Non-average Stability Analysis for a Boost Converter
ANDRES GARCIA, EDUARDO GUILLERMO
GESE, Departamento de Ingeniería Eléctrica,
Universidad Tecnológica Nacional. Facultad Regional Bahía Blanca,
11 de abril 461, Bahía Blanca, Buenos Aires,
ARGENTINA
Abstract: - The purpose of this paper is to present a method for analyzing the stability of DC-DC boost
converters providing a universal transfer function. This method relies on the application of the Laplace
transform without any ad-hoc linear approximation or system linearization. A common methodology for
evaluating the stability of DC-DC converters is the average method, which is essentially a linearization. In this
paper, the Laplace transform is applied directly, resulting in a universal Z-transform model being used to design
controllers and stability analyses as a discrete unifying transfer function. The examples cover both power and
low voltage converters. Matlab simulations have been conducted to verify the theoretical findings. In particular,
the second example considers a small 5V to 12V boost converter, which was previously examined in a Texas
Instruments application note. The transfer functions and the Bode plot are provided.
Key-Words: - Power converters, Z-transform, Stability analysis, Average model, Switching model, Boost
converter, State-Space.
Received: March 7, 2023. Revised: December 5, 2023. Accepted: December 21, 2023. Published: December 31, 2023.
1 Introduction
Renewable energy technologies always search for
new and optimized DC-DC battery chargers and
also DC-AC converters, [1]. While the well-known
DC-DC buck is well studied and the design process
is straightforward, for DC-DC boost converters, the
case is of a much more complex nature, proving in
general to be an unstable system, [2], [3].
One of the main limitations in the analysis and
design of control algorithms and techniques for
closed-loop stabilization of DC-DC converters lies
on its switching nature, [4]. Unlike linear and non-
linear continuous control systems, switching
systems, even piecewise-linear ones, offer a very
challenging control scenario, [5].
For this reason, the well-known average method
was introduced in the late 70s as a method to obtain
a linearized version of switching DC-DC converters
as a unique transfer function, [6], [7]. With a clear
advantage of having a unique model, continuous and
time-invariant. However, the great disadvantage of
this method lies in its local and linearized nature,
not allowing for a ripple of heavy transient analysis,
[8].
On the other hand, the available literature about
boost converter control makes use of switching
models either applying Laplace transform and then
applying Z-transform or working with switching
transfer functions, besides non-linear and complex
control techniques, [9], [10].
In this paper, considering an asynchronous
boost converter topology, under the assumption that
the inductor’s current is complementary to the
switching MOS-FET current: when the inductor is
charged, the switching is in on-mode and vice versa.
In the same manner, the diode’s current is
complementary and is in line with the inductor’s
one: both conduct current to the load at the same
time.
The remainder of the paper is organized as
follows:
Section 2 presents the main theorem and its
corollary to harvest and obtain a true transfer
function without linearization and applying
Laplace’s ttransform to a given power converter:
DC-DC or DC-AC, Section [Matlab] applies the
results to the well-known boost converter to design
a lead compensator providing stability along with
some Matlab simulations using the Power Systems
toolbox in Simulknk. Finally, Section [Conclusions]
depicts some conclusions and future work.
2 The Boost Converter’s Model
The most common asynchronous DC-DC Boost
converter topology is shown in Figure 1.
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Fig. 1: Asynchronous boost converter topology
As it is well known, the PWM signal is the
control modeled as a duty variation.
Therefore, the electrical equations can be modelled
to be:
(1)
Where the PWM signal is modeled as as
shown in Figure 2 and the inductance’s current is
modeled to be complementary to the diode’s
current.
Fig. 2: PWM signal
Using a resistive load along with the
capacitor’s law, Kirchoff’s law (1) can be
transformed to be:
(2)
Where is a constant voltage depending on the
duty . In the case of boost converters:
(3)
defining a current control scheme over the
inductance’s current , the complete model,
plugging (3) into (2) yields:
(4)
In the next subsections, a complete, non-
linearized model is developed using the Laplace
transform considering an internal voltage as an
auxiliary input variable parametrized by the duty .
