# An Output Feedback Discrete-Time Controller for the DC-DC Buck Converter

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## Abstract

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## 1. Introduction

- The presentation of a discrete-time output feedback controller based on the robust exact filtering differentiator for the DC-DC buck converter with saturated input.
- The introduction of a rigorous stability analysis of the discrete-time controller considering the asymmetric saturation of the system.
- An extensive experimental analysis that includes various operating conditions and disturbance scenarios to validate the performance of the controller.

## 2. Robust Exact Filtering Differentiator

**Assumption**

**1.**

**Definition**

**1**

#### Implicit Discretization of the Robust Exact Filtering Differentiator

**Assumption**

**2.**

**Definition**

**2.**

**Lemma**

**1**

- If ${b}_{k}>{a}_{0}$, then ${\xi}_{k}=\left(\right)open="\{"\; close="\}">-1$ and ${w}_{1,k+1}=-{\left(\right)}^{{r}_{0}}m+1$, where ${r}_{0}$ is the unique positive root of the following polynomial:$$\begin{array}{c}\hfill \begin{array}{c}\hfill p\left(r\right)={r}^{m+1}+{a}_{m}{r}^{m}+\cdots +{a}_{1}r+\left(\right)open="("\; close=")">-{b}_{k}+{a}_{0}.\end{array}\end{array}$$
- If ${b}_{k}\in [-{a}_{0},{a}_{0}]$, then ${w}_{1,k+1}=0$ and ${\xi}_{k}=\left(\right)open="\{"\; close="\}">-\frac{{b}_{k}}{{a}_{0}}$.
- If ${b}_{k}<-{a}_{0}$, then ${\xi}_{k}=\left\{1\right\}$ and ${w}_{1,k+1}={r}_{0}^{m+1}$, where ${r}_{0}$ is the unique positive root of the following polynomial:$$\begin{array}{c}\hfill \begin{array}{c}\hfill p\left(r\right)={r}^{m+1}+{a}_{m}{r}^{m}+\cdots +{a}_{1}r+\left(\right)open="("\; close=")">{b}_{k}+{a}_{0}.\end{array}\end{array}$$

**Remark**

**1.**

**Theorem**

**1**

## 3. DC-DC Buck Converter Analysis

## 4. Proposed Control Strategy

- ${t}_{{k}_{0}}$ is the lowest sampling time such that ${D}_{c}\left(t\right)$ is saturated for any measurement time greater than ${t}_{{k}_{0}}$ and previous to the instant of time ${t}_{{k}_{2}}$, i.e., ${u}_{k}\ge {u}_{{k}_{max}}$ or ${u}_{k}\le {u}_{{k}_{min}}$ for any ${t}_{k}$ with ${t}_{{k}_{0}}\le {t}_{k}<{t}_{{k}_{2}}$.
- ${t}_{{k}_{2}}$ is the time instant such that ${u}_{k}$ is unsaturated and for the previous measurement time ${u}_{k}$ was saturated, i.e., ${u}_{{k}_{min}}<{u}_{k}<{u}_{{k}_{max}}$ at ${t}_{{k}_{2}}$, and ${u}_{k}\ge {u}_{{k}_{max}}$ or ${u}_{k}\le {u}_{{k}_{min}}$ for any ${t}_{k}$ with ${t}_{{k}_{0}}\le {t}_{k}<{t}_{{k}_{2}}$.
- $\overline{u}\left(t\right)$ is the continuous-time function analogous to ${u}_{k}$, defined as$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \overline{u}\left(t\right)& =\mathit{K}{\left(\right)}^{{\overline{z}}_{I}}T,\hfill \end{array}\hfill {\overline{z}}_{I}\left(t\right)& ={z}_{I,k},\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\mathrm{for}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}t\in [{t}_{k},{t}_{k+1}).\hfill \end{array}$$
- ${t}_{f}$ is the time instant after ${t}_{{k}_{0}}$ and ${t}_{{k}_{1}}$ such that ${u}_{{k}_{min}}<\overline{u}\left({t}_{f}\right)<{u}_{{k}_{max}}$.

**Theorem**

**2.**

**Proof.**

**Theorem**

**3.**

**Proof.**

#### Numeric Example

## 5. Experimental Results

#### 5.1. System Implementation

#### 5.2. Experimental Results

#### 5.2.1. Output Voltage Tracking Performance

#### 5.2.2. Load Variation Performance

#### 5.2.3. Input Voltage Variation

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

UPS | Uninterruptible power supply |

GPI | Generalized proportional integral |

PID | Proportional integral derivative |

FPGA | Field-programmable gate array |

DSP | Digital signal processor |

SM | Sliding mode |

PIAW | Proportional integral with anti-windup |

ESR | Equivalent series resistance |

CCM | Continuous conduction mode |

DPWM | Digital pulse-width modulator |

DCM | Discontinuous conduction mode |

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**Figure 1.**DC-DC buck converter topology. The ESR of the capacitor C and the inductor ${L}_{i}$ are included.

