Consensus Formation of Multi-agent Systems with Obstacle Avoidance based on Event-triggered Impulsive Control

Abstract

This paper utilizes the distributed event-triggered impulsive control (ETIC) scheme to solve the consensus formation problem of leader-follower multi-agent systems (MASs) subject to obstacles. In order to solve the problem of the high cost of continuous control, an impulsive control strategy combined with the event triggering function is applied. By Lyapunov theory, the sufficient conditions for the strategy to achieve global exponential consensus are obtained, which implies that the formation of the system can be completed successfully. This research proposes a novel event-triggered condition which reduces the system’s energy loss when performing formation tasks. The proposed scheme not only realizes the system formation but also avoids the occurrence of Zeno behavior. Furthermore, the influence of the external environment with obstacles to consensus formation is considered. An improved artificial potential field (APF) function is proposed, which enables the MASs to avoid obstacles during the formation process. Finally, the effectiveness of the proposed method is verified by simulations.

Appendix

Proof of Theorem 1

The Lyapunov function that we create is as follows:

$$\begin{aligned} {V_k}(\vartheta (t)) = {\vartheta ^\textrm{T}}(t)S\vartheta (t),k \in {\mathrm{\mathfrak {N} }^ + }. \end{aligned}$$

(21)

Combined Eq. 10, the derivative of ${V_k}(\vartheta (t))$ is

$$\begin{aligned} {{\dot{V}}_k}(\vartheta (t))&= {\vartheta ^{\textrm{T}}}(t)[{({I_N} \otimes A)^{\textrm{T}}}S + S({I_N} \otimes A)\vartheta (t)] \\&\mathrm{{}} \le - \gamma {\left\| {\vartheta (t)} \right\| ^2}. \end{aligned}$$

According to ${\lambda _{min}}(S){\left\| {\vartheta (t)} \right\| ^2} \le {\vartheta ^{\textrm{T}}}(t)S\vartheta (t) \le {\lambda _{max}}(S)$${\big \Vert {\vartheta (t)} \big \Vert ^2}$,

$$\begin{aligned} {\lambda _{min}}(S){\left\| {\vartheta (t)} \right\| ^2}&\le {V_k}(\vartheta (t))\\ \nonumber&\le {V_k}(\vartheta ({t_k})) + \int _{{t_k}}^t { - \gamma {{\left\| {\vartheta (s)} \right\| }^2}ds}\\ \nonumber&\le {\lambda _{\max }}(S){\left\| {\vartheta ({t_k})} \right\| ^2} + \int _{{t_k}}^t { - \gamma {{\left\| {\vartheta (s)} \right\| }^2}ds}. \end{aligned}$$

(22)

From the Lemma 1, we can get

$$\begin{aligned} {\left\| {\vartheta (t)} \right\| ^2}&\le \frac{{{\lambda _{\max }}(S)}}{{{\lambda _{\min }}(S)}}{\left\| {\vartheta ({t_k})} \right\| ^2} + \int _{{t_k}}^t {\frac{{ - \gamma }}{{{\lambda _{\min }}(S)}}{{\left\| {\vartheta (s)} \right\| }^2}ds } \\ \nonumber&\le \frac{{{\lambda _{\max }}(S)}}{{{\lambda _{min}}(S)}}{\left\| {\vartheta ({t_k})} \right\| ^2}{\exp ({\int _{{t_k}}^t {\frac{{ - \gamma }}{{{\lambda _{\min }}(S)}}ds} })}. \end{aligned}$$

(23)

