# Performance of the Boost Converter Controlled with ZAD to Regulate DC Signals

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## Abstract

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## 1. Introduction

- The state-space model is presented in a compact form and variable change is performed to depend only on the parameter $\gamma $.
- The ZAD control technique is designed using a new sliding surface, where all the variables of the system are considered.
- It is analytically demonstrated that the piecewise linear approximation of the switching surface $s\left(\mathrm{x}\right(t\left)\right)$ is a good technique because the error can be considerably reduced. In addition, the maximum and minimum errors in the approximation occur precisely at the ends of the sub-intervals.
- It is shown numerically that the system presents good voltage and current regulations and low relative error with respect to the reference signals.
- The existence of the birth of a chaotic attractor is shown numerically for the boost converter controlled with ZAD.
- A large regulation area with low errors is found based on a parametric diagram that depends on ${k}_{1}$ and ${k}_{2}$, confirming the good performance of the boost converter controlled with ZAD.

## 2. Materials and Methods

#### 2.1. Boost Converter

- For the topology 1 ($u=1$),$${x}_{1}\left(t\right)={x}_{1}\left({t}_{0}\right){e}^{-\gamma (t-{t}_{0})},$$$${x}_{2}\left(t\right)={x}_{2}\left({t}_{0}\right)+(t-{t}_{0}).$$
- For the topology 2 ($u=0$),$${x}_{1}\left(t\right)={e}^{-\frac{\gamma}{2}(t-{t}_{0})}[\left({x}_{1}\left({t}_{0}\right)-1\right)cos\omega (t-{t}_{0})$$$$-\frac{\gamma}{2\omega}\left({x}_{1}\left({t}_{0}\right)-1-\frac{2}{\gamma}{x}_{2}\left({t}_{0}\right)\right)sin\omega (t-{t}_{0})]+1,$$$${x}_{2}\left(t\right)={e}^{-\frac{\gamma}{2}(t-{t}_{0})}[\left({x}_{2}\left({t}_{0}\right)-\gamma \right)cos\omega (t-{t}_{0})$$$$+\left(\frac{\gamma}{2\omega}\left({x}_{2}\left({t}_{0}\right)-\gamma \right)-\frac{1}{\omega}\left({x}_{1}\left({t}_{0}\right)-1\right)\right)sin\omega (t-{t}_{0})]+\gamma ,$$
- If $i\le 0$, Topology 3 is obtained as follows:$${x}_{1}\left(t\right)={x}_{1}\left({t}_{0}\right){e}^{-\gamma (t-{t}_{0})},$$$${x}_{2}\left(t\right)=0.$$

#### 2.2. Pulse-Width Modulation

#### 2.3. Zero Average Dynamics Technique

- Define the switching surface $s\left(\mathrm{x}\left(t\right)\right)={k}_{1}({x}_{1}\left(t\right)-{x}_{1ref})+{k}_{2}({x}_{2}\left(t\right)-{x}_{2ref})$.
- Fix a period T.
- Force s to have zero average in each cycle (ZAD):$${\int}_{nT}^{(n+1)T}s\left(\mathrm{x}\left(t\right)\right)dt=0}.$$

#### 2.4. Piecewise Linear Approximation of the Switching Surface

- The dynamics of the error or switching surface $s\left(\mathrm{x}\right(t\left)\right)$ behaves like a piecewise linear function.
- The slopes of the dynamic error at each section are determined by the slopes calculated at the switching time, assuming that the slope at the beginning of the period, noted as ${\dot{s}}_{1}$, is the same as at the end. That means that in the sections between $[nT,nT+\frac{d}{2}]$ and $[(n+1)T-\frac{d}{2},(n+1)T]$, the slope of $s\left(\mathrm{x}\right(t\left)\right)$ is ${\dot{s}}_{1}$, which corresponds to the derivative of the switching surface with respect to time for the case $u=1$. Furthermore, in the section $[nT+\frac{d}{2},(n+1)T-\frac{d}{2}]$ the slope of $s\left(\mathrm{x}\right(t\left)\right)$ is ${\dot{s}}_{2}$ and corresponds to the derivative of the switching surface with respect to time for the case $u=0$.

