This research article presents the design of a hybrid controller for a two-interleaved boost converter aimed at renewable energy and automotive applications, focusing on fast transient response and low input current ripple. The proposed controller uses a hybrid automaton formalism to stabilize a predefined optimal limit cycle and includes real-time load estimation for adaptive control. Simulations demonstrate the controller's effectiveness and robustness against disturbances, highlighting its application in fuel cell systems.

1. Research Article
Load estimator-based hybrid controller design for two-interleaved
boost converter dedicated to renewable energy and
automotive applications
Mohamed Bougrine a,b
, Mohammed Benmiloud a
, Atallah Benalia a
, Emmanuel Delaleau c
,
Mohamed Benbouzid b,d,n
a
University of Laghouat, LACoSERE Lab, Laghouat, Algeria
b
University of Brest, FRE CNRS 3744 IRDL, Brest, France
c
Ecole Nationale d'Ingénieurs de Brest, Mechatronics Department, Plouzané, France
d
Shanghai Maritime University, Shanghai, China
a r t i c l e i n f o
Article history:
Received 8 June 2016
Received in revised form
14 August 2016
Accepted 6 September 2016
Available online 16 September 2016
This paper was recommended for publica-
tion by Dr. Jeff Pieper
Keywords:
Interleaved boost converter
Hybrid dynamical system
Optimal limit cycle
Stabilization
Adaptive control
Fuel cell source
a b s t r a c t
This paper is devoted to the development of a hybrid controller for a two-interleaved boost converter
dedicated to renewable energy and automotive applications. The control requirements, resumed in fast
transient and low input current ripple, are formulated as a problem of fast stabilization of a predefined
optimal limit cycle, and solved using hybrid automaton formalism. In addition, a real time estimation of
the load is developed using an algebraic approach for online adjustment of the hybrid controller.
Mathematical proofs are provided with simulations to illustrate the effectiveness and the robustness of
the proposed controller despite different disturbances. Furthermore, a fuel cell system supplying a
resistive load through a two-interleaved boost converter is also highlighted.
& 2016 ISA. Published by Elsevier Ltd. All rights reserved.
1. Introduction
Climate change in the last century, mainly due to the increasing
carbon dioxide (CO2) released through human activities as shown
in Fig. 1, brings to light serious issues like global temperature and
sea level augmentation, ocean warming and acidification, shrink-
ing ice sheets, etc. These facts have pushed the scientific com-
munity for renewable and clean energy solutions to supply various
technological applications via switched converters.
Due to the power source and load constraints, the converter
structures have to meet some practical challenges such as relia-
bility, high power density, high efficiency, and low current/voltage
ripples. Parallel connection of switched converters or specifically
the interleaving approach meets the above requirements with
better power scalability characteristics compared to the classical
ones [2,3]. Fig. 2a depicts the topology of a DC–DC two-interleaved
boost converter, the two-phase boost topology was proposed as an
alternative to the classical single phase boost converter [4], and
the two-phase buck topology as an alternative to the classical buck
converter [5]. These interleaved topologies are widely used in
varieties of applications and systems that incorporate solar panels
or fuel cell sources, as reflected in the literature [6–11]. The
interleaving technique is also investigated for the microprocessors
power supply to achieve better computing performance using the
topology in Fig. 2b, which is known as multiphase/multi-channel
synchronous/interleaved buck converter [12–15]. The interleaved
bidirectional topology, obtained by changing each diode with a
controlled switching device in the interleaved boost converter, is
also explored in systems with rechargeable energy storage ele-
ments like batteries or supercapacitors [16–20].
Research studies have discussed these topologies from different
practical and fundamental points of view. The authors in [21] have
discussed the number of phases that can be used to obtain a tra-
deoff among some indexes such as: the switching losses, the
inductor volume, the input current ripples and the switches cost.
Power management in fuel cell hybrid vehicles involving battery/
Contents lists available at ScienceDirect
journal homepage: www.elsevier.com/locate/isatrans
ISA Transactions
http://dx.doi.org/10.1016/j.isatra.2016.09.001
0019-0578/& 2016 ISA. Published by Elsevier Ltd. All rights reserved.
n
Corresponding author at: University of Brest, FRE CNRS 3744 IRDL, Brest, France.
E-mail addresses: m.bougrine@lagh-univ.dz (M. Bougrine),
med.benmiloud@lagh-univ.dz (M. Benmiloud), a.benalia@lagh-univ.dz (A. Benalia),
delaleau@enib.fr (E. Delaleau), mohamed.benbouzid@univ-brest.fr (M. Benbouzid).
ISA Transactions 66 (2017) 425–436

2. supercapacitor as an auxiliary regenerative source is thoroughly
explored in [16,22]. A maximum power point tracking (MPPT)
approach in systems with solar module or a fuel cell source
associated to interleaved boost converter is performed in [23,24].
Other works deal with the energetic efficiency improvement by
considering variants of the interleaved topology such as soft
switching and resonant techniques, or coupled inductors [25,26].
Furthermore, the control of interleaved converters has been
dealt using several control strategies. Linear control techniques
based on the average model are effective only around a specific
operation points due to the neglected nonlinearities of the con-
verter model and the power source [27]. In [14,27], a sliding mode
and adaptive sliding mode controllers are designed and defended
to be robust ones. Model predictive control is developed for the
interleaved boost topology in [28]. Regardless of the fact that
several control techniques have been proposed, the transient and
steady-state control problem of the interleaved converter still pose
challenging issues. More precisely, the transient control deals with
a minimal response time and number of commutations require-
ments, which is beneficial for a fast MPPT achievement in a solar
panel based system for example. In addition, in electric vehicles, a
fast tracking of the supercapacitor reference current, may also
require an interleaved bidirectional converter to guarantee the
instantaneous peak power demands or to recover a maximum
energy through regenerative braking [29]. Moreover, information
processing in computers is directly related to the response time of
the microprocessors power supply in fast current variations
tracking [12]. The existing controllers may tackle the problem of a
fast transient with different approaches. Nevertheless, the steady-
state behavior, represented by states ripples, is not directly con-
trolled. Indeed, ripples amplitude and harmonic content of the
current is one of the various phenomena influencing fuel cell
lifespan as well as battery lifetime [11,22]. Besides, a small input
current ripple is advantageous for MPPT to operate around a
maximum power point without too much fluctuation [30].
