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 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 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 π·π‘πΌπ₯(π‘)=π(π‘) where π₯=0
is the equilibrium point,thefunction π(π‘) 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 systemβs 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 π>0Ο>0 and πΎ>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<πΌ<10<Ξ±<1, it functions as a
fractional-order sliding mode controller, and when πΌ=1, it behaves as a
conventional sliding mode controller.