What are the key differences and similarities between fractional and fractional-order control of converters?

1 Fractional and fractional-order control

Fractional and fractional-order control are two related but distinct concepts that involve the use of non-integer derivatives or integrals in the control design. Fractional control is a general term that refers to any control method that employs fractional calculus, which is a branch of mathematics that extends the notion of differentiation and integration to arbitrary (non-integer) orders. Fractional-order control is a specific type of fractional control that applies fractional calculus to the modeling and analysis of dynamical systems. A fractional-order system is a system whose behavior is described by a fractional-order differential equation, which contains fractional derivatives or integrals of the system variables.

2 Why use fractional and fractional-order control?

Fractional and fractional-order control offer potential benefits over conventional integer-order control for power electronics applications, such as better performance and robustness for complex systems, reduced complexity and dimensionality of the controller design, and enhanced flexibility and adaptability. This can lead to simpler implementation, lower cost, lower power consumption, improved trade-off between stability, accuracy, and speed of response.

3 How to design fractional and fractional-order controllers?

Various methods and techniques exist for designing fractional and fractional-order controllers for power electronics converters, such as Fractional PID (proportional-integral-derivative) controller, Fractional-order sliding mode controller, and Fractional-order adaptive controller. The Fractional PID controller can be tuned by adjusting the gains and fractional orders to achieve the desired performance. The Fractional-order sliding mode controller reduces the chattering phenomenon and improves robustness and accuracy of the control. The Fractional-order adaptive controller can adapt to system variations and uncertainties, optimizing control performance.

4 How to implement fractional and fractional-order controllers?

The implementation of fractional and fractional-order controllers for power electronics converters requires some special considerations and challenges, such as the computation of fractional derivatives or integrals which are more complex and time-consuming than integer derivatives or integrals. Therefore, efficient and accurate numerical methods and algorithms must be used to approximate the fractional operators and reduce the computational burden. Additionally, the hardware realization of fractional and fractional-order controllers may require additional components or circuits to implement the fractional operators, such as capacitors, resistors, operational amplifiers, microcontrollers, field-programmable gate arrays (FPGAs), or digital signal processors (DSPs). Finally, the stability and performance analysis of fractional and fractional-order controllers may require more advanced tools and criteria than integer-order controllers. For instance, classical stability theorems and frequency-domain methods may not be applicable or sufficient for fractional-order systems; therefore, new methods based on fractional calculus, time-domain, or state-space approaches must be utilized.

5 Examples of fractional and fractional-order control applications

Fractional and fractional-order control have been applied to various power electronics converters for different purposes and domains. For instance, a fractional PID controller can be utilized to regulate the output voltage of a DC-DC buck converter under load and input voltage variations, which can improve the transient response, steady-state error, and robustness of the converter in comparison to an integer PID controller. Additionally, a fractional-order sliding mode controller can be used to control the output voltage and frequency of a DC-AC inverter for grid-connected or standalone applications, reducing total harmonic distortion (THD), enhancing power quality, and rejecting load disturbances in comparison to an integer-order sliding mode controller. Finally, a fractional-order adaptive controller can be applied to maximize the power extraction and efficiency of a wind turbine system, adapting to wind speed variations and optimizing the converter parameters and fractional order based on the system error.

6 Here’s what else to consider

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