**Figure 2.**Block diagram representation of the system to evaluate the proposed control for the DC-DC buck converter.

**Figure 4.**Output voltage tracking performance of the proposed controller and a saturated discrete−time PID with $\tau =25$ $\mathsf{\mu}$s. (

**a**) Output voltage and reference; (

**b**) control output; (

**c**) reference tracking error and its estimation; (

**d**) reference tracking error detail.

**Figure 5.**Output voltage tracking performance of the proposed controller and a saturated discrete−time PID with $\tau =250$ $\mathsf{\mu}$s. (

**a**) Output voltage and reference; (

**b**) reference tracking error and its estimation.

**Figure 6.**Load variation performance of the proposed controller and a saturated discrete−time PID with $\tau =25$ $\mathsf{\mu}$s. (

**a**) Output voltage and reference; (

**b**) control output; (

**c**) reference tracking error and its estimation; (

**d**) reference tracking error detail.

**Figure 7.**Load variation performance of the proposed controller and a saturated discrete−time PID with $\tau =250$ $\mathsf{\mu}$s. (

**a**) Output voltage and reference; (

**b**) reference tracking error and its estimation.

**Figure 8.**Input voltage variation performance of the proposed controller and a saturated discrete−time PID with $\tau =25$ $\mathsf{\mu}$s. (

**a**) Output voltage and reference; (

**b**) control output; (

**c**) reference tracking error and its estimation; (

**d**) reference tracking error detail.

**Figure 9.**Input voltage variation performance of the proposed controller and a saturated discrete−time PID with $\tau =250$ $\mathsf{\mu}$s. (

**a**) Output voltage and reference; (

**b**) reference tracking error and its estimation.

$Vs=12.7$ V | ${R}_{{L}_{i}}=0.32\phantom{\rule{3.33333pt}{0ex}}\Omega $ | ${\mathsf{\mu}}_{0}=3$ | ${u}_{{k}_{max}}=0.99$ | ${k}_{d}=-0.00002$ |

$R=120\Omega $ | $C=998\mathsf{\mu}\mathrm{F}$ | ${\mathsf{\mu}}_{1}=4$ | ${u}_{{k}_{min}}=0.01$ | |

${R}_{c}=0.041\Omega $ | ${\xi}_{max}=200$ | $\tau =25\mathsf{\mu}s$ | ${k}_{i}=-3$ | |

${L}_{i}=255.81\mathsf{\mu}\mathrm{H}$ | ${\xi}_{min}=-200$ | $\rho =0.0001$ | ${k}_{p}=-0.185$ |

Description | Symbol | Nominal Value |
---|---|---|

Input voltage | ${V}_{s}$ | 12.3 V–24.7 V |

Capacitance | C | 998 $\mathsf{\mu}$F |

Capacitor ESR | ${R}_{C}$ | 0.041 $\mathbf{\Omega}$ |

Inductance | ${L}_{i}$ | 255.81 $\mathsf{\mu}$H |

Inductor resistance | ${R}_{{L}_{i}}$ | 0.32 $\mathbf{\Omega}$ |

Switching frequency | ${F}_{s}$ | 40 KHz |

Minimum load resistance | ${R}_{min}$ | 5 $\mathbf{\Omega}$ |

Maximum load resistance | ${R}_{max}$ | 124 $\mathbf{\Omega}$ |

Desired output voltage | ${V}_{o}$ | 2 V–10.5 V |

Parameter | Value |
---|---|

${k}_{i}$ | −3.35 |

${k}_{p}$ | −0.15 |

${k}_{d}$ | −0.00002 |

${u}_{{k}_{min}}$ | 0.01 |

${u}_{{k}_{max}}$ | 0.99 |

L | 2500 |

$\tau $ | 25 $\mathsf{\mu}$s, 250 $\mathsf{\mu}$s |

${\lambda}_{0}$ | 1.1 |

${\lambda}_{1}$ | 2.12 |

${\lambda}_{2}$ | 2 |

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## Share and Cite

**MDPI and ACS Style**

Alarcón-Carbajal, M.A.; Carvajal-Rubio, J.E.; Sánchez-Torres, J.D.; Castro-Palazuelos, D.E.; Rubio-Astorga, G.J.
An Output Feedback Discrete-Time Controller for the DC-DC Buck Converter. *Energies* **2022**, *15*, 5288.
https://doi.org/10.3390/en15145288

**AMA Style**

Alarcón-Carbajal MA, Carvajal-Rubio JE, Sánchez-Torres JD, Castro-Palazuelos DE, Rubio-Astorga GJ.
An Output Feedback Discrete-Time Controller for the DC-DC Buck Converter. *Energies*. 2022; 15(14):5288.
https://doi.org/10.3390/en15145288

**Chicago/Turabian Style**

Alarcón-Carbajal, Martin A., José E. Carvajal-Rubio, Juan D. Sánchez-Torres, David E. Castro-Palazuelos, and Guillermo J. Rubio-Astorga.
2022. "An Output Feedback Discrete-Time Controller for the DC-DC Buck Converter" *Energies* 15, no. 14: 5288.
https://doi.org/10.3390/en15145288