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Feedback Linearization Control of an H-Bridge
Multi Level Inverter
Abir Rehaoulia, Habib Rehaoulia, and Farhat Fnaiech ENSIT/Research Lab SIME, Tunis, Tunisia
Email: [email protected], {habib.rahaoulia, farhat fnaiech}@esstt.rnu.tn
Mahir Dursun Faculty of Technology, Department of Electrical and Electronics Engineering, Gazi University, Ankara, Turkey
Email: Email: [email protected]
Abstract—The present paper concerns the control of a multi
level inverter in autonomous operation. The inverter is a 5
level H-bridge type associated to a suitable LC filter. To
control the system output voltages, a feedback linearization
algorithm is developed with unknown load. Simulation is
carried out for either passive or active loads. The ability of
the controller to follow perfectly the desired reference was
tested on a passive load. For a load sensitive to voltage
change, like an induction motor, the study shows that the
controller maintains the output voltage to its rated value
under severe conditions.
Index Terms—multi level inverter, H-bridge, LC filter,
autonomous operation, feedback linearization
I. INTRODUCTION
Multi level inverters appeared in the early eighties.
Since, their advantages are well assumed [1], [2]. Their
particular topologies limit stresses across switches and
provide output voltages with reduced harmonic distortion
[2], [3]. Among the popular multi level inverters, the H-
bridge inverter is distinguished by its simple structure.
This latter consists of connecting in series single phase
full bridge cells.
In this study, we choose the five-level structure
associated to a low pass filter. It allows to profit from
previously mentioned multi-level benefits without
resorting to excessive use of H cells and DC sources. The
combination of a low pass filter, whose cut-off frequency
is well chosen, reduces the total harmonic distortion
(THD) of the output voltage to better values than those
obtained by inverters which number of levels is higher
than five.
In addition, to control the output voltage of the system,
we have developed an algorithm based on feedback
linearization techniques. Such control algorithm is stable
with very fast response time, compared to conventional
controllers.
Finally, two useful applications are detailed and
reported. They concern either passive or active loads.
Manuscript received January 7, 2015; revised April 17, 2015.
Load
VOA
VOB
VOC
C
ilB
ilA
ilC
iOA
iOB
iOC
r,l
5L H-bridge
Multilevel
Inverter
Figure 1. 5L H-bridge inverter with LC filter and load.
0 0.01 0.02 0.03 0.04 0.05 0.06-600
-400
-200
0
200
400
600
time (s)
Van
(V
)
Figure 2. Phase voltage of 5L H-bridge multilevel inverter.
0 0.01 0.02 0.03 0.04 0.05 0.06-1000
-500
0
500
1000
time (s)
Uab
(V
)
Figure 3. Line voltage of 5L H-bridge multilevel inverter.
II. MATHEMATICAL BACKGROUND
The power circuit of the H-bridge multi level inverter
operating in autonomous mode is depicted in Fig. 1.
Selecting the number of levels and filter components
International Journal of Electronics and Electrical Engineering Vol. 4, No. 1, February 2016
©2016 Int. J. Electron. Electr. Eng. 24doi: 10.18178/ijeee.4.1.24-28
design is a trade off which depends on the system cost
and bulkiness. The study assumes that the parameters of
the considered load are unknown.
The structure of the 5L H-bridge inverter is made by
series connection of two bridges in each arm. Fig. 2 and
Fig. 3 show the line and phase voltages of the considered
inverter without filter.
The inverter is going to be controlled by means of
input output feedback linearization control in order to
regulate the output filter voltages (VOA, VOB, VOC).
The three phase inverter model regarding the AC side
is expressed as follows:
(1)
A lA lA OA
B lB lB OB
C lC lC OC
dU r i l i V
dt
dU r i l i V
dt
dU r i l i V
dt
(1)
where (UA, UB, UC) and (ilA, ilB, ilC) are respectively the
multilevel inverter output voltages and currents, r and l
the filter parameters.
The dynamics of load voltages are obtained through
voltage-current relations across capacitors, so that:
1 1
1 1(2)
1 1
OA lA OA
OB lB OB
OC lC OC
dV i i
dt C C
dV i i
dt C C
dV i i
dt C C
(2)
where (iOA, iOB, iOC) denote load currents and C is the
filter capacitor.
