Download Controller design of DCDC buck converter basedon feedback linearization with integrator

Controller design of DC/DC buck converter based
on feedback linearization with integrator
Miklós Csizmadia
Miklós Kuczmann
Department of Automation,
Research Centre for Vehicle Industry
Széchenyi István University
Győr, Hungary
Email: [email protected]
Department of Automation,
Research Centre for Vehicle Industry
Széchenyi István University
Győr, Hungary
Email: [email protected]
Abstract—The paper presents the feedback linearization technique of voltage mode DC/DC buck converter. The nonlinear
state-space model is presented and has been stabilized by pole
placement technique and finally the voltage control of output
has been realized by integral control. The designed controller
has been simulated by MATLAB. Results are shown that the
developed controller has good dynamic and static behavior.
I.
II.
S TATE S PACE M ODEL OF B UCK C ONVERTER
The simplest form of SMPS (Switched-mode power supply) is the buck converter which can step down the voltage.
The circuit diagram of a buck converter is shown in Figure
1, where SW and D are representing the power electronics
switch and the diode, respectively. The input DC voltage is
presented by Vi , the load resistance is shown by R, and the
output voltage is UR [10].
I NTRODUCTION
An integral part of electric systems are DC/DC converters
which and widely used in lot of applications. These circuits
are provides regulated DC voltage in the output(s) at different
levels. Nowadays most power electronics system are based on
GaN (Gallium nitride) or SiC (Silicon carbide) semiconductors. Their positive advantages and properties: lower switching
power loss, lower on-state resistance, greater bandwidth, which
to allow to apply the high switching frequency [1]. At the
same time the high switching frequency delivers many positive
advantages (smaller switching loss, reducing size and volume
of circuits (through the reactive elements) etc.) but influence
the stability of the system (bifurcation and chaos), therefore
it is need to take into account the nonlinear behavior. The
current research are focused on the low-frequency region
(some hundred kilohertz) [2, 3]. Due to the switching frequency of system are increasing (up to megahertz region)
the investigation between switching frequency and systems
properties (e.g stability) is needed.
The converters are nonlinear and time-variant system which
can be derived from variation of load value and input voltage
[4, 5]: For control purposes, many linearized models have been
developed that facilitate the designing procedure of DC/DC
converter. Widely used for this aim the State Space Averaging
method (SSA). The goal of this method obtain the small signal
function of the DC/DC converter (linearization around equilibrium point). There are some limitations of this technique: (i)
the linearized model is valid just for the half of the switching
period, (ii) the chaotic behavior or subharmonic behavior can
not handle [6]. At the same time the nonlinear controllers
shows good results: sliding mode [7, 8] or backstepping [9]
etc. The presented controller in [5] use feedback linerization
(small overshot in output voltage and inductor current) but not
contain an integrator.
Fig. 1: Circuit diagram of buck converter without parasitic
elements
In this case, the state variables of the system are the
voltage across the capacitor (vC ) and the current through
the inductor (iL ). In CCM (Continuous Conduction Mode),
it can be distinguished two operating modes of DC/DC buck
converter depending on the SW operation:
•
on state (0 < t < DT ) and
•
off state (DT < t < T).
The first mode equivalent circuit is shown in Fig. 2:
Accordingly the state space equations can be express as:
1
u̇C = − RC
uC +
i̇L = − L1 uC +
In matrix form:
"
# "
iL
0
d
= 1
u
C
dt
C
− L1
1
− RC
1
C iL ,
1
L Vi .
# "1#
iL
+ L Vi .
uC
0
(1)
#"
(2)
The another operating mode during the switch (SW) is not
conducting (off), the equivalent circuit (Fig. 3):
III.
E XACT L INEARIZATION AND CONTROLLER DESIGN
APPLYING EXACT FEEDBACK LINEARIZATION
The control is said a Static State Feedback Control if it
is depending on the state of x (state variable) and external
reference signal (v). The input of SISO (Single Input Single
Output) system in case of static state feedback can be written
[11]:
u = α(x) + β(x)v,
(9)
where v is the external reference input (see Fig. 5).
Furthermore, the non-linear system can be represented in
the following form [11]:
Fig. 2: Buck converter equivalent circuit - on state
ẋ = f (x) + g(x)u,
y = h(x).
(10)
As indicated above, the closed loop function [11]:
ẋ = f (x) + g(x)α(x) + g(x)β(x)v,
y = h(x).
Fig. 3: Buck converter equivalent circuit - off state
Fig. 5: Static state feedback control [11]
The state space equations are:
1
u̇C = − RC
uC +
1
C iL ,
(3)
i̇L = − L1 uC .
In matrix form:
" # "
0
d iL
= 1
u
dt C
C
− L1
1
− RC
The goal is to determine α and β which characterize the
control, i.e. convert the nonlinear system into linear one.
