# Optimization and Its Implementation Impact of Two-Modes Controller Fractional Approximation for Buck Converters

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. DC-DC Buck Converter

#### 2.2. Biquadratic Modules to Fractionally Approximate Laplacian Operator

#### 2.3. Definition of Minimization Criteria: An Error-Based Approach

## 3. Results

#### 3.1. Two-Modes Controller Structure

- If desired phase margin is ${\varphi}_{d}=\pi /3={60}^{\xb0}$, from ${\varphi}_{c}+{\varphi}_{p}=-\pi +{\varphi}_{d}$ it is deduced that controller phase contribution has to be ${\varphi}_{c}=37.33$°.

#### 3.2. Numerical Results

- Particle swarm optimization algorithm (PS)
- Iterations: 600
- Population: 90
- Inertia coefficient w: 1
- Cognition constant ${c}_{1}$: 2
- Social constant ${c}_{2}$: 2

- Genetic optimization algorithm (GA)
- Iterations: 600
- Population: 90
- Mutation rate: 0.25
- Random recombination.
- Scale population coefficient: 1

#### 3.3. Experimental Results

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**Classical control diagram to regulate voltage in the buck converter of Figure 1.

**Figure 4.**Frequency response of fractional-order approximation (3) for ${s}^{\pm 0.6}$ where dashed lines represent the theoretical response and solid lines the approximation. (

**a**) Derivative effect. (

**b**) Integral effect.

**Figure 5.**Possible combinations for integer-order (IO) and fractional-order (FO) approaches for the system-controller duo when introducing Fractional Calculus into Control Theory.

**Figure 8.**(

**a**) Buck converter output voltage ${v}_{o}\left(t\right)$ (black) and inductor current ${i}_{L}\left(t\right)$ (green) to corroborate regulation and continuous conduction mode operation. (

**b**) Control law $\overline{d}\left(t\right)=0.6$ and its effect on the PWM signal.

**Figure 10.**(

**a**) Buck output voltage ${v}_{o}\left(t\right)$ (black) and inductor current ${i}_{L}\left(t\right)$ (green) to determine effective regulation and converter continuous conduction mode operation. (

**b**) Control law $\overline{d}\left(t\right)=0.6$ and its effect on the PWM signal.

**Figure 11.**Control laws comparison for both optimization algorithms when minimizing criteria ${J}_{1}$ and ${J}_{2}$.

**Figure 12.**(

**a**) Closed-loop step response of plant’s transfer function (2) with classical PD controller ${G}_{c}\left(s\right)=kp(1+{T}_{d}s)$. (

**b**) Frequency response corroborating closed-loop phase margin ${\varphi}_{d}\approx {60}^{\xb0}$.

**Figure 13.**Electrical arrangement for implementation of control diagram from Figure 2 where plant, controller, comparator and PWM blocks can be identified.

**Figure 14.**(

**a**) Oscilloscope view for measurements of output voltage ${v}_{o}\left(t\right)$ (yellow), output current ${I}_{o}\left(t\right)$ (green), input current ${I}_{i}\left(t\right)$ (purple) and PWM signal $d\left(t\right)$ (cyan). (

**b**) Alternative view of exported experimental data preserving scale of 5 V/unit for ${v}_{o}\left(t\right)$, 100 mV/unit for ${I}_{o}\left(t\right)$ and ${I}_{i}\left(t\right)$.

**Figure 15.**(

**a**) Oscilloscope view for measurement of output voltage ${v}_{o}\left(t\right)$ and load current ${I}_{o}\left(t\right)$ in the presence of load variation. (

**b**) Alternative view of output voltage ${v}_{o}\left(t\right)$ and load current ${I}_{o}\left(t\right)$ from exported data.

**Figure 16.**(

**a**) Oscilloscope view for measurement of reference tracking characteristic from output voltage ${v}_{o}\left(t\right)$ and the corresponding evolution of load current ${I}_{o}\left(t\right)$. (

**b**) Alternative view of output voltage ${v}_{o}\left(t\right)$ and load current ${I}_{o}\left(t\right)$ from exported data, preserving scale of 5 v/unit for ${v}_{o}\left(t\right)$.