Based on Eqs. 10 and 15, we have

$$\begin{aligned} \left\| {\vartheta (t)} \right\|&\le \sqrt{\frac{{{\lambda _{\max }}(S)}}{{{\lambda _{min}}(S)}}} \left\| ({I_N} - \alpha \tilde{L}\chi (k) )\vartheta (t_k^ - ) \right\| \\&\quad \times \ {\exp ({\int _{{t_k}}^t {\frac{{ - \gamma }}{{{2\lambda _{\min }}(S)}}ds} })} \\&\le \left\| {\vartheta (t_k^ - )} \right\| \sqrt{\frac{{{\lambda _{\max }}(S)}}{{{\lambda _{min}}(S)}}} \left\| {{I_N} - \alpha \tilde{L}\chi (k)} \right\| \\&\quad \times {\exp ({\int _{{t_k}}^t {\frac{{ - \gamma }}{{{2\lambda _{\min }}(S)}}ds} })}\\&\qquad \qquad \vdots \\ \qquad \quad&\le \left\| {\vartheta (t_{k - m + 1}^ - )} \right\| {\left( {\sqrt{\frac{{{\lambda _{\max }}(S)}}{{{\lambda _{min}}(S)}}} } \right) ^m} \\&\quad \times \left\| {{I_N} - \alpha \tilde{L}\chi (k)} \right\| \times \left\| {{I_N} - \alpha \tilde{L}\chi (k-1)} \right\| \\&\quad \times \cdots \times \left\| {{I_N} - \alpha \tilde{L}\chi (k-m+1)} \right\| \\&\quad \times {\exp ({ - (\int _{{t_k}}^t {\frac{\gamma }{{{2\lambda _{\min }}(S)}}ds} + \int _{{t_{k - 1}}}^{{t_k}} {\frac{\gamma }{{{2\lambda _{\min }}(S)}}ds} )})} \\&\qquad \qquad \vdots \\ \qquad \quad&\le \left\| {\vartheta (t_0^- )} \right\| \mathop \prod \limits _{m = 1}^k \sqrt{\frac{{{\lambda _{\max }}(S)}}{{{\lambda _{min}}(S)}}} ({I_N} - \alpha \tilde{L}\chi (m)) \\&\quad \times \exp (\sum \limits _{m = 0}^{k - 1} - (\int _m^{m + 1} {\frac{\gamma }{{{2\lambda _{\min }}(S)}}ds}\\&\quad + \int _{{t_k}}^t {\frac{\gamma }{{{2\lambda _{\min }}(S)}}ds} )). \end{aligned}$$

By the integral median theorem, we can get

$$\begin{aligned} \left\| {\vartheta (t)} \right\|&\le \left\| {\vartheta (t_0^ - )} \right\| \mathop \prod \limits _{m = 1}^k \sqrt{\frac{{{\lambda _{\max }}(S)}}{{{\lambda _{min}}(S)}}} ({I_N} - \alpha \tilde{L}\chi (m)) \\ \nonumber&\quad \times {\exp ({ - \frac{\gamma }{{2{\lambda _{\min }}(S)}}(t - {t_0})})}\\ \nonumber&\le \left\| {\vartheta (t_0^ - )} \right\| \exp (\sum \limits _{m = 1}^k {\ln (\sqrt{\frac{{{\lambda _{\max }}(S)}}{{{\lambda _{min}}(S)}}} ({I_N} - \alpha \tilde{L}\chi (m)))} \\ \nonumber&\quad - \frac{\gamma }{{2{\lambda _{\min }}(S)}}(t - {t_0})) \\ \nonumber&\le \left\| {\vartheta (t_0 )} \right\| \exp (- \omega t). \end{aligned}$$

(24)

Theorem 1 is proved.

Proof of Theorem 2

For time $t \in ({t_k},{t_{k + 1}}]$, we consider the time derivative of ${\iota _i}(t)$ as follows

$$\begin{aligned} \frac{{d\left\| {{{\iota }_i}(t)} \right\| }}{{dt}}&= \frac{d}{{dt}}\left\| {{q_i}(t_k^{i - }) - {q_i}(t)} \right\| \nonumber \\ \le&\sum \limits _{j \in {{\textrm{N}}_i}} {{a_{ij}}\left\| {(A{\xi _{ij}}(t)+ \alpha {q_{ij}}(t))} \right\| } + {b_i}\left\| {A{\xi _{i0}}(t) - {q_i}(t_k^{i - })} \right\| \nonumber \\&+ \left\| {{{\iota }_i}(t)} \right\| \nonumber \\&\le \left\| {{{\iota }_i}(t)} \right\| \nonumber \\&\times (\frac{{\sum \limits _{j \in {{\textrm{N}}_i}} {{a_{ij}}\left\| {(A{\xi _{ij}}(t) \!+\! \alpha {q_{ij}}(t))} \right\| } \!+\! {b_i}\left\| {A{\xi _{i0}}(t) \!-\! {q_i}(t_k^{i - })} \right\| }}{{\left\| {{{\iota }_i}(t)} \right\| }} \!+\! 1) \end{aligned}$$

