Dynamic switching event-triggered fixed-time cooperative control for nonlinear multi-agent systems subject to non-affine faults

Abstract

In this article, the cooperative control for nonlinear multi-agent systems (NMSs) subject to non-affine faults is investigated under directed graphs. A new distributed fixed-time event-triggered control protocol is proposed based on the leader–follower communication architecture, which saves network resources effectively and avoids Zeno behavior. A novel dynamic switching event-triggered mechanism (DSEM) is presented for each follower to ensure the smooth switching of two dynamic threshold strategies, and a dynamic internal trigger variable is designed based on the control input signals and errors, which can dynamically adjust the update frequency of the controller to reduce the number of system triggers. A new DSEM fixed-time distributed controller is designed based on local information to guarantee that all signals of the closed-loop system are fixed-time bounded. It is also proved that the consensus tracking errors of NMSs can converge to an explicitly given bound independent of the initial state in a fixed time. Finally, the validity of the presented control strategy is illustrated by two simulation examples.

Appendix

There exists a constant $0<\aleph _a<1$ such that ${\dot{L}}(s)\le -k_aL(s)-k_bL^{\alpha }(s)-k_cL^{\beta }(s)+k_d$ can be rewritten by

$$\begin{aligned} {\dot{L}}(s)\le & {} -\aleph _ak_aL(s)-(1-\aleph _a)k_aL(s)-k_bL^{\alpha }(s)\nonumber \\{} & {} -k_cL^{\beta }(s)+k_d \end{aligned}$$

(39)

$$\begin{aligned} {\dot{L}}(s)\le & {} -k_aL(s)-\aleph _ak_bL^{\alpha }(s)-(1-\aleph _a)k_bL^{\alpha }(s)\nonumber \\{} & {} -k_cL^{\beta }(s)+k_d \end{aligned}$$

(40)

or

$$\begin{aligned} {\dot{L}}(s)\le & {} -k_aL(s)-k_bL^{\alpha }(s)-\aleph _ak_cL^{\beta }(s)\nonumber \\{} & {} -(1-\aleph _a)k_cL^{\beta }(s)+k_d. \end{aligned}$$

(41)

From (39), if $L(s)>\frac{k_d}{(1-\aleph _a)k_a}$, ${\dot{L}}(s)$ satisfies ${\dot{L}}(s)\le -\aleph _ak_aL(s)-k_bL^{\alpha }(s)-k_cL^{\beta }(s)$. According to Lemma 1, s will be driven to the region

$$\begin{aligned} s\in \bigg \{s\Big |L(s)\le \frac{k_d}{(1-\aleph _a)k_a}\bigg \} \end{aligned}$$

(42)

in a fixed time $T_a\le \frac{\ln (1+\frac{\aleph _ak_a}{k_b})}{\aleph _ak_a(1-\alpha )} +\frac{\ln (1+\frac{\aleph _ak_a}{k_c})}{\aleph _ak_a(\beta -1)}$.

From (40), if $L^{\alpha }(s)>\frac{k_d}{(1-\aleph _a)k_b}$, ${\dot{L}}(s)$ satisfies ${\dot{L}}(s)\le -k_aL(s)-\aleph _ak_bL^{\alpha }(s)-k_cL^{\beta }(s)$. Similarly, s will be driven to the region

$$\begin{aligned} s\in \bigg \{s\Big |L^{\alpha }(s)\le \frac{k_d}{(1-\aleph _a)k_b}\bigg \} \end{aligned}$$

(43)

in a fixed time $T_a\le \frac{\ln (1+\frac{k_a}{\aleph _ak_b})}{k_a(1-\alpha )} +\frac{\ln (1+\frac{k_a}{k_c})}{k_a(\beta -1)}$.

