**1. Introduction**

DC–DC (direct current) conversion is one of the most studied and applied functionalities from power electronics. DC–DC elementary conversion modes step down and up the converter input voltage using power semiconductor devices, operated with high-speed control, to produce the well-known buck and boost topologies, respectively. By cascading both elementary conversion modes the buck–boost topology is obtained. With the same quantity of elements, the resulting converter produces a smaller or greater output voltage than its input power source, with inverse polarity.

The versatility of buck–boost converter to transform its supply voltage into a higher/lower output makes it an alternative for applications requiring DC voltage regulation, ranging from light-emitting diode (LED) lighting [1] to renewable energy sources [2,3], microgrids [4], and battery charging [5,6], to mention the most relevant. Biomedicine applications [7–9] take special relevance due to their impact in human's health, since the need for appropriate energy management/harvesting/storage strategies to be applied in micro/nano-scale is one of the main drawbacks of emerging cardiac technologies, for instance. Some of the most recent and significant results on strategies to regulate voltage in a buck–boost converter are the following: in [10], a deep learning-based approach was used to stabilize voltage in the converter. A sliding mode-based observer combined with an optimization algorithm, a deep reinforcement learning technique, and a neural network were suggested to estimate converter unknown dynamics, while controller gains were adjusted online. Good transient behavior and output-robust stabilization against reference

**Citation:** S. Sánchez, A.G.; Soto-Vega, J.; Tlelo-Cuautle, E.; Rodríguez-Licea, M.A. Fractional-Order Approximation of PID Controller for Buck–Boost Converters. *Micromachines* **2021**, *12*, 591. https://doi.org/10.3390/ mi12060591

Academic Editor: Young-Ho Cho

Received: 22 April 2021 Accepted: 18 May 2021 Published: 21 May 2021

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

changes were the major improvements. In [11], an intelligent control with metaheuristic optimization was proposed. A fish-swarm algorithm is used to optimally tune a PI controller to regulate voltage in the converter. The simplicity of the approach is the strongest advantage of this contribution. The effectiveness of the controller was determined experimentally, where a fast-tracking characteristic was the main improvement. A fuzzy logic controller to stabilize voltage through a buck–boost converter in a turbine generation unit was proposed in [12]. Takagi–Sugeno-type rules were employed due to their wider range for gain variations and versatility. Even though authors compare open-loop vs. closed-loop performance to determine effectiveness, the latter exhibited acceptable time constants, and good tracking performance.

A passivity-based control was suggested in [13] to stabilize voltage fed to a microgrid. The proposed strategy achieved Lyapunov asymptotic stability through a state-feedback control law, which required an online invariance and immersion power estimator. In spite of authors compared their results with a pure PI to determine controller effectiveness, acceptable tracking characteristic, good transient performance and negligible steady-state error for both conversion modes were the improvements. With a similar approach but targeting a different application, a passivity-based controller with active disturbance rejection was proposed in [14]. A generalized proportional integral observer was used to provide accurate estimations to the controller. Good tracking characteristic and disturbances rejected effectively are the main contribution.

In [15], a modified sliding mode controller was proposed. The control strategy was divided into two portions, a linear approach based on a PI controller for voltage control loop and a nonlinear one for current loop based on hysteresis. The resulting regulated output voltage described a smooth response with overshoot absence, acceptable robustness against load variations, and good tracking performance. In [16], a predictive control approach was proposed. The authors combined a quadratic programming optimization algorithm, to predict the control signal at every sampling time, with a predictive controller, to consider load variations in the model, thus ensuring robustness and stability. The appropriate control law was generated by predicting the future behavior of the plant, resulting in a fast response, minimum overshoot and good tracking characteristic. On the other hand, due to the inherent closed-loop instability of non-minimum phase systems, PI/PID controllers are used in combination with some of the above-described techniques or some optimization algorithms. The above derived in control strategies, although efficient, with high computational/implementation complexity [17–19].

In this paper, a fractional-order PID controller approximation to regulate voltage in a buck–boost converter is proposed. In addition to the accuracy modeling real systems, robustness against parameter variations, and noise-level reduction through lower-order derivatives from fractional calculus, exploring its effectiveness controlling non-minimum phases systems is the main reason to consider a non-integer approach in the control strategy. The controller synthesis is achieved through a biquadratic module that exhibits a flat phase response. Its design considers both robustness and closed-loop performance, while a simple tuning method allows us to determine appropriate gains to achieve design requirements. The controller structure to operate in both conversion modes is generalized to generate its electrical representation directly. Numerical and experimental results are provided to corroborate effectiveness of the proposed approach.

The paper is organized as follows: in Section 2 preliminaries on buck–boost converter and the methodology to approximate fractional-order differentiator are provided. In Section 3 the algorithm to synthesize the controller is described step by step. Numerical simulations, including a comparison with a typical PID controller, a generalization of the synthesized controller, and experimental results of the obtained electrical arrangement are presented in this section as well. A discussion on the most significant results, including future work and some conclusions/remarks, are given in Sections 4 and 5, respectively.
