# Adaptive Fixed-Time Neural Network Tracking Control of Nonlinear Interconnected Systems

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## Abstract

**:**

## 1. Introduction

- (1)
- The combination of fixed-time control and neural network adaptive control for nonlinear interconnected systems.
- (2)
- A fixed-time low pass filter is designed to solve the “explosion of complexity” based on backstepping control technology.
- (3)
- A fixed-time controller is designed, which contains the convergence time of the error system, weights of neural networks, and a low pass filter system.

## 2. Problem Formation and Preliminaries

**Remark**

**1:**

## 3. Adaptive Fixed-Time Tracking Control System Design

#### 3.1. Control System Design

**Remark**

**2:**

#### 3.2. Control System Analysis

**Theorem**

**1.**

**Proof.**

**Remark**

**3:**

**Remark**

**4:**

## 4. Numerical Examples

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Gao, W.; Jiang, Y.; Jiang, Z.P.; Chai, T. Output-Feedback Adaptive Optimal Control of Interconnected Systems Based on Robust Adaptive Dynamic Programming; Automatica Oxford: Oxford, UK, 2016. [Google Scholar]
- Gao, W.; Jiang, Z.P. Adaptive Dynamic Programming and Adaptive Optimal Output Regulation of Linear Systems. IEEE Trans. Autom. Control.
**2016**, 61, 4164–4169. [Google Scholar] [CrossRef] - Yan, X.G.; Edwards, C.; Spurgeon, S.K. Decentralised robust sliding mode control for a class of nonlinear interconnected systems by static output feedback. Automatica
**2004**, 40, 613–620. [Google Scholar] [CrossRef] - Zhang, J.H.; Li, Y.; Fei, W.B. Neural Network-Based Nonlinear Fixed-Time Adaptive Practical Tracking Control. for Quadrotor Unmanned Aerial Vehicles. Complexity
**2020**, 2020, 13. [Google Scholar] [CrossRef] - Tong, S.C.; Min, X.; Li, Y.X. Observer-Based Adaptive Fuzzy Tracking Control. for Strict-Feedback Nonlinear Systems with Unknown Control. Gain Functions. IEEE Trans. Cybern.
**2020**, 50, 3903–3913. [Google Scholar] [CrossRef] - Zhang, J.; Zhu, Q.; Li, Y.; Wu, X. Homeomorphism Mapping Based Neural Networks for Finite Time Constraint Control of a Class of Nonaffine Pure-Feedback Nonlinear Systems. Complexity
**2019**, 2019, 1–11. [Google Scholar] [CrossRef] - Zhang, J.; Li, Y.; Fei, W.; Wu, X. U-Model Based Adaptive Neural Networks Fixed-Time Backstepping Control for Uncertain Nonlinear System. Math. Probl. Eng.
**2020**, 2020, 1–7. [Google Scholar] [CrossRef] [Green Version] - Zhu, Q.M.; Zhao, D.; Zhang, J. A general U-block model-based design procedure for nonlinear polynomial control systems. Int. J. Syst. Sci.
**2016**, 47, 3465–3475. [Google Scholar] [CrossRef] - Li, R.; Zhu, Q.; Narayan, P.; Yue, A.; Yao, Y.; Deng, M. U-Model-Based Two-Degree-of-Freedom Internal Model Control of Nonlinear Dynamic Systems. Entropy
**2021**, 23, 169. [Google Scholar] [CrossRef] - Yu, X.; Man, Z. Fast terminal sliding-mode control design for nonlinear dynamical systems. IEEE Trans. Circuits Syst. I Fundam. Theory Appl.
**2009**, 49, 261–264. [Google Scholar] - Muñoz, D.; Sbarbaro, D. An adaptive sliding-mode controller for discrete nonlinear systems. IEEE Trans. Ind. Electron.
**2000**, 47, 574–581. [Google Scholar] [CrossRef] - Da, F. Decentralized sliding mode adaptive controller design based on fuzzy neural networks for interconnected uncertain nonlinear systems. IEEE Trans. Neural Netw.
