Introduction
”Non-integer calculus,” frequently referred to as ”fractional calculus,”
includes not only integer orders but also generalized functional orders
such as fractional, irrational, and complex orders. This broadens its
scope, positioning it as a form of generalized calculus [1]. The
historical roots of fractional calculus date back to 1695, when Leibniz
proposed half-order derivatives in correspondence with L’Hospital. The
ability of fractional calculus to model the dynamics of various natural
phenomena provides more accurate descriptions than traditional
integer-order dynamic systems with the advantage of a high degree of
freedom, nonlocality, flexibility, nonuniformity, and memory effect. Due
to these advantages, its applications span multiple fields, including
control engineering, biology, biomedical engineering, financial markets,
and signal processing. In electrical engineering, fractional calculus is
increasingly recognized for its utility in modeling electrical
equipment, wireless power transmission systems, and studying chaotic
behaviors in fractional-order dynamic systems[2][3]. Recent
advancements have led to practical designs in areas such as
electrode-electrolyte polarization, viscoelastic fluids, and power
converters. However, conventional integer-order models often fail to
adequately represent the behavior of inductors and capacitors and
uniform definitions of orders of differentiation [4][5]. DC-DC
converters have demonstrated that output voltage gain can be controlled
only by the duty cycle while modeling using integer order calculus. The
stability and differentiability characteristics of integer-order systems
further enhance their appeal, as some non-differentiable functions. As
well as integer-order models (IOM) can capture some characteristics,
they often lack the necessary accuracy. These unique properties motivate
ongoing research into the applications of fractional calculus across
various physical and natural phenomena[6].
Recent studies have illustrated the effectiveness of fractional calculus
in controller design and modeling different nonlinear dynamic systems,
particularly DC-DC converters[7][8][11].
DC/DC (Direct Current to Direct Current) converters are power
electronics devices composed of passive components like inductors,
capacitors, resistors, and transistors functioning as switches. These
converters are extensively used in various fields such as
communications, industrial applications, and as power supplies for
personal computers. Buck converters, boost converters, buck-boost
converters, Cuk converters, and Single-Ended Primary Inductor Converter
(SEPIC) converters are some of the most well-known DC/DC converters.
Disturbances in most DC/DC converters arise from load variations, input
voltage uncertainty, and electromagnetic interference generated from
transistors, which complicate their control. Among these, the SEPIC
converter is non-inverting, maintaining a positive output when the input
is positive, and functions as both a step-up (BOOST) and step-down
(BUCK) converter. It is particularly useful in applications requiring
low ripple current at both input and output terminals and off-grid
photovoltaic systems such as batteries and PV[9]. However, the SEPIC
converter exhibits a non-linear variable structure with non-minimum
phase characteristics, and instability, and is a time-varying system.
Due to its non-minimum-phase nature (with zeros or poles at the origin),
direct output voltage control is not feasible[10]. Consequently, to
manage the output voltage of a SEPIC converter, the approach involves
directly controlling the input current through the duty cycle and
indirectly influencing the output voltage. Various controllers, such as
PWM, PID, fuzzy logic controller, and sliding mode controls, are
employed to regulate the SEPIC converter’s operation[9].
A type of controller design known as a nonlinear variable structure
(NVS) utilizes a nonlinear control rule based on a set of switching
variable signals[1][11] [12]. Sliding mode control is
closely related to NVS, as it can be used to design NVS controllers that
provide stable control and smooth transitions between different control
behaviors[13]. This approach can be applied to SEPIC converters to
improve stability, adaptability, and efficiency. Given the intricate
behaviors of the SEPIC converter, a sliding mode controller is selected
among the possible options. The nonlinear behavior of the SEPIC
converter[14], indicated by a specific characteristic, aligns with
the requirements met by the sliding mode controller, making it an ideal
choice. This control method allows the converter to operate in two
modes: step-up (boost) when the reference voltage is greater than the
input voltage, and step-down (buck) when the reference voltage is less
than the input voltage, all with a single controller. Which give new
circuit topologies based on these elements continue to emerge, although
their characteristics remain an active area of research
[15][16][11][17] and, enhance the analysis of their
dynamic behavior[18][12].
Sliding Mode Control (SMC) is a highly effective strategy for
Controlling uncertainties in both linear and nonlinear systems. Known
for its rapid dynamics and excellent transient response, SMC offers
robustness against external disturbances and parameter variations. The
primary goal of SMC is to guide system states to a predefined manifold,
called the sliding surface, and to maintain this state despite
uncertainties [19][20]. The design of SMC consists of two
phases: (i) the Reaching Phase and (ii) the Sliding Phase. In the
Reaching Phase, system states are driven to the sliding manifold in
finite time, but this phase can be sensitive to disturbances and
parameter variations. To mitigate this sensitivity, various methods have
been proposed to minimize or eliminate the reaching phase. During the
Sliding Phase, the closed-loop system enters a sliding motion where
robustness and order reduction become critical. In this phase,
trajectories are less sensitive to disturbances and parameter
variations, enhancing the robustness of SMC. However, it is important to
note that robustness is not guaranteed during the Reaching
Phase[21][22][23].
