## 3 Problem Formulation  
For clarity, the decentralized event-triggered output feedback saturated control system is illustrated in Figure 1 at first, and the specific details will be given later. Consider the following linear time-invariant plant with saturated input  
\[\dot{x} =Ax+Bsat(\dot{u}), \tag{1}\] \[y =Cx,\]  
Figure 1: Decentralized event-triggered output feedback control system with saturated inputs. where \(x\in\mathbb{R}^{n}\), \(\bar{u}\in\mathbb{R}^{m}\), and \(y\in\mathbb{R}^{q}\) are, respectively, the state vector, the control input, and the output. \(A,B,C\) are constant matrices of appropriate dimensions. \((A,B)\) is supposed to be controllable, and \((A,C)\) is supposed to be observable. For briefness, we assume that in the following analysis, the upper bound for the saturation function is \(c=1\) and the subscript \(c\) is dropped. As shown in Figure 1, both control signal \(u(t)\) and output \(y(t)\) are implemented in a decentralized manner. Hence, they are, respectively, partitioned into \(p\) actuator nodes and \(r\) sensor nodes, i.e., \(u=(u_{1}^{\mathrm{T}},\ldots,u_{p}^{\mathrm{T}})^{\mathrm{T}}\) and \(y=(y_{1}^{\mathrm{T}},\ldots,y_{r}^{\mathrm{T}})^{\mathrm{T}}\), where \(u_{s}\in\mathbb{R}^{m_{s}},s\in\{1,2,\ldots,p\}\) and \(y_{l}\in\mathbb{R}^{q_{l}},l\in\{1,2,\ldots,r\}\). \(m_{s}\) (\(q_{l}\)) denotes the dimension of the \(s\)th (\(l\)th) actuator (sensor) node and satisfies \(p_{s=1}^{p}m_{s}=m\) (\(\sigma_{l=1}^{r}q_{l}=q\)). To identify the nodes, we introduce the matrices \(\Pi_{u}^{s}\in\mathbb{R}^{m_{s}\times m}\) and \(\Pi_{y}^{l}\in\mathbb{R}^{q_{l}\times q}\) defined as  
\[\Pi_{u}^{s} =(0_{m_{s}\times m_{1}},...,0_{m_{s}\times m_{s-1}},I_{m_{s}},0_{ m_{s}\times m_{p+1}},...,0_{m_{s}\times m_{p}}),\] \[\Pi_{y}^{l} =(0_{q_{l}\times q_{1}},...,0_{q_{l}\times q_{l-1}},I_{q_{l}},0_{ q_{l}\times q_{l+1}},...0_{q_{l}\times q_{l}}).\]  
Thus, \(u_{s}=\Pi_{u}^{s}u\) and \(y_{l}=\Pi_{y}^{l}y\). Because of the communication constraints, either between sensors and observer, or between observer and actuators, the information cannot be transmitted continuously. Denote \(y_{l}(t_{k_{l}}^{l}),l=1,\ldots,r,k_{l}\in\mathbb{N}_{\geq 0}\) as the latest sampled value at the \(l^{th}\) sensor node, which is available for the observer-based controller to calculate the control signal. \(u_{s}(t_{j_{s}}^{s}),s=1,\ldots,p,j_{s}\in\mathbb{N}_{\geq 0}\) is the latest sampled actuator signal applied to the system. The triggering instants \(t_{k_{1}}^{1},\ldots,t_{k_{r}}^{l},t_{j_{1}}^{1},\ldots,t_{j_{r}}^{p}\) are decided by the designed event condition. It is worth noting that different sensor nodes and different actuator nodes are sampled asynchronously. Thus, at time \(t\), the latest available output \(\tilde{y}(t)\) is  
\[\tilde{y}(t)=(y_{1}^{\mathrm{T}}(t_{k_{1}}^{1}),\ldots,y_{r}^{\mathrm{T}}(t_{ k_{r}}^{r}))^{\mathrm{T}},\]  
and the input \(\bar{u}(t)\) is  
\[\bar{u}(t)=(u_{1}^{\mathrm{T}}(t_{j_{1}}^{1}),\ldots,u_{p}^{\mathrm{T}}(t_{j_{ p}}^{p}))^{\mathrm{T}}.\]  
Moreover, it is supposed that the first triggering instants of all the nodes are the initial instant, i.e., \(t_{0}^{l}=t_{0}^{s}:=t_{0}\), for \(l=1,\ldots,r\) and \(s=1,\ldots,p\). Therefore, the observer-based controller can be described as  
\[\dot{\hat{x}} =A\hat{x}+L(\tilde{y}-C\hat{x})+Bsat(\bar{u}),\] \[u =K\hat{x}, \tag{2}\]  
where \(\hat{x}\) is the state of observer. The matrices \(K\) and \(L\) are gain matrices to be designed. Denote the observer error by \(z=\hat{x}-x\). Then, its dynamic becomes  
\[\dot{z}=\dot{\hat{x}}-\dot{x}=Az+L(\tilde{y}-C\hat{x}). \tag{3}\]  
Define the sampling error as \(E=(e_{u}^{\mathrm{T}},e_{y}^{\mathrm{T}})^{\mathrm{T}}\), where \(e_{u}=\bar{u}-u\) and \(e_{y}=\bar{y}-y\). Then, we introduce the following decentralized integral-based event condition for each sensor/actuator node:  
\[y =Cx,\]0 \[y =Cx,\]1  
where \(\sigma_{y}^{l},\tau_{y}^{l},\sigma_{u}^{s}\) and \(\tau_{u}^{s}\) are positive constants to be designed. **Remark 1**. : _In Ref.