9th International Conference on Power Electronics and Motion Control - EPE-PEMC 2000 Košice
ROBUST SLIDING MODE CONTROL OF MATRIX CONVERTERS
WITH UNITY POWER FACTOR
S. Ferreira Pinto, J. Fernando Silva
Instituto Superior Técnico (IST), Máquinas Eléctricas e Electrónica de Potência, Av. Rovisco Pais
1096, Lisboa Codex, Portugal
fax: +351-21 841 71 67; e-mail: pcsoniafp@alfa.ist.utl.pt fernandos@alfa.ist.utl.pt
Košice
Slovak Republic
2000
Abstract. This work presents the design of robust sliding mode controllers for three-phase ac-ac
matrix converters, in order to guarantee output sinusoidal waveforms and near unity input power
factor.
First, the sliding mode controller, based on the state-space vectors modulation technique, is designed
for the matrix converter and the obtained results are compared to those obtained using the classical
Alesina / Venturini method.
Then, a new sliding mode controller is designed, considering the matrix converter with a high
frequency ‘LC’ filter. A new integrated approach is necessary to design the controller, using the
matrix converter model in ‘αβ’ coordinates and the state-space vectors modulation technique.
The obtained results show that sliding mode controllers guarantee a robust on-line control of the
matrix converter output voltages, with near unity input power factor.
Keywords: Matrix converters, Sliding mode control.
connection of each one of the three output phases to one of
the three input phases, as shown in Fig. 1.
1. INTRODUCTION
With the recent progress of power devices and the
development of large power integrated circuits, matrix
converters have become extremely attractive as they are
capable of providing simultaneously sinusoidal output
voltages with varying amplitude and frequency as well as
sinusoidal input currents with near unity input power factor.
Besides, they allow bi-directional power flow, have noise
above the audio range and have minimum energy storage
requirements, that results in a higher efficiency, when
compared with the previously used rectifier-inverter
structures (which have an intermediate DC link) [4,6,8].
Matrix converters are useful in many applications,
especially ac drives and, after the introduction of their high
frequency control in 1980 by Alesina and Venturini [1,2]
several
control
techniques
have
been
studied
[3,4,5,9,11,12,13]. One of the most recent is the Space
Vector Modulation technique [6,7,8,10,14,15,16], with
several advantages over the previous control methods such
as immediate comprehension of the required commutation
processes, simplified control algorithm and maximum
voltage transfer ratio without adding third harmonic
components. In this paper, the association of sliding mode
controllers, known for their robustness and system order
reduction, together with the State-Space vector modulation
technique, will allow the design of a robust controller for
the matrix converter.
2. MATRIX CONVERTER MODEL
The matrix converter is made by an association of nine bidirectional switches, with turn-off capability, that allow the
i
va a
vb
vc
ib
S11
S21
S31
S12
S22
S32
S13
S23
S33
ic
iB
iA
vAB
iC
vBC
vCA
Fig. 1. Matrix converter topology
However, not all the switches combinations are possible:
there are constraints to guarantee that the input phases are
never short-circuited and that the output currents are never
interrupted (due to the presence of inductive loads) [6].
Considering these constraints, only 27 switching
combinations are possible [6] and are schematically
represented according to a nine-element matrix (1). Each
one of this matrix elements represents one switch with two
possible states: ‘1’ if it is closed and ‘0’ if it is open and the
referred constraints are verified if the sum of each one of
this matrix rows elements is always equal to '1' [6].
 S11 S12 S13 
S =  S 21 S 22 S 23 
(1)
 S 31 S 32 S 33 
The matrix which relates the input phase to the output line
currents and voltages [6] (2) is obtained from (1):
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9th International Conference on Power Electronics and Motion Control - EPE-PEMC 2000 Košice
 S11 − S 21
Sc =  S 21 − S 31
 S 31 − S11
S12 − S 22
S 22 − S 32
S 32 − S12
S13 − S 23 
S 23 − S 33 
S 33 − S13 
(2)
In order to guarantee that the output voltages follow the
predefined reference, it is necessary that, in a certain
switching period ‘T’, the output voltages average values
(5,6) are equal to their reference average values voα
and
ref
The matrix which relates the output line voltages to the
input line voltages (3) can be obtained from (2) [16].
 S11 − S21 − S12 + S22 S12 − S22 − S13 + S23 S13 − S23 − S11 + S21 