3 Output Stage Modelling
It is worth to develop a separate Laplace’s transform
analysis for the output voltage using the voltage
source as shown in Figure 3.
Fig. 3: Output circuit
Then, it is clear the two-stage switching equations:
0
0
11
()
( ) 1 discharge
(0) ( ) Previous: charge
1 1 1 1
( ) ( )
( ) 0 charge diode modeled as a resistance
(0) ( ) Previous: discharge
L
o
L
o
o
D L D
D
DT
RC
o
v
C v t R
t
v V D
v t v V D
R R C R C
tR
v e V D
Finally:
(5)
Where
. Next subsection groups
this output equation along with the boost’s circuit
equations (4).
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4 The Complete Model in the s and z
Domains
4.1
Taking Laplace’s transform on model (4) along with
the output equations (5):
(6)
Where means the Laplace’s transform.
Then, the focus is on :
According to (4):
and
integrating by parts:
Taking into account the definition of :
Then:
( 1) )
0
0
1
()
1 ( ) ( ) ( ) (( 1) )
(( ) )
()
sK
s K T
LL
K
L
K
st
KT in
K D T
DT
e
t I t s I K T s
e
I K D T s
V V D
edt
sL
L
Considering high frequency switching, the
following assumption will be used along the paper:
Assumption 1: , then:
.
In this way:
This infinite sum involving exponentials is
exactly the definition of the Z-transform replacing
[11]:
Moreover, recalling the formal definition of the
Z-transform:
along with the change of variables ,
yields:
Then, considering the usual analysis with zero
initial conditions :
Finally, rearranging terms and simplifying:
(7)
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4.2
The output equations (5) yields:
Following the same methodology as for :
Where
and
.
Rearranging terms:
As for , in order to collect a Z-transform
expression, the repalcement leads:
To simplify the analysis, we can recall the
Assumption 1:
In this way:
Considering the transformation:
Finally:
(8)
4.3 Complete Transfer’s Function
The electrical circuit’s equations (4]) along with (7)
and (8) leads:
(9)
Where . In the next section, the
main results will be presented obtaining both: a
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simplified asymptotic transfer function and a
stability theorem to ensure stability for .
5 Main Results
The Boost converter along with its dynamical model
in z-transform (9), can be represented as a transfer
function in the Z domain:
For the purpose of collecting a complete expression
in the domain of z:
(10)
Then, considering the asymptotic behavior of
when the sample’s number tends to infinity,[11]:
Then it is clear that, asymptotically, the
dominating part of the transfer function (10) is
([12], [13]):
Theorem 1: Given the z-transfer’s function (10), an
asymptotic equivalent transfer function is as
follows:
(11)
As a consequence, the next corollary shows a
sufficient condition for stability allowing the
available LTI stability tools:
Corollary 1: Given the transfer’s function (11), the
complete closed-loop system, after some
controller’s addition, will be asymptotically stable if
is asymptotically stable for .
Proof 1: Considering that the duty changes over
time but onyl from interval to interval:
, then if for every , the closed-loop system
is stable, each interval is decreasing exponentially
(LTI system) and so, because of the inductor’s
current continuity (either for continuos or
discontinuous modes), the current will set to zero or
continues from the previous state in previous
switching, the complete current evolution over time
is decreasing to some equilibrium, so asymptotic
stability is proved. This completes the proof.
It is worth noticing that Corollary 1 allows the
use of any LTI stability criteria as long as is applied
for the complete range .
Using these results, the next section presents the
design of a Lead compensator for the closed-loop
stability of the Boost converter.
6 Sensitivity and Error Analysis
According to model (10) and its asymptotic
approximation (11), the following analysis can be
depicted:
Analyses of sensitivity: An analysis of the
asymptotic transfer function’s parameter
dependence is not difficult:
As the model relies on Assumption 1, it is valid
for high frequency switching PWM signals. In
contrast, since the inductance value acts as a gain,
it is important to keep it close to reality in order to
design a more reliable controller.