The overall system state space model can be
transformed in (d, q) frame as:
1 1
1 1
(3)1 1
1 1
od oq od ld
oq od oq lq
ld ld lq od d
lq lq ld oq q
dV V i i
dt C C
dV V i i
dt C C
d ri i i V U
dt l l l
d ri i i V U
dt l l l
(3)
III. INPUT OUTPUT FEEDBACK LINEARIZATION
CONTROL
This strategy was developed to simplify non linear
control by eliminating non linearities. Contrary to
conventional Jacobean linearization, whose idea is based
on the linear approximation of dynamics around a point,
the feedback concept is an exact state transformation in
closed loop [4]-[7].
Its control principle includes two steps:
To generate a linear input/output relationship by
handling algebraic equations of the state space
model associated to the nonlinear system.
To design controller that enables the application of
linear control.
Since the studied system is a nonlinear multi input
multi output one, its state space model can be written in
matrix form such as:
( ) ( ). (4)x f x g x u (4)
Notice that x is the state vector, u the control vector
and f(x) and g(x) are vector fields. They are defined as
follows:
1
2
3
4
, (5)
od
oq d
qld
lq
Vx
V Uxx u
Ux i
x i
(5)
2 3
1 4
1 3 4
2 3 4
1 1
0 0
1 1 0 0
1( ) , ( ) (6)01
10
1
od
oq
x x ic c
x x ic c
f x g xr lx x x
l l
r lx x xl l
(6)
The main purpose of this paper is to control output
filter voltages to fulfill the desired references whatever
the load is. Accordingly, Vod and Voq are the model
outputs and the output vector y is made up by the first and
second states, equation (7).
1 1
2 2
(7)od
oq
Vy xy
Vy x
(7)
Applying the input output feedback linearization
control, each output is differentiated just one time and the
control variables appear as in (8), [8], [9]:
2 2 2
1 0 1 0 3 4
2
0 1
2 2 2
2 0 2 3 0 4
2
0 2
2 1
(8)2
1
od
oq
od
oq
diy x r x x
C C dt
i uC
y x x r x iC C
diu
C dt
(8)
where 01
l C is the filter corner frequency.
Hence, the relative degree of the outputs is two. From
the above equation, we derive the matrix form of (9):
1 1 1
2 2 2
( ). (9)y a u
E xy a u
(9)
The decoupling matrix E(x) is deduced so that:
20
20
0( ) (10)
0E x
(10)
International Journal of Electronics and Electrical Engineering Vol. 4, No. 1, February 2016
©2016 Int. J. Electron. Electr. Eng. 25
Equation (11) illustrates the new linear relationship
between inputs and outputs.
1 1
2 2
(11)y v
y v
(11)
where v1 and v2 are the new linear control laws.
It can be noticed that E(x) is clearly non singular,
consequently we can calculate the inverter modulation
index in (d, q) frame md and mq via equation (12).
2 2 2
1 0 1 0 3 4
2
0
2 2 2
2 0 2 0 3 4
2
0
2
1
1
(12)2
1
1
d
oddcoq
q
oqdc
od
v x r x xC
mdiV
iC dt C
v x r x xC
mdiV
iC C dt
(12)
Finally, to ensure a zero error between the controlled
dynamics and their references, the appropriate controller
is designed according to equation (13), [10].
2
11 12 12 12
2
21 22 22 12
0
(13)
0
d d d I d
q q q I q
d de k e k e k k e dt
dtdt
d de k e k e k k e dt
dtdt
(13)
where ed = Vod* - Vod, eq = Voq* - Voq and Vod*, Voq*
denote the desired reference voltage signals in (d, q)
frame.
Substituting (3) in (13) leads to:
2* * *
1 11 12 122
*11 1111 12 1
2* * *
2 21 22 212
*21 2122 22 2
-
(14)
od od od Od
Oq ld od I od od
oq oq oq Od
Oq lq Oq I oq oq
d dv V k V k V k V
dtdt
k kk V i i k k V V dt
C C
d dv V k V k V k V
dtdt
k kk V i i k k V V dt
C C
(14)
where kij are the controller parameters determined using
pole placement method.