3
3
3
S − S − S + S

21
31
22
32 S 22 − S 32 − S 23 + S33 S 23 − S33 − S 21 + S 31 

S' =
3
3
3


 S31 − S11 − S32 + S12 S32 − S12 − S33 + S13 S33 − S13 − S31 + S11 


3
3
3


(3)
Considering the previously defined matrices (1,2,3), it is
possible to establish relations for the output line voltages
[16] and for the input phase currents (4).
v ab 
v AB 
 


v BC  = S ' vbc 
vCA 
v ca 
ia 
i  = S T
 b
ic 
i A 
i 
 B
iC 
ref
. This is valid if the sliding mode controller
commutation frequency is much higher than the controlled
voltage frequency.
Making the derivatives of (5,6) we can write the system
equations in the phase canonical form (7,8).
d voα 1
= voα
(7)
dt
T
d voβ
dt
=
1
vo
T β
±2, ±5, ±8
±1, ±2, ±3
The sliding surfaces are expressed as linear combinations of
all the phase canonical state variables that do not depend
directly on the command variables (9,10).
− voα  = 0
S ( evo , t ) = K α  v oα
(9)
α
 ref

− voβ  = 0
S ( evo , t ) = K β  voβ
β
 ref

β
±3, ±6, ±9
±1, ±4, ±7
(10)
The two necessary sliding surfaces (11,12), expressed as
functions of the output voltages and their reference values
are easily obtained from (9,10).
K T
− voα ) dt = 0
S ( evo , t ) = α ∫ ( v oα
(11)
ref
α
T 0
S ( evo , t ) =
±7, ±8, ±9
Output line voltage vectors
(8)
(4)
In order to use the State-Space Vector modulation
technique, it is necessary to express the 27 previously
referred switching states in αβ co-ordinates, using the
Concordia transformation [6]. However, as six of these
states have time varying phase and will not be used [6] and
three other combinations correspond to the null state, this
leaves only 18 possible non-null switching combinations
that can be represented in the ‘αβ’ plane according to Fig.
2. [6].
±4, ±5, ±6
voβ
Kβ T
− voβ ) dt = 0
∫ ( vo
T 0 β ref
(12)
According to the number of state-space vectors available at
each time instant, two three level comparators are
necessary. The results of S(evo ) and S(evo ) are applied to
Input phase current vectors
α
Fig. 2. Representation in the ‘αβ’ plane of the output line
voltage and the input current state-space vectors
3. DESIGN OF THE SLIDING MODE
CONTROLLER FOR THE MATRIX CONVERTER
Output voltage control
The Concordia transformed output voltages (voα, voβ) are
discontinuous functions of the control variables and, in the
specific case of the matrix converter, with no associated
dynamics. The average values of these voltages ( voα , voβ )
may be calculated, in ‘αβ’ co-ordinates, according to (5,6).
voα =
1T
∫ v o dt
T 0 α
(5)
voβ =
1T
∫ v o dt
T 0 β
(6)
β
the comparators, and according to each one of the nine
possible error voltages, one of the two nearest and highest
amplitude state-space vectors available at that time instant is
applied to the matrix converter (Fig. 2., Tab. 1.). The
existence of these two vectors, capable of guaranteeing the
desired output voltage, introduces an extra degree that will
be further used to control the input power factor.
Input power factor control
Using the Blondel-Park transformation, when the rotating
frame is synchronized with the mains, the ‘q’ component is
zero. In order to guarantee the near unity input power factor,
it is necessary that the three input currents have exactly the
same phase as the corresponding input voltages (in ‘abc’
coordinates). Using the Blondel-Park transformation, this
condition is verified when the input current component ‘ i q ’
is equal to zero. The ‘ id ’ current component is
automatically adjusted in order to guarantee the input/output
power constraint.
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9th International Conference on Power Electronics and Motion Control - EPE-PEMC 2000 Košice
evα
φu >-π/6
φu <π/6
evβ
-1
-1
-1
0
0
0
1
1
1
-1
0
1
-1
0
1
-1
0
1
φu >π/6
φu <3π/6
φu >3π/6
φu <5π/6
φu >5π/6
φu <7π/6
φu >7π/6
φu <9π/6
φu >9π/6
φu <11π/6
eiq=1