Error analysis: taking into account models
(10) and (11), the following error analysis
can be carried out:
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Both models converge as the system runs out
under Assumption 1 and for small values of the
capacitor .
7 Other Methods Comparison
7.1 Computational Intelligence
A boost controller can be designed using methods
derived from computational intelligence, such as
fuzzy logic, without the need for a model or transfer
function. As a negative aspect, some knowledge
about the boost under study must be provided
regarding the case-dependent nature of the method,
[14], [15].
Alternatively, it is possible to employ neural
networks with some very recent references, [16], in
which a controller with nonlinear function based on
Back Propagation (BP) neural networks combined
with PID is developed. Because the model is never
used or derived, it is always necessary to train a
neural network, even for simple boost designs
without the possibility of determining asymptotic
stability. Moreover, even with realistic modeling
using real data, training a neural network for every
controller design is not advantageous for some
control objectives, [17].
7.2 Artificial Intelligence
A genetic algorithm is used in [18], even when a
model and backstepping are considered to provide
asymptotic stability using Lyapunov’s method, an
average model is finally utilized with the obvious
drawback of linearization and without any universal
model (transfer function) as presented in this paper.
In the recent paper [19], a black-box model is
presented using real time measurements and a
NARX-ANN structure, however, as for the previous
analysis, its case-dependent nature and the lack of
asymptotic stability tools, make the contribution
limited when compared to the universal and non-
approximated model in this paper.
7.3 Classical Methods
Boost converters are switching models, so a
switching strategy or average (linearized) model can
be used to obtain a stabilizing controller (see, for
instance [20], and [21]). Several advantages can be
enumerated in this paper, including:
No linearization is involved
A universal asymptotic model is provided
with a very simple structure
A transfer function in the z-domain with no
discrete approximations is presented.
The universal and ready-to-use transfer function
is a very important contribution in this paper when
compared to the available literature where a case-
dependent study must be carried out under a linear
and local approximation.
8 Matlab Simulations
8.1 Example 1
In this example, a boost converter with in
and with an expected DC value of will
be considered. Using the simplified transfer function
(11) for a boost converter, a lead compensator can
be designed starting, for instance, from a root locus
plot using a simple proportional controller:
, with
as shown in Figure 4.
Fig. 4: Root locus for a boost converter with a
proportional controller
Where the proportional was considered only
for the present real case of .
Clearly, as expected, a simple proportional
controller can not stabilize a boost converter within
this range of . Apllying a lead compensator with
:
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Then, the root locus plot is shown in Figure 5.
Now the closed-loop system is stable and can be
applied to the power systems model in Simulink
(Figure 6), where the current versus a set point
of 100A is shown in Figure 7. As usual, overpeaks
in internal voltage can be as high as 500V.
Fig. 5: Root locus for a boost converter with a lead
compensator
Fig. 6: Boost converter Simulink’s model
Fig. 7: versus a 100A setpoint
8.2 Example 2
The asymptotic stability analysis in this paper can
be also applied to small booost converters, let’s say:
in and with and expected DC value of
, [22]. Then, the asymptotic transfer’s function
(11) leads:
For comparison, let’s consider the high
frequency used in [21]: and with
for an output power of about using
, the following controller stabilizes the DC-
DC converter:
It turns out that this controller is a z-transform
version of the one in [22]. Notice the reduced phase
margin due to the discrete conversion when
compared to the continuous model (Figure 8).
Fig. 8: Bode plot for the compensated boost
converter
9 Conclusion
As a result of applying Laplace transforms, the z-
transform naturally follows without any local
approaches or linearization, rather the switching
nature of the model is captured by the z-transform,
and the complete model is transformed into a unique
transfer function (universal transfer function for any
boost converter).