When traditional integer-order SMC methods are applied to
fractional-order systems, they effectively reject disturbances but often
suffer from chattering a significant drawback. The fundamental feature
of SMC is that the state slides along the sliding surface. Conventional
SMC typically employs a fixed, predefined sliding surface, which can
lead to extended reaching times if the initial state is far from the
surface, ultimately degrading control performance. Increasing the
discontinuous control gain may shorten the reaching phase but can also
exacerbate chattering issues [19][24][25].
Fractional calculus has shown promise in addressing the chattering
problem and improving control performance. By incorporating
fractional-order elements into SMC design, the chattering issue can be
mitigated, and the response time of the closed-loop system can be
enhanced. Fractional-order controllers provide additional design
parameters, such as adjustable non-integer differentiator and integrator
orders, allowing for the tuning of the fractional order to optimize
dynamic response while preserving the advantages of conventional sliding
mode control(SMC) [26][27][28][29][30].
Despite the demonstrated advantages of fractional calculus in various
applications, its specific implementation in controlling DC-DC
converters, particularly the SEPIC converter, remains underexplored.
While existing studies have utilized fractional calculus in controller
design, few have focused on its ability to manage the complex
nonlinearities and non-minimum phase characteristics inherent in SEPIC
converters. This research aims to fill this gap by demonstrating how an
Indirect Sliding Mode Adaptive Fractional Order Controller (FOSMC) can
effectively enhance the control performance and reliability of the SEPIC
converter, ultimately contributing to more efficient power conversion
systems.
Proposed Methods
In this study, the proposed a Fractional Order Sliding Mode Controller
(FOSMC) designed specifically for a Fractional Order SEPIC Converter.
The proposed method enhances traditional SMC by incorporating fractional
calculus principles, which address the limitations of conventional
integer-order controllers, particularly chattering and performance
degradation during the reaching phase.
Grunwald-Letnikov Fractional-Order Derivative:The
Grunwald-Letnikov fractional-order derivative is defined as:
When the sign of α is negative, this equation becomes a fractional-order
integral[31][32].
Riemann-Liouville (RL) Fractional-Order Integral:Cauchy’s formula for repeated integration reduces n-fold integration
of a function 𝑓(𝑡)f(t) to a single integral:
This can be generalized to a fractional-order integral:
where Γ(𝑛 is the Euler’s
The left Riemann-Liouville fractional-order derivative of a function
𝑓(𝑡) is defined as:
where:n=⌈α⌉ is the smallest integer greater than or equal to 𝛼, Γ is the
gamma function,𝑓(𝑛) (τ) is the n-th derivative of 𝑓 is the lower limit
of the integral.The right Riemann-Liouville derivative is:
Where b is the upper limit of the integral.
Caputo Fractional Derivative: The Caputo
fractional-order derivative modifies the Riemann-Liouville definition
to allow for broader applications: The left Caputo fractional-order
derivative is given by:
Where n=⌈α⌉ is the smallest integer greater than or equal to 𝛼,Γ is the
gamma function,𝑓(n)(𝜏) is the n-th derivative of f,𝑎 a is the lower
limit of the integral.The right Caputo derivative is defined similarly:
where is the upper limit of the integral.
Mittag-Leffler Function
The one-parameter Mittag-Leffler function is crucial for modeling
physical processes using fractional calculus:
Properties of GL, RL, and Caputo Fractional Order Derivatives
Essential characteristics of fractional-order operators consist of:
- Semigroup and Commutative Property
- Consistency Property:
- Constant Property:
- Initial Conditions:
If 𝑓(𝑠)(0)=0f (s) (0)=0 for 𝑠=𝑛,+1,…,𝑞s=n,n+1,…,q.
Fractional Order Controller Design
Fractional-order systems, despite lacking traditional state variables,
can be represented similarly to integer-order systems. This section
focuses on both multiple-input, multiple-output (MIMO) systems and
single-input, single-output (SISO) systems, specifically within the
context of the fractional-order SEPIC converter model.
Let 𝛼 represent the fractional order, constrained by
0<𝛼<1. The state variables for the fractional-order
system are denoted as 𝑥1(𝑡),𝑥2(𝑡),𝑥3(𝑡), and 𝑥4(𝑡) with 𝑎,𝑏,𝑑,𝑓and 𝑔
serving as system parameters.
While fractional-order systems do not possess conventional state
variables, it is feasible to derive representations akin to those of
integer-order systems. This section first examines the general case of
MIMO systems before addressing SISO systems. The fractional-order SEPIC
(FOSEPIC) converter model is described using fractional calculus.
Notably, when 𝛼=1, the model corresponds to a conventional SEPIC
converter.Table 1 .Description for variables on the block diagram