Adaptive RBF Neural Network Tracking Control of Stochastic Nonlinear Systems with Actuators and State Constraints
Abstract
:1. Introduction
- Compared with the existing stochastic nonlinear systems with fixed numerical constraints on the system output, the control strategy designed in this paper ensures that the system output satisfies the time-varying function constraints, which is more in line with the state requirements of the actual physical system.
- By using the properties of the hyperbolic tangent function, this paper ensures that the intermediate virtual controllers required to realize the control task also meet the corresponding constraints. It is mathematically ensured that any state of the controlled system satisfies the constraint requirements at any moment.
- With the control method used in this paper, the control inputs and intermediate states are consistent with the constraints, which meets the realistic requirements of the actual physical control process, and the output tracking error can be quickly converged to within a bounded and adjustable tight set in fixed time.
2. Preliminaries
2.1. Stochastic Theory
- 1.
- The system is semi-globally finite-time stable in probability.
- 2.
- Mathematical expectation of the settling time function is bounded and the upper bound is a positive constant , which is independent of the initial state of system (1). That is to say, .
2.2. Related Lemmas
2.3. Practical Virtual Controller
2.4. System Description
3. Adaptive NN Control Design and Stability Analysis
4. Simulation
5. Conclusions and Future Work
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
Appendix A
- If , which means is bounded and .
- If , It is not difficult to derive that . As a result, is bounded.
References
- Imran, I.H.; Stolkin, R.; Montazeri, A. Adaptive Control of Quadrotor Unmanned Aerial Vehicle with Time-Varying Uncertainties. IEEE Access 2023, 11, 19710–19724. [Google Scholar] [CrossRef]
- Li, Z.; Chen, X.; Xie, M.; Zhao, Z. Adaptive fault-tolerant tracking control of flying-wing unmanned aerial vehicle with system input saturation and state constraints. Trans. Inst. Meas. Control. 2022, 44, 880–891. [Google Scholar] [CrossRef]
- Zhang, X.; Zhuang, Y.; Zhang, X.; Fang, Y. A Novel Asymptotic Robust Tracking Control Strategy for Rotorcraft UAVs. IEEE Trans. Autom. Sci. Eng. 2023, 20, 2338–2349. [Google Scholar] [CrossRef]
- Ba, D.; Li, Y.X.; Tong, S. Fixed-time adaptive neural tracking control for a class of uncertain nonstrict nonlinear systems. Neurocomputing 2019, 363, 273–280. [Google Scholar] [CrossRef]
- Sun, H.; Tu, L.; Yang, L.; Zhu, Z.; Zhen, S.; Chen, Y.H. Adaptive Robust Control for Nonlinear Mechanical Systems with Inequality Constraints and Uncertainties. IEEE Trans. Syst. Man Cybern. Syst. 2023, 53, 1761–1772. [Google Scholar] [CrossRef]
- Li, Y.; Zhu, Q.; Zhang, J. Distributed adaptive fixed-time neural networks control for nonaffine nonlinear multiagent systems. Sci. Rep. 2022, 12, 8459. [Google Scholar] [CrossRef]
- Zhao, K.; Chen, L.; Meng, W.; Zhao, L. Unified Mapping Function-Based Neuroadaptive Control of Constrained Uncertain Robotic Systems. IEEE Trans. Cybern. 2023, 53, 3665–3674. [Google Scholar] [CrossRef]
- Hu, Y.; Yan, H.; Zhang, H.; Wang, M.; Zeng, L. Robust Adaptive Fixed-Time Sliding-Mode Control for Uncertain Robotic Systems with Input Saturation. IEEE Trans. Cybern. 2023, 53, 2636–2646. [Google Scholar] [CrossRef] [PubMed]
- Jin, X. Adaptive fault tolerant tracking control for a class of stochastic nonlinear systems with output constraint and actuator faults. Syst. Control. Lett. 2017, 107, 100–109. [Google Scholar] [CrossRef]
- Wu, J.; He, F.; He, X.; Li, J. Dynamic Event-Triggered Fuzzy Adaptive Control for Non-strict-Feedback Stochastic Nonlinear Systems with Injection and Deception Attacks. Int. J. Fuzzy Syst. 2023, 25, 1144–1155. [Google Scholar] [CrossRef]