**Table 1.**Parameter values for the implementation of buck converter in Figure 1.

Component/Element | Notation | Value | Generals |
---|---|---|---|

Capacitor | C | 7 $\mathsf{\mu}$F | C4AQCBU4700A1YJ, 650 V, ±5% |

Inductor | L | 2.7 mH | 1140-272K-RC, 555 m$\Omega $, 2.2/3.9 A, ±10% |

Resistance | R | 10 $\Omega $ | CB25JB10R0, 25 W, ±5% |

Power supply | ${V}_{i}$ | 25 V | Programmable BK Precision 9129B |

MOSFET | Q | PSMN022-30PL, N-Ch, 30 V , 22 m$\Omega $, 30 A | |

Diode | D | MUR840, 50–600 V, 8 A, 1 V | |

MOSFET driver | Optocoupler TLP250 | ||

Switching frequency | ${f}_{sw}$ | 20 kHz |

${\mathit{J}}_{1}$ | ${\mathit{J}}_{2}$ | |||||
---|---|---|---|---|---|---|

$\mathit{\alpha}$ | ${\mathit{k}}_{\mathit{p}}$ | ${\mathit{T}}_{\mathit{d}}$ | $\mathit{\alpha}$ | ${\mathit{k}}_{\mathit{p}}$ | ${\mathit{T}}_{\mathit{d}}$ | |

GA | 0.4148 | 1.2839 | 4.9995 | 0.4148 | 1.9331 | 2.5497 |

0.4148 | 1.2847 | 4.9995 | 0.4148 | 1.9225 | 2.5553 | |

0.4148 | 1.2842 | 4.9995 | 0.4148 | 1.7595 | 3.0316 | |

0.4148 | 1.2851 | 4.9994 | ||||

0.4148 | 1.2856 | 4.9927 | ||||

0.4148 | 1.2844 | 4.9989 | ||||

PS | 0.4148 | 1.2851 | 5 | 0.4148 | 1.8759 | 2.7445 |

0.4148 | 1.2852 | 5 | 0.4148 | 1.8773 | 2.7404 | |

0.4148 | 1.2854 | 5 | 0.4148 | 1.8733 | 2.7522 |

**Table 3.**Computed values for coefficients of controller’s transfer function (19).

${\mathit{J}}_{1}$ | ${\mathit{J}}_{2}$ | |||||
---|---|---|---|---|---|---|

${\mathit{\beta}}_{1}$^{1}/${\mathit{\beta}}_{2}$^{2} | ${\mathit{\beta}}_{3}$^{3}/${\mathit{\beta}}_{4}$^{4} | ${\mathit{k}}_{\mathit{c}}$ | ${\mathit{\beta}}_{1}$^{1}/${\mathit{\beta}}_{2}$ ^{2} | ${\mathit{\beta}}_{3}$^{3}/${\mathit{\beta}}_{4}$ ^{4} | ${\mathit{k}}_{\mathit{c}}$ | |

GA | 7.461/6.739 | 1.813/3.46 | 18.7218 | 8.121/8.461 | 1.813/3.46 | 15.3230 |

7.461/6.739 | 1.813/3.46 | 18.7334 | 8.118/8.454 | 1.813/3.46 | 15.2683 | |

7.461/6.739 | 1.813/3.46 | 18.7261 | 7.916/7.926 | 1.813/3.46 | 16.2504 | |

7.461/6.739 | 1.813/3.46 | 18.7389 | ||||

7.462/6.741 | 1.813/3.46 | 18.7228 | ||||

7.461/6.739 | 1.813/3.46 | 18.7270 | ||||

PS | 7.461/6.739 | 1.813/3.46 | 18.7410 | 8.03/8.225 | 1.813/3.46 | 15.8624 |

7.461/6.739 | 1.813/3.46 | 18.7425 | 8.032/8.229 | 1.813/3.46 | 15.8533 | |

7.461/6.739 | 1.813/3.46 | 18.7454 | 8.027/8.216 | 1.813/3.46 | 15.8796 |

^{1}×10

^{4}.

^{2}×10

^{8}.

^{3}×10

^{5}.

^{4}×10

^{9}.

**Table 4.**Computed values for components of controller’s partial fraction expansion (20) and corresponding electrical circuit of Figure 6, where ${C}_{1}={C}_{2}=2.2$ $\mathsf{\mu}$F and $R=1$ k$\Omega $. Units for ${R}_{1}$ and ${R}_{2}$ are given in $\Omega $. Units for ${R}_{3}$, ${R}_{4}$ and ${R}_{5}$ are given in k$\Omega $.

