1
2022 International Electrical Engineering Congress (iEECON) | 978-1-6654-0206-4/22/$31.00 ©2022 IEEE | DOI: 10.1109/iEECON53204.2022.9741689
Application of port-Hamiltonian Approach in
Controller Design for Buck Boost Converter
Walarcheth Thongrailuck
Electrical and Computer Engineering,Faculty of Engineering
King Mongkut’s University of Technology North Bangkok
Bangkok, Thailand
s6101011811519@email.kmutnb.ac.th
Abstract—Controlling output voltage of the DC-DC BuckBoost converter has been a challenging topic for many years,
since its averaged dynamics is a non-minimum phase system
with highly load variation. Several methods avoid this difficulty
by controlling the voltage via the inductor current. This paper
presents a nonlinear energy-based controller design approach,
which can directly control the output voltage without internal
current loop. Due to damping assignment technique, the
damping of the closed-loop system can be determined, and the
problem with load variation is solved. To verify the performance
of the proposed approach, the controller is test with a simulation
and with existing converter. The result confirms the
achievement of the proposed controller.
Keywords—Port Hamiltonian, Buck-Boost DC/DC Converter,
Energy Shaping (ES), Current Controller
I. INTRODUCTION
Controlling the output voltage of DC-DC Buck-Boost
converter is a challenging topic. Firstly, the converter is a
nonlinear variable structure system and the average model of
the converter has non-minimum zero [1]. The highly load
variation affects system stability, and it is, therefore,
troublesome to control output voltage directly [2]. By using an
inner loop or current controller, the problem with nonminimum phase system can be avoided, and the output voltage
can be controlled through the regulation of inductor current
[3]. However, the load variation must be concerned, because
many study show the load variation has considerable effect on
the performance of the closed-loop system [3], [4]. The portHamiltonian system theory is developed on the basis of portbased modeling and Hamiltonian dynamics. This combination
yields an efficient approach for modeling and analysis of
multi-physics system, which uses energy for communication
among the components. Besides, based on port-Hamiltonian
system concept, many energy-based methods in controller
design e.g., damping assignment-passivity based control or
energy shaping are developed [8]. The advantage of energybased approach is the ability to manage the variable structure
system in a tidy manner [6], [9]. In this paper, energy
balancing method and damping assignment are utilized to
solve the non-minimum phase and load variation issue. The
developed controller is tested by simulation and experiments.
The result certifies the success of the proposed method.
II. PORT-HAMILTONIAN CONTROLLER DESIGN APPROACH
A. Port Hamiltonian System
According to the concept of port-Hamiltonian, a physical
system has three elements: (1) energy-storage, (2) energydissipative and (3) energy-routing, as shown in Fig.1. All
energy-storage components in the system are merged as a
Witthawas Pongyart
Electrical and Computer Engineering,Faculty of Engineering
King Mongkut’s University of Technology North Bangkok
Bangkok,Thailand
witthawas.p@eng.kmutnb.ac.th
single unit indicated by S, and in the same manner, all energydissipative components are lumped together as a single unit
indicated by R. The energy exchange among the components
occurs via a pair of equally dimensioned vectors of flow and
effort variables, shown by (f,e) in Fig. 1. Each pair of (f,e)
vectors establishes a port joining to the system, and the inner
product of flow and effort is the power flowing through the
port. The internal ports (fS,eS) and (fR,eR) connect the energystorage element and the energy-dissipative element to the
system respectively. The external port (fP,eP) connects the
system to exterior components for instance controller, voltage
source including disturbance. Finally, the coupling of all the
energy-routing is represented as a single energy-routing
structure designated by D, to represent the geometric notion
of Dirac structure which determine energy flow-path of the
system [6]. In addition, the Dirac structure provides a power
continuous interconnection among the ports [5]. Assuming
H(x) is total energy in the system and x is the vector of state
variables, the dynamic of energy in the system can be
expressed by the port-Hamiltonian concept as in (1). The flow
and effort of the system are shown by the vector x& and
∂H ( x) / ∂x respectively. The interconnection matrix J(u)
describes the interaction of the stored energy and the damping
matrix R represents the energy dissipation of the system. Both
matrixes, J(u) and R, are defined in (2). The input matrix g(u)
defines the connection to external source E. The variable u
play an important role in (1), since it controls the energy flow
in and into the system via the matrix J(u) and g(u)
respectively. The model in (1) is chosen for this research,
since it is suitable for variable structure system [6], [9].
S (storage)
eR
eS
D
fS
fR
eP
R (dissipation)
fP
Fig. 1 Structure of port-Hamiltonian System with Dissipation
=( ( )− )
ℎ
( )
+ ( )
( ) = − ( ),
≥0
(1)
(2)
The 2022 International Electrical Engineering Congress (iEECON2022), March 9 - 11, 2022, Khon Kaen, THAILAND
978-1-6654-0206-4/22/$31.00 ©2022 IEEE
Authorized licensed use limited to: King Mongkut's University of Technology North Bangkok. Downloaded on September 08,2022 at 09:56:15 UTC from IEEE Xplore. Restrictions apply.