Motivated by the above facts, we aim to address the control
problem of the transient and steady-state of the two-interleaved
boost converter. For this purpose, an exact instantaneous model of
the converter will be investigated in the next section using the
theory of hybrid dynamical systems. An adaptive transient and
steady-state controllers will be developed in Section 3 to solve the
problem of fast stabilization of a predefined optimal limit cycle.
Simulation results will be discussed in Section 4 and followed by
conclusions and future works that outline the main contributions
of the present paper.
2. Hybrid model of 2IBC and control formulation
We consider the 2IBC presented in Fig. 2a, supplying a resistive
load (R) and operating in continuous conduction mode (CCM).
Phase k consists of a non-ideal inductor ðLk; rkÞ, a diode ðDkÞ, and a
controlled semiconductor device ðswkÞ by a binary input signal
uk Af0; 1g. The instantaneous model of the 2IBC is given by:
diLk
dt
¼
1
Lk
ðvi ÀrkiLk
ÀukvoÞ; k ¼ 1; 2
dvo
dt
¼
1
Co
X2
k ¼ 1
ukiLk
À
vo
R
!
;
8
>>>><
>>>>:
ð1Þ
where iLk
is the inductor current of phase k. The voltages vi and vo
are the input and output voltages respectively. Á : f0; 1g↦f0; 1g is
the Not function.
The input signal ðu1; u2Þ combinations offer four different con-
verter operations without redundancy, known as discrete modes
in hybrid systems vocabulary. From (1), each discrete mode can be
represented by the following affine differential equation:
_x ¼ AqðtÞxþBqðtÞ ¼ f qðtÞðxÞ ð2Þ
with x ¼ ½iL1
iL2
voŠT
AX is the continuous state vector defined in a
physical operating region X DR3
. qðtÞ : Rþ ↦Q is the switching
signal, with Q ¼ fq1; q2; q3; q4g is the set of discrete modes. The
state matrices Aqi
AR3Â3
and Bqi
AR3Â1
are given by:
Aqi
¼
À r1
L1
0 Àu1
L1
0 Àr2
L2
Àu2
L2
u1
Co
u2
Co
À 1
RCo
0
B
B
B
@
1
C
C
C
A
; Bqi
¼
vi
L1
vi
L2
0
0
B
@
1
C
A
For each discrete mode, the corresponding values of the inputs
are given in Table 1.
The control design consists in the adequate state feedback
switching low design q(x) that orchestrates the switching among
the discrete modes to meet the control requirements. In closed
loop, the hybrid automaton H of the 2IBC, operating in CCM, can be
1980 1985 1990 1995 2000 2005 2010 2015
330
350
370
390
410
Years
CO2
(ppm)
Seasonal
Monthly
Fig. 1. Recent global monthly and seasonal mean CO2 in parts per million (ppm)
over marine surface sites (October 2015) [1].
Fig. 2. Topology of interleaved converter.
M. Bougrine et al. / ISA Transactions 66 (2017) 425–436426

3. represented by the following 6-tuple:
H ¼ ðQ; X; Sc; T; G; InitÞ ð3Þ
where Sc : ðQ Â XÞ↦R3
is the application that assigns to every discrete
mode a continuous dynamic given by (2). T ¼ fTij; i; jAf1; …; 4gg
represents a set of all possible transitions between discrete modes,
G : T-2X
associates to each transition a continuous set where the
transition is valid (called a guard condition), and InitDX Â Q gives the
initial states.
Based on (3) the switching design corresponds to the free
elements definition of the hybrid automaton ðT; G; InitÞ in order to
solve the stabilization problem discussed in the next subsection. In
the remainder of the paper, we will consider r1 ¼ r2 ¼ r and
L1 ¼ L2 ¼ L.
The control of interleaved boost converters addresses the reg-
ulation of the output voltage vo around the desired voltage Vref by
controlling the input current Iin around an average reference
generated by an outer loop as follows:
Iref ðtÞ ¼ INom
ref þ
Z t
0
K sgnðVref ÀvoðτÞÞ Vref ÀvoðτÞ
α
dτ ð4Þ
where K is a sufficiently large positive real number, αAð0; 1Þ is a
small coefficient to ensure a finite time settling, INom
ref ¼
V2
ref
Rvi
is the
nominal value of Iref in static steady-state out of model uncer-
tainties. To guarantee the balancing of the converter, the input
current must be shared between the two phases. Hence for the
2IBC, the phase currents must be regulated around the average
reference Irefk
¼ 0:5Iref .
On the other hand, it is clear from the hybrid nature of the
converter that the regulation of the phase currents can be
obtained in a cyclic manner. This means that in steady-state, the
converter trajectory converges to a limit cycle. The optimal limit
cycle is characterized by low input ripple which corresponds to the
following condition on the inputs:
1. The input signals have the same duty cycle d.
2. The phase shift between the two phases is equal to π.
The duty cycle d depends on the required output voltage as fol-
lows:
d ¼ 1À
vi ÀrIref
Vref
ð5Þ
Fig. 3 illustrates the hybrid trajectory (x,q) of the 2IBC in the
currents phase plane ðiL1
; iL2
Þ for different duty cycle values and a
fixed desired phase shift ðϕ ¼ πÞ of the input signals. One can
remark that the periodic behavior of the input signals leads to a
cyclic hybrid trajectory, which is one of the important nonlinear
phenomena that may be exhibited by DC–DC power converters
[31]. Note that each phase shift leads to a specific limit cycle where
the optimal one, with respect to input current ripple, is shown in
Fig. 3. Therefore, it is more natural to consider the stabilization
problem of the 2IBC as follows.