eiq= -1
eiq=1
eiq= -1
eiq=1
eiq= -1
eiq=1
eiq= -1
eiq=1
eiq= -1
eiq=1
eiq= -1
+3
+3; -6
-6
-9
0
+9
+6
-3; +6
-3
-1
-1; +4
+4
+7
0
-7
-4
+1; -4
+1
-2
-2; +5
+5
+8
0
-8
-5
+2; -5
+2
+3
+3; -6
-6
-9
0
+9
+6
- 3; +6
-3
+1
+1; -4
-4
-7
0
+7
+4
-1; +4
-1
-2
- 2; +5
+5
+8
0
-8
-5
+2;- 5
+2
-3
-3; +6
+6
+9
0
-9
-6
+3;-6
+3
+1
+1; -4
-4
-7
0
+7
+4
-1; +4
-1
+2
+2; -5
-5
-8
0
+8
+5
-2; +5
-2
-3
-3; +6
+6
+9
0
-9
-6
+3; -6
+3
-1
-1; +4
+4
+7
0
-7
-4
+1; -4
+1
+2
+2; -5
-5
-8
0
+8
+5
-2; +5
-2
Tab. 1. Output voltages and input currents Space Vector control (φu represents the input voltage instantaneous phase in
‘αβ’ coordinates)
As for the output voltages, the input currents are also
discontinuous variables with no associated dynamics. For
this reason, the current controller design laws are similar to
those obtained for the output voltage: assuming that it is
possible to obtain an input current whose average value in a
certain switching period ‘T’ is equal to the reference
current, the sliding surface to the ‘ i q ’ current is designed
according to (13) [15].
T
S ( ei q ) = Kq ∫ eiq dt = 0
(13)
0
At each time instant, the measured ‘ i q ’ current is compared
to a zero reference, then the sliding surface is applied to this
current error ( ei q ) and the final result is applied to a twolevel comparator. According to the comparator output, it is
chosen the best of the two state-space vectors available.
Simulation results
Two control methods were simulated and compared: the
proposed sliding mode controller and the classical Alesina
and Venturini [1,2]. In order to test the robustness of the
sliding mode controller, the results were obtained to a 20%
increase in the mains voltages at t=0.05s. The reference
voltage was chosen to have an amplitude of 2 × 150 V and
a frequency of fo=60Hz. An inductive L=10mH and
resistive load R=10Ω were considered. The MATLAB/
SIMULINK simulations show that both methods present
good results in steady-state operation (t<0.05s) (the output
line voltages and input phase current follow the references,
guaranteeing the near unity input power factor). The
proposed controller (Fig. 3. a, b, c) presents a good response
to the increase in the mains voltages. The input phase
current amplitude decreases at t=0.05s (Fig. 3.a) as a
consequence of the input-output power constraint.
Also, the input power factor as well as the output line
average voltages and the output phase currents (Fig. 3.b, c)
keep unchanged, as expected. The results obtained using
Venturini controller (Fig. 3.d, e, f) were not satisfactory.
This controller does not have a good line rejection factor as
a consequence of the predefined controller parameters. At
the instant t=0.05s the amplitude of the output voltages and
currents, increase (Fig. 3.e, f), when it should keep
40
40
600
30
30
400
20
20
vAB [V]
iia [A]
0
-10
iA [A], iB [A], iC [A]
200
10
0
-200
-20
10
0
-10
-20
-400
-30
-30
-600
-40
0.02
0.03
0.04
0.05
t [s]
0.06
0.07
0.08
0.02
0.03
0.04
a)
0.05
t [s]
0.06
0.07
-40
0.02
0.08
0.03
0.04
b)
40
0.05
t [s]
0.06
0.05
t [s]
0.06
0.07
0.08
c)
40
600
30
30
400
20
20
vAB [V]
iia [A]
0
-10
iA [A], iB [A], iC [A]
200
10
0
-200
-20
10
0
-10
-20
-400
-30
-30
-600
-40
0.02
0.03
0.04
0.05
t [s]
0.06
0.07
0.08
0.02
d)
0.03
0.04
0.05
t [s]
0.06
0.07
-40
0.02
0.08
e)
0.03
0.04
0.07
0.08
f)
Fig. 3. AC-AC matrix converter waveforms obtained for a 20% increase in the mains voltages with an output reference
voltage of 2 × 150 V and fo = 60Hz, using the proposed sliding mode controller a) b) c) and the Alesina and
Venturini controller d) e) f); a) d) Input phase current 'ia'; b) e) Output line voltage 'vAB'; c) f) Output phase
currents 'iA', 'iB', 'iC'
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9th International Conference on Power Electronics and Motion Control - EPE-PEMC 2000 Košice
0 
 0
1
 0