2
B. Buck-Boost Converter Model
The circuit of DC-DC Buck-Boost converter is shown in
Fig.2. The inductor and capacitor serve as energy storage
elements and the variable resistor r is the energy dissipative
element of the system. The energy interaction in the system
and with the external source E is controlled by the transistor.
Based on port-Hamiltonian principle, the mathematical model
of the converter can be written in (3). The state variable and
total energy, or Hamiltonian function are defined in (4) and
(5) respectively. The interconnection matrix J(u) is shown in
(6) and the damping matrix R is in (7). According to (6) the
converter is a variable structure system. Besides the load
variation has direct effect on transient response of the system.
TR
E
ir(t)
D
vC(t)
+
L
iL(t)
C vr(t)
+
iC(t)
ℎ
( )=
( )
r
=[
# % $
( )
$ &
( )=
)*+
+
0
1−
0
+
(3)
!]
(4)
# ' $
( ) ,
$ (
( ( )−
0
.(
/
(6)
(7)
III. ENERGY BALANCING CONTROLLER DESIGN
ℎ
.
)*+
= −0
.(
.
)=
=
.(
)
.(
( )+
+
1
/
.
)
/(
≥0
0
.(
)
=8
),
)
1
1
.
0
/
0
/
2( ) ≜
−
1
2
2( ) = < # = = >
2$
0
#:,∴
.(
( )
)
+ ( )
4( )
15
=
.
9
#
6 ∗
/
0
(12)
(13)
=0
(14)
0
0
(15)
/ #
@
=
+
−
$
?
/ $
?
2# ( )
$
+
−
A
(16)
(17)
The control law in (18) is obtained by solving (16). The
parameter α in (19) is used for tuning purpose and the constant
C1 in (21) is determined at the equilibrium point. The
completed control law is presented in (22), and it is to be noted
that the parameter u* in (23) depends on the command v*.
Hence the developed controller has adaptive feedforward
gain. The structure of the control system is shown in Fig.3.
+
+
$
=
B
$
B =1−
=
?# =
∗
(8)
(9)
(10)
−
2$ ( )
To avoid the effect of load variation and improve the
system behavior the desired closed-loop system is determined
in (8), where the closed loop system energy Hd(x) in (9), and
the closed loop damping matrix Rd is defined and (10). The
controller shapes the closed loop system energy Hd(x) by
adding additional energy Ha(x) to the H(x) as in (8). Besides,
the controller adjusts the system damping matrix by adding Ra
to the existing matrix R, and the effect of load variation is
reduced. Actually the energy-based controller design
approach can rearrange the interconnection also, but in this
paper the matrix J(u) is left untouched. Since the energy
function Hd(x) is a positive definite function, and its derivative
is negative as in (11), the stability of the closed-loop system is
ensured.
.)
)=
ℎ
(5)
−
0
0 0
1=,
0
=( ( )−
. )2(
By substituting the damping matrix Ra and Rd are in (12)
and (16) is derived. A necessary and sufficient condition for
the K(x) in (13) to be a gradient function is given in (17) [7].
Since the inductor current x1 is not fixed, the control law u
depends on the x2 only, the equation (16) becomes ordinary
differential equation in (18).
Fig.2 Circuit of DC-DC Buck-Boost converter
=( ( )− )
(12) is solvable for Ha(x) with the constraint as shown in (14)
that the resulting Hd(x) has minimum at x*. To avoid the load
variation, the injected damping matrix Ra is defined in (15).
vC*
=
+
=
H∗
( $)
/
@
E
$ ?#
∗
(18)
?
(19)
(20)
∗
( $∗ )E
F ∗G
$
$
(21)
E
, where H ∗ =
Controller
1
?
u(t)
(22)
(23)
∗
$
Plant
vC(t)
(11)
To achieve the desired closed loop system the control law
u must be determined. The energy-shaping and damping
assignment can be obtained, when the matching equation in
Fig.3 Structure of the control system
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3
IV. SIMULATION AND EXPERIMENTAL RESULT
To verify the performance of the developed controller, the
controller is tested with MATLAB simulation. Then the
controller is tested with the existing DC-DC Buck-Boost
converter and converter parameters are s in Table I. Those
parameters are used in the simulation as well. The controller
is tuned by letting α = 0.1 in every test. The closed-loop
system becomes faster, when this parameter is reduced.
TABLE I.
dropping of load resistance, the inductor current increases to a
new level. Even under load fluctuation the system is still stable
and has pretty good transient behavior.