- For Topology 1,$$\frac{{d}^{2}}{d{t}^{2}}}s\left(\mathrm{x}\left(t\right)\right)={k}_{1}{\gamma}^{2}{x}_{1}\left(t\right).$$
- For Topology 2,$$\frac{{d}^{2}}{d{t}^{2}}}s\left(\mathrm{x}\left(t\right)\right)=({k}_{1}({\gamma}^{2}-1)+\gamma {k}_{2}){x}_{1}\left(t\right)+(-\gamma {k}_{1}-{k}_{2}){x}_{2}\left(t\right)+{k}_{1}.$$

- For Topology 1, given $\u03f5>0$, if the following expression is chosen:$$t>ln\left({\displaystyle \frac{1}{\sqrt[\gamma ]{\u03f5{\gamma}^{2}\left|{k}_{1}{x}_{1}\left(nT\right){e}^{\gamma nT}\right|}}}\right),$$$$\left|{\displaystyle \frac{{d}^{2}}{d{t}^{2}}}s\left(\mathrm{x}\left(t\right)\right)\right|\hspace{0.17em}<\hspace{0.17em}\u03f5.$$
- For Topology 2, the following holds:$$\frac{{d}^{2}}{d{t}^{2}}}s\left(\mathrm{x}\left(t\right)\right)={e}^{\frac{-\gamma}{2}(t-{t}_{0})}\xb7\left[cos\left(\omega (t-{t}_{0})\right)\xb7a+sin\left(\omega (t-{t}_{0})\right)\xb7b\right],$$$$a\hspace{0.17em}=\hspace{0.17em}\left({k}_{1}({\gamma}^{2}-1)+\gamma {k}_{2}\right)\left({x}_{1}\left({t}_{0}\right)-1\right)-\left(\gamma {k}_{1}+{k}_{2}\right)\left({x}_{2}\left({t}_{0}\right)-\gamma \right),$$$$b=\frac{1}{\omega}\left[\left({k}_{1}(1-{\gamma}^{2})-\gamma {k}_{2}\right)\left({x}_{1}\left({t}_{0}\right)-1\right)\hspace{0.17em}\xb7\hspace{0.17em}\frac{\gamma}{2}\hspace{0.17em}+\hspace{0.17em}\left({x}_{2}\left({t}_{0}\right)-\gamma \right)\hspace{0.17em}\xb7\hspace{0.17em}\left(1-\frac{\gamma}{2}\left(\gamma {k}_{1}+{k}_{2}\right)\right)\right],$$$${t}_{0}=nT+\frac{d}{2}.$$

#### 2.5. Duty Cycle

- If $d<0$, the system is forced to evolve according to Topology 1.
- If $d>T$, the system is forced to evolve according to Topology 2.
- If denominator of Equation (51), defined now as ${k}_{1}{x}_{2}\left(nT\right)-{k}_{2}{x}_{1}\left(nT\right)=0$, is equal to zero, then two possibilities are defined: (a) the system is forced to evolve according to Topology 1 if the numerator $2s\left(\mathrm{x}\left(nT\right)\right)+T{\dot{s}}_{2}\left(\mathrm{x}\left(nT\right)\right)>0$; (b) the system is forced to evolve according to Topology 2 if $2s\left(\mathrm{x}\left(nT\right)\right)+T{\dot{s}}_{2}\left(\mathrm{x}\left(nT\right)\right)<0$.

## 3. Results and Analysis

#### 3.1. ZAD with Piecewise Linear Approximation

#### 3.2. Regulation

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 2.**Validation of the mathematical model proposed in Equation (15).

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**MDPI and ACS Style**

Trujillo, S.C.; Candelo-Becerra, J.E.; Hoyos, F.E.
Performance of the Boost Converter Controlled with ZAD to Regulate DC Signals. *Computation* **2021**, *9*, 96.
https://doi.org/10.3390/computation9090096

**AMA Style**

Trujillo SC, Candelo-Becerra JE, Hoyos FE.
Performance of the Boost Converter Controlled with ZAD to Regulate DC Signals. *Computation*. 2021; 9(9):96.
https://doi.org/10.3390/computation9090096

**Chicago/Turabian Style**

Trujillo, Simeón Casanova, John E. Candelo-Becerra, and Fredy E. Hoyos.
2021. "Performance of the Boost Converter Controlled with ZAD to Regulate DC Signals" *Computation* 9, no. 9: 96.
https://doi.org/10.3390/computation9090096