From (41), if $L^{\beta }(s)>\frac{k_d}{(1-\aleph _a)k_c}$, ${\dot{L}}(s)$ satisfies ${\dot{L}}(s)\le -k_aL(s)-k_bL^{\alpha }(s)-\aleph _ak_cL^{\beta }(s)$. Similarly, s will be driven to the region

$$\begin{aligned} s\in \bigg \{s\Big |L^{\beta }(s)\le \frac{k_d}{(1-\aleph _a)k_c}\bigg \} \end{aligned}$$

(44)

in a fixed time $T_a\le \frac{\ln (1+\frac{k_a}{k_b})}{k_a(1-\alpha )} +\frac{\ln (1+\frac{k_a}{\aleph _ak_c})}{k_a(\beta -1)}$. $\ln (\cdot )$ satisfies $\ln (1+\breve{\lambda })<\breve{\lambda }$ when $\breve{\lambda }>0$. According to the property of $\ln (\cdot )$, $\frac{\ln (1+\frac{k_a}{\aleph _ak_b})}{k_a(1-\alpha )}>\frac{\ln (1+\frac{k_a}{k_b})}{k_a(1-\alpha )}$ and $\frac{\ln (1+\frac{k_a}{\aleph _ak_c})}{k_a(\beta -1)}>\frac{\ln (1+\frac{k_a}{k_c})}{k_a(\beta -1)}$.

Define $\daleth (\aleph _a)=\frac{\ln (1+\frac{\aleph _ak_a}{k_b})}{\aleph _ak_a(1-\alpha )}-\frac{\ln (1+\frac{k_a}{k_b})}{k_a(1-\alpha )} =\frac{\ln (1+\frac{\aleph _ak_a}{k_b})}{\aleph _ak_a(1-\alpha )}-\frac{\aleph _a\ln (1+\frac{k_a}{k_b})}{\aleph _ak_a(1-\alpha )}$ and $\daleth _a(\aleph _a)=\ln (1+\frac{\aleph _ak_a}{k_b})-\aleph _a\ln (1+\frac{k_a}{k_b})$. $\dot{\daleth }_a(\aleph _a)=\frac{k_a/k_b}{1+\aleph _ak_a/k_b}-\ln (1+k_a/k_b)$ and $\ddot{\daleth }_a(\aleph _a)=-\frac{(k_a/k_b)^2}{(1+\aleph _ak_a/k_b)^2}<0$, where $k_a/k_b>0$ and $0<\aleph _a<1$. ${\lim _{\aleph _a \rightarrow 0^{+}}}\dot{\daleth }_a(\aleph _a)>0$ and ${\lim _{\aleph _a \rightarrow 1}}\dot{\daleth }_a(\aleph _a)<0$. When $0<\aleph _a<1$, $\daleth (\aleph _a)$ increases monotonically and then decreases monotonically. Since $\daleth (0)=0$ and $\daleth (1)=0$, $\daleth (\aleph _a)>0$ for $0<\aleph _a<1$. Therefore, $\frac{\ln (1+\frac{\aleph _ak_a}{k_b})}{\aleph _ak_a(1-\alpha )} >\frac{\ln (1+\frac{k_a}{k_b})}{k_a(1-\alpha )}$. Similarly, $\frac{\ln (1+\frac{\aleph _ak_a}{k_c})}{\aleph _ak_a(\beta -1)} >\frac{\ln (1+\frac{k_a}{k_c})}{k_a(\beta -1)}$.

When $0<\aleph _a<1$, the relationship between $\frac{\ln (1+\frac{\aleph _ak_a}{k_b})}{\aleph _ak_a(1-\alpha )}$ and $\frac{\ln (1+\frac{k_a}{\aleph _ak_b})}{k_a(1-\alpha )}$ is impossible to judge. Similarly, $\frac{\ln (1+\frac{\aleph _ak_a}{k_c})}{\aleph _ak_a(\beta -1)}$ and $\frac{\ln (1+\frac{k_a}{\aleph _ak_c})}{k_a(\beta -1)}$ cannot be determined. Corollary 1 can be confirmed. $\blacksquare$