# Edge-Event-Triggered Synchronization for Multi-Agent Systems with Nonlinear Controller Outputs
## 2 Preliminaries and Problem Formulation
### Problem Formulation  
We consider a multi-agent system with a group of agents with nonlinear dynamics, which are described as  
\[x_{i}^{\prime}(t)=f(x_{i})+u_{i}(t),i\in\mathcal{V} \tag{1}\]  
where \(x_{i}(t)\in\mathbb{R}^{n}\) is the state vector of the \(i\)th agent, \(u_{i}(t)\in\mathbb{R}^{n}\) is the control input, and \(f(x_{i})\in\mathbb{R}^{n}\) is the inherent nonlinear dynamics.  
**Definition 1**.: _The nonlinear function \(f(x)\) in system (1) satisfies the Lipschitz condition in \(x\in\mathbb{R}^{n}\), i.e.,_  
\[\|f(x_{i})-f(x_{j})\|\leq\mathcal{L}\|x_{i}-x_{j}\|,x_{i},x_{j}\in\mathbb{R}^{ n}, \tag{2}\]  
_where \(\mathcal{L}\) represents the Lipschitz constant._  
In this article, the main goal is to design edge-based event- and self-triggered policies to solve the synchronization problem of nonlinear multi-agent systems. Assume that for each agent the given event conditions are constantly monitored to determine whether to update, and the event time sequence of each agent \(v_{i}\) is set as \(t_{0}^{i},t_{1}^{i},t_{2}^{i},\cdots\). For agent \(v_{i}\), denote the latest sampling edge states of \(\varepsilon_{p}\) by \(\tilde{y}_{p}^{i}(t)=\tilde{y}_{p}^{i}(t_{k}^{i})=y_{p}(t_{k}^{i})\), \(t\in[t_{k}^{i},t_{k+1}^{i})\). For each agent \(v_{i}\), \(i=1,2,\cdots,N\), we design the control protocol with nonlinear controller outputs as follows,  
\[u_{i}(t)=-g\left(c\sum_{p\in\mathcal{E}_{i}}h_{ip}w_{p}\tilde{y}_{p}^{i}(t) \right),t\in[t_{k}^{i},t_{k+1}^{i}), \tag{3}\]  
where the control gain \(c>0\). Here, the nonlinear vector function \(g(\mathcal{Y}):\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}\) has the form \(g(\mathcal{Y})=\left[g_{1}(\mathcal{Y}_{1})^{T},g_{2}(\mathcal{Y}_{2})^{T}, \cdots,g_{N}(\mathcal{Y}_{N})^{T}\right]^{T}\). Here, for any \(q=1,2,\cdots,N\), \(g_{q}(\mathcal{Y}_{q})\) satisfies the following properties.  
1. \(g_{q}(\mathcal{Y}_{q})=0\) if and only if \(\mathcal{Y}_{q}=0\).
2. \(-g_{q}(\mathcal{Y}_{q})=g_{q}(-\mathcal{Y}_{q})\).
3. \(g_{1}\mathcal{Y}_{q}\leq g_{q}(\mathcal{Y}_{q})\leq k_{2}\mathcal{Y}_{q}\) with the positive scalars \(\kappa_{1}<\kappa_{2}\).  
Besides, the following inequalities can be derived based on the properties P1-P3.  
1. \(k_{1}^{2}\mathcal{Y}_{q}^{2}\leq g_{q}(\mathcal{Y}_{q})^{2}\leq k_{2}^{2} \mathcal{Y}_{q}^{2}\).
2. \(k_{1}\mathcal{Y}_{q}^{2}\leq\mathcal{Y}_{q}g_{q}(\mathcal{Y}_{q})\leq k_{2} \mathcal{Y}_{q}^{2}\).
3. \(\frac{k_{1}}{k_{2}}g_{q}(\mathcal{Y}_{q})^{2}\leq\mathcal{Y}_{q}g_{q}(\mathcal{ Y}_{q})\leq\frac{k_{2}}{k_{1}}g_{q}(\mathcal{Y}_{q})^{2}\).  
For system (1), its graph \(\mathcal{G}\) satisfies the following assumption and lemma.  
**Assumption 1**.: _The interaction topology \(\mathcal{G}\) of system (1) is connected._  
**Lemma 1**.: _For a connected graph \(\mathcal{G}\), its Laplacian matrix \(L\) and edge Laplacian matrix \(L_{E}\) satisfy \(\lambda_{2}\xi^{T}\xi\leq\xi^{T}L\xi\leq\lambda_{n}\xi^{T}\xi\) and \(\bar{\lambda}_{2}\xi^{T}\zeta\leq\xi^{T}L_{E}\xi\leq\bar{\lambda}_{n}\xi^{T}\zeta\), for any real vectors \(\xi,\zeta\in\mathbb{R}^{n}\), where \(\lambda_{2}\), \(\bar{\lambda}_{2}\)and \(\lambda_{n}\), \(\bar{\lambda}_{n}\) are the minimum and maximum non-zero eigenvalues of \(L\), \(L_{E}\), respectively, with the properties \(\xi^{T}1_{n}=0\) and \(\zeta\)._  
**Remark 1**.: _The proposed controller output nonlinearities mainly include two aspects: one is the inherent nonlinear characteristics of controllers, such as the nonlinear oscillation characteristics of the control circuit. The second is the nonlinear encoding required for network transmission of information, such as quantization transmission. These influencing factors can be described by nonlinear functions that satisfy the properties P1-P3._