Therefore, all classical tools apply to the design
of compensators and stabilizers for boost
converters/power converters using the simplified
asymptotic transfer function in this paper.
Moreover, an interesting new transfer function
showed up including terms like deserving a
complete separate study. In this paper, an
asymptotic model was also derived to provide a
simple methodology to design a lead compensator
using a universal discrete transfer function.
To validate the analysis, a boost converter was
evaluated in Simulink using the Power Systems
toolbox showing the asymptotic convergence of the
current to some desired setpoint of about 100A
(plus ripples) and also a small 5V to 12V (24W) is
analyzed and compared with a Texas Instruments’
design.
This research opens the door to more convenient
and sophisticated models using complementary
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switching for a wider range of DC-DC and also DC-
AC converters, which comprises a complete future
research line. The universal structure of the
asymptotic transfer, which facilitates asymptotic
stability, renders the utilization of these strategies
highly appealing.
As future work, more DC-DC converters will be
explored with the ideas presented in this paper along
with real cases and test measurements.
Acknowledgement:
The authors would like to acknowledge Universidad
Tecnológica Nacional, Departamento de Ingeniería
Eléctrica, and GESE.
References:
[1] Al-Baidhani, H.; Corti, F.; Reatti, A.;
Kazimierczuk, M.K. Robust Sliding-Mode
Control Design of DC-DC Zeta Converter
Operating in Buck and Boost Modes.
Mathematics, vol. 11, Issue 3791, 2023,
https://doi.org/10.3390/math11173791.
[2] Brian T. Lynch. Under the Hood of a DC/DC
Boost Converter. Texas Instruments.
[3] A. P. Podey and J. O. Chandle. Stability
Analysis of DC-DC Boost Converter using
Sliding Mode Controller. IEEE PES/IAS
PowerAfrica, Nairobi, Kenya, pp. 1-5, 2020,
10.1109/PowerAfrica49420.2020.9219954.
[4] Praful V. Nandankar and Prashant P. Bedekar
and Prashantkumar V. Dhawas. Efficient DC-
DC converter with optimized switching
control: A comprehensive review. Sustainable
Energy Technologies and Assessments, vol.
48, pp. 101670. 2021,
https://doi.org/10.1016/j.seta.2021.101670.
[5] M. di Bernardo, C. J. Budd, A. R.
Champneys, P. Kowalczyk. Piecewise-smooth
dynamical systems: theory and applications.
Springer. 2008.
[6] R. D. Middlebrook and S. Cuk. A general
unified approach to modeling switching
converter stages. IEEE Power Electronics
Specialists Conf., pp. 18-34, 1976.
[7] C. T. Rim, G. B. Joung and G. H. Cho. A
state-space modeling of nonideal DC-DC
converters. PESC ’88 Record., 19th Annual
IEEE Power Electronics Specialists
Conference, Kyoto, Japan, vol.2, pp. 943-950.
1988. doi: 10.1109/PESC.1988.18229.
[8] Marey, A.; Bhaskar, M.S.; Almakhles, D.;
Mostafa, H. Analytical Solution for Transient
Reactive Elements for DC-DC Converter
Circuits. Electronics, vol 11, issue 3121,
2022,
https://doi.org/10.3390/electronics11193121.
[9] Anup Marahatta and Yaju Rajbhandari and
Ashish Shrestha and Sudip Phuyal and Anup
Thapa and Petr Korba. Model predictive
control of DC/DC boost converter with
reinforcement learning. Heliyon, Vol. 8, issue
11, pp. 1-13, 2022,
https://doi.org/10.1016/j.heliyon.2022.e11416.
[10] Sanakhan, Safoora and Mahery, H. Mashinchi
and Babaei, E. and Sabahi, M. Stability
Analysis of Boost DC-DC Converter Using Z-
Transform. IEEE 5th India International
Conference on Power Electronics (IICPE),
pp: 1-6, 2012. 10.1109/IICPE.2012.6450432.