**2000**, 11, 1471–1480. [Google Scholar] - Moreno, J.A.; Osorio, M. Strict Lyapunov Functions for the Super-Twisting Algorithm. IEEE Trans. Autom. Control
**2012**, 57, 1035–1040. [Google Scholar] [CrossRef] - Zhang, J.; Zhu, Q.; Li, Y. Convergence Time Calculation for Supertwisting Algorithm and Application for Nonaffine Nonlinear Systems. Complexity
**2019**, 2019, 1–15. [Google Scholar] [CrossRef] [Green Version] - Ge, S.S.; Cong, W. Adaptive neural control of uncertain MIMO nonlinear systems. IEEE Trans. Neural Netw.
**2004**, 15, 674–692. [Google Scholar] [CrossRef] - Zhang, J.; Zhu, Q.; Wu, X.; Li, Y. A generalized indirect adaptive neural networks backstepping control procedure for a class of non-affine nonlinear systems with pure-feedback prototype. Neurocomputing
**2013**, 121, 131–139. [Google Scholar] [CrossRef] - Pan, J.; Qu, L.; Peng, K. Sensor and Actuator Fault Diagnosis for Robot Joint Based on Deep CNN. Entropy
**2021**, 23, 751. [Google Scholar] [CrossRef] [PubMed] - He, W.; Huang, H.; Ge, S.S. Adaptive Neural Network Control of a Robotic Manipulator With Time-Varying Output Constraints. IEEE Trans. Cybern.
**2017**, 47, 3136–3147. [Google Scholar] [CrossRef] [PubMed] - Wu, Y.; Huang, R.; Li, X.; Liu, S. Adaptive neural network control of uncertain robotic manipulators with external disturbance and time-varying output constraints. Neurocomputing
**2019**, 323, 108–116. [Google Scholar] [CrossRef] - Ge, S.S.; Hang, C.C.; Lee, T.H.; Zhang, T. Stable Adaptive Neural Network Control; Springer: New York, NY, USA, 2002. [Google Scholar]
- Polyakov, A.; Fridman, L. Stability notions and Lyapunov functions for sliding mode control systems. J. Frankl. Inst.
**2014**, 351, 1831–1865. [Google Scholar] [CrossRef] [Green Version] - Li, G.; Ji, H. A three-dimensional robust nonlinear terminal guidance law with ISS finite-time convergence. Int. J. Control
**2016**, 89, 938–949. [Google Scholar] [CrossRef] - Du, P.; Liang, H.; Zhao, S.; Ahn, C.K. Neural-Based Decentralized Adaptive Finite-Time Control for Nonlinear Large-Scale Systems With Time-Varying Output Constraints. IEEE Trans. Syst. Man Cybern. Syst.
**2021**, 51, 3136–3147. [Google Scholar] [CrossRef] - Li, Y.; Zhang, J.; Ye, X.; Chin, C. Adaptive Fixed-Time Control of Strict-Feedback High-Order Nonlinear Systems. Entropy
**2021**, 23, 963. [Google Scholar] [CrossRef] [PubMed] - Wang, H.; Liu, W.; Qiu, J.; Liu, P.X. Adaptive Fuzzy Decentralized Control for a Class of Strong Interconnected Nonlinear Systems with Unmodeled Dynamics. IEEE Trans. Fuzzy Syst.
**2018**, 26, 836–846. [Google Scholar] [CrossRef] - Li, X.; Yang, G. Neural-Network-Based Adaptive Decentralized Fault-Tolerant Control for a Class of Interconnected Nonlinear Systems. IEEE Trans. Neural Netw. Learn. Syst.
**2018**, 29, 144–155. [Google Scholar] [CrossRef] [PubMed] - Si, W.; Dong, X.; Yang, F. Decentralized adaptive neural control for high-order interconnected stochastic nonlinear time-delay systems with unknown system dynamics. Neural Netw.
**2018**, 99, 123–133. [Google Scholar] [CrossRef]

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**MDPI and ACS Style**

Li, Y.; Zhang, J.; Xu, X.; Chin, C.S.
Adaptive Fixed-Time Neural Network Tracking Control of Nonlinear Interconnected Systems. *Entropy* **2021**, *23*, 1152.
https://doi.org/10.3390/e23091152

**AMA Style**

Li Y, Zhang J, Xu X, Chin CS.
Adaptive Fixed-Time Neural Network Tracking Control of Nonlinear Interconnected Systems. *Entropy*. 2021; 23(9):1152.
https://doi.org/10.3390/e23091152

**Chicago/Turabian Style**

Li, Yang, Jianhua Zhang, Xinli Xu, and Cheng Siong Chin.
2021. "Adaptive Fixed-Time Neural Network Tracking Control of Nonlinear Interconnected Systems" *Entropy* 23, no. 9: 1152.
https://doi.org/10.3390/e23091152