Fast optimal limit cycle stabilization: Given the set of discrete
modes Q ¼ fq1; q2; q3; q4g, how should they be selected to guaran-
tee the fast stabilization of the predefined optimal limit cycle
depicted in Fig. 3 with a fast transient property?
This problem is not directly discussed using the continuous
dynamical systems theory. In the next section, we will solve the
above problem efficiently using hybrid systems theory.
3. Hybrid control design: main results
The formulated stabilization problem in the previous section
can be restated based on the closed loop hybrid model (3) as fol-
lows: (1) For a given initial condition x0, how one should select the
convenient discrete modes among the set Q to reach fast the
neighborhood of the optimal limit cycle shown in Fig. 3, and fast
balance of the input current among the two phases. (2) When the
neighborhood of the optimal limit cycle is reached, how to select
the discrete modes to ensure its local stabilization.
In the following, we design an adaptive hybrid controller, illu-
strated in Fig. 4, with two parts: A transient state automaton for
fast transition and a steady-state automaton for local asymptotic
Table 1
The discrete modes of the two interleaved boost converter and their equilibrium
points.
Discrete
modes
Input signals Currents
evolution
Equilibrium point
u1 u2 iL1
iL2 x
qi
eq
q1 0 0 ↘ ↘ vi
2Rþr
vi
2Rþr
2Rvi
2Rþr
!T
q2 1 0 ↗ ↘ vi
r
vi
Rþr
Rvi
Rþr
!T
q3 0 1 ↘ ↗ vi
Rþr
vi
r
Rvi
Rþr
!T
q4 1 1 ↗ ↗ vi
r
vi
r
0
h iT
Iref1
Iref2
1q
1q
2q
3q
Iref1
Iref2
2q
3q
Iref1
Iref2
3q
2q
4q
4q
Fig. 3. Projection of the desired limit cycle in the currents plane.
Fig. 4. The proposed adaptive hybrid control scheme for the interleaved boost
converter.
M. Bougrine et al. / ISA Transactions 66 (2017) 425–436 427

4. stability of the desired limit cycle. Besides, an adaptive rule for the
load resistance will be proposed using an algebraic approach.
3.1. Transient automaton controller design
In order to achieve a fast convergence of the inductance cur-
rents to a neighbor of their references Iref1
and Iref2
, one can dis-
tinguish two cases:
x0 AΩ2 with Ω2 ¼ fxAX∣iL2
4iL1
g: The selected discrete mode in
this region should ensure a decreasing value of the current iL2
and an increasing one for the current iL1
to direct the converter
trajectory toward the balancing surface S1ðxÞ ¼ 0, with
S1ðxÞ ¼ iL2
ÀiL1
. From the currents evolution in Table 1, it can be
noticed that mode q2 is the only appropriate choice in this
region.
x0 AΩ3 with Ω3 ¼ fxAX∣iL2
oiL1
g: One should select the discrete
mode that guarantees an increasing value of the current iL2
and
a decreasing one for the current iL1
to reach the balancing sur-
face S1ðxÞ ¼ 0. From Table 1, the mode q3 is the only suitable one
for this purpose.
Once the converter trajectory reaches the surface S1ðxÞ ¼ 0, a
transition should be made to another discrete mode to bring the
converter currents (iL1
; iL2
) toward the reference surface S2ðxÞ ¼ 0,
with S2ðxÞ ¼ iL1
ÀIref1
. Two cases can be distinguished:
xðtsÞAΩ4 with ts is the reaching time of the converter trajectory
to the balancing surface and Ω4 ¼ fxAX∣iL1
ðtsÞoIref1
g: In order
to force the converter currents to reach the surface S2ðxÞ ¼ 0 and
preserving the invariance of the balancing surface, the discrete
mode q4 is the one suitable.
xðtsÞAΩ1 with Ω1 ¼ fxAX∣iL1
ðtsÞ4Iref1
g: In this case, the dis-
crete mode q1 is the appropriate one to reach S2ðxÞ ¼ 0 while
preserving the invariance of the balancing surface as listed in
Table 1.
Fig. 5 shows a graphical representation, in the currents phase
plane, of the transient automaton controller. The converter tra-
jectory evolution can be highlighted as follows: the first initial
condition x01 belongs to the region Ω3 where mode q3 should be
activated. The converter trajectory x(t) evolves through the
dynamics _x ¼ f q3
ðxÞ until hitting the balancing surface S1ðxÞ ¼ 0 at
the instant ts. In this case, the state xðtsÞ belongs to the region Ω4
and hence mode q4 is selected to charge the inductors currents
until reaching the reference surface S2ðxÞ ¼ 0. After that, a steady-
state controller must be activated to ensure the local stability of
the desired cyclic behavior.