- Kanellakopoulos, I.; Kokotovic, P.V.; Morse, A.S. Systematic Design of Adaptive Controllers for Feedback Linearizable Systems. In Proceedings of the 1991 American Control Conference, Boston, MA, USA, 26–28 June 1991; pp. 649–654. [Google Scholar] [CrossRef]
- Qian, Y.C.; Miao, Z.H.; Zhou, J.; Zhu, X.J. Leader-follower consensus of nonlinear agricultural multiagents using distributed adaptive protocols. Adv. Manuf. 2023. [Google Scholar] [CrossRef]
- Wang, F.; Chen, B.; Sun, Y.; Gao, Y.; Lin, C. Finite-Time Fuzzy Control of Stochastic Nonlinear Systems. IEEE Trans. Cybern. 2020, 50, 2617–2626. [Google Scholar] [CrossRef]
- Wang, F.; Chen, B.; Sun, Y.; Lin, C. Finite time control of switched stochastic nonlinear systems. Fuzzy Sets Syst. 2019, 365, 140–152, Theme: Control Engineering. [Google Scholar] [CrossRef]
- He, W.; Mu, X.; Zhang, L.; Zou, Y. Modeling and trajectory tracking control for flapping-wing micro aerial vehicles. IEEE/CAA J. Autom. Sin. 2021, 8, 148–156. [Google Scholar] [CrossRef]
- Lv, Y.; Fu, J.; Wen, G.; Huang, T.; Yu, X. Distributed Adaptive Observer-Based Control for Output Consensus of Heterogeneous MASs with Input Saturation Constraint. IEEE Trans. Circuits Syst. I Regul. Pap. 2020, 67, 995–1007. [Google Scholar] [CrossRef]
- Liu, Z.; Han, Z.; Zhao, Z.; He, W. Modeling and adaptive control for a spatial flexible spacecraft with unknown actuator failures. Sci. China Inf. Sci. 2021, 64, 152208. [Google Scholar] [CrossRef]
- Li, Y.; Liu, L.; Feng, G. Robust adaptive output feedback control to a class of non-triangular stochastic nonlinear systems. Automatica 2018, 89, 325–332. [Google Scholar] [CrossRef]
- Wang, F.; You, Z.; Liu, Z.; Chen, C.L.P. A Fast Finite-Time Neural Network Control of Stochastic Nonlinear Systems. IEEE Trans. Neural Netw. Learn. Syst. 2023, 34, 7443–7452. [Google Scholar] [CrossRef]
- Li, Z.; Wang, F.; Wang, J. Adaptive Finite-Time Neural Control for a Class of Stochastic Nonlinear Systems with Known Hysteresis. IEEE Access 2020, 8, 123639–123648. [Google Scholar] [CrossRef]
- Lu, C.; Pan, Y.; Liu, Y.; Li, H. Adaptive fuzzy finite-time fault-tolerant control of nonlinear systems with state constraints and input quantization. Int. J. Adapt. Control. Signal Process. 2020, 34, 1199–1219. [Google Scholar] [CrossRef]
- Liang, Y.; Li, Y.X.; Hou, Z. Adaptive fixed-time tracking control for stochastic pure-feedback nonlinear systems. Int. J. Adapt. Control. Signal Process. 2021, 35, 1712–1731. [Google Scholar] [CrossRef]
- Jin, X. Adaptive Fixed-Time Control for MIMO Nonlinear Systems with Asymmetric Output Constraints Using Universal Barrier Functions. IEEE Trans. Autom. Control. 2019, 64, 3046–3053. [Google Scholar] [CrossRef]
- Pan, Y.; Du, P.; Xue, H.; Lam, H.K. Singularity-Free Fixed-Time Fuzzy Control for Robotic Systems with User-Defined Performance. IEEE Trans. Fuzzy Syst. 2021, 29, 2388–2398. [Google Scholar] [CrossRef]
- Zhang, J.X.; Yang, G.H. Fault-Tolerant Fixed-Time Trajectory Tracking Control of Autonomous Surface Vessels with Specified Accuracy. IEEE Trans. Ind. Electron. 2020, 67, 4889–4899. [Google Scholar] [CrossRef]
- Jin, X.; Li, Y.X. Adaptive fuzzy control of uncertain stochastic nonlinear systems with full state constraints. Inf. Sci. 2021, 574, 625–639. [Google Scholar] [CrossRef]
- Xu, B.; Li, Y.X.; Tong, S. Neural learning fixed-time adaptive tracking control of complex stochastic constraint nonlinear systems. J. Frankl. Inst. 2023, 360, 13671–13691. [Google Scholar] [CrossRef]
- Min, H.; Xu, S.; Zhang, Z. Adaptive Finite-Time Stabilization of Stochastic Nonlinear Systems Subject to Full-State Constraints and Input Saturation. IEEE Trans. Autom. Control. 2021, 66, 1306–1313. [Google Scholar] [CrossRef]
