Switching Fuzzy Adaptive Finite-Time Control of Nonlinear Systems via Event-Triggered Output Feedback Strategy

Abstract

This paper presents a study on the finite-time tracking control problem for a class of nonstrict-feedback nonlinear systems with immeasurable states and unknown nonlinearities. To model the unknown functions, fuzzy logic system technique is employed and a state observer is proposed. Using this observer, an event-triggered finite-time output feedback control scheme is designed based on the backstep** method. The proposed controller guarantees finite-time convergence of the tracking error and boundedness of all states in the closed-loop system. Additionally, it reduces the computing burden and communication resources. A $C^1$ switching function is introduced to overcome the singularity problem, and the Zeno behavior caused by the event-triggered strategy is avoided. Simulation results are presented to validate the effectiveness of the proposed controller.

Appendix

Proof of Lemma 5

$$\begin{aligned}{} & {} -\sum _{i=2}^{n}z_{i}\frac{\partial \alpha _{i-1}}{\partial \hat{\Theta }} \dot{\hat{\Theta }} \nonumber \\\,=\, & {} -\sum _{i=2}^{n}z_{i}\frac{\partial \alpha _{i-1}}{\partial \hat{\Theta }} \left( \sum _{i=1}^{n}\frac{h_{1}z_{i}^{2}}{2\eta _{i}^{2}\phi _{i}^{T}\phi _{i}}-\sigma _{11}\hat{\Theta }-\sigma _{12}\hat{\Theta }^{r}\right) \nonumber \\\,=\, & {} \sum _{i=2}^{n}z_{i}\frac{\partial \alpha _{i-1}}{\partial \hat{\Theta }} \left( \sigma _{11}\hat{\Theta }+\sigma _{12}\hat{\Theta }^{r}\right) -\sum _{i=2}^{n}z_{i}\frac{\partial \alpha _{i-1}}{\partial \hat{\Theta }} \left( \sum _{i=1}^{n}\frac{h_{1}z_{i}^{2}}{2\eta _{i}^{2}\phi _{i}^{T}\phi _{i}}\right) \nonumber \\\,=\, & {} \sum _{i=2}^{n}z_{i}\frac{\partial \alpha _{i-1}}{\partial \hat{\Theta }} \left( \sigma _{11}\hat{\Theta }+\sigma _{12}\hat{\Theta }^{r}\right) \nonumber \\{} & {} -\sum _{i=2}^{n}z_{i}\frac{\partial \alpha _{i-1}}{\partial \hat{\Theta }} \left( \sum _{j=1}^{i-1}\frac{h_{1}z_{j}^{2}}{2\eta _{j}^{2}\phi _{j}^{T}\phi _{j}}+\sum _{j=i}^{n}\frac{h_{1}z_{j}^{2}}{2\eta _{j}^{2}\phi _{j}^{T}\phi _{j}}\right) \nonumber \\\,=\, & {} \sum _{i=2}^{n}z_{i}\frac{\partial \alpha _{i-1}}{\partial \hat{\Theta }} \left( \sigma _{11}\hat{\Theta }+\sigma _{12}\hat{\Theta }^{r}\right) -\sum _{i=2}^{n}z_{i}\frac{\partial \alpha _{i-1}}{\partial \hat{\Theta }} \left( \sum _{j=1}^{i-1}\frac{h_{1}z_{j}^{2}}{2\eta _{j}^{2}\phi _{j}^{T}\phi _{j}}\right) \nonumber \\{} & {} -\sum _{i=2}^{n}z_{i}\frac{\partial \alpha _{i-1}}{\partial \hat{\Theta }} \left( \sum _{j=i}^{n}\frac{h_{1}z_{j}^{2}}{2\eta _{j}^{2}\phi _{j}^{T}\phi _{j}}\right) \end{aligned}$$

(105)