SPECIFICATIONS OF THE CONVERETER
Component
Value
Inductor : L
25 mH
Capacitor : ?
Nominal value of Load Resistance : r
30 Ω
Input Voltage
15 V
Switching Frequency : Hz
20 kHz
Sampling Period : NO
Fig.5 Response for step input 10-20 V
47 µF
22
20
50PQ
18
16
14
0.05
A. Simulation Result
To observe the tracking performance of the controller, a
step command v* 0-10 V is applied. The result is shown in Fig.
4. Obviously, the capacitor voltage vC(t) tracks the command
and reaches to steady state with in 0.004 second. The response
has some overshoot but no steady state error. Also, the
inductor current iL(t) shows the same behavior.
0.06
0.07
0.08
0.09
0.1
0.11
0.09
0.1
0.11
Time in ms
7
6
5
4
0.05
0.06
0.07
0.08
Time in ms
Fig.4 Response for step input 0-10 V
Then after reaching steady state, another test is performed
by stepping the input from 10 to 20 V, and the result is shown
in Fig. 5. The capacitor voltage vC(t) reaches to equilibrium
with some overshoot, but the transient period is shorter than
the one in the previous test, since it takes only 0.6 millisecond.
There is no steady state error in the response, but a small
undershoot, due to non-minimum phase zero, can be seen. The
inductor current shows the same behavior but no undershoot.
Finally, the disturbance rejection is investigated. The load
resistance is reduced by 20 %, and the output voltage and
inductor current are shown in Fig. 6. The voltage vC(t) drops
and recovers in two millisecond, and iL(t) increases to
maintain the output voltage. Finally, to investigate system
performance under highly load variation, the load resistance is
decreased by 50 %. The output voltage and inductor current
are shown in Fig. 7. The output voltage drops more and returns
to the set point value within two millisecond. Due to the
iL (t) [A]
vC (t) [V]
Fig.6 Load resistance reduced by 20 %
Fig.7 Load resistance reduced by 50 %
B. Experimental Result
The developed controller is realized on a microcontroller
and tested with the existing converter in the laboratory. The
process is similar to the test in previous section. The step
response of a step command from 0-10 V is shown in Fig. 8.
The blue line shows the inductor current and the red one is the
capacitor voltage. The result has similar characteristic as the
simulation result in Fig. 4 does. The vC(t) tracks the command
with some overshoot and requires about 0.004 second to reach
the steady state. There is no steady state error in the output,
and the inductor current shows the same behavior.
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4
v C (t) [V]
30
20
10
0
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
0.3
0.35
0.4
0.45
0.5
Time in ms
6
iL (t) [A]
4
2
0
0
0.05
0.1
0.15
0.2
0.25
Time in ms
v C (t) [V]
Fig.8 Experiment: response for step command 0-10 V
voltage vC(t), the red line, reaches to equilibrium with very
small overshoot, and the transient period is much less than the
one in the previous test, since it is about 0.6 millisecond. There
is no steady state error, but a tiny under shoot, due to nonminimum zero, can be clearly observed. Since the output is
increased, the inductor current raises. The behavior of both
signal is similar to the simulation result in Fig. 5. The load
variation is occurred by connecting a resistor in parallel to the
load to reduce the load resistance by 20%. The result is shown
in Fig. 10. Obviously, the capacitor drops and recovers within
0.2 millisecond and, to maintain the output voltage, the current
iL(t) raises. To observe system performance under highly load
variation, the load resistance is reduced by 50%. The output
voltage and inductor current are shown in Fig. 11. The vC(t),
the red line, drops and recovers in 0.25 millisecond. Due to the
dropping of load resistance, iL(t) rearises rapidly to almost six
Amps to maintain the output constant.
iL (t) [A]
V. CONCLUSTION
v C (t) [V]
Fig.9 Experiment: response for step command 10-20 V
An output feedback controller for Buck-Boost DC-to-DC
converter is presented. The proposed controller is robust to
load fluctuation, since the closed-loop damping is free from
the load resistance. Besides, the controller has very good
disturbance rejection behavior. The feedforward gain u*
adapts to new operating point and supports the feedback
controller in command tracking. The system can track the
command rapidly. With special inductor from the factory.
Simulation and experimental results confirm the good
performance in tracking the command and disturbance
rejection. It is to be noted that there is no integrator in the
controller but the steady state error is zero.
REFERENCES
[1]
iL (t) [A]
[2]
[3]
Fig.10 Experiment: Load resistance reduced by 20 %
[4]
vC (t) [V]
30
20
[5]
10
0
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
[6]
Time in ms
6
iL (t) [A]
4
[7]
2
0
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
Time in ms
[8]
Fig.11 Experiment: Load resistance reduced by 50 %
[9]
Then another test is performed by stepping the command
from 10 to 20 V, and the result is in Fig. 9. The capacitor
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