Fig. 6 illustrates the hybrid automaton of the developed hybrid
controller with both transient and steady control parts, where
each node represents a discrete mode and the arrows indicate the
possible discrete transitions. The transient automaton controller
corresponds to the above analysis where the elements ðG; T; InitÞ
are defined in the following proposition:
Proposition 1. Under the transient automaton controller shown in
Fig. 6 and defined by the guard conditions (6)–(8), the input current
ðIinÞ of the 2IBC reaches its desired value ðIref Þ in finite time and
within two commutations: q3q4; q3q1; q2q4, or q2q1. The initial dis-
crete mode depends on the initial continuous state as follows:
Init ¼ ðInitðq1Þ Â q1Þ [ ðInitðq2Þ Â q2Þ [ ðInitðq3Þ Â q3Þ
[ðInitðq4Þ Â q4Þ ð6Þ
with
Initðq1Þ ¼ fxAΩ1 ðS1ðxÞ ¼ 0Þg; Initðq2Þ ¼ Ω2
Initðq4Þ ¼ fxAΩ4 ðS1ðxÞ ¼ 0Þg; Initðq3Þ ¼ Ω3
The guard conditions corresponding to the fast balance of the input
current are given by:
Gt
ðT21Þ ¼ fxAX∣ðiL2
riL1
Þ4ðiL1
4Iref1
Þg
Gt
ðT24Þ ¼ fxAX∣ðiL2
riL1
Þ4ðiL1
oIref1
Þg
Gt
ðT31Þ ¼ fxAX∣ðiL2
ZiL1
Þ4ðiL1
4Iref1
Þg
Gt
ðT34Þ ¼ fxAX∣ðiL2
ZiL1
Þ4ðiL1
oIref1
Þg ð7Þ
The guard conditions corresponding to the convergence of the
inductor currents to their references and the transition to the steady-
state controller are defined as follows:
Gts
1 ðT12Þ ¼ fxAX∣ðiL1
rIref1
Þ4ðVref o2viÞg
Gts
2 ðT12Þ ¼ fxAX∣ðiL1
rIref1
Þ4ðVref ¼ 2viÞg
Gts
3 ðT12Þ ¼ fxAX∣ðiL1
rIref1
Þ4ðVref 42viÞg
Gts
1 ðT42Þ ¼ fxAX∣ðiL1
ZIref1
Þ4ðVref o2viÞg
Gts
2 ðT42Þ ¼ fxAX∣ðiL1
ZIref1
Þ4ðVref ¼ 2viÞg
Gts
3 ðT42Þ ¼ fxAX∣ðiL1
ZIref1
Þ4ðVref 42viÞg ð8Þ
Proof. See the Appendix.
3.2. Steady state automaton controller design
The steady-state controller must operate in a cyclic fashion to
guarantee the periodic hybrid motion depicted in Fig. 3, which
corresponds to a three distinct converter operations depending on
the desired output voltage Vref as follows:
Vref o2vi: The limit cycle is characterized by the discrete
sequence q2q1q3q1 where the spent time in mode qi before the
transition to mode qj is denoted tÃ
ij and given by:
tÃ
21 ¼ tÃ
31 ¼ dT; tÃ
12 ¼ tÃ
13 ¼ ð0:5ÀdÞT ð9Þ
with T ¼ tÃ
21 þtÃ
12 þtÃ
31 þtÃ
13 is the period of the limit cycle, which
can be generally fixed depending on the switching frequency
f¼1/T of the controlled semiconductor devices (sw1 and sw2).
Fig. 7a illustrates the hybrid trajectory waveforms of the 2IBC in
this case where the current ripples are listed in Table 2.
Vref ¼ 2vi: This case is a transition one between case 1 and case
3 where the discrete sequence is q2q3. The input current ripple
is equal to zero and the other ripples are provided in Table 2.
balancing
Fig. 5. State space partition of the 2IBC under the proposed transient state
controller.
M. Bougrine et al. / ISA Transactions 66 (2017) 425–436428

5. Fig. 6. The proposed hybrid automaton controller of the 2IBC.
Fig. 7. Projection of the desired limit cycle in the currents plane.
M. Bougrine et al. / ISA Transactions 66 (2017) 425–436 429

6. The spent time in each mode is given by:
tÃ
23 ¼ tÃ
32 ¼ 0:5T; ð10Þ
The converter trajectory in steady-state is presented in Fig. 7b.
Vref 42vi: The converter trajectory waveforms are illustrated in
Fig. 7c where the limit cycle is defined by the discrete sequence
q2q4q3q4 with the following spent times:
tÃ
24 ¼ tÃ
34 ¼ ð1ÀdÞT; tÃ
42 ¼ tÃ
43 ¼ ðdÀ0:5ÞT ð11Þ
The corresponding current ripples are reported in Table 2.
From the above analysis, we note that the desired limit cycle is
completely defined by the discrete sequence and the spent times,
which depend on the desired switching frequency f, the input
voltage and the desired output voltage.
From the fact that the desired limit cycle has three geometric
shapes, one can conclude that the steady-state controller must
have three parts depending on the desired voltage Vref for a given
switching frequency and input voltage as proposed in Fig. 6. In
each part, we have the desired discrete sequence defined by a
closed unidirectional automaton where the guard conditions,
denoted Gs
ðTijÞ, should be designed to guarantee the local
asymptotic stability of the desired limit cycle. Transitions between
the three parts, denoted ci with iAf1; 2; 3g, occur if the reference
voltage is changed and does not meet the actual part. The condi-
tions c1; c2 and c3 are valid if the reference output voltage verifies
the conditions Vref o2vi; Vref ¼ 2vi and Vref 42vi respectively.
Proposition 2 outlines the steady-state controller of the 2IBC.
Proposition 2. Under the proposed steady-state controller in Fig. 6
with the guard conditions (12)–(14), the desired limit cycle of the
2IBC depicted in Fig. 3 is locally asymptotically stable.
Part 1 ðVref oviÞ:
Gs
ðT21Þ ¼ fxAX∣iL1
ZIref1
þΔi1g
Gs
ðT13Þ ¼ fxAX∣Iin rIref ÀΔig
Gs
ðT31Þ ¼ fxAX∣iL2
ZIref2
þΔi2g
Gs
ðT12Þ ¼ fxAX∣Iin rIref ÀΔig ð12Þ
Part 2 ðVref ¼ viÞ:
Gs
ðT23Þ ¼ fxAX∣iL2
rIref2
ÀΔi2g
Gs
ðT32Þ ¼ fxAX∣iL1
rIref1
ÀΔi1g ð13Þ
Part 3 ðVref 4viÞ:
Gs
ðT24Þ ¼ fxAX∣iL2
rIref2
ÀΔi2g
Gs
ðT43Þ ¼ fxAX∣Iin ZIref þΔig
Gs
ðT34Þ ¼ fxAX∣iL1
rIref1
ÀΔi1g
Gs
ðT42Þ ¼ fxAX∣Iin ZIref þΔig ð14Þ
Proof. See Appendix.
