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:
  1. Semigroup and Commutative Property
  2. Consistency Property:
  3. Constant Property:
  4. 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