Figure 1: Block diagram for proposed FOSMC controller and FOSEPIC converter
SEPIC Converter Models
The conventional SEPIC converter is described by the following equations:
The fractional-order SEPIC converter syst em is defined as:
The output voltage is indirectly controlled by the input inductor current through the x1state variable[33].
Sliding Mode Control
To achieve control, the sliding surface function is defined as:
where π‘₯π‘Ÿ denotes the reference for the π‘₯1 state variable. The inductor current reference can be generated using a proportional-integral (PI) controller without a compensation term:
Here, Ξ» and πœ– are the proportional and integral gains, respectively. The derivative can be expressed as:
By applying the properties of fractional calculus, we can obtain:
Stability of the Sliding Dynamics
In a fractional differential system described by 𝐷𝑑𝛼π‘₯(𝑑)=𝑓(𝑑) where π‘₯=0 is the equilibrium point,thefunction 𝑓(𝑑) is assumed to be Lipschitz continuous. Under these conditions, the solutions of the system exhibit Mittag-Leffler stability. This means that if the initial state is close to the equilibrium, the solution will remain close over time, highlighting the system’s ability to return to equilibrium despite perturbations. This behavior is particularly important in applications involving fractional dynamics, where traditional stability concepts may not apply.The reaching law is designed to ensure the stability of the closed-loop system:
where 𝜌>0ρ>0 and 𝛾>0Ξ³>0.
Control Law
Equating the fractional-order system dynamics with the reaching law leads to the control law:
From the SEPIC converter dynamics, the obtained equqtion:
The control law can be expressed as:
\(u(t)=1-\frac{\alpha(x3(t)+x4(t)}{\text{vin}}\)
This controller is applicable for both fractional-order and conventional SEPIC converters, depending on the value of Ξ±. When 0<𝛼<10<Ξ±<1, it functions as a fractional-order sliding mode controller, and when 𝛼=1, it behaves as a conventional sliding mode controller.