${\mathit{J}}_{1}$ | ${\mathit{J}}_{2}$ | |||
---|---|---|---|---|

${\mathit{R}}_{1}$/${\mathit{R}}_{2}$ | ${\mathit{R}}_{3}/{\mathit{R}}_{4}/{\mathit{R}}_{5}$ | ${\mathit{R}}_{1}$/${\mathit{R}}_{2}$ | ${\mathit{R}}_{3}/{\mathit{R}}_{4}/{\mathit{R}}_{5}$ | |

GA | 13.78/101.54 | 12.11/2.96/18.72 | 13.78/101.54 | 9.30/2.28/15.32 |

13.78/101.54 | 12.12/2.97/18.73 | 13.78/101.54 | 9.27/2.27/15.27 | |

13.78/101.54 | 12.12/2.96/18.73 | 13.78/101.54 | 10.07/2.46/16.25 | |

13.78/101.54 | 12.12/2.97/18.74 | |||

13.78/101.54 | 12.11/2.96/18.72 | |||

13.78/101.54 | 12.12/2.96/18.73 | |||

PS | 13.78/101.54 | 12.13/2.97/18.74 | 13.78/101.54 | 9.72/2.38/15.86 |

13.78/101.54 | 12.13/2.97/18.74 | 13.78/101.54 | 9.71/2.38/15.85 | |

13.78/101.54 | 12.13/2.97/18.75 | 13.78/101.54 | 9.7/2.38/15.87 |

${\mathit{J}}_{\mathit{1}}$ | ${\mathit{J}}_{\mathit{2}}$ | |||||

Optimization results | ||||||

$\alpha $ | ${k}_{p}$ | ${T}_{d}$ | $\alpha $ | ${k}_{p}$ | ${T}_{d}$ | |

GA | 0.9936 | 29.728 | 29.8142 | 0.975 | 29.9437 | 7.8793 |

0.9936 | 29.9927 | 29.9945 | 0.9396 | 29.9912 | 3.2028 | |

0.9937 | 29.9999 | 29.9975 | 0.846 | 29.9346 | 1.2147 | |

0.9937 | 29.9931 | 29.9971 | 0.8917 | 29.9948 | 1.7824 | |

0.9936 | 30 | 29.9904 | 0.9762 | 29.9868 | 8.4766 | |

0.9936 | 29.9959 | 29.9357 | 0.9489 | 29.978 | 3.9742 | |

PS | 0.9937 | 30 | 30 | 0.8847 | 30 | 1.7319 |

0.8843 | 30 | 1.7255 | ||||

0.8786 | 30 | 1.607 | ||||

0.8889 | 30 | 1.8075 | ||||

Controller coefficients | ||||||

${\beta}_{1}$^{1}/${\beta}_{2}$ ^{2} | ${\beta}_{3}$^{2}/${\beta}_{4}$ ^{3} | ${k}_{c}$^{1} | ${\beta}_{1}$^{1}/${\beta}_{2}$^{2} | ${\beta}_{3}$^{4}/${\beta}_{4}$ ^{5} | ${k}_{c}$^{6} | |

