Preprints of the 18th IFAC World Congress
Milano (Italy) August 28 - September 2, 2011
Adaptive Control of the Boost Converter in
Discontinuous Conduction Mode
Aya Alawieh, ∗ Romeo Ortega, ∗ Harish Pillai, ∗∗
Alessandro Astolfi ∗∗∗ and Eric Berthelot ∗∗∗∗
∗
Laboratoire des Signaux et Systèmes, Supelec, Plateau du Moulon,
91192 Gif-sur-Yvette, France (e-mail:aya.alawieh@lss.supelec.fr).
∗∗
Department of Electrical Engineering, IIT Bombay, Mumbai 400076,
India(e-mail:hp@ee.iitb.ac.in) .
∗∗∗
Department of Electrical and Electronic Engineering, Imperial
College, London, SW7 2AZ, UK, and DISP, University of Roma, Tor
Vergata, Via del Politecnico 1, 00133 Rome, Italy (e-mail:
a.astolfi@imperial.ac.uk).
∗∗∗∗
Laboratoire de Genie Electrique de Paris, Supelec, Plateau du
Moulon, 91192 Gif-sur-Yvette, France
(e-mail:eric.berthelot@lgep.supelec.fr).
Abstract: In this brief note we are interested in the problem of voltage regulation of a classical
boost converter operating in discontinuous conducting mode. The system does not admit a
(continuous–time) average model approximation, hence is a bona fide hybrid system where the
control objective is the generation of a periodic orbit and the actuator commands are switching
times. The standard procedure to solve the problem relies on approximations of the solution of
the differential equations, which may lead to below par performances (or, as shown here, even
instability). Our main contribution is a simple adaptive algorithm that gives explicit formulas
for the switching times without approximations and estimates all the uncertain parameters.
Simulation and experimental results that illustrate the robustness of the scheme, as well as
performance comparisons with current practice, are presented.
Keywords: Hybrid systems, power converters, stabilization of nonlinear systems, stabilization
of limit cycles, adaptive control.
1. INTRODUCTION
nonlinear controller design techniques, e.g., sliding mode
and passivity–based, are directly applicable.
Ideal switches in power converters are typically implemented using unidirectional semiconductor devices that
may lead to a new operation mode generically called
discontinuous conduction mode (DCM). The DCM arises
when the ripple is large enough to cause the polarity of the
signal (current or voltage) applied to the switch to reverse,
violating the unidirectionality assumptions made in the
realization of the switch. In classical converter topologies
DCM appears very frequently in low load operating modes.
More interestingly, to achieve high performance some new
converters are purposely design to operate all the time in
DCM Mohr (2010).
In DCM the converter is a bona fide hybrid system, which
exhibits behaviors of continuous–time and discrete–time
dynamical systems. Distinguishing features of converters
in DCM include the following.
The converter dynamics are significantly altered in DCM.
The existing techniques for controller analysis and design,
in particular, the approximations of averaging dynamics
(valid under fast switching) or small ripple, are not valid
anymore. With the latter approximations the converter
can be treated as a continuous (possibly nonlinear) dynamical system, with the action of the switches assimilated to
continuous signals ranging in some closed interval, e.g.,
[0, 1]. Neglecting the presence of the ripple permits to
recast the converter regulation problem as a classical equilibrium stabilization problem for which, by–now standard,
Copyright by the
International Federation of Automatic Control (IFAC)
(i) The control is not a continuous signal, but directly
the switches positions, that take values in the binary
set {0, 1}, and decide the commutation among the
various converter topologies.
