Distributed adaptive neural control for uncertain multi-agent systems with unknown actuator failures and unknown dead zones

Abstract

In actuality, the dead zones and failures often occur in actuators, but the existing algorithms have difficulty simultaneously tolerating dead zones and actuator failures in multi-agent systems. In this paper, the directed topology, uncertain dynamics, unknown dead zones and actuator failures are simultaneously taken into account for the multi-agent systems. By introducing distributed backstepping technique, the radial basis function neural networks and a bound estimation approach, the distributed fault-tolerant tracking controllers and relative adaptive laws for each follower are proposed, which guarantee all followers reach the synchronization and obtain the ideal tracking performance. Comparing with the existing results, it is a new attempt for strict-feedback multi-agent system to take unknown dead zones and unknown actuator failures into consideration. Moreover, the basis function vectors in RBF NNs are no longer required for controllers to decrease computational burden significantly. In the end, the efficiency of our proposed algorithm is verified by comparison simulation results.

Appendix: Proof of Theorem 1

The following overall Lyapunov function candidate function V is employed to analyze the stability in the total closed-loop system:

$$\begin{aligned} V = \sum _{i=1}^{N}V_{i,n_{i}}. \end{aligned}$$

(60)

Then, (56) is rewritten as

$$\begin{aligned}&{\dot{V}} \le -\sum _{i=1}^{N}\sum _{m=1}^{n_{i}}c_{i,m}z_{i,m}^{2} \nonumber \\&\quad +\sum _{i=1}^{N}\varphi _{i,n_i} +\sum _{i=1}^{N}\left( {\eta _{i} +{\bar{\lambda }}_{i}\varrho }\right) \mu _{i} \nonumber \\&\quad + \sum _{i=1}^{N}\dfrac{k_{i,0}}{2r_i}{\theta }_i^2. \end{aligned}$$

(61)

If the following compact set holds, ${\dot{V}} < 0$,

$$\begin{aligned} \varOmega= & {} \left\{ \sum _{i=1}^{N}\sum _{m=1}^{n_{i}}c_{i,m}{z_{i,m}^{2}}\right. \nonumber \\> & {} \left. \sum _{i=1}^{N}\varphi _{i,n_{i}} +\sum _{i=1}^{N}\left( {\eta _{i} +{\bar{\lambda }}_{i}\varrho }\right) \mu _{i}\right. \nonumber \\&\left. + \sum _{i=1}^{N}\dfrac{k_{i,0}}{2r_i}{\theta }_i^2\right\} , \end{aligned}$$

(62)

which implies that

$$\begin{aligned}&\lim _{t\rightarrow +\infty } \sum _{i=1}^{N}c_{i,1}{z_{i,1}^{2}}\le \sum _{i=1}^{N}\varphi _{i,n_{i}}\nonumber \\&+\sum _{i=1}^{N}\left( \eta _{i} +{\bar{\lambda }}_{i}\varrho \right) \mu _{i} \nonumber \\&+ \sum _{i=1}^{N}\dfrac{k_{i,0}}{2r_i}{\theta }_i^2. \end{aligned}$$

(63)

According to the similar results in [14], all the signals in the closed-loop system are bounded. Based on the work in [34], we have $\left\| y-y_0\right\| \le \Vert z_{.1}\Vert /\left( {\underline{\sigma }}\right. \left. (L + B)\right)$, where ${\underline{\sigma }}(L + B)$ is the minimum singular value of $L + B$, $z_{.1} = [z_{1,1}, z_{2,1}, \ldots , z_{N,1}]^{\mathrm{T}}$. It can be shown that, for $\forall {\bar{\varepsilon }} > 0$,

$$\begin{aligned} \left\| y-y_0\right\| \le {\bar{\varepsilon }},~~if ~ {z_{. 1}^{2}} \le {\bar{\varepsilon }}^2 \left( {\underline{\sigma }}(L + B)\right) \end{aligned}$$

(64)

It is worth mentioning that the desired tracking error $\left\| y-y_0\right\|$ can be controlled in a small neighborhood by tuning the parameters $k_{i,0}$, $r_i$, $a_{i,m}$, $c_{i,m}$$(i=1, \ldots , N, ~ m=1, \ldots , n_i)$. In order to obtain the desired tracking error, the parameters $k_{i,0}$, $r_i$, $a_{i,m}$, $c_{i,1}$, $\mu _{i}$, ${\bar{\varepsilon }}_{i,m}$ would be tuned in a appropriate set. According to Definition 1, the distributed consensus tracking error $\left\| y-y_0\right\|$ in the closed-loop system is CSUUB.

The proof is completed.