[11] Charles Phillips and Troy Nagle. Digital
Control System Analysis and Design. Pearson,
3th Edition. 2014.
[12] N. G. de Bruijn, Asymptotic Methods in
Analysis. Dover ed Edition. 2010.
[13] D. Murray. Asymptotic analysis: (Applied
Mathematical Sciences, 48). Springer; Reprint
edition. 1984.
[14] R. J. Wai, M. W. Chen and Y. K. Liu. Design
of Adaptive Control and Fuzzy Neural
Network Control for Single-Stage Boost
Inverter. IEEE Transactions on Industrial
Electronics, Vol. 62, issue 9, pp. 5434-5445,
2015, doi: 10.1109/TIE.2015.2408571.
[15] S. Chimplee and S. Khwan-On, Fuzzy
Controller Design for Boost Converter Based
on Current Slope. 2022 International
Electrical Engineering Congress (iEECON),
Khon Kaen, Thailand, pp: 1-4, 2022,
10.1109/iEECON53204.2022.9741597.
[16] H. Xi and Q. Wang, Design of Back
Propagation Neural Network PID Control for
Boost Converter. IEEE Sustainable Power
and Energy Conference (iSPEC), Nanjing,
China, pp. 3889-3893, 2021,
10.1109/iSPEC53008.2021.9735583.
[17] O. Asvadi-Kermani, B. Felegari, H. Momeni,
S. A. Davari and J. Rodriguez. Dynamic
Neural-Based Model Predictive Voltage
Controller for an Interleaved Boost Converter
With Adaptive Constraint Tuning. IEEE
Transactions on Industrial Electronics, vol.
70, no. 12, pp. 12739-12751, 2023, doi:
10.1109/TIE.2023.3234138.
[18] I. A. Ayad, E., Elmostafa, A. Umaid, Q.
Saeed, W. Asad, B. Mohamed and A. Ahmad.
Optimized Nonlinear Integral Backstepping
Controller for DC–DC Three-Level Boost
Converters. IEEE Access, Vol. 11, pp. 49794-
WSEAS TRANSACTIONS on CIRCUITS and SYSTEMS
DOI: 10.37394/23201.2023.22.29
Andres Garcia, Eduardo Guillermo
E-ISSN: 2224-266X
296
Volume 22, 2023
49805, 2023,
10.1109/ACCESS.2023.3274773.
[19] A. Zilio, D. Biadene, T. Caldognetto and P.
Mattavelli. Black-Box Large-Signal Average
Modeling of DC-DC Converters Using
NARX-ANNs. IEEE Access, vol. 1, pp.
43257-43266, 2023. doi:
10.1109/ACCESS.2023.3271731.
[20] Ahmed Chouya. Adaptive Linearizing Control
with MRAC Regulator for DC-DC Boost
Converter. International Journal on Applied
Physics and Engineering, vol. 1, pp. 25-30,
2022, doi:10.37394/232030.2022.1.4.
[21] M. Cherif Daia Eddine Oussama, C. Ali, K.
Abdelhalim. Robust Integral Linear Quadratic
Control for Improving PV System Based on
Four Leg Interleaved Boost Converter.
WSEAS Transactions on Systems, vol. 21, pp.
39-48, 2022,
https://doi.org/10.37394/23202.2022.21.4.
[22] AN-1994 Modeling and Design of Current
Mode Control Boost Converters. Application
Report SNVA408B, Texas Instruments, 2013,
[Online].
https://www.ti.com/lit/an/snva408b/snva408b.
pdf?ts=1710084702459 (Accessed Date:
March 1, 2024).
Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting
Policy)
The authors equally contributed in the present
research, at all stages from the formulation of the
problem to the final findings and solution.
No funding was received for conducting this study.
Sources of Funding for Research Presented in a
Scientific Article or Scientific Article Itself
No funding was received for conducting this study.
Conflict of Interest
The authors have no conflicts of interest to declare.
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