l
0


i
1
1   α 
+ −
+ 0
 2C 2 3C  iβ  

0
1
1 
−
−

0
2C 

 2 3C
unchanged. As a result, the input currents amplitude also
increases in order to guarantee the input-output power
constraint (Fig. 3.d).
4. INTEGRATED SLIDING MODE CONTROLLER
FOR A MATRIX CONVERTER WITH AN INPUT
‘LC’ FILTER
The introduction of the ‘LC’ filter is responsible for a new
approach to control ac-ac matrix converters (Fig. 4.). The
controller is now designed based on the integration of the
switches with the input filter [16]. The output voltage
controller will no longer depend directly on the mains
voltages – it will depend on the voltages applied to the
switches (the capacitor voltages) (Fig. 4.). Also, the input
current controller will have to be designed considering the
effect of the filter, that introduces some differences both in
phase and amplitude between the currents actually
consumed from the mains iia , iib , ii c and the currents at
(15)
The sliding mode controllers will be designed based on this
model.
via
iia
r
l
ia
S 11
iA
S 12
vab
vib
Cab
S 21
iib
r
l
ib
r
l
S 31
S 32
ic
S 33
iB
Load
S 23
Cbc
iic
vCA
vAB
S 22
vca
vbc
vic
Load
S 13
Cca
the switches inputs ia , ib , i c (Fig. 4.).
Matrix converter model
The state-space model (14) of the input filter can be directly
obtained from Fig. 4.
 diia   r
1
1 

 −
0
0 −
0
dt
3l
3l 

  l

di
r
1
1
i
 b   0 −
−
0  iia 
0
 dt  
l
3l
3l
 
 di  
r
1
1  iib 
i
c
−
−  i 
0
0

  0
l
3l
3l  ic  +
 dt  = 

1
1
 dv ab  
−
0
0
0
0  v ab 

  3C
3C
 v bc 
 dvdt  
1
1
 
bc
−
0
0
0
0

 
 v ca 
C
C
3
3
 dt  

1
 dv ca   − 1
0
0
0
0 
 dt   3C
3C

1

0
0 
 0
 l 0 0
 0

0
0 
 1 

0 0 
 0
0
0  i
 
 1
  a   l  via 
1


−
0
1




+ 3C 3C
ib +
(14)
vi 

   0 0 l   b 
1
1


v
i
 0 −
  c  0 0 0   i c 



3C 3C 
0 0 0 
 1
1 
0 −




3C 
 3C
0 0 0 
The state-space model can be written according to (15),
using the Concordia transformation [16].
 diiα   r
1
1 
−
−
0

  −

l
2l
2 3l 
 dt  
di
r
1
1  iiα 
 iβ  


−
−

  0
l
2l  iiβ 
dt
l
2
3

=
 v  +
 dv c α   1 − 1
0
0   cα 
 dt   2C
 v cβ 
2 3C

 dv   1

1
c
β  

0
0 
2C
 dt   2 3C


0
1  viα 
 
l  viβ 
0
0 
v
BC
iC
Load
Fig. 4. AC-AC Matrix Converter with input ‘LC’ filter
Output voltage sliding mode control
The output voltage sliding mode controller is designed as in
section 3, considering that the switching vectors are
calculated based on the capacitors voltages [16].
Input power factor control
Following the same line of thought of section 3, in order to
control of the ii q current, it is necessary to obtain the state
space model (15) in ‘dq’ co-ordinates (16), using the
Blondel-Park transformation.
 diid   r
1
1 
ω
−
−

