## 5 Stability Analysis
Consequently,  
\[\dot{R}(\xi;F(\xi,w)) \leq -|x|^{2}+\eta|w|^{2}, \tag{24}\] \[\text{where }F(\xi,w):=(\mathcal{A}_{1}x(t)+\mathcal{B}_{1}e(t)+ \mathcal{E}_{1}w(t),\mathcal{A}_{2}x(t)+\mathcal{E}_{2}w(t),1).\] \[\text{When }\xi\in C\text{ and }\phi(\tau)>0,\text{ we have }R(\xi)=V(x)+\tilde{\gamma}\phi(\tau)W^{2}(e).\text{ As with Assumptions 1}\] and (17), we obtain \[\dot{R}(\xi;F(\xi,w)) \leq -(1+\varepsilon)|x|^{2}-H^{2}(x,w)+\gamma^{2}W^{2}(e)+\eta|w|^{2} +2\lambda\phi(\tau)W(e)H(x,w)\] \[-\tilde{\gamma}^{2}\phi^{2}(\tau)W^{2}(e)-\tilde{\gamma}^{2}W^{2 }(e)\]Using the fact that \(2\tilde{\gamma}^{2}\phi(\tau)W(\epsilon)H(x,w)\leq\tilde{\gamma}^{2}\phi^{2}(\tau)W ^{2}(\epsilon)+H^{2}(x,w)\), we obtain  
\[\dot{R}(\xi;F(\xi,w)) \leq -(1+\epsilon)|x|^{2}+\gamma^{2}W^{2}(\epsilon)+\eta|w|^{2}-\tilde{ \gamma}^{2}W^{2}(e) \tag{26}\]  
Recall that \(\tilde{\gamma}^{2}=\gamma^{2}+\
u\), it holds that  
\[\mathcal{D} = \big{\{}\xi(t):|e(t)|\geq\sigma|x(t)|\;\;\text{and}\;\;\tau\geq T \big{\}}.\]0  
As a result, in view of (24) and (27), for all \(\xi\in\mathcal{C}\), it holds that  
\[\mathcal{D} = \big{\{}\xi(t):|e(t)|\geq\sigma|x(t)|\;\;\text{and}\;\;\tau\geq T \big{\}}.\]1  
which implies \(\mathcal{L}_{2}\)-gain stability. The developed ETC strategy in this study is based on the following assumptions:  
* The hybrid wind-diesel system is described by the linearized model (3). * The full state vector \(x(t)\) is available for feedback. * Plant matrices \(A,B\) are controllable/stabilizable. * The communication network is noiseless and delay-free. * The event-triggering rule \(|e|\leq\sigma|x|\) can be continuously verified by the ETC. The first assumption is valid when the HWDPS system operates around the equilibrium point or when a small signal analysis is used to derive the dynamic model of the HWDPS [46]. The second statement is used to simplify the design of the feedback law. In practice, this requirement might be difficult to meet, and in this case, output feedback methods can be employed, such as the observer-based control and the ETC approach, which can be adapted by modifying the closed-loop matrices \(\mathcal{A}_{1},\mathcal{B}_{1},\mathcal{E}_{1},\mathcal{A}_{2},\mathcal{B}_{ 2},\mathcal{E}_{2}\) in the hybrid model (21), as shown in the next section. The third assumption is standard in order to design state-feedback controllers of the form \(u=-Kx\), as in (3). This assumption practically implies that sufficient actuators are available in the HWDPS system to control the required state variables, which, in practice, is often the case. The fourth assumption is introduced to facilitate the analysis since the presence of communication delays can significantly affect the ETC performance. In particular, if the communication delays are larger than the sampling intervals, then all the control updates will be outdated, which can destabilize the closed-loop system. In this context, it is often assumed that communication delays are small compared to sampling intervals, and the concept of the maximal allowable delay (MAD) is employed, see, e.g., [44]. Addressing such an issue is far from being trivial and requires separate analysis to derive the MAD bound. The fifth assumption means that the sensors are equipped with some special circuits (smart sensors) to continuously verify the event-triggering rule and it decides when a new measurement should be transmitted to the controller. This requirement is feasible when the feedback signal is analog and the sensor can be equipped with such a circuit. If the sensors are digital devices, periodic event-triggering conditions can be used alternatively, in which the triggering rule is only checked at periodic time instants, as in [47; 48]. However, the application of the developed ETC method in this study cannot be directly applied to this scenario and a complete new analysis is required to ensure the closed-loop stability.