(ii) Due to technological considerations, the activation of
the switches is submitted to a minimal dwell time
that has to be taken into account in the design.
(iii) Besides the commutations induced by (designer selected) switch positions, there are forced commutations due to the aforementioned violation of the unidirectionality assumption, e.g., the presence of diodes.
(iv) As the ripple cannot be neglected, the control objective is not stabilization of an equilibrium but generation of a periodic orbit (with “minimal amplitude”)
around the desired operating point.
Interested readers are referred to the classical reference
Cuk and Middlebrook (1976) and the excellent monograph
Erikson and Maksimovic (2000) for a detailed explanation
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Preprints of the 18th IFAC World Congress
Milano (Italy) August 28 - September 2, 2011
x1
of the physical phenomenon and some rules for controller
design that rely on some approximations. More recent
developments may be found in Sun (2009), Qiao and
Zhang (2005), Viswanathan (2002). It is widely recognized by the power electronics community that the use
of these approximations leads to below par performances
of the device (or, as shown here, even instability). This
situation makes the problem a challenging new task for
control systems theorists.
A burgeoning literature on hybrid systems has emerged
in the control community in the last few years, see e.g.,
Goebel, Sanfelice and Teel (2009) for a recent tutorial
account. As usual in control theory, the main thrust
of the research has been towards the development of
general methodologies, mainly for analysis Liberzon (2003)
but also for controller design, of classes of theoretically–
motivated, hybrid dynamical systems.
To the best of our knowledge, none of the existing results
reported in the literature is applicable to address the
problem at hand. 1 Motivated by this fact, in this paper a
different perspective is adopted, namely the development
of a solution for a specific example of practical relevance.
Towards this end, we consider a boost converter operating
in DCM, which exhibits the four distinguishing features
discussed above. Our main contribution is a simple robust
algorithm that, in contrast with current practice, gives
explicit formulas for the switching times without approximations and, thanks to the incorporation of an estimator
for all uncertain parameters, is robust vis–à–vis parameter
uncertainty. Although the design relies on the specific
topology of the boost converter, this device constitutes a
basic building block in power converter design, hence it is
our belief that the main ideas may be applicable to other
converters.
rL
1 The second author thanks Panos Anstaklis, Daniel Liberzon and
Dragan Nesic for looking into this problem.
1
x1
rL
x2
C
R0
x2
0
L
vin
R0
1
x1
rL
C
0
L
vin
1
C
x2
R0
Fig. 1. Ideal representation of the boost converter in the
topologies: Ω1 (top), Ω2 (middle) and Ω3 (bottom).
topology Ω2 , corresponding to the DCM, is also depicted
in Fig. 1. Moreover, due to technological considerations, a
minimal dwell time, denoted tD ∈ R+ , is imposed to the
switch. If tD is large, with respect to the time constant
of the R0 C tank, the inductor current x1 becomes zero
and a forced transition from Ω1 to Ω2 is induced. On the
other hand, the transitions from Ω2 to Ω3 and from Ω3
to Ω1 , determined by the switch position, are decided by
the controller. Of course, these switching transition times
should respect the minimal dwell time constraint.
The dynamics of the system is described by a piece–wise
affine model
ẋ = Ai x + bi vin , i = 1, 2, 3,
where the pairs (Ai , bi ) for the three topologies are given
by

