SCYR 2010 - 10th Scientific Conference of Young Researchers – FEI TU of Košice
State space controller for bidirectional DC/DC
converter-buck mode
1
Matúš OCILKA, 2Tomáš BÉREŠ
1
Dept. of Electrical, Mechatronic and Industrial Engineering FEI TU of Košice, Slovak Republic
1
ocilka.matus@gmail.com, 2tomas.beres@tuke.sk
Abstract—This paper deals with designing of cascade state
space controller for buck mode of bidirectional DC/DC
converter. The control of converter is decomposed into
outer voltage control and inner current control. First, the
simple circuit of buck converter is created using the state
space
averaging
method
and
simulated
in
Matlab/Simulink. The pole placement method is used to
design the controller.
Keywords—Buck converter, controller, state space model, pole
placement
Fig. 1 The buck converter
First, let the switch position function to be u=1. Using
Kirchhoff’s laws we obtain set of equations
di L
= −u c + V g .u
dt
du
u
C c = iL − c
dt
R
L
I. INTRODUCTION
DC/DC converters are electronic systems which transfers
input voltage to output load. There are many topologies of
non-isolated DC/DC converters as buck, boost, Zeta, Ćuk. In
this paper a buck mode of bidirectional cascade buck/boost
converter is introduced which provides bidirectional flow of
energy. Proposed control of converter is divided into three
modes; buck mode, boost mode and buck/boost mode.
The aim of this paper is to design the controller for buck
mode of converter. There are many control structures as
voltage mode or current mode control, sliding mode control,
delta sigma control, which provide output voltage independent
of load or input voltage variations.
This paper deals with cascade state space controller for
proposed mode of operation.
When diode is in non-conduction mode u=0 the equation
results in
di L
= −u c
dt
du
u
C c = iL − c
dt
R
L
(2)
By comparing the obtained particular dynamic systems
descriptions, we immediately obtain the following unified
dynamic system model. This result in
di L
= −u c + V g .u
dt
du
u
C c = iL − c
dt
R
L
II. DC/DC BUCK CONVERTER
A. Modeling of buck converter
The simple circuit of converter is shown in Fig. 1. The
converter consists of input voltage source, inductor, capacitor,
switch and load resistor. For the sake of clarity the
consideration of the following details shall be omitted: the
influence of ESR (equivalent series resistance) of the output
capacitor and the ohmic contribution of the inductive storage
element on the control behavior; ideal switches are assumed.
This can be done because the basic dynamic system quality is
not affected [5].
(1)
(3)
We usually refer to this model as a switched model
u ∈ 0,1 . The average converter model would be
{ }
represented exactly by the same mathematical model, possibly
by renaming the state variables with different symbols and by
redefining the control variable u as a sufficiently smooth
function taking values in the compact interval of the real line
[0, 1]. In order to simplify the exposition, we shall refer to the
model, with u replaced by uav, as the average model. We shall
however distinguish between the average control input,
denoted by uav and the switched control input, denoted by u.
The average model of the Buck converter is then described by
[3]
SCYR 2010 - 10th Scientific Conference of Young Researchers – FEI TU of Košice
di L
= −u c + V g .u av
dt
du
u
C c = iL − c
dt
R
L
(4)
Modeling using state space averaging is well known method
since many years [4]. The state space averaged model of the
converter can be expressed as
y = Cx
e 
b  x   0 
   +  u 2 +   z
0  v 2   K 2 
0
(10)
C. Voltage loop
According to Fig.2. the voltage control loop can be written
as follows
B. State space model
x& = Ax + bu
 x&   A − br2 T
v&  = 
T
 2  − K 2 c2
(6)
T
T
u1 = v1 − r1 x − d 1v 2 ; v&1 = K 1 (u1 − c1 x)
(11)
State space equations of the new system result in
T
A − br1
 x&  

 v&  = − K c T − K r T
2 2
2 1
 1 
T

− K 1c1
v&2  
b
K 2 d1
0
0  x   0 
e

K 2  v 2  +  0 u1 + 0 z
0   v1   K 1 
0
(12)
The state variables are the capacitor voltage and inductor
current. Vector of state variables is then
x = [u c , i L ] ; y = u c
T
(7)
The matrices of the system can be written as follows
 −1