(25)

so we get

$$\begin{aligned}{} & {} \frac{d\left\| \iota _{i} (t) \right\| /dt}{\left\| \iota _{i}(t)\right\| } - 1\nonumber \\\le & {} \frac{{\sum \limits _{j \in {{\textrm{N}}_i}} {{a_{ij}}\left\| {(A{\xi _{ij}}(t) \!+\! \alpha {q_{ij}}(t))} \right\| } \!+\! {b_i}\left\| {A{\xi _{i0}}(t) \!-\! {q_i}(t_k^{i - })} \right\| }}{{\left\| {{{\iota }_i}(t)} \right\| }}. \end{aligned}$$

(26)

In addition, we have

$$\begin{aligned} \begin{aligned} \begin{array}{l} \frac{1}{{{\exp ({t - t_k^i})} - 1}} \le \frac{{\sum \limits _{j \in {{\textrm{N}}_i}} {{a_{ij}}\left\| {(A{\xi _{ij}}(t) + \alpha {q_{ij}}(t))} \right\| } + {b_i}\left\| {A{\xi _{io}}(t) - {q_i}(t_k^{i - })} \right\| }}{{\left\| {{{\iota }_i}(t)} \right\| }} \end{array} \end{aligned} \end{aligned}$$

(27)

then

$$\begin{aligned} \begin{aligned} \left\| {{{\iota }_i}(t)} \right\| \le&(\sum \limits _{j \in {{\textrm{N}}_i}} {{a_{ij}}\left\| {(A{\xi _{ij}}(t) + \alpha {q_{ij}}(t))} \right\| } \\&+ {b_i}\left\| {A{\xi _{io}}(t) - {q_i}(t_k^{i - })} \right\| )({\exp ({t - t_k^i})} - 1). \end{aligned} \end{aligned}$$

(28)

When the event-triggering function ${U_i}(t)$ tends to zero, according to Eq. 8

$$\begin{aligned} \begin{aligned} \sqrt{\beta \left\| {{q_i}(t_k^{i - })} \right\| } <&(\sum \limits _{j \in {{\textrm{N}}_i}} {{a_{ij}}\left\| {(A{\xi _{ij}}(t) + \alpha {q_{ij}}(t))} \right\| } \\&+\! {b_i}\left\| {A{\xi _{io}}(t) \!-\! {q_i}(t_k^{i - })} \right\| )({\exp ({t \!-\! t_k^i})} \!-\! 1). \end{aligned} \end{aligned}$$

(29)

Defining $T_k^i = t_{k + 1}^i - t_k^i$, we obtain

$$\begin{aligned} \begin{aligned}&{\exp ({T_k^i})}> \frac{{\sqrt{\beta \left\| {{q_i}(t_k^{i - })} \right\| } }}{{\sum \limits _{j \in {{\textrm{N}}_i}} {{a_{ij}}\left\| {(A{\xi _{ij}}(t) \!+\! \alpha {q_{ij}}(t))} \right\| } \!+\! {b_i}\left\| {A{\xi _{io}}(t) - {q_i}(t_k^{i - })} \right\| }} \!+\! 1. \end{aligned} \end{aligned}$$

(30)

Thus

$$\begin{aligned} T_k^i >&\ln&(\frac{{\sqrt{\beta \left\| {{q_i}(t_k^{i - })} \right\| } }}{{\sum \limits _{j \in {{\textrm{N}}_i}} {{a_{ij}}\left\| {(A{\xi _{ij}}(t) + \alpha {q_{ij}}(t))} \right\| } + {b_i}\left\| {A{\xi _{io}}(t) - {q_i}(t_k^{i - })} \right\| }} + 1) \nonumber \\\ge & {} 0. \end{aligned}$$

(31)

The Proof of Theorem 2 is completed.