The guard conditions (12)–(14) are obtained from the limit
cycle trajectory computation based on the dynamics of each dis-
crete mode. For example, let us consider the guard conditions
design in part 1 where the discrete sequence is q2q1q3q1.
For the discrete mode q2, the inductor current iL1
increases
(charging case) where an upper constraint ðIref1
þΔi1Þ is chosen.
Besides, the inductor current iL2
decreases (discharging case) but it
is not used due to the necessity of other intermediate ripple cal-
culation, which can be remarked in Fig. 7a. Furthermore, the input
current (Iin) is avoided because its dynamic depends on the con-
verter state. Using the same reasoning and by symmetry, the guard
condition Gs
ðT31Þ is obtained. In mode q1, the input current
decreases, which allows us to add the lower constraint ðIref ÀΔiÞ
for input current ðIinÞ in Gs
ðT12Þ and Gs
ðT13Þ. The same thought is
applied to part 2 and part 3 for the guards design.
The suggested hybrid controller shown in Fig. 6 and defined by
the guard conditions (6)–(8) for the transient part and (12)–(14)
for the steady part requires only one condition to be verified at
each instant to determine the next discrete mode, which is
attractive for practical implementations.
3.3. Adaptive rule based on an algebraic approach
From the guard conditions, the hybrid controller can be clas-
sified as a current control loop with the aim to maintain the
output voltage at a desired level. In fact, the used reference current
depends on the load resistance, which may affect the output vol-
tage regulation. For that purpose, an online estimation of the load
resistance should be developed for online adjustment of the
hybrid control scheme. We opt for algebraic techniques for the
load resistance estimation, which can be reliably achieved in real
time that depends only on the arithmetic precision of the used
processor [32–34].
Let us consider the output capacitor voltage dynamic:
voðtÞ
1
R
¼ ð1Àu1ðtÞÞiL1
ðtÞþð1Àu2ðtÞÞiL2
ðtÞÀCo
dvo
dt
ðtÞ
Using the Laplace transformation for each discrete mode and
taking into account that the control inputs are constant (u1ðtÞ ¼ u1
and u2ðtÞ ¼ u2), one can obtain:
1
R
voðsÞ ¼ ð1Àu1ÞiL1
ðsÞþð1Àu2ÞiL2
ðsÞÀCoðsvoðsÞÀvoð0ÞÞ
with s is the Laplace variable. To get rid of the initial condition
voð0Þ, we differentiate both sides with respect to s as follows:
1
R
dvo
ds
ðsÞ ¼ ð1Àu1Þ
diL1
ds
ðsÞþð1Àu2Þ
diL2
ds
ðsÞÀCoðvoðsÞþs
dvo
ds
ðsÞ
Time differentiations in this expression, i.e. multiplication by s in
the operational form, can be avoided by multiplying both sides by
a give power of sÀ1
. Moreover, as the integral plays the role of a
low-pass filter, one seeks relation in which every time-function
appears inside an integral in the time domain. By multiplication by
sÀ2
, one obtains:
1
s2
dvo
ds
ðsÞ
|fflfflfflfflffl{zfflfflfflfflffl}
nðsÞ
1
R
¼ ð1Àu1Þ
1
s2
diL1
ds
ðsÞþð1Àu2Þ
1
s2
diL2
ds
ðsÞÀCo
1
s2
voðsÞþ
1
s
dvo
ds
ðsÞ
|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
dðsÞ
Returning to time domain using the inverse Laplace transforma-
tion, one gets an algebraic estimator of the load resistance:
Table 2
Different state ripples of 2IBC.
Ripples Δi1 Δi1 Δi
Case 1 vi ÀrIref1
2L
dT
vi ÀrIref2
2L
dT
Vref Àvi ÀrIref1
L
ð0:5ÀdÞT
Case 2 vi ÀrIref1
4L
T
vi ÀrIref2
4L
T
0
Case 3 Vref þrIref1
Àvi
2L
ð1ÀdÞT
Vref þrIref2
Àvi
2L
ð1ÀdÞT
vi ÀrIref1
L
ðdÀ0:5ÞT
M. Bougrine et al. / ISA Transactions 66 (2017) 425–436430

7. identification:
~RðtÞ ¼
nðtÞ
dðtÞ
ð15Þ
with:
nðtÞ ¼
Z T
0
ðT ÀσÞðÀσÞvoðtÀσÞdσ ð16aÞ
dðtÞ ¼
Z T
0
ðT ÀσÞ ðÀσÞ ð1Àu1ÞiL1
ðσÞþð1Àu2ÞiL1
ðσÞ
À Á
ÀCovoðσÞ
 ÃÂ
þCoσvoðσÞŠdσ ð16bÞ
This estimator is not asymptotic, it can be evaluated for any
t4T and give a real-time estimation of R. In practice, the integrals
appearing in (16) are calculated by discrete methods (e.g. the
trapezium rule) using a finite number of samples of iL1
, iL1
and vo in
the time-interval ½tÀT; tŠ.
4. Simulation results
For practical purposes, designing the converter parameters,
especially passive components, is of utmost importance. To this
end, it is necessary to reveal the existing links between the
switching frequency, the output voltage, the current and voltage
ripples, and the passive components. This has already been dis-
cussed in the literature. For brevity reasons, only the most
important equations will be mentioned.