A =  RC
−1

 L
1
 Vg 
C  ; b =   ; c = [1 0]

L

0
0

(8)
D. Pole placement
The main advantage of state space controllers is pole
placement. This method allows placing poles of the system to
obtain desired outputs. In this case we choose the damping
factor ξ and time of regulation tr. The natural frequency can be
obtained from equation below
ω0 =
[
1
3 − 0,5 ln(1 − ξ 2 )
ξ .t r
]
(13)
The poles of the system
III. DESIGN OF CONTROLLER
A. Control structure
The general block diagram of cascade state space controller
is shown in Fig. 2.
s12 = −ξ .ω 0 ± jω 1 − ξ 2
(14)
If system order is higher than two the rest of the poles are
chosen as
si = − Nξω 0 ; i=3…..n
(15)
The desired polynomial is then
P( s) = s n + f n −1s n −1 + ... + f1s + f 0
Characteristic polynomial of the system can be obtained
from state space model of system (10), (12)
Fig.2 General block diagram of controller
Vector rT realizes proportional gain of state vector and
constants Kj are gain of the integrators. The control of buck
mode is decomposed in outer voltage control loop and inner
current control loop.
B. Current loop
First, consider the current equation of converter as inner loop
of structure
T
x& = A.x + b.u + ez = Ax + b(v 2 − r1 x) + ez
T
v&1 = K 2 (u 2 − c 2 x)
State space equation of current loop
(16)
(9)
P(λ ) = det( sI − A + br T )
P(λ ) = λn + f n −1 (r )λn −1 + .. + f 1 (r )λ + f 0 (r )
(17)
By comparing between the characteristic polynomial (17) of
system with the desired polynomial (16) gives the design
equations of the feedback coefficients [1],[2].
E. Controller scheme
The designed controller was created in Matlab/Simulink
according to equations (10), (12). Vref is reference value of
output voltage.
SCYR 2010 - 10th Scientific Conference of Young Researchers – FEI TU of Košice
Fig.3 State space controller of buck converter
IV. SIMULATION RESULTS
The controller was simulated in Matlab/Simulink to verify
the properties of proposed controller and parameters of
converter used in simulations are L=47 µH,, C=40 µF,
R=1.2Ω.
Damping factor and time of regulation for voltage loop
ξ=0.85, tr=1ms and for current loop ξ=0.85, tr=0,5ms. The
desired poles for current loop are s12 =-3.6410.104 ±
j2.2565.104, s3 =-2.5487.105 and poles of the voltage loop s12
= -3.3367.103 ± j3.4041.103, s3 = -1.0010.104, s4 = 1.3347.104.
The coefficients of current loop are then K2 = 3.5932.103,
r2 = 0.1426 and voltage loop coefficients K1 =367.2795, d1 =
-12.9531, r12 = 0.9113, r11 = 0.6583.
The performance of the converter with state space
controller is shown in Fig.4.
Fig.5 Performance of converter with load step (Iz = 2A)
V. CONCLUSION
In this paper the cascade state space controller for buck
mode of bidirectional DC/DC converter was designed using
state space averaging and pole placement method. The state
space controllers advantages are pole placement and easy
implementation by setting the time of regulation and damping
factor. The proposed control structure was finally tested in
Matlab/Simulink. Future work will be to design controllers for
boost and buck/boost mode and compare all three modes with
state space controllers to PI regulators.
ACKNOWLEDGMENT
This work was supported by Slovak Research and
Development Agency under project APVV-0095-07 and by
Scientific Grant Agency of the Ministry of Education of
Slovak Republic under the contract VEGA No.1/0099/09.
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Fig.4 Performance of converter (Vref=12V)
[7]
The reference of output voltage was set to 12V with
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voltage and current loop. The simulation of the load step is
shown in Fig.5. The load step causes a variation 1V in the
output voltage and voltage settle down to its reference value in
1ms.
[8]
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