"
−rL −1

 L
L
Ω1 : A1 =  1
−1  , b1 =
C CR0
#
"
0 0
0
−1
, b2 =
Ω2 : A2 =
0
0
CR0


−rL
"
0 
 L
Ω3 : A3 = 
−1  , b3 =
0
CR0

2. MATHEMATICAL MODEL AND PROBLEM
FORMULATION
In most practical applications the switch is implemented
with a Mosfet device and a diode. Since the diode is
a single quadrant switch, a new operation mode—the
DCM—appears if its current crosses through zero. The
L
vin
The remaining of the paper is organized as follows. The
converter model and the problem formulation are presented in Section 2. The basic control algorithm given in
Section 3, with an adaptive version presented in Section 4.
Some simulation results are shown in Section 5. The paper
is wrapped–up with some concluding remarks in Section
6.
The boost converter is a switched electronic device that
amplifies a dc voltage commuting between two basic
topologies depicted in Fig. 1. In the topology Ω3 , with the
switch in the position 1, the external battery vin ∈ R+ supplies magnetic energy to the (leaky) inductor L while the
electric energy of the capacitor C is discharged in the load,
represented here by a resistance R0 . When the switch is in
the position 0 the circuit takes the topology Ω1 , then the
energy flows from the inductor to the capacitor, building
up its voltage x2 and achieving the desired amplification.
0
1
L
0
#
1
L
0
#
.
The system evolves in the set {x1 > 0, x2 > 0} =: R2+
for the topologies Ω1 and Ω3 , while for the topology Ω2 ,
trajectories are restricted to the set {x1 = 0, x2 > 0}. 2
A typical trajectory in the phase plane x1 − x2 is shown
in Fig. 2, where the ti ∈ R+ , i = 1, 2, 3, denote the time
spent on each of the topologies.
Given t1 —the time it takes for the current to go to zero—
the designer has to decide the switching times t2 and t3 ,
which should satisfy the dwell time constraints
t3 ≥ tD , t1 + t2 ≥ tD .
(1)
2 The possibility of total discharge of the capacitor, e.g., x = 0,
2
is ruled out because below a critical value of x2 the diode starts
conducing again commuting to the topology Ω1 and driving the
trajectory inside the positive quadrant.
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Preprints of the 18th IFAC World Congress
Milano (Italy) August 28 - September 2, 2011
x1
x1
x2
x01
x12
t3
x01
t1
x02
x02
t2
x12
t3
t2
t1
t2
t3
Fig. 3. Time evolution of x1 (t) and x2 (t) along a periodic
orbit.
Fig. 2. Typical periodic orbit in the phase plane.
The control objective is to generate an attractive limit
cycle (of period t1 + t2 + t3 ) contained in the band
x2 (t) ∈ [x∗2 − ǫ1 , x∗2 + ǫ2 ],
where x∗2 ∈ R+ is the desired average value for x2 and—to
minimize the voltage ripple—the constants ǫi ∈ R+ are as
small as possible. Although the range of values of x1 (t) is
not the main concern, for practical reasons, this current is
also restricted to satisfy an upperbound x1 (t) ≤ xM
1 . As
discussed below, in the standard approximated algorithms
Cuk and Middlebrook (1976); Erikson and Maksimovic
(2000), xM
1 is fixed to some given value.
Notice that, associated to the topologies Ω1 and Ω3 , there
is a unique stable equilibrium, namely,