$\square$

With the help of the sequence rearrangement, we have

$$\begin{aligned}{} & {} -\sum _{i=2}^{n}z_{i}\frac{\partial \alpha _{i-1}}{\partial \hat{\Theta }} \left( \sum _{j=i}^{n}\frac{h_{1}z_{j}^{2}}{2\eta _{j}^{2}\phi _{j}^{T}\phi _{j}}\right) \nonumber \\\le & {} \left| z_{2}\frac{\partial \alpha _{1}}{\partial \hat{\Theta }} \right| \frac{h_{1}z_{2}^{2}}{2\eta _{2}^{2}\phi _{2}^{T}\phi _{2}} +\left| z_{2}\frac{\partial \alpha _{1}}{\partial \hat{\Theta }} \right| \frac{h_{1}z_{3}^{2}}{2\eta _{3}^{2}\phi _{3}^{T}\phi _{3}} \nonumber \\{} & {} +...+\left| z_{2}\frac{\partial \alpha _{1}}{\partial \hat{\Theta }} \right| \frac{h_{1}z_{n}^{2}}{2\eta _{n}^{2}\phi _{n}^{T}\phi _{n}} \nonumber \\{} & {} +\left| z_{3}\frac{\partial \alpha _{2}}{\partial \hat{\Theta }} \right| \frac{h_{1}z_{3}^{2}}{2\eta _{3}^{2}\phi _{2}^{T}\phi _{2}} +\left| z_{3}\frac{\partial \alpha _{2}}{\partial \hat{\Theta }} \right| \frac{h_{1}z_{4}^{2}}{2\eta _{4}^{2}} \nonumber \\{} & {} +...+\left| z_{3}\frac{\partial \alpha _{2}}{\partial \hat{\Theta }} \right| \frac{h_{1}z_{n}^{2}}{2\eta _{n}^{2}\phi _{n}^{T}\phi _{n}} \nonumber \\{} & {} +...+\left| z_{n-1}\frac{\partial \alpha _{n-1}}{\partial \hat{\Theta }} \right| \frac{h_{1}z_{n-1}^{2}}{2\eta _{n-1}^{2}\phi _{n-1}^{T}\phi _{n-1}} \nonumber \\{} & {} +\left| z_{n-1}\frac{\partial \alpha _{n-1}}{\partial \hat{\Theta }} \right| \frac{h_{1}z_{n}^{2}}{2\eta _{n}^{2}\phi _{n}^{T}\phi _{n}} +\left| z_{n}\frac{\partial \alpha _{n}}{\partial \hat{\Theta }} \right| \frac{h_{1}z_{n}^{2}}{2\eta _{n}^{2}\phi _{n}^{T}\phi _{n}}\nonumber \\{} & {} =\sum _{i=2}^{n}\frac{h_{1}z_{i}^{2}}{2\eta _{i}^{2}\phi _{i}^{T}\phi _{i}} \left( \sum _{j=2}^{i}\left| z_{j}\frac{\partial \alpha _{j-1}}{\partial \hat{\Theta }}\right| \right) \end{aligned}$$

(106)

which indicates

$$\begin{aligned}{} & {} -\sum _{i=2}^{n}z_{i}\frac{\partial \alpha _{i-1}}{\partial \hat{\Theta }} \dot{\hat{\Theta }} \nonumber \\\le & {} \sum _{i=2}^{n}z_{i}\frac{\partial \alpha _{i-1}}{\partial \hat{\Theta } }\left( \sigma _{11}\hat{\Theta }+\sigma _{12}\hat{\Theta }^{r}\right) -\sum _{i=2}^{n}z_{i}\frac{\partial \alpha _{i-1}}{\partial \hat{\Theta }} \left( \sum _{j=1}^{i-1}\frac{h_{1}z_{j}^{2}}{2\eta _{j}^{2}\phi _{j}^{T}\phi _{j}}\right) \nonumber \\{} & {} +\sum _{i=2}^{n}\frac{h_{1}z_{i}^{2}}{2\eta _{i}^{2}\phi _{i}^{T}\phi _{i}} \left( \sum _{j=2}^{i}\left| z_{j}\frac{\partial \alpha _{j-1}}{\partial \hat{\Theta }}\right| \right) \nonumber \\\,=\, & {} \sum _{i=2}^{n}z_{i}\Bigg (\frac{\partial \alpha _{i-1}}{\partial \hat{\Theta }} \left( \sigma _{11}\hat{\Theta }+\sigma _{12}\hat{\Theta }^{r}\right) -\frac{\partial \alpha _{i-1}}{\partial \hat{\Theta }}\left( \sum _{j=1}^{i-1}\frac{h_{1}z_{j}^{2}}{2\eta _{j}^{2}\phi _{j}^{T}\phi _{j}} \right) \nonumber \\{} & {} +\frac{h_{1}z_{i}}{2\eta _{i}^{2}\phi _{i}^{T}\phi _{i}}\left( \sum _{j=2}^{i}\left| z_{j}\frac{\partial \alpha _{j-1}}{\partial \hat{\Theta }}\right| \right) \Bigg ) \nonumber \\\,=\, & {} -\sum _{i=2}^{n}z_{i}\sigma _{i} \end{aligned}$$

(107)

Therefore, it can be concluded that $\sum _{i=2}^{n}z_{i}\left( \sigma _{i}- \frac{\partial \alpha _{i-1}}{\partial \hat{\Theta }}\dot{\hat{\Theta }}\right)$ is non-positive.