# Novel Active Disturbance Rejection Control Based on Nested Linear Extended State Observers

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## Abstract

**:**

## Featured Application

## Abstract

## 1. Introduction

- (1)
- For the LESO to increase the estimation accuracy, the bandwidth of the LESO has to be increased, which tolerates noise and leads to hardware difficulties. Additionally, the LESO suffers from a peaking phenomenon due to large gain values.
- (2)
- For the nonlinear ESO, the performance will abruptly deteriorate when the amplitude or derivative of the generalized disturbance goes large to a certain degree [20]. Moreover, stability analysis and performance analysis are very complicated for the nonlinear ESO.
- (3)
- For other classes of observers like the AESO, the parameter tuning process becomes more time-consuming as the observer order goes higher.

## 2. Problem Description and Contribution

#### 2.1. Problem Description

#### 2.2. Paper Contribution

## 3. Conventional Active Disturbance Rejection Control Problem

#### 3.1. Tracking Differentiator (TD)

_{1}is the tracking signal of the input r, and r

_{2}is the tracking signal of the derivative of the input r. To speed up or slow down the system during transient effects, the coefficient R is adapted, making it application dependent [27]. Other versions of enhanced TD are proposed in [28,29,30,31].

#### 3.2. Nonlinear State Error Feedback (NLSEF)

#### 3.3. Extended State Observer

## 4. Main Results

**Assumption**

**A1.**

**Assumption**

**A2.**

**Theorem**

**1.**

**Proof.**

**Lemma**

**1.**

**Proof.**

**Corollary**

**1.**

**Proof.**

**Assumption**

**A3.**

**Assumption**

**A4.**

**Theorem**

**2.**

- where ${\U0001d4c0}_{i}:\mathbb{R}\to {\mathbb{R}}^{+}$ is an even nonlinear gain function.
- where ${\tilde{e}}_{i}={r}_{i}-{\widehat{z}}_{i}\text{},i\in \left\{1,2,\dots ,n\right\}$ is the tracking error. Assuming that Assumptions A3 and A4 hold true, then, the closed-loop system is asymptotically stable, i.e., $\underset{t\to \infty}{\mathrm{lim}}|{\tilde{e}}_{i}|=0,\text{}i\in \left\{1,2,\dots ,n\right\}$.

**Proof.**

## 5. Simulations Results

#### 5.1. Hypothetical Model

- (a)
- LESO:$$\{\begin{array}{c}{\dot{\widehat{x}}}_{1}={\widehat{x}}_{2}+{\beta}_{1}\left(y-{\widehat{x}}_{1}\right)\\ {\dot{x}}_{2}={\widehat{x}}_{3}+{\beta}_{2}\left(y-{\widehat{x}}_{1}\right)\\ {\dot{\widehat{x}}}_{3}={\beta}_{3}\left(y-{x}_{1}\right)\end{array}$$

- (b)
- The NLSEF control law:$$u=fal\left({\tilde{e}}_{1}.{\alpha}_{1}.{\delta}_{1}\right)+fal\left({\tilde{e}}_{2}.{\alpha}_{2}.{\delta}_{2}\right)-\frac{{\widehat{x}}_{3}}{{b}_{0}}$$

- (c)
- The TD is given as [11]:$$\{\begin{array}{c}{\dot{r}}_{1}={r}_{2}\\ {\dot{r}}_{2}=-R\text{}sign\left({r}_{1}-r\left(t\right)+\frac{{r}_{2}\text{}|{r}_{2}|}{2R}\right)\end{array}$$

- (a)
- Inner-loop LESO

- (b)
- Outer-loop LESO$$\{\begin{array}{c}{\dot{\widehat{z}}}_{1}={\widehat{z}}_{2}+{l}_{1}\left(y-{\widehat{z}}_{1}\right)\\ {\dot{\widehat{z}}}_{2}={\widehat{z}}_{3}+{l}_{2}\left(y-{\widehat{z}}_{1}\right)\\ {\dot{\widehat{z}}}_{3}={l}_{3}\left(y-{\widehat{z}}_{1}\right)\end{array}$$
^{T}is the observer gain vector. The design parameters where selected as ${a}_{1}=0.1305$, ${a}_{2}=0.0922$, ${a}_{3}=0.5119$, and ${b}_{0}=1$, and ${\omega}_{0}^{\prime}=22.83.$ - (c)
- The control law is selected as in Equation (52) with the same parameter values and tracking error vector defined as ${\tilde{e}}_{i}={r}_{i}-{\widehat{z}}_{i}$, $i=1,\text{}2$ as illustrated in Figure 2.
- (d)
- The TD for the N-ADRC is identical to Equation (53) with the same parameter values.