R0 vin
vin



 ( rL + R0 , rL + R0 ) for Ω1
x̄ =


( vin , 0) for Ω ,

3
rL
which correspond to a focus and a sink, respectively. For
Ω2 the equilibrium is the subspace {x2 = 0}.
3. AN EXACT FEEDBACK–BASED SWITCHING
ALGORITHM
The objective of the note is to propose a switching algorithm that guarantees the existence of the desired periodic
orbit. That is, the derivation of a rule to compute the
switching instants t3 and t2 . 3 To ensure robustness it
should be a feedback strategy—that is, based on measurements of the system state—and should not be very
sensitive to variations of the system parameters.
An obvious procedure to compute the switching times
relies on the solution of the differential equations along the
periodic orbit, which leads to a set of nonlinear algebraic
equations that are very hard, if at all possible, to solve.
As explained in Section 5, to obtain explicit formulas in
practice the state transition matrices of the LTI systems
are approximated by a linear function, that is, eAi t ≈ I2 +
Ai t. The approximation induces an error that, in high
performance applications, may be inadmissible and, as
shown below, may even lead to instability.
In the present work no approximations are made and exact
explicit formulas for the switching times are given. Instrumental to this end are the following key computations—
3
t
t
t1
x2
As shown in Section 5 this is equivalent to computing the duty
ratios for operation in a synchronous mode with a fixed sampling
time—as done in Cuk and Middlebrook (1976); Erikson and Maksimovic (2000).
along a closed trajectory.
Computation of t2 + t3 . Define the time interval I1 :=
[t1 , t1 + t2 + t3 ]. For all t ∈ I1 the capacitor voltage evolves
1
according to ẋ2 = − CR
x2 , whose solution is
0
1
x2 (t) = e− CR0 (t−s) x2 (s),
∀t ≥ s, ∀t, s ∈ I1 .
Hence, along the orbit depicted in Fig. 2 and from Fig.3,
one has
− 1 (t +t )
x02 = e CR0 2 3 x12 ,
yielding
1
x2
t2 + t3 = R0 C ln
.
(2)
x02
Since x12 > x02 we have that t2 + t3 > 0. Furthermore, since
x02 is the smallest value of x2 (t) along the periodic orbit,
it should be selected to satisfy x02 < x∗2 .
Computation of t3 . Define the time interval I2 := [t1 +
t2 , t1 + t2 + t3 ]. For all time t ∈ I2 the inductor current
evolves according to ẋ1 = − rLL x1 + vLin , whose solution
with zero initial conditions is
v
rL
in
, ∀t ∈ I2 .
x1 (t) = 1 − e− L t
rL
Hence, fixing x1 (t3 ) = x01 —again referring to Figs. 2 and
3—and solving for t3 yields
L
rL x01
t3 = − ln 1 −
.
(3)
rL
vin
r x0
The time t3 is also positive, because 0 < 1 − vLin1 < 1. Indeed, x01 > 0 ensures the upper bound. On the other hand,
recall that the equilibrium associated to this topology is
( vrin
, 0), which is only reached at infinity, hence x01 < vrin
.
L
L
The computations above lead naturally to the following
control algorithm.
Step 1. Fix a point (x01 , x02 ) ∈ R2+ , such that t3 in (3)
yields t3 ≥ tD and x02 < x∗2 .
Step 2. At a time t0 ≥ 0 when x(t0 ) = (x01 , x02 ) switch
from u = 0 to u = 1.
Step 3. Wait (in mode Ω1 ) until x1 (t) = 0 and measure
the corresponding x12 .
Step 4. Compute t2 + t3 from (2). If t2 > 0 go to Step 5,
else define t2 := tD − t1 , then go to Step 5.
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Preprints of the 18th IFAC World Congress
Milano (Italy) August 28 - September 2, 2011
Step 5. Wait (in mode Ω2 ) during t2 units of time and
then switch from u = 1 to u = 0.
7
6
Step 6. Wait (in mode Ω3 ) during t3 units of time and
then measure the state, call it (x01 , x02 ). Check whether, for
the new (x01 , x02 ), (3) yields t3 ≥ tD and x02 < x∗2 . If so, go
to Step 2, else wait for a longer time until the state meets
the requirements, then assign the value t3 to this new time
and go to Step 2.
As indicated in the introduction, in normal operation the
circuit operates in DCM, hence Step 3 will, in general,
yield t2 > 0. Also, it is clear that, in ideal conditions, when
the parameters are exactly known and there are no disturbances, the point (x01 , x02 ) obtained in Step 6 will coincide
with the initial point. However, in practice there will be
a discrepancy, hence the need to recompute the switching
times as a function of the measurements. Finally, regarding
Step 6, in the topology Ω3 , x2 (t) decreases (tending to
converge to zero) and x1 (t) increases towards vrin
—both
L
evolutions being monotonic. Consequently, there is indeed
a (sufficiently long) time such that the new values of the