GA | 3.717/4.545 | 1.674/5.925 | 41.241 | 4.142/17.24 | 42.9/14.85 | 27.534 |

3.717/4.519 | 1.674/5.925 | 41.859 | 5.005/42.22 | 17.79/5.901 | 4.4812 | |

3.716/4.514 | 1.7/6.02 | 42.543 | 7.426/107.2 | 7.007/2.086 | 0.6255 | |

3.716/4.514 | 1.7/6.02 | 42.533 | 6.197/74.93 | 9.946/3.119 | 1.3394 | |

3.717/4.52 | 1.674/5.925 | 41.864 | 4.104/16.05 | 45.05/15.62 | 31.198 | |

3.717/4.527 | 1.674/5.925 | 41.782 | 4.731/34.19 | 21.02/7.049 | 6.6248 | |

PS | 3.716/4.514 | 1.7/6.02 | 42.547 | 6.298/77.16 | 9.345/2.907 | 1.2161 |

6.309/77.43 | 9.313/2.896 | 1.2071 | ||||

6.5/82.77 | 8.878/2.742 | 1.0682 | ||||

6.184/74.08 | 9.697/3.031 | 1.3206 | ||||

Controller component values | ||||||

${R}_{1}$/${R}_{2}$ | ${R}_{3}/{R}_{4}/{R}_{5}$ | ${R}_{1}$/${R}_{2}$ | ${R}_{3}/{R}_{4}/{R}_{5}$ | |||

GA | 0.13/62.01 | 412.364/10.19/412.405 | 0.52/63.05 | 27.491/10.38/27.534 | ||

0.13/62.01 | 418.551/10.34/418.594 | 1.26/65.07 | 4.439/9.81/4.481 | |||

0.13/62 | 425.389/10.18/425.432 | 3.29/70.62 | 0.585/8.52/0.626 | |||

0.13/62 | 425.287/10.18/425.329 | 2.29/67.87 | 1.298/9.28/1.339 | |||

0.13/62.01 | 418.596/10.34/418.638 | 0.49/62.98 | 31.156/10.66/31.198 | |||

0.13/62.01 | 417.775/10.32/417.818 | 1.06/64.54 | 6.582/10.41/6.625 | |||

PS | 0.13/62 | 425.426/10.18/425.468 | 2.44/68.29 | 1.174/9.53/1.216 | ||

2.45/68.31 | 1.165/9.52/1.207 | |||||

2.57/68.65 | 1.027/9.24/1.068 | |||||

2.35/68.04 | 1.279/9.63/1.321 |

^{1}× 10

^{4}.

^{2}× 10

^{7}.

^{3}× 10

^{11}.

^{4}× 10

^{5}.

^{5}× 10

^{10}.

^{6}× 10

^{3}.

${\mathit{J}}_{1}$ | ${\mathit{J}}_{2}$ | |||
---|---|---|---|---|

Optimization results | ||||

${k}_{p}$ | ${T}_{d}$ | ${k}_{p}$ | ${T}_{d}$ | |

GA | 9.9988 | 2.98 $\times {10}^{-6}$ | 9.9993 | 2.85 $\times {10}^{-6}$ |

9.9983 | 2.98 $\times {10}^{-6}$ | 9.999 | 2.95 $\times {10}^{-6}$ | |

PS | 10 | 2.99 $\times {10}^{-6}$ | 10 | 2.95 $\times {10}^{-6}$ |

10 | 2.98 $\times {10}^{-6}$ | 10 | 2.97 $\times {10}^{-6}$ | |

Controller component values | ||||

${R}_{i}$/${R}_{f}$ | ${C}_{i}/{R}_{f}$ | ${R}_{i}$/${R}_{f}$ | ${C}_{i}/{R}_{f}$ | |

GA | 1 k$\Omega $/9.9 k$\Omega $ | 1 $\mathsf{\mu}$F/2.98 $\Omega $ | 1 k$\Omega $/9.9 k$\Omega $ | 1 $\mathsf{\mu}$F/2.85 $\Omega $ |

1 k$\Omega $/9.9 k$\Omega $ | 1 $\mathsf{\mu}$F/2.98 $\Omega $ | 1 k$\Omega $/9.9 k$\Omega $ | 1 $\mathsf{\mu}$F/2.95 $\Omega $ | |

PS | 1 k$\Omega $/10 k$\Omega $ | 1 $\mathsf{\mu}$F/2.99 $\Omega $ | 1 k$\Omega $/10 k$\Omega $ | 1 $\mathsf{\mu}$F/2.95 $\Omega $ |

1 k$\Omega $/10 k$\Omega $ | 1 $\mathsf{\mu}$F/2.98 $\Omega $ | 1 k$\Omega $/10 k$\Omega $ | 1 $\mathsf{\mu}$F/2.97 $\Omega $ |

Parameter | Symbol | FOPD | PD |
---|---|---|---|

Rise time | ${t}_{r}$ | 8.72 $\mathsf{\mu}$s | 9.83 $\mathsf{\mu}$s |

Settling time | ${t}_{s}$ | 76.66 $\mathsf{\mu}$s | 144.01 $\mathsf{\mu}$s |

Peak time | ${t}_{p}$ | 22.33 $\mathsf{\mu}$s | 24.78 $\mathsf{\mu}$s |

Overshoot | % M_{p} | 39.6% | 51.7% |

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**MDPI and ACS Style**

S. Sánchez, A.G.; Perez-Pinal, F.-J.; Espinosa-Calderón, A.
Optimization and Its Implementation Impact of Two-Modes Controller Fractional Approximation for Buck Converters. *Micromachines* **2022**, *13*, 1600.
https://doi.org/10.3390/mi13101600

**AMA Style**

S. Sánchez AG, Perez-Pinal F-J, Espinosa-Calderón A.
Optimization and Its Implementation Impact of Two-Modes Controller Fractional Approximation for Buck Converters. *Micromachines*. 2022; 13(10):1600.
https://doi.org/10.3390/mi13101600

**Chicago/Turabian Style**

S. Sánchez, Allan G., Francisco-Javier Perez-Pinal, and Alejandro Espinosa-Calderón.
2022. "Optimization and Its Implementation Impact of Two-Modes Controller Fractional Approximation for Buck Converters" *Micromachines* 13, no. 10: 1600.
https://doi.org/10.3390/mi13101600