In CCM operation, the minimal required values for the passive
components are obtained by considering the worst case in terms
of ripples, which corresponds to the duty cycles d¼0.25 and
d¼0.75. These are given by
Lmin ¼
vo
8ΔIin;CCM;max f
ð17aÞ
Cmin ¼
vo
4RminΔvo;CCM;max f
ð17bÞ
with the output voltage vo, the desired switching frequency f ¼ 1
T,
and the maximum allowed current ripple ΔIin;CCM;max and voltage
ripple Δvo;CCM;max. A typical converter with the parameters in
Table 3 will be studied in closed loop under the hybrid controller
presented in Fig. 6 and defined by the two propositions alongside
the adaptive rule equations (16)–(17). The initial value of the load
estimator R0 is fixed at 15 Ω where the evaluation starts after
ϵ¼0.5 ms and re-initialized each period Te ¼ 2 ms. Performance
and robustness of the adaptive hybrid controller will be checked
through different scenarios and tested with a fuel cell source in the
next subsections.
4.1. Variable reference output voltage
Fig. 8 illustrates the continuous state evolution of the 2IBC in
closed loop for a variable reference output voltage. At startup,
the reference input current Iref that corresponds to Vref ¼ 35 V is
calculated using (4), which is perturbed by the wrong value of
the estimated resistor load ðRe ¼ R0Þ. The hybrid controller forces
the input current to converge rapidly to Iref ¼ 4:08 A, as it can be
remarked from Fig. 8b, which justifies the overshoot in the
output voltage. After ϵ ¼ 0:5 ms, the algebraic estimator pro-
vides the real value of the load resistance, as illustrated in Fig. 9,
to the hybrid controller. The latter overcomes the situation and
stabilizes quickly the converter state around its reference
xref ¼ ½1:53 A 1:53 A 35 VŠT
. The reached limit cycle is depicted in
Fig. 10a, which has the discrete sequence q2q1q3q1. The current
ripples are obtained from Table 2 for a fixed frequency f¼10 Hz.
At t¼0.01 s, the reference voltage Vref is decreased by 5 V. The
hybrid controller allows us to bring the input current in finite time
to the new desired value Iref ¼ 2:25 A and forces the converter
trajectory to converge to the optimal limit cycle using part 1 of the
steady-state controller as it can be observed from the used discrete
modes in Fig. 10b. In the interval [0.02 s, 0.03 s], the reference
voltage is set to 40 V. The hybrid controller uses part 2 of the
steady-state controller to stabilize the optimal limit cycle in this
case as shown in Fig. 10c, which contains only the discrete modes
q2 and q3. It should be noted that the input current ripple is equal
to zero as it can be remarked from the currents evolution in
Table 3
Converter parameters.
Parameter vi L; r Co f R0
Value 20 V 3.3 mH, 1 mΩ 23 μF 10 kHz 15 Ω
Fig. 8. State evolution of the 2IBC for a variable load resistance.
Fig. 9. Load estimation based on the algebraic approach.
M. Bougrine et al. / ISA Transactions 66 (2017) 425–436 431

8. Fig. 8b. In the last time interval, the hybrid controller drives the
converter state to its new reference xref ¼ ½2:53 A 2:53 A 45 VŠT
quickly where the reached limit cycle is depicted in Fig. 10d. The
used discrete sequence is q2q4q3q4 because the reference output
voltage Vref verifies the part 3 condition ðVref 42viÞ of the steady-
state controller.
In summary, the adaptive hybrid controller exhibits interesting
properties such as: minimum setting time, zero steady-state error
and a fixed operating frequency in steady-state (f¼10 kHz). Next,
we will proceed with the load resistance variations for adaptive
property test of the hybrid controller.
4.2. Variable load resistance
Fig. 11 shows the obtained results for a desired output voltage
Vref ¼ 35 V and a variable load resistance. The latter is increased by
Fig. 10. Projection of the reached limit cycle in the currents plane ðiL1
; iL2
Þ and the corresponding discrete mode evolution.
Fig. 11. State evolution of the 2IBC for a variable load resistance.
Fig. 12. Evolution of the real and the estimated load resistance value.
Fig. 13. PEM fuel cell characteristics V–I and P–I.
M. Bougrine et al. / ISA Transactions 66 (2017) 425–436432

9. 5 Ω from 20 Ω to 30 Ω each 10 ms and decreased by 10 Ω at the
instant 30 ms. Fig. 9 shows the algebraic estimator response under
these disturbances. The algebraic estimator tries to detect and
reject any load variation by the re-initialization of the algebraic
formulae each 2 ms, which is clear from the periodic small over-
shoots in Fig. 12. Once the load resistance is estimated, the hybrid
controller will be updated by a new corrected reference input
current to reject the load perturbation. The obtained results
illustrate the effectiveness and the robustness of the hybrid con-
troller in spite of these fast variations.
4.3. Association of the 2IBC with a fuel cell source
Fuel cell systems are increasingly used in the last decade in
automotive applications, which are classified as clean energy with
zero CO2 emission. In this part, we consider an example of 1.26 kW
proton exchange membrane (PEM) fuel cell with the P–I and V–I
characteristics shown in Fig. 13, which is provided in MATLAB
s
SIMPOWER toolbox.
Fig. 14 illustrates the 2IBC state evolution for the output voltage
reference and a variable load resistance. One can remark that the
fuel cell Vref ¼ 80 V has a minimal ripple in different phases of load
variations. The adaptive hybrid controller is able to drive the
converter state x to its reference quickly and with a good tracking.
The robustness of the controller despite load variations is clearly
observed (Fig. 16). The closed loop maintains the control require-
ments in spite of fuel cell stack voltage variations presented in Fig.
15.
Table 4 shows a qualitative comparison between the proposed
control scheme based on hybrid automata theory and some other
existing control strategies for the conventional boost converter
and the interleaved one. One may read the proposed controller as
an extension for the interleaved case of the controller presented in
[35] for the conventional boost converter. The problem of the
phase shifting between the control signals that does not show up
in the conventional boost has been solved implicitly for the two-
interleaved converter by considering a limit cycle stabilization
problem in CCM. However, the case of DCM is beyond the scope of
this paper and will be dealt with in future work.