LoadH-bridge
Multilevel
Inverter
VOA
VOB
VOC
C
r,l
ilB
ilA
ilC
iOA
iOB
iOC
PWM
mA mB mC
3
2
UA UB UC
Linearizing Control
Law
Ud Uq
ABCd q
Feedback
linearization Control
Fd
Fq
ABC
d q
32
ildilq
Vod
Voq* Vod*
Voq
θ
θ
θ
iodioq
θ
Figure 4. Input output feedback linearization control of an H-bridge
multilevel inverter operating in islanded mode.
The block diagram of Fig. 4 depicts the H-bridge/LC
filter power circuit and its control scheme using input
output feedback linearization.
IV. SIMULATION AND RESULTS
We shall test the algorithm developed on two common
cases for voltage control. The first deals with the
deliberate variation of the voltage supplying a passive
load, resistor bench for heating, as example. The second
deals with maintaining the voltage to a nominal value, it
is the case of an induction motor, for instance.
The parameters of simulated system are:
H-bridge multilevel inverter: Vdc= 200V, fsw=
5KHz
LC filter: r= 0.15Ω, L= 2 mH, C= 12µF
Resistive load: R= 30 Ω
Induction motor: 2 pole pairs, Rs=8.21 Ω, Rr=6.73
Ω, Lm=0.9 H, ls=lr=0.0607 H.
All simulations were performed with Matlab Simulink.
A. Case of Passive Load
The effectiveness of the designed controller is verified
for a passive load, a resistance in this case. Fig. 5 shows
that the filter output voltages are perfectly regulated since
they track their references even if they are changed.
This is also valid for Vod and Voq. It can be observed
that the dynamics reach rapidly their references and
present small oscillations, Fig. 6.
(VOA, VOB, VOC) are poor on harmonics and their
waveforms are almost sinusoidal. This is illustrated in Fig.
7 which presents a low total harmonic distortion of 0.81%.
Fig. 8 depicts the evolution of d component linear
control laws used to modulate the multilevel inverter.
0 0.05 0.1 0.15-400
-200
0
200
400
Time (s)
Vo
(A,B
,C)
Figure 5. Output filter voltages.
0 0.05 0.1 0.15-100
0
100
200
300
400
500
Time (s)
Vod
Voq
Vod*
Figure 6. Vod and Voq versus time.
International Journal of Electronics and Electrical Engineering Vol. 4, No. 1, February 2016
©2016 Int. J. Electron. Electr. Eng. 26
0 0.05 0.1 0.15
-200
0
200
Selected signal: 7.5 cycles. FFT window (in red): 1 cycles
Time (s)
0 20 40 60 80 100 120 1400
20
40
60
80
100
Harmonic order
Fundamental (50Hz) = 325.4 , THD= 0.81%
Mag
(%
of
Fu
nd
am
en
tal)
Figure 7. Harmonic spectrum of load voltage.
0 0.05 0.1 0.150
100
200
300
400
500
Time (s)
Ud
(V
)
Figure 8. Dynamic response of Ud.
B. Case of Active Load
It is well known that the mechanical torque of an
induction motor is proportional to the square of the
supply voltage. A slight voltage drop results in a
significant degradation of its torque, hence the necessity
to operate at its rated voltage.
To this end, we imposed the system a constant
reference for obtaining a phase voltage of 230V. Then,
the induction motor is started without load from the rest,
Fig. 9. At time t = 0.35S, the motor is suddenly loaded by
a load torque of 4N. Fig. 10 illustrates the relatively high
inrush current after inserting the load torque and
especially during the first moments of starting. Despite
the voltage drop caused by the filter elements r and l, the
voltage at the motor terminals is maintained perfectly
constant at 230V, Fig. 11. This demonstrates, once again,
the robustness of the control algorithm based on feedback
linearization. It is able to fulfill the requirements even in
severe mode of operations.
0 0.1 0.2 0.3 0.4 0.5 0.6-5
0
5
10
15
Time(s)
Tem
(N)
Figure 9. Evolution of the motor torque.