state guarantee that (3) yields t3 ≥ tD and x02 < x∗2 .
The computations involved in (2) and (3) are extremely
simple, and can be programmed with a few lines of code.
Moreover, they only depend on the input voltage, which
is reasonable to assume known, and the time constants rLL
and CR0 .
4. ESTIMATION OF THE TIME CONSTANTS
The computations involved in (2) and (3) are extremely
simple, and can be programmed with a few lines of code.
Moreover, they only depend on the input voltage, which
is reasonable to assume known, and the time constants
rL
L and CR0 . The algorithm can be robustified vis–à–
vis uncertainty in the time constants incorporating an
adaptation stage.
Let us illustrate a simple way to estimate the parameter
CR0 . As explained above, during the interval I1 , x2 (t)
1
x2 . Discretizing this equation with a
satisfies ẋ2 = − CR
0
(fast) sampling time 4 Tf yields the difference equation
−
1
T
x2 (k) = θx2 (k − 1), θ := e CR0 f ,
where the standard notation x2 (k) = x2 (t) for t ∈
((k − 1)Tf , kTf ], with k ∈ Z+ , is used. Now, sample
x2 (t) with the sampling rate Tf ∈ R+ and take N
3
samples, where N := ⌊ t2T+t
⌋, with ⌊·⌋ the floor operator
f
that takes the largest previous integer. Define the N –
dimensional vectors X2 := col(x2 (1), . . . , x2 (N )) and Φ :=
col(x2 (0), . . . , x2 (N − 1)). Since X2 = θΦ, it is clear that
θ can be computed from
1 ⊤
θ=
Φ X2 ,
(4)
|Φ|2
with | · | the Euclidean norm. Note that |Φ| is bounded
away from zero because x2 (t) ∈ R+ . From the knowledge
of θ the time constant CR0 is directly obtained.
Since Φ is a shifted version of X2 the rule (4) can be
easily implemented in firmware. The parameter estimation
4
Clearly, Tf should be selected much smaller that t2 + t3 .
4
1
x [A]
5
3
2
1
0
73.98
74
74.02
74.04
74.06
x [V]
74.08
74.1
74.12
74.14
2
Fig. 4. A periodic orbit in the phase plane obtained in the
ideal case.
algorithm need not be repeated in every period, but only
when performance degradation is detected. Finally, it is
obvious that a similar procedure is possible to estimate
the second time constant rLL , and eventually also vin , but
they are omitted for brevity.
5. SIMULATION AND EXPERIMENTAL RESULTS
The usefulness of the proposed control scheme was evaluated through numerical simulations and experiments.
For the simulations the considered circuit parameters are
L = 0.0001H, R0 = 100Ω, rL = 0.001Ω and C =
0.00025F . The eigenvalues of the system matrix A1 —
associated to the focus equilibrium of topology Ω1 —are
located in λ1,2 = −25 ± i6324.537, which corresponds to
a highly undamped response. On the other hand, the sink
equilibrium of topology Ω3 is placed high up in the x2 = 0
axis at (the unrealistic value of) x̄1 = 25000A.
To carry out an extensive evaluation, the simulations have
been developed in two steps, the first devoted to illustrate the performance under nominal (ideal) conditions,
where the dynamics exactly coincide with the ones used
in the theoretical developments with all the parameters
known, and was done in Matlab–Simulink. The second
set of simulations have been done with the circuit toolbox
Simpower systems, which includes realistic models of the
Mosfet and the diode, and is intended to exhibit the
operation under dynamic and parametric uncertainty. In
Fig. 4 a periodic orbit obtained via the application of the
algorithm in an ideal simulation is depicted. Since the
calculations are exact a perfect orbit is obtained. Fig. 5
show the result of the realistic simulation where, due to the
presence of unmodeled effects, the orbit is slightly shifted
and the algorithm takes some time to converge. To test
the robustness of the algorithm, the same simulation was
repeated with ±50% changes in the load resistance and it
was observed that the algorithm still generates (shifted)
periodic orbits.
Figs. 6 and 7 shows the result of the simulation when a
step in the value of the load resistance Ro occur in the
nonadaptive case. Notice that the value of the regulated
voltage exhibits an undesirable shift. On the other hand,
in Figs. 8 and 9 we show the behavior of the adaptive
algorithm, which estimates (almost instantaneously) the
new value of Ro , avoiding the performance degradation.
3313
Preprints of the 18th IFAC World Congress
Milano (Italy) August 28 - September 2, 2011
7
74.25
6
74.2
5
74.15
x2 [V]
x1 [A]
4
3
74.1
74.05
2
1
74
0
73.95
−1
73.95
74
74.05
74.1
74.15
0
0.5
1
1.5
2
time [sec]
2.5
3
3.5
4
−3
x 10
74.2
x2 [V]
