC converter with high frequency transformer.
The switches S1 and S2 are commanded with fixed
frequency and duty-cycle equal to
0.5,
so producing a
high
frequency square wave v,, at the input of the cycleconverter.
On
the other hand, the switches S 3 and S4 are commanded by
the following law:
S4= sign(s) sign@,)
S3=
-sign(s)
.
sign(v,)
,where -1 is for switch opened and 1 for switch closed. The
inverter and the clycleconverter topology can be implemented
using other configurations, for instance, half and full-bridge
respectively.
EXPERIMENTALESULTS
It has
been
implemented a prototype of the single
phase inverter shown in Fig.1. The implemented inverter
controller block &gram with reduced order observer and
current limiter is shown in Fig.8.
%-
Fig.8: Block diagram of the implemented controller.
The controller and output filter parameters have been
derived from the following input datas:
Rated output power Po=600VA
DC input voltage E =lOOV
Maximum switchmg frequency F,,,,,= 15KHz
Current Ripple(limiting current mode) AM.4
Current transducer gain
k,,,=. 1V/A
2=1/1oooo
s.
RMS fundamental output voltage v0=55v
Output voltage Ripple v0,=4v
Desired time constant of system
The calculated parameter values are:
Hysteresis width AS.65
Output filter inductance
M . 2 5
mH
Output filter capacitance C=85.5
JJF
Gain C,
C,=0.0855
Mmimum switchmg frequency Fm,,=2.85KHZ
Gain c c;=o. 1
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The experimental results shown in Figures 9, lO and
11 validate previous analysis and simulation.
Fig.9: Output voltage v, wth variable frequency operation
for step input.Voltage Scale: 20 V/div.
Time scale:0.2ms/dlv.
Fig.10: Output voltage v, and inductor current il ( variable
frequency operation mode) with sinusoidal input and when a
temporary output short-circuit occurs.Voltage Scale:
20
V/div. Current Scale:lONdw. Time scale: Sms/dv.
Fig. 11: Output voltage
v,
and
PWM
signal with constant
switching frequency operation (15 Khz) .Voltage Scales:
upper 50 V/dlv,lower 100 V/div. Time scale:0.2ms/&v.
CONCLUSIONS
The s l i m mode control techmque is a powerfull tool
for the controller design of static converters. The resulting
system is robust, simple and posses
h h
perfomance.
Furthermore, it can create the switching signals in same
framework.
The suggested controller has an observer, whch
becomes the system less sensitive to noise when compared
with the one that use hfferentiator and avoids the
measurment of the load current. The use of the observer does
not increase significantly the circuitry implementation.
The prototype has been sucessfully implemented with
both variable and constant frequency operation. The power
semiconductor currents
are
limited during overload
and
short-
circuit on the load.
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Sliding Modes
and
Their Applications
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