^{®}/Simulink

^{®}ODE45 solver for models with continuous states. The reference input (r(t)) to the system was $\mathrm{cos}\left(0.5t\right)$ applied at t = 0 sec. Two test conditions were considered for this work. In the first case, the output of the proposed system did not include any measurement noise, while in the second test case, a Gaussian noise was applied with variance ($\sigma $) equal to ${10}^{-4}$ and the mean $\mu =0$. The simulation results of both conventional ADRC and N-ADRC are shown in Figure 3. The estimated states ${\widehat{x}}_{2}$ and ${\widehat{x}}_{3}$ of the nonlinear system given in Equation (50) using C-ADRC scheme are shown in Figure 4. These states are also estimated using the N-ADRC scheme and are depicted in Figure 5. Moreover, the control signals for both schemes are illustrated in Figure 6.

#### 5.2. The Nonlinear Mass–Spring–Damper Model

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**The Novel Active Disturbance Rejection Control (N-ADRC) structure with nested Linear Extended State Observers (LESOs).

**Figure 3.**The output response curves (y). (

**a**) Conventional ADRC (C-ADRC) (without measurement noise), (

**b**) C-ADRC (with measurement noise), (

**c**) N-ADRC (without measurement noise), (

**d**) N- ADRC (with measurement noise).

**Figure 4.**Estimated states using C-ADRC with measurement noise, (

**a**) ${\widehat{x}}_{2}$, (

**b**) ${\widehat{x}}_{3}$.

**Figure 5.**Estimated states using outer-loop LESO for the N-ADRC with measurement noise, (

**a**) ${\widehat{\mathrm{z}}}_{2}$, (

**b**) ${\widehat{\mathrm{z}}}_{3}$.

**Figure 14.**The results for the mass spring dumper model using C-ADRC, (

**a**) output (y), (

**b**) control signal (u), (

**c**) the generalized disturbance.

**Figure 15.**The result for the mass spring dumper model using N-ADRC, (

**a**) output (y), (

**b**) control signal, (

**c**) the generalized disturbance.

Symbol | Without Noise | With Noise | ||
---|---|---|---|---|

ITAE | ISU | ITAE | ISU | |

C-ADRC | 1.71 | 7.17 | 7.07 | 457.30 |

N-ADRC | 1.33 | 6.63 | 2.13 | 310.91 |

Reduction (%) | 22.32 | 7.51 | 69.87 | 32.01 |

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

$M$ | $1.0$ |

$D$ | $1.0$ |

${d}_{1}$ | $0.01$ |

${d}_{2}$ | $0.1$ |

${d}_{3}$ | $0.01$ |

${d}_{4}$ | $0.67$ |

${d}_{5}$ | $0$ |

$a$ | $1.5$ |

$b$ | $1.5$ |

Controller | ITAE | ISU |
---|---|---|

C-ADRC | 0.10 | 0.17 |

N-ADRC | 0.05 | 0.16 |

Reduction (%) | 50 | 6 |

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**MDPI and ACS Style**

Abdul-Adheem, W.R.; Azar, A.T.; Ibraheem, I.K.; Humaidi, A.J.
Novel Active Disturbance Rejection Control Based on Nested Linear Extended State Observers. *Appl. Sci.* **2020**, *10*, 4069.
https://doi.org/10.3390/app10124069

**AMA Style**

Abdul-Adheem WR, Azar AT, Ibraheem IK, Humaidi AJ.
Novel Active Disturbance Rejection Control Based on Nested Linear Extended State Observers. *Applied Sciences*. 2020; 10(12):4069.
https://doi.org/10.3390/app10124069

**Chicago/Turabian Style**

Abdul-Adheem, Wameedh Riyadh, Ahmad Taher Azar, Ibraheem Kasim Ibraheem, and Amjad J. Humaidi.
2020. "Novel Active Disturbance Rejection Control Based on Nested Linear Extended State Observers" *Applied Sciences* 10, no. 12: 4069.
https://doi.org/10.3390/app10124069