  −

l
2l
2 3l 
 dt  
di
r
1
1  iid 
 iq  


−
ω
−
−

 
l
2l  iiq 
dt
l
2
3
 dv  = 
 v  +
 cd   1 − 1
0
ω   cd 
 dt   2C
 v cq 
2 3C

 dv   1

1
 cq  
−ω
0 
2C
 dt   2 3C

0 
 0
1 
 0

 l 0
0


1
1  id   1  vid 
(16)
+ −
+ 0   
 2C 2 3C  i q   l  viq 
 

0 0
1
1 
−
−

0 0
2C 


 2 3C
To design the sliding mode controller, the system equations
are written in the phase canonical form (17).
 diiq

=θ
 dt

 dθ = − r θ +  − ω2 − 1  i + ωr i + 1 i +
i
i
q
 dt
l
l d 3lC
3lC  q

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9th International Conference on Power Electronics and Motion Control - EPE-PEMC 2000 Košice
1 dv iq
ω
ω
ω
v cd +
v cq − v id +
(17)
l
l
l dt
3l
Knowing that the system must follow a predefined
reference, with an error as close to zero as possible, new
state error variables, such as the input current error eii are
+
q
defined (18).
eii = ii q
q
ref
− ii q
(18)
As the input current error eii command variable is the iq
q
current, the sliding surface is expressed as a linear
combination of all the phase canonical state variables that
do not depend directly on the referred command variable.
As a consequence, the sliding mode controller (16) will be
one order lower than the system order written in the phase
canonical form (19).
S ( eii ) = eii + k θ eθ
(19)
q
q
Doing all the necessary algebra, it is possible to express the
sliding surface (19) as a combination of the input filter
currents and voltages (20).
r