0 0.1 0.2 0.3 0.4 0.5 0.6-10
-5
0
5
10
Time(s)
ph
ase
cu
rren
t (A
)
Figure 10. Phase motor current.
0 0.1 0.2 0.3 0.4 0.5 0.6-400
-200
0
200
400
Time(s)
ph
ase
volt
age(V
)
Figure 11. Phase motor voltage.
V. CONCLUSION
The paper has presented the control of a 5 level H-
bridge inverter by means of a feedback linearization
algorithm. The developed control algorithm can be
applied with any load without changing the controller
design. The investigation proves the ability and the
effectiveness of the selected algorithm to maintain the
output voltage to the desired value under severe
conditions, either for passive or active loads.
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Habib Rehaoulia received the B.Sc. degree in 1978, the M.Sc. degree in 1980, the PhD degree in 1983, and the habilitation degree in 2007, all
in Electrical Engineering and from the ENSET (Institute of Technical
Sciences), University of Tunis, Tunisia. He joined the teaching staff of the ENSET in 1978.
Pr Habib Rehaoulia is currently professor at the department of Electrical Engineering at ENSIT (National School of Engineers of Tunis),
University of Tunis.
During his career, he was on leave for several months at WEMPEC (University of Madison Wisconsin USA), ENSIEG (University of
Grenoble France), “Lab. d’´electrotechnique” (University of Paris VI France), and the CREA (University of Picardie France). His main
research interests are Analysis, Modeling and Simulation of electrical
machines, Power Electronics, Renewable Energy…
Farhat Fnaiech was born in 1955 in Chebba (Tunisia), he received the
BSc degree in Mechanical Engineering in 1978 from the ENSET High
school of sciences and techniques of Tunis and the master degree in 1980, the Doctorate of 3 cycle degree from the same school in Electrical
Engineering in 1983, and the Doctorate Es Science in Physics from
Faculte des Sciences of Tunis in 1999. He is currently Professor at the Ecole Nationale Superieure des Ingénieurs de Tunis. Pr Fnaiech has
published More than 250 research papers in many journals and
international conferences. He has organized many national and international conferences and has been the general chairman and
member of the international Board committee of many International
Conferences, ICIT 2004, ICELIE 2006-2012, ISIE 2006-2012, IECON 2005- 2012. He is Associate Editor of IEEE Transactions Industrial
Electronics. He has served as IEEE Chapter committee coordination
sub-committee delegate of Africa Region 8 and Vice Chair of IEEE Tunisia Section. He has been appointed as an AdCom member in IEEE
Industrial Electronics Society. He is the head of a big research
laboratory in Signal Image and Energy Management in University of Tunis (100 researchers). His main interest research areas are nonlinear
adaptive Signal processing, nonlinear control of power electronic.
International Journal of Electronics and Electrical Engineering Vol. 4, No. 1, February 2016
©2016 Int. J. Electron. Electr. Eng. 28
Abir Rehaoulia was born in 1986 at Tunis, Tunisia. She received her mathematical baccalaureate in 2004, the BSc degree in 2008, the MSc
degree in 2010 and she is working on the PHD degree since 2011.
She is currently an assistant professor at the department of Electrical Engineering of ENSIT (Ecole Nationale Supérieure des Ingénieurs de
Tunis), University of Tunis, Tunisia.Mrs Abir research interests are Modeling and Simulation of electrical
systems, Power Electronics including Multi Level Inverters (H-bridge,
NPC, FC), Electric Machines and Drives, Advanced Control…
Mahir Dursun was born in 1970, Corum, Turkey. He received the BS degree in 1993, the MSc degree in 1996, and the PhD degree 2002 from
Gazi University, Ankara, Turkey. He is currently an associate professor
at the Department of Electric Machinery Education, Gazi University. His research interests include, Motor Design, Modeling, Motor Control,
Switched Reluctance Motors, Linear Switched Reluctance Motors, Brushless DC motors, DC-DC converters, Matrix Converters, FLC,
Artificial Neural Network, Elevator motors, Motor and Centrifugal
Pump Drivers, DSP, PLC, microprocessors and microcontroller programming, serial and parallel active power filters, and photovoltaic
systems, photovoltaic irrigating systems, RF control and communications, and distance education material design.