# Complete Synchronization and Partial Anti-Synchronization of Complex Lü Chaotic Systems by the UDE-Based Control Method

^{1}

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## Abstract

**:**

## 1. Introduction

- (a)
- Firstly, the dynamic gain feedback control method and linear feedback control method are presented to solve the synchronization and partial anti-synchronization problems of the nominal chaotic system, respectively;
- (b)
- Secondly, the controller of the nominal system is combined with the UDE controller to deal with the synchronization and partial anti-synchronization problems of a given chaotic system with both uncertainty and disturbance;
- (c)
- Finally, take the example of the complex Lü system and the numerical simulation verifies the effectiveness and feasibility of the proposed control method.

## 2. Preliminary Knowledge

#### 2.1. Control Method of the Nominal System

**Definition**

**1.**

**Lemma**

**1**

**.**Consider system (3), where $W={({W}_{ij})}_{n\times r}$ and ${W}_{ij}=1$ or ${W}_{ij}=0$, $i=1,2,\dots ,n$, $j=1,2,\dots ,n$, where $(M(t),W)$ is controllable, then the dynamic gain feedback controller is designed as follows:

**Definition**

**2.**

**Lemma**

**2**

**.**For sum system (10), $-K\left(N\right)$ satisfies the matrix $\left(A\right(N)+BK(N\left)\right)$ which is Hurwitz no matter what $N$ is, and then the master and slave subsystems are partially anti-synchronized under the action of the controller.

#### 2.2. UDE-Based Control Method

**Lemma**

**3**

**.**Consider system (6) and reference (7), if the designed controller meets the following conditions:

**Remark**

**1.**

## 3. Problem Formation

## 4. Main Results and Discussion

#### 4.1. Dynamic Gain Feedback Control for Synchronization

**Theorem**

**1.**

**Proof.**

#### 4.2. UDE-Based Dynamic Gain Feedback Control Method for Synchronization

**Theorem**

**2.**

**Proof.**

#### 4.3. Partial Anti-Synchronization of the Nominal System

**Theorem**

**3.**

**Proof.**

**Theorem**

**4.**

**Proof.**

#### 4.4. UDE-Based Linear-like Feedback Control Method for Partial Anti-Synchronization

**Theorem**

**5.**

**Proof.**

## 5. Numerical Simulations

#### 5.1. Synchronous Numerical Simulation of the Complex Lü System

#### 5.2. A UDE-Based Dynamic Feedback Control Synchronous Numerical Simulation

#### 5.3. An Anti-Synchronous Numerical Simulation of the Nominal System

#### 5.4. UDE-Based Linear-like Feedback Control Anti-Synchronous Numerical Simulation

#### 5.5. Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**The error system is asymptotically stable. (

**a**) ${M}_{1},{M}_{2},{M}_{3}$ are asymptotically stable; (

**b**) ${M}_{4},{M}_{5},{M}_{6}$ are asymptotically stable.

**Figure 2.**The state of the master and slave systems are synchronized, respectively. (

**a**) The states ${H}_{1},{H}_{2},{H}_{3}$ synchronize the states ${Q}_{1},{Q}_{2},{Q}_{3}$, respectively. (

**b**) The states ${H}_{4},{H}_{5},{H}_{6}$ synchronize the states ${Q}_{4},{Q}_{5},{Q}_{6}$, respectively.

**Figure 4.**The error system is asymptotically stable. (

**a**) ${M}_{1},{M}_{2},{M}_{3}$ are asymptotically stable; (

**b**) ${M}_{4},{M}_{5},{M}_{6}$ are asymptotically stable.

**Figure 5.**The state of the master and slave systems are synchronized, respectively. (

**a**) The states ${H}_{1},{H}_{2},{H}_{3}$ synchronize the states ${Q}_{1},{Q}_{2},{Q}_{3}$; (

**b**) the states ${H}_{4},{H}_{5},{H}_{6}$ synchronize the states ${Q}_{4},{Q}_{5},{Q}_{6}$.

**Figure 6.**(

**a**) $\widehat{{U}_{d1}}$ tends to ${U}_{d1}$; (

**b**) $\widehat{{U}_{d2}}$ tends to ${U}_{d2}$.

**Figure 8.**The sum system is asymptotically stable. (

**a**) ${E}_{1},{E}_{2}$ are asymptotically stable; (

**b**) ${E}_{3},{E}_{4}$ are asymptotically stable.

**Figure 9.**The states of the master and slave systems are anti-synchronized, respectively. (

**a**) The states ${Y}_{1},{Y}_{2}$ anti-synchronize states ${P}_{1},{P}_{2}$, respectively; (

**b**) the states ${Y}_{3},{Y}_{4}$ anti-synchronize states ${P}_{3},{P}_{4}$, respectively.

**Figure 10.**The sum system is asymptotically stable. (

**a**) ${E}_{1},{E}_{2}$ are asymptotically stable; (

**b**) ${E}_{3},{E}_{4}$ are asymptotically stable.

**Figure 11.**The states of the master and slave systems are anti-synchronized, respectively. (

**a**) ${Y}_{1},{Y}_{2}$ anti-synchronizes ${P}_{1},{P}_{1}$, respectively; (

**b**) ${Y}_{3},{Y}_{4}$ anti-synchronizes ${P}_{3},{P}_{4}$, respectively.

**Figure 12.**(

**a**) $\widehat{{U}_{d1}}$ tends to ${U}_{d1}$; (

**b**) $\widehat{{U}_{d2}}$ tends to ${U}_{d2}$.

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**MDPI and ACS Style**

Wang, Z.; Song, C.; Yan, A.; Wang, G.
Complete Synchronization and Partial Anti-Synchronization of Complex Lü Chaotic Systems by the UDE-Based Control Method. *Symmetry* **2022**, *14*, 517.
https://doi.org/10.3390/sym14030517

**AMA Style**

Wang Z, Song C, Yan A, Wang G.
Complete Synchronization and Partial Anti-Synchronization of Complex Lü Chaotic Systems by the UDE-Based Control Method. *Symmetry*. 2022; 14(3):517.
https://doi.org/10.3390/sym14030517

**Chicago/Turabian Style**

Wang, Zuoxun, Cong Song, An Yan, and Guijuan Wang.
2022. "Complete Synchronization and Partial Anti-Synchronization of Complex Lü Chaotic Systems by the UDE-Based Control Method" *Symmetry* 14, no. 3: 517.
https://doi.org/10.3390/sym14030517