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Observer-Based Adaptive Fuzzy Tracking Control for MIMO Nonlinear Systems with Asymmetric Time-Varying Constraints

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Abstract

This article studies the adaptive tracking control for a type of multi-input multi-output nonlinear systems, which still is an open issue coming from considerations of system uncertainties, unmeasurable states, and asymmetric time-varying full-state constraints. Fuzzy logic systems (FLSs) are trained online to estimate and compensate for unknown nonlinear system dynamics, effectively improving uncertainties rejection capabilities. To solve unmeasurable states, a state observer, including FLSs, is built and does provide a fairly good observation accuracy. Notably, an asymmetric time-varying barrier Lyapunov function is constructed to realize full-state constraint satisfactions. Based on the above theoretical findings, an adaptive fuzzy tracker is designed in a recursive manner. A Lyapunov analysis demonstrates the stability of the closed-loop system, and particularly, the constrained conditions will not be violated during the whole operation. Simulation results show the usefulness of the developed tracker.

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Acknowledgements

This work was supported by the Shanghai Commission of Science and Technology under Grant 23010500100.

Author information

Authors and Affiliations

  1. School of Mechatronic Engineering and Automation, Shanghai University, Shanghai, 200444, China

    Xiang Liu, Yang Wu, Yueying Wang & Hengyu Li

  2. East China Branch, State Grid Corporation of China, Shanghai, 200120, China

    Feng Zhu

  3. School of Information Science and Engineering, East China University of Science and Technology, Shanghai, 200237, China

    Huaicheng Yan

Authors
  1. Xiang Liu
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  2. Feng Zhu
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  4. Huaicheng Yan
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Corresponding author

Correspondence to Yang Wu.

Appendix

Appendix

To give the comparative result, the existing adaptive NN controller is briefly shown in this section. In addition, to ensure the strict fairness of the comparison result, according to the design ideal of this paper, the following controller can be obtained by amending the control algorithm proposed in [42]. Meanwhile, the initial values, the nationals and the designed parameters used in this section are exactly the same as the controller developed in this paper.

Similar to the designed controller in our results, the following adaptive NN controller with linear observer can be obtained as

$$\begin{aligned}{} & {} \begin{aligned} \alpha _{m,1}=&-\frac{1}{2a_{m,1}^2}z_{i,1}W_{m,1}\Phi _{m,1}^\text{T}(Z_{m,1})\Phi _{m,1}(Z_{m,1})\\ {}&-(c_{m,1}+0.5)z_{m,1}+{\dot{y}}_{m,d}, \end{aligned} \end{aligned}$$
(78)
$$\begin{aligned}{} & {} \begin{aligned} {\dot{\theta }}_{m,1}=-\gamma _{m,1}z_{m,1}\varphi _{m,1} (\hat{x}_{m,1})-\sigma _{m,1}\theta _{m,1}, \end{aligned} \end{aligned}$$
(79)
$$\begin{aligned}{} & {} \begin{aligned} {\dot{W}}_{m,1}=&-\rho _{m,1}W_{m,1}+\frac{\beta _{m,1}}{2a_{m,1}^2}z_{m,1}^2\\ {}&\Phi _{m,1}^\text{T}\times (Z_{m,1})\Phi _{m,1}(Z_{m,1}), \end{aligned} \end{aligned}$$
(80)
$$\begin{aligned}{} & {} \begin{aligned} u_m=&-\frac{1}{2a_{m,2}^2}z_{m,2}W_{m,2}\Phi _{m,2}^\text{T}(Z_{m,2})\Phi _{m,2}(Z_{m,2})\\ {}&-(c_{m,2}+0.5)z_{m,2}, \end{aligned} \end{aligned}$$
(81)
$$\begin{aligned}{} & {} \begin{aligned} {\dot{\theta }}_{m,2}=-\gamma _{m,2}z_{m,2}\varphi _{m,2} (\hat{x}_{m,2})-\sigma _{m,2}\theta _{m,2}, \end{aligned} \end{aligned}$$
(82)
$$\begin{aligned}{} & {} \begin{aligned} {\dot{W}}_{m,2}=&-\rho _{i,2}W_{m,2}+\frac{\beta _{m,2}}{2a_{m,2}^2}z_{m,2}^2\\ {}&\Phi _{m,2}^\text{T}(Z_{m,2})\Phi _{m,2}(Z_{m,2}). \end{aligned} \end{aligned}$$
(83)

Due to the fact that there is no consider the constraint problem in the controller design, the stability analysis of the closed-loop system in [42] can be revealed by selecting \(V=\sum _{i=1}^{m}\{ \frac{1}{2}z_{i,1}^2+\frac{1}{2\gamma _{i,1}}{\tilde{\theta }}_{i,1}^\text{T}{\tilde{\theta }}_{i,1} +\frac{1}{2\beta _{i,1}}{\tilde{W}}_{i,1}^\text{T}{\tilde{W}}_{i,1}\}\), which is the general quadratic Lyapunov function and it can be seen that the difference between this one and previous Lyapunov function in Theorem 1.

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Liu, X., Zhu, F., Wu, Y. et al. Observer-Based Adaptive Fuzzy Tracking Control for MIMO Nonlinear Systems with Asymmetric Time-Varying Constraints. Int. J. Fuzzy Syst. 26, 777–794 (2024). https://doi.org/10.1007/s40815-023-01634-7

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