ISSN: 1693-6930
TELKOMNIKA Vol. 14, No. 2, June 2016 : 497 – 506
498 in order to obtain improved dynamic model for effective controller design. The digital controllers
are designed using the direct digital design and digital redesign approach using the Matlab- SISO_tool control design tools. The effectiveness of the controllers on the boost converter
demonstrated fast transient response with improves dynamic performance necessary for eliminating voltage fluctuations associated to highly variable voltage source like solar
photovoltaic system.
2. Design and Modelling of Boost Converter
A DC-DC boost converter is a non-isolation power converter that produces output voltage larger than the input voltage. Figure 1 shows the power circuit module and the controller
unit which can be implemented using digital signal controller DSC. The two basic control methods are the voltage and current control mode. The voltage mode control is a single loop
control system with only output voltage used as control signal, while current control mode is a multi-loop control with inner current feedback loop in addition to the outer voltage feedback [5,
14]. The schematic of the boost converter shown in Figure 1 is a voltage mode control, the output voltage
which serves as feedback is sensed by opt-isolator and transformed to
discrete values via analogue to digital converter ADC. The sensed voltage is compared with the desired reference voltage
to generate error signal . The error is then processed by
a compensator block to determine the duty cycle of pulse width modulation PWM which controls the converter switching
‘on’ and ‘off’ period.
i
ct Vct
+ -
i
outt
C D
L
Compensator Hybri d
DPWM
H
Digital Signal Processor
1
Q
PWM
V
out
t
V
ref
e
ADC V
o
t
R
L
i
Lt
V
in
dt
i
int
Figure 1. Boost converter with digital controller
2.1. Steady State Design
To study the dynamic behaviour of dc-dc boost converter with respect to the digital controllers, the steady state and dynamic modelling for continuous current mode CCM
operation of non-ideal boost converter are presented in sub-section 2.1 and 2.2. Table 1 shows the summary of the steady state design example used as case study in the project.
Table 1. Steady state parameters
Parameters Symbol Value
Input voltage
in
V
12 [V] Output voltage
out
V
48 [V] Output voltage ripple
o
V
500 [mV] Rated power
o
P
500 [W] Average output current
o
I
9.6 [A] Switch on resistance
on
R
0.1 [ Ω]
Parasitic inductor resistance
L
r
0.01 [ Ω]
Parasitic capacitor resistance
C
r
0.01 [ Ω]
Load resistance
L
R 5 [
Ω] Duty cycle
d 0.75
Inductor
L
90 [µH] Output capacitor
C
100 [µF] Switching frequency
s
f
100 [kHz]
TELKOMNIKA ISSN: 1693-6930
State-space Modelling and Digital Controller Design for DC-DC Converter Oladimeji Ibrahim 499
The details of the steady state design equations for dc-dc boost converters have been presented in [4-6]. A 500W, 48V boost converter was design with 12V nominal input and
50 input variation. The switching frequency of pulse width switching control is 100 kHz to ensure high power density by reducing filter component size and the load was modelled as
resistance. The on resistance of the power switch, the internal resistance of filter inductor and
capacitor equivalent resistance were all considered for improved modelling and analysis of the boost converter.
2.2. Dynamic State Model Switching converter is a periodic time-variant system that can be modelled using circuit
averaging or state space averaging technique [6]. The dynamic models of converters are useful for controller design, predicting the system stability margin, and for studying transient response
to supplyload perturbation [6]. It helps to predict dynamic behaviour of the converter prior to system prototyping to reduce design cycle time and cost. The state space average model
method is used to model the non-ideal boost converter considering components parasitic parameters [15, 16]. This modelling technique gives a complete converter model with both
steady state DC and dynamic AC quantities and the transfer functions of the converter are readily obtainable for dynamic analysis of the system.
Based on block diagram in Figure 1, the switching period for complete cycle is denoted is
s
f T
1
and
d
is the duty cycle. The switch total on-time is
dT t
on
and off-time is
T d
t
off
1
.
When switch is
on at time interval
dT
, by applying KVL and KCL, the differential state variables for boost the converter are derived as follows:
L r
R t
i L
t V
dt t
di
L on
L in
L
1
c L
L c
out
r R
R t
V t
V
2
c L
r R
C t
Vc dt
t dVc
3
The state space representation for the on-state is:
1 1
t V
L t
V t
i
r R
C L
r R
dt t
dV dt
t di
in c
L
c L
L on
c L
4
t V
t i
r R
R t
V
c L
c L
L out
5
Similarly, when switch
1
Q
is in of-state at time interval
T D
1
, applying KVL and KCL, the differential state variables for boost the converter are derived as follows:
t V
r t
i t
V dt
t di
L
out L
L in
L
6
c
r
L L
c L
L c
out
R t
i r
R R
t V
t V
7
ISSN: 1693-6930
TELKOMNIKA Vol. 14, No. 2, June 2016 : 497 – 506
500
L r
R R
t V
L R
r t
i L
t V
dt t
di
c L
L c
L L
L in
L
r
c
8
t i
t i
t i
out L
c
9
c L
c L
c L
L
r R
C t
V t
i r
R C
R dt
t dVc
10
The state space representation for the off state is:
1 1
t V
L t
V t
i
r R
C r
R C
R r
R L
R L
r R
r
dt t
dVc dt
t di
in c
L
C L
c L
L c
L L
c L
L L
11
t V
t i
r R
R r
R t
V
c L
c L
L c
L out
12 For dynamic state analysis and controller design for the boost converter, the converter
continuous-time transfer function s
G is obtained by performing state averaging on the state
equations using the duty cycle
d
as weighting factor. The averaged state space equations for a converter are given by 13 and 14 [3], [17-18].
t u
B t
x A
t x
13
t x
C t
y
14 Where, the state averaged matrix are defined as
1
2 1
d A
d A
A
,
1
2 1
d B
d B
B
and
1
2 1
d C
d C
C
. The term
t x
is the converter DC state vector defined as inductors currents and capacitors voltages,
t u
is converter DC input vector and
t y
is the converter DC output vector.
The averaged matrix based on steady state parameters in Table 1 is derived as:
1
2 1
d A
d A
A
15
23 .
1996 01
. 2495
23 .
2772 17
. 972
A
16
1
2 1
d B
d B
B
17
1
. 11111
B
18
TELKOMNIKA ISSN: 1693-6930
State-space Modelling and Digital Controller Design for DC-DC Converter Oladimeji Ibrahim 501
1
2 1
d C
d C
C
19
9985 .
002495 .
C
20 Using the Matlab state-space
ss to transfer-function tf, the converter control to output continuous-time transfer function is obtained as 21:
07 2
06
27.72 2.774
2968 8.857
s e
G s s
s e
21
3. Digital Compensator
