Feedback Linearization Control of an H-Bridge Multi Level ... · Abstract—The present paper concerns the control of a multi level inverter in autonomous operation. The inverter

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)


©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.



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.



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.

Documents

Feedback Linearization Control of an H-Bridge Multi Level ... · Abstract—The present paper concerns the control of a multi level inverter in autonomous operation. The inverter