Fuzzy observer-based command filtered tracking control for uncertain strict-feedback nonlinear systems with sensor faults and event-triggered technology

Abstract

For the trajectory tracking problem of nth-order uncertain nonlinear systems with sensor faults, a fuzzy controller based on command filtered and event-triggered technology is designed to improve the tracking error of the system. Concurrently, a fault-tolerant control scheme is introduced to effectively solve the problem of sudden output sensor failure. Additionally, the proposed controller can also greatly avoid complexity explosion problem of derivations of virtual control laws, which makes the design of the controller simpler. Furthermore, an effective observer is designed to solve the problem of system state immeasurability. Therefore, the proposed control scheme makes the design of the controller more convenient and flexible. According to Lyapunov stability theory, it is proved that all closed-loop signals are uniformly and ultimately bounded. Finally, two simulation examples of second-order nonlinear system and single-link robot show the effectiveness of the proposed scheme.

Appendix A

Proof of Proposition 1:

The proof is carried out from 2 cases for ${\tilde{\iota }}\ge 0$ and ${\tilde{\iota }}<0.$

Case 1: For ${\tilde{\iota }}\ge 0,$ $\iota >0$ and ${\hat{\iota }}\ge 0,$ then we have

$$\begin{aligned} \left( {\tilde{\iota }}-\iota \right) ^{2}\left( 2{\tilde{\iota }}+\iota \right) >0. \end{aligned}$$

(97)

It follows from (97) that

$$\begin{aligned} 2{\tilde{\iota }}^{3}-3{\tilde{\iota }}^{2}\iota +\iota ^{3}>0. \end{aligned}$$

(98)

Then, from ${\tilde{\iota }}=\iota -{\hat{\iota }},$ the following results hold.

$$\begin{aligned} {\tilde{\iota }}^{2}\left( \iota -{\tilde{\iota }}\right) ={\tilde{\iota }}^{2}\hat{ \iota }<-\frac{1}{3}{\tilde{\iota }}^{3}+\frac{1}{3}\iota ^{3}. \end{aligned}$$

(99)

From ${\tilde{\iota }}\ge 0,$ it gives $sgn\left( {\tilde{\iota }}\right) =1$, therefore, we can obtain

$$\begin{aligned} {\tilde{\iota }}^{2}{\hat{\iota }}sgn\left( {\tilde{\iota }}\right) <-\frac{1}{3}| {\tilde{\iota }}|^{3}+\frac{1}{3}\iota ^{3}. \end{aligned}$$

(100)

Case 2: For ${\tilde{\iota }}<0,$ $\iota >0$ and ${\hat{\iota }}\ge 0,$ we start from the following inequality:

$$\begin{aligned} 2{\tilde{\iota }}^{3}-3{\tilde{\iota }}^{2}\iota -\iota ^{3}<0. \end{aligned}$$

(101)

Then, it gives that

$$\begin{aligned} -3{\tilde{\iota }}^{2}\iota +3{\tilde{\iota }}^{3}=-3{\tilde{\iota }}^{2}\left( \iota -{\tilde{\iota }}\right) <{\tilde{\iota }}^{3}+\iota ^{3}. \end{aligned}$$

(102)

From ${\tilde{\iota }}<0$ and ${\tilde{\iota }}=\iota -{\hat{\iota }}$, it produces that

$$\begin{aligned} -{\tilde{\iota }}^{2}{\hat{\iota }}<\frac{1}{3}{\tilde{\iota }}^{3}+\frac{1}{3}\iota ^{3}. \end{aligned}$$

(103)

For ${\tilde{\iota }}<0,$ it implies $sgn\left( {\tilde{\iota }}\right) =-1$; thus, we have

$$\begin{aligned} {\tilde{\iota }}^{2}{\hat{\iota }}sgn\left( {\tilde{\iota }}\right) <-\frac{1}{3}| {\tilde{\iota }}|^{3}+\frac{1}{3}\iota ^{3}. \end{aligned}$$

(104)

$\square$