5. Conclusions and future works
In this paper, a new adaptive hybrid control scheme is devel-
oped for the asymptotic stabilization of the desired limit cycle of
two phase interleaved boost converter. From practical point of
view, this corresponds to fast transient and low input current
achievements, which is advantageous for many real world appli-
cations such as: fast MPPT with low fluctuations around the
maximum power point, extended lifetime of fuel cell system and
solar panels, high efficiency and enhanced thermal properties of
the converter, etc.
The controller is developed via a hybrid automaton formalism
with two parts: (1) a transient state automaton for fast transient
and (2) a steady-state automaton for local limit cycle stabilization.
An online adaptive rule based on the algebraic approach is also
investigated for robustness purpose. Mathematical proofs and
simulation results confirm the effectiveness and the robustness of
Fig. 14. State evolution of the 2IBC associated with fuel cell source under a variable
load resistance.
Fig. 15. Fuel cell stack voltage.
Table 4
Qualitative comparison.
Previous works Control Model nature Phase shift Studied
modes
Complexity Applicability
Giral et al. [36] Sliding mode-based Instantaneous model Enforced in OL with latch CCM Medium Two-interleaved
Hubber et al. [37] Open loop control Instantaneous model Enforced in OL with delay CCM/DCM Simple Two-interleaved
Thammasiriroj et al. [38] Flatness-based Average model Enforced in OL with PWM CCM Very high N-interleaved
Sreekumar and Agarwal
[35]
Limit cycle
stabilization
Hybrid model – CCM/DCM Simple Conventional DC–DC
boost
Proposed controller Limit cycle
stabilization
Hybrid model Controlled in CL by the limit cycle
stabilization
CCM Medium Two-interleaved
M. Bougrine et al. / ISA Transactions 66 (2017) 425–436 433

10. the suggested controller under varying load and input voltage. A
PEM fuel cell source associated with the 2IBC is also highlighted
where satisfactory results are obtained.
As a future work, a generalization of the proposed hybrid
controller to n-phase interleaved boost converter and other
multilevel topologies will be considered. A complete study of
systems including different renewable and clean energy hybrid
sources will be investigated. An adaptive hybrid controller for
continuous conduction mode (CCM) and discontinuous conduc-
tion mode (DCM) operation of the converter for light loads will
also be discussed.
Appendix A
Proposition 1
Proof. From the transient state automaton configuration and the
developed guard conditions, the input current Iin reaches its
reference Iref in finite time if the following conditions are verified:
Finite time reachability of the balancing surface ðS1ðxÞ ¼ 0Þ for
x0 AΩ2 [ Ω3.
Invariance of the balancing surface.
Finite time reachability of the reference surface ðS2ðxÞ ¼ 0Þ for
xðtsÞAðS1ðxÞ ¼ 0Þ.
Roughly speaking, for any continuous initial condition in X, the
input current has to reach the balancing surface in finite time
(condition 1) and remain there (the invariance condition). Besides,
in the balancing surface, we must choose a discrete mode that
exhibits finite time reachability of the inductor currents to their
references.
By taking into consideration that all the state matrices of the
discrete modes are Hurwitz, the first and the last conditions are
equivalent to the following statements proof:
x
q2
eq AΩ3; x
q3
eq AΩ2; x
q1
eq AΩ4 ðS1ðxÞ ¼ 0Þ;
x
q4
eq AΩ1 ðS1ðxÞ ¼ 0Þ:
Indeed, if for example an initial condition x0 belongs to Ω2 then
the discrete mode q2 will be activated, and from the fact that the
stable equilibrium point is in the complementary region of Ω2, the
converter trajectory must pass through the balancing surface,
which is equivalent to finite time reachability of the balancing
surface ðS1ðxÞ ¼ 0Þ from one side. The same thought is valid in the
other side for x0 AΩ3. Once the converter trajectory hits the bal-
ancing surface a transition to mode q1 or mode q4 will be done.
Fortunately, from the dynamics of these modes, it is clear that the
balancing surface is invariant. It remains to prove that the equili-
brium point of mode q1 is on the balancing surface and belongs to
the opposite region (Ω4) with the same thought for mode q4 to
conclude with the finite time attractivity of the reference surface.
Let us start by the first statement:
x
q2
eq AΩ3⟹ieq
L2
oieq
L1
From Table 1, we have:
ieq
L2
¼
vi
Rþr
oieq
L1
¼
vi
r
which proves the first statement. The second statement x
q3
eq AΩ2
can be verified with same manner.
The third statement can be proved as follows:
x
q1
eq AΩ4 ðS1ðxÞ ¼ 0Þ⟹ieq
L2
¼ ieq
L1
oIref1
From Table 1, one can remark that ieq
L2
¼ ieq
L1
¼ vi
2R þr and it remains to
prove only the following condition:
vi
2Rþr
oIref1
¼
1
2
V2
ref
Rvi
Let us begin by the boost property Vref 4vi and multiplying both
sides by Vref
2Rvi
, one can get:
1
2
V2
ref
Rvi
4
Vref
2R
4
Vref
2Rþr
4
vi
2Rþr
Hence the statement x
q1
eq AΩ4 ðS1ðxÞ ¼ 0Þ is proved. The last
statement can be checked by taking into consideration that the
parasitic resistance r is negligible where a reference input current
cannot be bigger than vi
r . □
0 0.01 0.02 0.03 0.04
40
45
50
Time (s)
LoadResistance()
R
R
e
Ω
Fig. 16. Evolution of the real and the estimated load resistance value.
Fig. 17. Eigenvalues of the Jacobian of the Poincaré map in case 1.