Fig. 5. A periodic orbit in the phase plane with realistic
components.
Fig. 8. Time evolution of x2 (t) for the realistic simulation
in the case of the adaptive algorithm.
7
6
74.3
5
74.25
x [A]
4
x2 [V]
1
74.2
3
2
74.15
1
74.1
0
74.05
−1
73.95
74
74.05
74
73.95
0
0.5
1
1.5
2
time [sec]
2.5
3
3.5
74.1
x2 [V]
74.15
74.2
74.25
Fig. 9. The orbit in the phase plane in the case of the
adaptive algorithm.
4
−3
x 10
Fig. 6. Time evolution of x2 (t) for the realistic simulation
with a step variation in R0 .
7
6
5
7
x [A]
4
1
6
1
x [A]
5
3
2
4
1
3
0
−1
73.7
2
73.9
74
74.1
74.2
2
Fig. 10. Unstable behavior in the phase plane using the
approximate method of Cuk and Middlebrook (1976).
0
−1
73.95
73.8
x [V]
1
74
74.05
74.1
74.15
74.2
74.25
74.3
x2 [V]
Fig. 7. The orbit in the phase plane with the step variation
in R0 .
Simulations showing that the approximate method of Cuk
and Middlebrook (1976) may lead to instability were
performed, and are shown in Fig. 10. In this method,
whose details are given for completeness in Appendix A, a
synchronous mode with a fixed sampling time is assumed
and the duty ratios are computed fixing the maximum
current x01 and approximating with eAi t ≈ I2 + Ai t.
Experiments were carried out in an setup located in the
Laboratoire de Genie Electrique de Paris. The circuit parameters are L = 0.00136H, R0 = 500Ω, C = 0.000094F
and a negligible rL . In Fig. 11 we show that the desired
periodic orbit is indeed obtained. Notice that, due to
presence of noise and modeling imperfections, the behavior
in DCM differs from the one predicted by our previous analysis—in particular, the measured current exhibits
negative values. It is fair to say that, in this particular
example, the approximated switching times of Cuk and
Middlebrook (1976), are very close to our exact values.
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Preprints of the 18th IFAC World Congress
Milano (Italy) August 28 - September 2, 2011
Appendix A
0.12
0.1
0.08
x1 [A]
0.06
0.04
0.02
0
−0.02
−0.04
13.788 13.79 13.792 13.794 13.796 13.798
x2 [V]
13.8
13.802 13.804 13.806 13.808
Fig. 11. A periodic orbit in the phase plane obtained in
the experimental rig.
6. CONCLUDING REMARKS
The switching times may, in principle, be derived integrating the differential equations describing the system along
the periodic orbit. Unfortunately, due to the “coupled
dynamics” in topology Ω1 , this leads to a complicated
set of nonlinear algebraic equations that is difficult, if
at all possible, to solve. The key observation that allows
to obtain explicit solutions is that the existence of the
periodic orbit can be guaranteed without the computation
of the flow associated to topology Ω1 , but just looking at
the intervals I1 and I2 , where the inductor and capacitor
dynamics are decoupled. Since this principle of operation
of the boost is the basis for all power converters it is our
belief—and fondest hope—that similar calculations can be
done for more complicated power circuits.
REFERENCES
R.W. Erikson and D. Maksimovic (2000). Fundamentals
of Power Electronics. Second Edition, Kluwer Academic
Publishers.
D. Liberzon (2003). Switching in Systems and Control. System and Control: Foundations and Applications,Birkhauser, 2003.
S. Cuk and R.D. Middlebrook (1976). A General Unified
Approach to Modelling Switching DC to DC Converters
in Discontinous Conduction Mode. PESC 76, Rec, 1976.
M. Mohr, W. T. Franke, B. Witting, F.W. Fuchs (2010).
Converter Systems for Fuel Cells in the Medium Power
Range - A Comparative Study. IEEE Transactions on
Industrial Electronics, 57, 2010, pp. 2024–2032.
A. Goebel, R. Sanfelice and A. Teel (2009). Hybrid
Dynamical Systems. IEEE Control Systems Magazine,
April 2009, pp. 28–93.
J. Sun, D. M. Mitchell, F. Greuel, P. T. Krein, and R. M.
Bass (2009). Averaged Modeling of PWM Converters
Operating in Discontinuous Conduction Mode. IEEE
Transactions on Power Electronics, Vol. 16, No. 4, 2001,
pp. 482-492.
C. Qiao and J. Zhang (2005). Control of boost type converter at discontinuous conduction mode by controlling
the product of inductor voltage-second. in Proc. IEEE
PESC, Brazil, Sep. 2005, pp. 1213-1219.
K. Viswanathan, R. Oruganti. and D. Srinivasan (2002).
A novel tri state converter with fast dynamics. IEEE
Trans. on Power Electronics, Vol. 17, No.5, Sept 2002.
pp 677-683.
In the approximate method of Cuk and Middlebrook
(1976) the equations for the three topologies of the boost,
namely,