- Yin, J.; Khoo, S.; Man, Z.; Yu, X. Finite-time stability and instability of stochastic nonlinear systems. Automatica 2011, 47, 2671–2677. [Google Scholar] [CrossRef]
- Yu, J.; Yu, S.; Li, J.; Yan, Y. Fixed-time stability theorem of stochastic nonlinear systems. Int. J. Control. 2019, 92, 2194–2200. [Google Scholar] [CrossRef]
- Yu, J.; Cheng, S.; Shi, P.; Lin, C. Command-Filtered Neuroadaptive Output-Feedback Control for Stochastic Nonlinear Systems with Input Constraint. IEEE Trans. Cybern. 2023, 53, 2301–2310. [Google Scholar] [CrossRef]
- Yuan, X.; Yang, B.; Pan, X.; Zhao, X. Fuzzy Control of Nonlinear Strict-Feedback Systems with Full-State Constraints: A New Barrier Function Approach. IEEE Trans. Fuzzy Syst. 2022, 30, 5419–5430. [Google Scholar] [CrossRef]
- Meng, Q.; Ma, Q.; Shi, Y. Adaptive Fixed-Time Stabilization for a Class of Uncertain Nonlinear Systems. IEEE Trans. Autom. Control. 2023, 68, 6929–6936. [Google Scholar] [CrossRef]
- Chen, M.; Wang, H.; Liu, X. Adaptive Fuzzy Practical Fixed-Time Tracking Control of Nonlinear Systems. IEEE Trans. Fuzzy Syst. 2021, 29, 664–673. [Google Scholar] [CrossRef]
- Song, X.; Sun, P.; Song, S.; Stojanovic, V. Event-driven NN adaptive fixed-time control for nonlinear systems with guaranteed performance. J. Frankl. Inst. 2022, 359, 4138–4159. [Google Scholar] [CrossRef]
- Li, Y.X. Command Filter Adaptive Asymptotic Tracking of Uncertain Nonlinear Systems with Time-Varying Parameters and Disturbances. IEEE Trans. Autom. Control. 2022, 67, 2973–2980. [Google Scholar] [CrossRef]
- Wang, H.; Kang, S.; Zhao, X.; Xu, N.; Li, T. Command Filter-Based Adaptive Neural Control Design for Nonstrict-Feedback Nonlinear Systems with Multiple Actuator Constraints. IEEE Trans. Cybern. 2022, 52, 12561–12570. [Google Scholar] [CrossRef]
Symbol | Definition | Symbol | Definition |
---|---|---|---|
Absolute value | Sample space | ||
Compact set around zero | -algebra | ||
P | The probability measure | The transpose of matrix x | |
The mathematical expectation of ∗ | Set of non-negative real numbers | ||
The Euclidean or Frobenius norm | Real n-dimension space | ||
The set of i-times continuous differentiable functions |
System Parameter |
---|
Constraint functions for system outputs and constraints for state |
; ; |
The constraint function or constraint value corresponding to the tracking error |
; |
Initial values and design parameters |
; ; ; ; ; ; |
Parameter | Description | Value |
---|---|---|
J | Torsion coefficient | 0.5 kg·m2 |
M | Mess of the link | 1 kg |
g | Acceleration of gravity | 9.8 m/s2 |
Length of the connecting rod | 1 m | |
B | Coefficient of viscous friction | 0.5 N·m·s |
Noise coefficient | 0.1 N·m·s |
System Parameter |
---|
Constraint functions for system outputs and constraints for state |
; ; |
The constraint function or constraint value corresponding to the tracking error |
; |
Initial values and design parameters |
; ; , ; ; ; |
Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |
© 2024 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).
Share and Cite
Zhang, J.; Li, Y. Adaptive RBF Neural Network Tracking Control of Stochastic Nonlinear Systems with Actuators and State Constraints. Mathematics 2024, 12, 1378. https://doi.org/10.3390/math12091378
Zhang J, Li Y. Adaptive RBF Neural Network Tracking Control of Stochastic Nonlinear Systems with Actuators and State Constraints. Mathematics. 2024; 12(9):1378. https://doi.org/10.3390/math12091378
Chicago/Turabian StyleZhang, Jianhua, and Yinguang Li. 2024. "Adaptive RBF Neural Network Tracking Control of Stochastic Nonlinear Systems with Actuators and State Constraints" Mathematics 12, no. 9: 1378. https://doi.org/10.3390/math12091378
APA StyleZhang, J., & Li, Y. (2024). Adaptive RBF Neural Network Tracking Control of Stochastic Nonlinear Systems with Actuators and State Constraints. Mathematics, 12(9), 1378. https://doi.org/10.3390/math12091378