M. Bougrine et al. / ISA Transactions 66 (2017) 425–436434

11. Proposition 2
Proof. The local stability of the desired limit cycle can be proved
using the following theorem [39], and the reader may refer to
[40,41,35] for more detail and for various applications to multi-
level and classical power converters.
Theorem 1. Consider a general piecewise linear system with the
form given by (1). Assume that there exists a limit cycle γ with period
T, and defined by the discrete sequence q1q2…qN. Guard condition
between mode qi to mode qj, denoted GðTijÞ, is given by a hyperplane
of dimension nÀ1 as follows:
GðTijÞ ¼ fxAX∣Cijxþdij ¼ 0g ð18Þ
with Cij AR1Ân
and dij AR.
The stability analysis of the limit cycle γ can be checked by looking
at the eigenvalues of the Jacobian of the Poincaré map P as follows:
dP ¼ dPk1…dP34dP23dP12 with dPij ¼ IÀ
f qi
ðxÃ
ijÞCij
Cijf qi
ðxÃ
ijÞ
!
eAitÃ
ij ð19Þ
where xÃ
ij is the switching state between mode qi and mode qj, which
belongs to the guard condition GðTijÞ. The local stability of the limit
cycle γ can be checked as follows:
If the map dP has all its eigenvalues inside the unit circle, then the
limit cycle γ is locally asymptotic stable.
If at least one of the eigenvalues is outside the unit circle, then the
limit cycle γ is unstable.
We will use this theorem to prove the local stability of the
desired limit cycle under the proposed hybrid steady-state con-
troller in the following three cases.
Case 01 ðVref o2viÞ: In this case, the Jacobian of the Poincaré
map that corresponds to the discrete sequence q2q1q3q1 is
dP ¼ dP12dP31dP13dP21. Transitions between the discrete modes of
the discrete sequence occurs in 2-dimensional hyperplanes
respecting (18), which can be defined using (12) as follows:
C12 ¼ ½1 1 0Š; d12 ¼ ÀðIref ÀΔiÞ
C31 ¼ ½0 1 0Š; d31 ¼ ÀðIref2
þΔi2Þ
C13 ¼ ½1 1 0Š; d13 ¼ ÀðIref ÀΔiÞ
C21 ¼ ½1 0 0Š; d21 ¼ ÀðIref1
þΔi1Þ
8
:
The switching points between the discrete modes of the desired
limit cycle are given as follows:
xÃ
12 ¼ ½Iref1
ÀΔi1; Iref2
þΔi1 ÀΔi; Vref þΔvŠT
xÃ
31 ¼ ½Iref1
ÀΔi2 þΔi; Iref2
þΔi2; Vref ÀΔvŠT
xÃ
13 ¼ ½Iref1
þΔi2 þΔi; Iref2
ÀΔi2; Vref þΔvŠT
xÃ
21 ¼ ½Iref1
þΔi1; Iref2
ÀΔi1 þΔi; Vref ÀΔvŠT
8
:
Replacing these results and the corresponding discrete mode
dynamic in (19), one can get the Jacobian of the Poincaré map.
Using the converter parameters in Table 3, the eigenvalues of dP
are evaluated numerically for a variable reference voltage ðVref o
2viÞ and load resistance, which are depicted in Fig. 17. All the
eigenvalues are inside the unit circle, which proves the local sta-
bility of the desired limit cycle under the proposed steady-state
controller in this case.
Case 02 ðVref ¼ 2viÞ: In this case, the discrete sequence q2q3 has
the Jacobian of the Poincaré map dP ¼ dP32dP23. The switching
surfaces in the form (18) and the switching points are given as
Fig. 18. Eigenvalues of the Jacobian of the Poincaré map in case 2.
Fig. 19. Eigenvalues of the Jacobian of the Poincaré map in case 3.
M. Bougrine et al. / ISA Transactions 66 (2017) 425–436 435

12. follows:
C32 ¼ ½1 0 0Š; d12 ¼ ÀðIref1
ÀΔi1Þ
C23 ¼ ½0 1 0Š; d31 ¼ ÀðIref2
ÀΔi2Þ
(
xÃ
32 ¼ ½Iref1
ÀΔi1; Iref2
þΔi2; Vref ŠT
xÃ
23 ¼ ½Iref1
þΔi1; Iref2
ÀΔi2; Vref ŠT
(
Fig. 18 illustrates that all the eigenvalues of the dP are inside
the unit circle for varying load resistance, which proves the local
stability of the desired limit cycle in this case.
Case 03 ðVref 42viÞ: The desired limit cycle has the discrete
sequence q2q4q3q4 where its Jacobian of the Poincaré map is
dP ¼ dP42dP34dP43dP24. From the guard conditions given by (14),
one can get the following switching surfaces parameters:
C42 ¼ ½1 1 0Š; d42 ¼ ÀðIref þΔiÞ
C34 ¼ ½1 0 0Š; d34 ¼ ÀðIref1
ÀΔi1Þ
C43 ¼ ½1 1 0Š; d43 ¼ ÀðIref þΔiÞ
C24 ¼ ½0 1 0Š; d24 ¼ ÀðIref2
ÀΔi2Þ
8
:
The switching points between the discrete modes of the desired
limit cycle are given as follows:
xÃ
42 ¼ ½Iref1
þΔi1; Iref2
ÀΔi1 þΔi; Vref ÀΔvŠT
xÃ
34 ¼ ½Iref1
ÀΔi1; Iref2
þΔi1 ÀΔi; Vref þΔvŠT
xÃ
43 ¼ ½Iref1
ÀΔi2 þΔi; Iref2
þΔi2; Vref ÀΔvŠT
xÃ
24 ¼ ½Iref1
þΔi2 ÀΔi; Iref2
ÀΔi2; Vref þΔvŠT
8
:
Fig. 19 shows that all the eigenvalues of the dP are inside the
unit circle for different values of Vref 42vi and varying load
resistance, which proves the local stability of the desired limit
cycle in this case.
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