Adaptive fixed-time containment control of MIMO nonlinear multiagent systems via dynamic event-triggered communication

Abstract

We address the adaptive fixed-time containment (FxTC) control problem of a class of uncertain multi-input multi-output nonlinear multiagent systems using event-triggered inter-agent communication under a directed network. Our primary contribution is to develop multi-leader estimation and dynamic event-triggering strategies to ensure practical fixed-time convergence in a distributed containment control framework. First, an event-triggered mechanism that uses distributed containment errors in the dynamic threshold is introduced to reduce needless data transmissions among agents according to decreasing errors. Subsequently, a multi-leader-estimation-based command-filtered backstepping strategy is established to design a distributed adaptive FxTC controller using only neighbors’ output information. Using the fixed-time stability theory, we show that all closed-loop signals are bounded and all followers can be steered to the convex hull spanned by the leaders within a fixed time, regardless of the initial state conditions. Finally, two simulation examples containing robotic manipulators demonstrate the merits of the suggested control strategy.

Appendices

Appendices

1.1 A. Proof of Lemma 9

The following well-known lemma is required to prove Lemma 9.

Lemma 10

[55] Consider $\alpha _{i,k,l}$ with input noises satisfying $|\alpha _{i,k,l}$ $-\alpha _{i,k,l0}|<\o _{i,k,l}$ with a constant $\o _{i,k,l}$. Then the following inequalities are satisfied: $|\sigma _{i,k,l,1}-\alpha _{i,k,l0}| \le K_{1}\o _{i,k,l} = \varDelta _{i,k,l}$ and $|\varUpsilon _{i,k,l,1}-\dot{\alpha }_{i,k,l0}| \le K_{2}\o ^{\frac{1}{2}}_{i,k,l} = \bar{\varDelta }_{i,k,l}$, where $K_{1}$, $K_{2}$, $\varDelta _{i,k,l}$, and $\bar{\varDelta }_{i,k,l}$ are positive constants.

Now, let us prove Lemma 9. Select a Lyapunov function $V_{\varXi }=\sum _{i=1}^{N}\sum _{k=1}^{n_{i}}\varXi ^{\top }_{i,k}\varXi _{i,k}/2$. The time derivative $\dot{V}_{\varXi }$ using (14) and Lemma 10 is obtained as

$$\begin{aligned} \dot{V}_{\varXi }&\le -\sum _{i=1}^{N}\sum _{k=1}^{n_{i}} \bigg [4C_{i,k1} \left( \frac{\Vert \varXi _{i,k}\Vert ^{2}}{2} \right) ^{2}\nonumber \\&\qquad \qquad \quad -\sum _{q=1}^{m}(\kappa _{i,k}-\varDelta _{i,k})|\varXi _{i,k,q}| \bigg ] \end{aligned}$$

(64)

where $\varDelta _{i,k}=\max \{\varDelta _{i,k,1},\dots ,\varDelta _{i,k,m}\}$.

By defining $\bar{C}=\min _{i \in \mathcal {\bar{V}}_{F}}\{C_{i,k1}\}$ and $\bar{\kappa }=\min _{i \in \mathcal {\bar{V}}_{F}}$ $\{\kappa _{i,k}-\varDelta _{i,k}\}$ with $k=1,\dots ,n_{i}$, choosing the proper parameter $\kappa _{i,k}$ such that $\kappa _{i,k}>\varDelta _{i,k}$, and using Lemmas 4 and 5, we have $-\sum _{i=1}^{N}\sum _{k=1}^{n_{i}}\sum _{q=1}^{m}(\kappa _{i,k}-\varDelta _{i,k})|\varXi _{i,k,q}| \le -\sqrt{2}\bar{\kappa } (\sum _{i=1}^{N}\sum _{k=1}^{n_{i}}$ $\Vert \varXi _{i,k}\Vert ^{2}/2)^{\frac{1}{2}}$ and $-\sum _{i=1}^{N}\sum _{k=1}^{n_{i}}4C_{i,k1}( \Vert \varXi _{i,k}\Vert ^{2}/2)^{2} \le - \frac{4\bar{C}}{Nn_{i}} ($ $\sum _{i=1}^{N}\sum _{k=1}^{n_{i}}(\Vert \varXi _{i,k}\Vert ^{2}/2))^{2}$. From Lemmas 4 and 10, we can obtain $\dot{V}_{\varXi } \le -\hat{C}V_{\varXi }^{2}-\hat{\kappa }V^{\frac{1}{2}}_{\varXi }$, where $\hat{C}=4\bar{C}/Nn_{i}$ and $\hat{\kappa }=\sqrt{2}\bar{\kappa }$. Then, Lemma 1 leads to the convergence of $\varXi _{i,k}$ to zero in the fixed time, and the bound of the settling time $T_{1}$ is represented by $T_{1}=(1/\hat{C})+(2/\hat{\kappa })$.

1.2 B. System matrices of two-link robot manipulators [56]

The system matrices are defined as

$$\begin{aligned} F_{i}(Q_{i})&= \left[ \begin{array}{cc} F_{i,11} &{} F_{i,12} \\ F_{i,12} &{} F_{i,22} \\ \end{array} \right] , \\ Z_{i}(Q_{i},\dot{Q}_{i})&= \left[ \begin{array}{cc} Z_{i,11} &{} Z_{i,12} \\ Z_{i,21} &{} 0 \\ \end{array} \right] \\ G_{i}(Q_{i})&= [G_{i,1}(Q_{i}), G_{i,2}(Q_{i})]^{\top } \end{aligned}$$

where $F_{i,11}=m_{i,1}l_{i,c1}^{2}+m_{i,2}l_{i,1}^{2}+m_{i,2}l_{i,c2}^{2} + 2m_{i,2}l_{i,1}l_{i,c2}\cos q_{i,2}$ $+ I_{i,1}+I_{i,2}$, $F_{i,12}=m_{i,2}l_{i,c2}^{2}+m_{i,2}l_{i,1}l_{i,c2}\cos q_{i,2} + I_{i,2}$, $F_{i,22}=m_{i,2}l_{i,c2}^{2} + I_{i,2}$, $Z_{i,11}=-h_{i}\dot{q}_{i,2}$, $Z_{i,12}=-h_{i}(\dot{q}_{i,1} + \dot{q}_{i,2})$, $Z_{i,21}=h_{i}\dot{q}_{i,1}$, $h_{i}=m_{i,2}l_{i,1}l_{i,c2}\sin q_{i,2}$, $G_{i,1}=(m_{i,1}l_{i,c1}+m_{i,2}l_{i,1})g\cos q_{i,1} + m_{i,2}l_{i,c2}g\cos (q_{i,1} + q_{i,2})$, and $G_{i,2}=m_{i,2}l_{i,c2}$ $g\cos (q_{i,1} + q_{i,2})$.