The output voltage - controlled quantity- can be defined as
[11, 12]:
y = h(x) = uC .
(12)
#"
# " #
iL
0
+
Vi .
uC
0
(4)
The output function need to differentiate by time until the input
function appears in expression [11, 12]:
ḣ = u̇C =
Averaging (1) and (2) by duty cycle (D), can be written as:
" # "
0
d iL
= 1
u
dt C
C
− L1
1
− RC
# "D#
iL
+ L Vi .
uC
0
1
1
iL −
uC ,
C
RC
(5)
1
1
1
1
uC +
Vi −
+ 2 2 uC .
(14)
LC
LC
RC 2
R C
It can be see that the second derivative (14) contains the input
function. Accordingly, the results in Lie algebra framework
[12]:
ḧ = −
ḣ = Lf h, ḧ = L2f h + Lg Lf hu.
where
f (x) =
1
− RC
uC
C iL
− L1 uC
g(x) =
(13)
#"
The state space model can be written into a general matrix
form:
ẋ = f (x) + g(x)u,
(6)
"1
(11)
0
Vi
L
#
,
(7)
,
(8)
The block diagram of control system are shown in Fig. 4
(see next page).
(15)
To apply a feedback linearization method to buck converter,
firstly need to check the rank of matrix, which can be done
easily via Matlab [11]. The rank of matrix is 2 which equal
the system order, therefore the system is controllable.
Using the following coordinate transformation:
"
# h(x)
uC
z = T (x) =
= 1
,
1
Lf h(x)
C iL − RC uC
(16)
Fig. 4: Input-output feedback linerization pole placement and integral conrol of buck converter
where z = T (x): z1 = h1 , z2 =Lf h1 i.e.
For dynamic testing two test has been applied:
z˙1 = z2 ,
z˙2 = L2f h + Lg Lf hu = v.
(17)
Accordingly, considering new state-variable T , system model
will be linear as:
0 1
0
Ṫ =
T+
v.
(18)
0 0
1
•
output load test (R) and
•
input voltage test. (Vi ).
The dynamic behavior of controller was investigated at steady
state, which are shown in Fig. 9. and Fig. 10, respectively. The
results are clearly shows that the controller is invariant to the
changing of the input voltage and has good transient behavior
to load variations.
In (18), v is a new control input. So, the controller of nonlinear
system (3) is the following:
u = α(x) + β(x)v,
(19)
where
L2 h
α(x) = − Lg Lf f h = − V1i uC −
β(x) =
1
Lg Lf h
IV.
=
L
Vi RC iL
+
L
V iR2 C uC ,
LC
Vi .
(20)
L INEAR CONTROLLER DESIGN
There are several ways to realize a linear regulator, e.g. PI,
PID regulator, pole placement, linear quadratic regulator etc
[1].
In this paper the pole placement with integrator has been
used. All the original poles have been moved to -0.25, as
results the obtained gain matrix is K = [k1 k2 kI ]T = [0.1875
0.75 0.0156]T (in coherent unit system).
V.
S IMULATION RESULTS
To investigate the properties of controller DC/DC buck converter has been designed with following parameters: Vi =52V,
Vo =12V, Io =4.8A, R=2.5Ω, L=50µH, C=50µF. The [1] contains in detail the design steps and procedures of DC/DC buck
converter. For better numerical handling coherent unit system
is used.
The start-up simulation shows that the inductor current
(IL ) is very high which may be related to the value of the
output capacity. To investigate this, the simulation was run
with two different C values which are shown in Fig. 6 and
Fig. 7. respectively. Besides that two solvers were compared
(Fig.8.). It can be see the type of solver is affects the shape
of current.
Fig. 6: State variables of the converter in start-up (C=50µF )
VI.
C ONCLUSIONS
This article is describe the exact feedback linearization with
integrator. The simulation results shows the extra integrator
means also better transient behavior. At the same time, the
inductor current is very high at start-up which can be depend
on: the presence of parasitic elements and/or the type of
numerical solver. These detailed examination is needed in
the future. Another future plan is to realize the controller by
microcontroller and investigate the possibility of applying wide
bandgap semiconductors (SiC or GaN) in buck converter.
ACKNOWLEDGMENT
The research presented in this paper was carried out as part
of the “Dynamics and Control of Autonomous Vehicles meet-
Fig. 7: State variables of the converter in start-up (C=25µF )
Fig. 9: Output resistance test at steady state
Fig. 8: Comparison of Runge-Kutta and Euler solver
(C=50µF )
Fig. 10: Input voltage test at steady state
[6]
ing the Synergy Demands of Automated Transport Systems
(EFOP-3.6.2-16-2017-00016)” project in the framework of the
New Széchenyi Plan. The completion of this project is funded
by the European Union and co-financed by the European Social
Fund.
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