# Adaptive Active Disturbance Rejection Control of Solar Tracking Systems with Partially Known Model

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## Abstract

**:**

## 1. Introduction

- 1.
- In contrast with the disturbance estimation approach proposed in [49], in this article, the disturbance is estimated in terms of both states and an additional constant term used to compensate possible offset errors and external components that are independent of the states.
- 2.
- The stability and robustness of the controller is mathematically tested by means of the second method of Lyapunov, and its effectiveness is experimentally tested in a robotic test bed.
- 3.
- Some numerical and experimental tests show that the proposed controller demands a low energy consumption, in contrast to a classic ADRC scheme, while keeping appropriate estimation and tracking results for the solar tracking application.

## 2. Controller Design

**Disturbance approximation:**In this article, the generalized disturbance input is proposed to be approximated by the time varying combination of the system states and an additionally constant term (to incorporate arising offset contributions):

## 3. Observer Design

#### Control Law

**Theorem**

**1.**

**Proof.**

## 4. Case Study: A Two Degrees of Freedom Solar Tracker

#### 4.1. System and Control Parameters

- The parameters of the robotic system are provided in Table 1.
- The reference trajectory is defined by the Cooper’s algorithm [56], given by$$\begin{array}{cc}\hfill {\delta}_{r}& =23.45\xb0sin\left(360\left({\displaystyle \frac{284+n}{365}}\right)\right)\hfill \\ \hfill {q}_{2}^{*}& =arcsin\left(cos({\varphi}_{r}cos\left({\delta}_{r}\right)cos\left({\sigma}_{r}\right))+sin\left({\varphi}_{r}\right)sin\left({\delta}_{r}\right)\right)\hfill \\ \hfill {q}_{1}^{*}& =arccos\left({\displaystyle \frac{sin\left({q}_{1}^{*}\right)sin\left({\varphi}_{r}\right)-sin\left({\delta}_{r}\right)}{cos\left({q}_{2}^{*}\right)cos\left({\varphi}_{r}\right)}}\right)\hfill \end{array}$$
- The controller gain parameters were set to be$$\begin{array}{c}\hfill {l}_{1}=0.1;\phantom{\rule{1.em}{0ex}}{K}^{\u22ba}=\left[\begin{array}{cccc}69& 0& 0.9& 0\\ 0& 69& 0& 0.9\end{array}\right];\\ \hfill {\left({L}^{*}\right)}^{\u22ba}=\left[\begin{array}{cccc}400& 0& 1.4\times {10}^{4}& 0\\ 0& 400& 0& 4.5\times {10}^{4}\end{array}\right]\end{array}$$The choice of ${L}^{*}$, ${l}_{1}$, ${K}^{\u22ba}$ was in the context of a set of a model matching with two decoupled, stable, second-order linear model references of the form ${s}^{2}+2{\zeta}_{i}{\omega}_{ni}s+{\omega}_{ni}^{2}$, $i=1,2$, ${\zeta}_{i},{\omega}_{ni}>0$. That is,$$\begin{array}{cc}\hfill {K}^{\u22ba}& =\left[\begin{array}{cccc}2{\zeta}_{1c}{\omega}_{n1c}& 0& {\omega}_{n1c}^{2}& 0\\ 0& 2{\zeta}_{2c}{\omega}_{n2c}& 0& {\omega}_{n2c}^{2}\end{array}\right]\hfill \\ \hfill {\left({L}^{*}\right)}^{\u22ba}& ={l}_{1}\left[\begin{array}{cccc}2{\zeta}_{1o}{\omega}_{n1o}& 0& {\omega}_{n1o}^{2}& 0\\ 0& 2{\zeta}_{2o}{\omega}_{n2o}& 0& {\omega}_{n2o}^{2}\end{array}\right]\hfill \end{array}$$

#### 4.2. Numerical Results

#### 4.3. Experimental Results

- As advantages, the proposal provides low energy consumption, achieving acceptable results in trajectory tracking for solar tracking. It showed low energy consumption with respect to both classic PID control and robust control of the LADRC nature. The adaptation rule is suitable for an implementation in embedded systems, which ensures low energy consumption in contrast with other strategies that are tested in a PC-based controller. The adaptive nature of the system may be suitable for noisy measurements with respect to high-gain state estimators.
- As possible drawbacks, even when the proposed tuning process is of the same nature as the classic PID and LADRC controllers, the process is not as natural as the former controllers. The robustness of the scheme is lower than that shown by the LADRC, but in the case of solar trackers, the mechanism design can contribute to avoiding aggressive robust actions. Besides, even when the system was successfully implemented in an embedded processor, the computational cost of the scheme was larger in comparison to classic schemes.

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 3.**Tracking error behavior comparison (non-disturbed case) for azimuthal axis (${q}_{1}$) and elevation axis (${q}_{2}$). LADRC stands for linear active disturbance rejection control and ASSC for adaptive state space combination.

**Figure 4.**Tracking error behavior comparison (disturbed case) for azimuthal axis (${q}_{1}$) and elevation axis (${q}_{2}$). LADRC stands for linear active disturbance rejection control and ASSC for adaptive state space combination.

**Figure 9.**Performance index behavior of the state estimation. The graphics on the left side denote the case without disturbance and the graphics on the right side show the performance in presence of disturbance.

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

${l}_{1}$ | 100 mm |

${l}_{2}$ | 120 mm |

${l}_{cm1}$ | 61 mm |

${l}_{cm2}$ | 104 mm |

${m}_{1}$ | $0.908$ Kg |

${m}_{2}$ | $0.290$ Kg |

${I}_{y1}$ | $0.01$ g·mm${}^{2}$ |

${I}_{x2}$ | $0.04$ g·mm${}^{2}$ |

${I}_{y2}$ | $0.01$ g·mm${}^{2}$ |

${I}_{z2}$ | $0.95$ g·mm${}^{2}$ |

${g}_{r}$ | $9.81$ Kg·m/s${}^{2}$ |

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## Share and Cite

**MDPI and ACS Style**

Palomino-Resendiz, S.I.; Lozada-Castillo, N.B.; Flores-Hernández, D.A.; Gutiérrez-Frías, O.O.; Luviano-Juárez, A.
Adaptive Active Disturbance Rejection Control of Solar Tracking Systems with Partially Known Model. *Mathematics* **2021**, *9*, 2871.
https://doi.org/10.3390/math9222871

**AMA Style**

Palomino-Resendiz SI, Lozada-Castillo NB, Flores-Hernández DA, Gutiérrez-Frías OO, Luviano-Juárez A.
Adaptive Active Disturbance Rejection Control of Solar Tracking Systems with Partially Known Model. *Mathematics*. 2021; 9(22):2871.
https://doi.org/10.3390/math9222871

**Chicago/Turabian Style**

Palomino-Resendiz, Sergio Isai, Norma Beatriz Lozada-Castillo, Diego Alonso Flores-Hernández, Oscar Octavio Gutiérrez-Frías, and Alberto Luviano-Juárez.
2021. "Adaptive Active Disturbance Rejection Control of Solar Tracking Systems with Partially Known Model" *Mathematics* 9, no. 22: 2871.
https://doi.org/10.3390/math9222871