ẇ = A1 w + b1 vin 0 ≤ t ≤ t1
ẏ = A2 y + b2 vin t1 ≤ t ≤ t1 + t2

ż = A3 z + b3 vin
t1 + t2 ≤ t ≤ t1 + t2 + t3
are solved, yielding

A t
w(t1 ) = e 1 1 w(0) + B1 (t1 )b1 vin
y(t + t2 ) = eA2 t2 y(t1 ) + B2 (t2 )b2 vin
 1
z(t1 + t2 + t3 ) = eA3 t3 z(t1 + t2 ) + B3 (t3 )b3 vin
Rt
where Bi (t) := 0 eAi τ dτ. Imposing the boundary conditions
w(t1 ) = y(t1 ), y(t1 + t2 ) = z(t1 + t2 ),
using the linear approximations eAi ti ≈ I2 +Ai ti , and after
retention of only first order terms yields
z(t1 + t2 + t3 ) = (I + A)w(0) + bvin ,
where we have defined
A := A1 t1 + A2 t2 + A3 t3 , b := t1 b1 + t2 b2 + t3 b3 .
In a closed cycle we have z(t1 + t2 + t3 ) = w(0), hence
Aw(0) + bvin = 0.
(A.1)
Using the expressions of Ai and bi , with the practical
consideration that rl = 0, yields


#
"
t1
t1 + t3
0
−


L
A=t
(A.2)
L
T , b =
1
0
−
C
RC
where T := t1 +t2 +t3 , is assumed fixed a priori. Replacing
(A.2) in (A.1) defines a set of two algebraic equations
−t1 x02 + (t1 + t3 )vin = 0
(A.3)
Rt1 x01 − T x02 = 0,
0
0
where we have adopted our notation w(0) = (x1 , x2 ). The
first equation can be rewritten as
(M − 1)t1 = t3 ,
(A.4)
x0
where we have defined M := vin2 . Since x02 is the smallest
value of the capacitor voltage and, in normal operation,
this should be larger than vin , it is reasonable to assume
that M > 1. At this step, another practical consideration
is made, namely, to fix x01 , which corresponds to the largest
value of the inductor current along the cycle, to
vin t3
x01 =
.
(A.5)
2L
This value is obtained assuming that in the interval t3 the
current grows as a ramp of slope vLin , which is indeed the
case if rl = 0, and taking half of the peak value.
Replacing (A.4) and (A.5) in the second equation of (A.3)
yields, after some simple manipulations,
s
r
2LT M
M (M − 1)2LT
t1 =
, t3 =
,
R(M − 1)
R
which are, precisely, the expressions (38) and (39) derived
in Cuk and Middlebrook (1976).
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