S ( eii ) = iiq
+ k θ θ ref − 1 − k θ  iiq +
q
ref
l

+ k θ ω iid −
kθ
nominal values but does not present a robust response
concerning line supply variations.
As a conclusion, the designed sliding mode controllers are
robust and can be implemented on-line without any special
circuits. Associated with the state space vectors they allow a
quick and efficient choice of the correct switches
combinations and can operate at near unity input power
factor.
v cd +
kθ
k
v cq − θ viq
l
2l
(20)
2 3l
As before (section 3), the result of S ( eii ) is applied to a
q
two-level comparator. According to this comparator output,
it is chosen the best of the two state space-vectors available
[14,15,16].
Simulation results
In order to test the robustness of the proposed sliding mode
controller, the results were obtained to a 20% increase in the
mains voltages at t=0.05s (Fig. 5.) with a reference voltage
of 2 × 150 V and fo=60Hz. An inductive L=10mH and
resistive load R=10Ω, and an input filter of l=1mH, r=3Ω
and C=10µF were considered.
The MATLAB/ SIMULINK simulations show that the
controller presents good results to the increase in the mains
voltages. The amplitude of the input phase current decreases
at t=0.05s (Fig. 5.a) in order to guarantee the input-output
power constraint and the capacitors voltages increase, as
expected (Fig. 5.b). Also, the input power factor keept
unchanged, as well as the output line average voltages and
the output phase currents (Fig. 5.d, e).
5. CONCLUSIONS
This paper presents the design of sliding mode controllers
for ac-ac matrix converters: first, for the matrix converter
alone and then for the matrix converter with an input ‘LC’
filter. The introduction of this filter is responsible for a new
integrated approach to control these converters.
The sliding mode on-line controllers were tested and the
results were the expected as they presented good and robust
responses. On the contrary, the classical Alesina/Venturini
method, based on predefined values, works well for the
6. REFERENCES
[1] Alesina, A.; Venturini, M.; "Solid-State Power
Conversion: A Fourier Analysis Approach to
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Transformer
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no 4, April 1981, pp. 319-330.
[2] Alesina, A.; Venturini, M.; "Analysis and Design of
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[3] Watthanasarn, C.; Zhang, L.; Liang, D. T. W.;
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[5] Holmes, D. G.; Lipo, T. A.; "Implementation of a
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[6] Huber, L.; Borojevic, D.; Burany, N.; "Analysis,
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[7] Wiechmann, E. P.; Espinoza, J. R.; Salazar, L. D.;
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[8] Casadei, D.; Grandi, G.; Serra, G.; Tani, A.; "Space
Vector Control of Matrix Converters with Unity Input
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[9] Kaserani, M., Ooi, B. T.; "Feasibility of Both Vector
Control and displacement Factor Correction by
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IEEE Transactions on Industrial Electronics, Vol. 42,
no 5, October 1995, pp. 524-530.
[10] Huber, L.; Borojevic, D.; "Space Vector Modulated
Three-Phase to Three-Phase Matrix Converter with
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[11] Oyama, J.; Xia X.; Higuchi, T.; Yamada, E.; "Effect of
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9th International Conference on Power Electronics and Motion Control - EPE-PEMC 2000 Košice
40
40
600
30
30
400
20
10
0
-10
-20
200
10
ia [A]
Capacitors voltages [V]
iia [A], via/10 [V]
20
0
0
-10
-200
-20
-400
-30
-30
-600
-40
0.02
0.03
0.04
0.05
t [s]
0.06
0.07
0.08
0.02
0.03
0.04
a)
0.05
t [s]
0.06
0.07
-40
0.02
0.08
0.03
0.04
b)
0.05
t [s]
0.06
0.07
0.08
c)
40
600
30
400
20
10
iA, iB, iC [A]
vAB [V]
200
0
0
-10
-200
-20
-400
-30
-600
0.02
0.03
0.04
0.05
t [s]
0.06
0.07
-40
0.02
0.08
d)
0.03
0.04
0.05
t [s]
0.06
0.07
0.08
e)
Fig. 5. AC-AC matrix converter waveforms for a 20% increase in the mains voltages with an output reference voltage
amplitude of 2 × 150 V and frequency fo = 60Hz; a) Input phase current iia and input phase voltage va / 10; b)
Input filter capacitors voltages vab, vbc, vca; c) Input filter intermediate current ia; d) Output line voltage vAB; e)
Output phase currents iA, iB, iC
[12] Matsuo, T.; Bernet, S.; Colby, R. S.; Lipo, T.;
"Modelling
and
Simulation
of
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1996, Saint-Nazaire, France.
[13] Oyama, Jun; Xia X.; Higuchi, T.; Yamada, E.;
"Displacement Angle Control of Matrix Converter";
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[14] Casadei, D.; Serra, G.; Tani, A.; "The use of Matrix
Converters in Direct Torque Control of Induction
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[15] Pinto, S.; Silva, J.; ‘Modeling, Simulation and Sliding
Mode Control of Matrix Converters with Sinusoidal
Input/Output Waveforms and Near Unity Input Power
Factor’; Proc. ELECTRIMACS'99 Conference, Vol. 1,
pp. 1.139-1.144, September 1999, Lisboa, Portugal.
[16] Pinto, S.; Silva, J.; ‘Sliding Mode Control of Space
Vector Modulated Matrix Converter with Sinusoidal
Input/Output Waveforms and near Unity Input Power
Factor’, Proc. EPE’99 Conference, September 1999,
Lausanne, Switzerland.
THE AUTHORS
Sónia Ferreira Pinto received the Dipl.
Ing. degree and the M.Sc. degree in
electrical and computer engineering in
1992 and 1995, respectively, from the
Instituto Superior Técnico, Universidade
Técnica de Lisboa, Lisbon, Portugal,
where she is currently working towards the Ph. D. degree.
She is currently an Assistant at Electrical Engeneering
Department and researcher at Centro de Automática,
Instituto Superior Técnico. Her main interests are modeling,
simulation and control of power converters.
J. Fernando A. Silva, born in Monção
Portugal in 1956, received the Dipl. Ing. in
Electrical Engineering (1980) and the Doctor
Degree in Electrical and Computer
Engineering (Power Electronics and Control)
in 1990, from Instituto Superior Técnico
(IST), Universidade Técnica de Lisboa
(UTL), Lisbon, Portugal. He is currently Associate
Professor of Power Electronics at IST, teaching Power
Electronics and Control of Power Converters, and
researcher at Centro de Automatica of UTL. His main
research interests include power semiconductor devices,
modelling and simulation, new converter topologies and
sliding mode control of power converters. He has written
more than a hundred papers.
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