# Quasi-Consensus of Disturbed Nonlinear Multiagent Systems with Event-Triggered Impulsive Control
## 2 Preliminaries and Model Description
### The Mathematical Model of Leader-Following MASs  
Consider the MASs consist of one leader and \(N\) followers, and each agent suffers from the ROUs, RONs and external disturbance. The dynamics of the \(i\)-th \((i=1,2,\ldots,N)\) follower is:  
\[\begin{cases}\dot{x}_{i}(t)=A(t)x_{i}(t)+\beta(t)f(t,x_{i}(t))+w_{i}(t)+u_{i}(t ),t\geq t_{0},\\ x_{i}(t_{0})=\varphi_{i},\end{cases} \tag{1}\]  
where \(t_{0}\geq 0\) is the initial instant and \(\varphi_{i}\in\mathbb{R}^{n}\) is the initial value of the \(i\)-th follower. \(x_{i}(t)\in\mathbb{R}^{n}\) is the system state, \(u_{i}(t)\in\mathbb{R}^{n}\) is the control input and \(w_{i}(t)\in\mathbb{R}^{n}\) denotes the external disturbance. \(A(t)=A+\alpha(t)MP(t)Q\), where \(A,M\) and \(Q\) are suitable constant matrices. \(P(t)\) is time-varying matrix and satisfies \(P(t)P^{\mathrm{T}}(t)\leq I_{n}\). \(f:\mathbb{R}\times\mathbb{R}^{n}\to\mathbb{R}^{n}\) is a nonlinear function. \(\beta(t)f(\cdot)\) and \(\alpha(t)MP(t)Q\) are used to describe the uncertain information of the system and nonlinearities that may occur randomly (i.e., RONs and ROUs) in practical application, respectively. Additionally, assume that \(\alpha(t)\) and \(\beta(t)\) satisfy Assumption 2.  
Let \(x_{0}(t)\in\mathbb{R}^{n}\) denote the state of the leader, whose dynamic is described as:  
\[\begin{cases}\dot{x}_{0}(t)=A(t)x_{0}(t)+\beta(t)f(t,x_{0}(t))+w_{0}(t),t\geq t _{0},\\ x_{0}(t_{0})=\varphi_{0},\end{cases} \tag{2}\]  
where \(\varphi_{0}\in\mathbb{R}^{n}\) is the initial value of the leader.  
To realize the quasi-consensus of the above MASs, the following impulsive protocol is designed:  
\[u_{i}(t)=\sum_{k=1}^{+\infty}\mu_{k}(\sum_{j\in N_{i}}a_{ij}(x_{i}(t)-x_{j}(t ))+b_{i}(x_{i}(t)-x_{0}(t)))\delta(t-t_{k}), \tag{3}\]  
where \(\mu_{k}\in\mathbb{R}\) is the impulsive gain, and \(\delta(t)\) is the Dirac delta function which is used to model the impulsive dynamic [35]. \(\{t_{k},k\in\mathbb{N}^{+}\}\) is the impulsive control sequence generated by the ETM to be designed.  
Combining (1) and (3), one can obtain that  
\[\begin{cases}\dot{x}_{i}(t)=A(t)x_{i}(t)+\beta(t)f(t,x_{i}(t))\ +w_{i}(t),\quad t\
eq t_{k},\\ \Delta x_{i}(t_{k})=x_{i}(t_{k}^{+})-x_{i}(t_{k}^{-})\\ \qquad\qquad=\mu_{k}\sum_{j\in N_{i}}a_{ij}(x_{i}(t_{k}^{-})-x_{j}(t_{k}^{-}) )+\mu_{k}b_{i}(x_{i}(t_{k}^{-})-x_{0}(t_{k}^{-})),\quad t=t_{k},\\ x_{i}(t_{0})=\varphi_{i},\end{cases} \tag{4}\]  
where \(x_{i}(t)\) at \(t_{k}\) is right continuous, with \(x_{i}(t_{k})=x_{i}(t_{k}^{+})\) being assumed. \(\Delta x_{i}(t_{k})\) represents the jump value at \(t=t_{k}\).  
Then, according to (4), we construct the following error system:  
\[\begin{cases}\dot{e}_{i}(t)=A(t)e_{i}(t)+\beta(t)g(t,e_{i}(t))+\bar{w}_{i}(t ),\quad t\
eq t_{k},\\ \Delta e_{i}(t_{k})=e_{i}(t_{k}^{+})-e_{i}(t_{k}^{-})\\ \qquad\qquad=\mu_{k}\sum_{j\in N_{i}}a_{ij}(e_{i}(t_{k}^{-})-e_{j}(t_{k}^{-}) )+\mu_{k}b_{i}e_{i}(t_{k}^{-}),\quad t=t_{k},\\ e_{i}(t_{0})=\varphi_{i}-\varphi_{0},\end{cases} \tag{5}\]  
where \(e_{i}(t)=x_{i}(t)-x_{0}(t)\), \(\bar{w}_{i}(t)=w_{i}(t)-w_{0}(t)\), \(g(t,e_{i}(t))=f(t,x_{i}(t))-f(t,x_{0}(t))\).  
Let \(e(t)=[e_{1}^{\mathsf{T}}(t),e_{2}^{\mathsf{T}}(t),\ldots,e_{N}^{\mathsf{T}}(t)]^{ \mathsf{T}},G(t,e(t))=[g^{\mathsf{T}}(t,e_{1}(t)),g^{\mathsf{T}}(t,e_{2}(t)), \ldots,g^{\mathsf{T}}(t,e_{N}(t))]^{\mathsf{T}},\tilde{w}(t)=[\tilde{w}_{1}^{ \mathsf{T}}(t),\tilde{w}_{2}^{\mathsf{T}}(t),\ldots,\tilde{w}_{N}^{\mathsf{T }}(t)]^{\mathsf{T}}\) and \(1_{N}\) be the \(N\)-dimensional vector with all elements equal to one. Using the Kronecker product, the error system (5) can be rewritten as the following compact form:  
\[\begin{cases}\dot{e}(t)=(I_{N}\otimes A(t))e(t)+\beta(t)G(t,e(t))+\tilde{w}(t),\quad t\
eq t_{k},\\ \Delta e(t_{k})=\mu_{k}(H\otimes I_{n})e(t_{k}^{-}),\quad t=t_{k},\\ e(t_{0})=1_{N}\otimes(\varphi_{i}-\varphi_{0}),\end{cases} \tag{6}\]