## 3 The Self-Triggered Controller Design  
### Function \(\sigma\) for Self Triggering  
The communication cost of self-triggered control depends on the triggering rate, which is directly related to the triggering time in self-triggered control. The system calculates the next triggering time and updating control output when the triggering time is up. The function \(\sigma\) for self-triggering is a crucial function. For obtaining the function \(\sigma\), some important results about exponential stability and input-to-state stability under self-triggered control are introduced. To investigate the relationship between the maximal triggering interval function \(\sigma\) and system performance, exponential stability and input-to-state stability from disturbance \(W\) to system state \(x\), Theorem 1 is proposed. **Theorem 1**. : _For the dynamic system (5), given scalars \(\beta>0\), \(\gamma>0\), if there exists an \(n\times n\) matrix \(P=P^{T}\), a positive scalar \(\gamma\) and a bounded function \(\sigma(x):R^{n}\to R_{+}\) are for all \(x\in R^{4n}\) and \(\tau\in[0,\sigma(x)]\):_  
\[x^{T}\Phi_{P,\beta}(\tau)x\leq\psi(\tau) \tag{9}\]  
where \(\psi\) is,  
\[\psi=\left\{ \begin{array}{ll}0&\tau\in R_{1}\cap R_{2}\\ (\gamma/2-(2\beta+1)(e^{\pi\tau}-1)r\epsilon)n\eta&\tau\in R_{1}/R_{2}\\ (\gamma/2-r\epsilon^{\pi\tau})n\eta&\tau\in R_{2}/R_{1}\\ (\gamma-r\epsilon^{\pi\tau}(2\beta+1)(e^{\pi\tau}-1)r\epsilon)n\eta&\tau\in R ^{+}/R_{1}/R_{2}\end{array} \right.\]  
with \(R_{1}=\{\tau|\gamma/2-r\epsilon^{\pi\tau}\}>0\), \(R_{2}=\{\tau|\gamma/2-(2\beta+1)(e^{\pi\tau}-1)r>0\}\), \(\alpha=\lambda_{max}(A^{T}+A)\), \(\epsilon=\lambda_{max}(P)\), \(r=\lambda_{max}(E^{T}E)\), where,  
\[\Phi_{P,\beta}(\tau)=\left( \begin{array}{cc}\Lambda(\tau)\\ I\end{array} \right)^{T}\left( \begin{array}{cc}A^{T}P+PA+2\beta P&-PBK\\ -K^{T}B^{T}P&0\end{array} \right)\left( \begin{array}{c}\Lambda(\tau)\\ I\end{array} \right) \tag{10}\]  
and,  
\[\Lambda(\tau)=I+\int_{0}^{\tau}e^{sA}ds(A-BK), \tag{11}\]  
then the origin system (2) is global \(\beta\)-stable and input-to-state stability from \(W\) to \(x\) for any triggering interval \(\sigma(x):R_{+}\times R^{n}\to R_{+}\) defines the triggering interval sequence by the law \(t_{k+1}=t_{k}+\sigma(x(t_{k}))\), \(k\in N\). \(||x||^{2}\leq n(\gamma+\vartheta)\eta/(\epsilon\beta)\) under a given zero initial state \(x(t_{k})=0\). **Proof. ** See Appendix A. The key of Theorem 1 is that the function \(\sigma\) must satisfy the inequality (9). However directly solving inequality (9) for obtaining \(\sigma\) is difficult. Approaches, such as in [30; 47], are employed to cut the triggering interval (sampling interval) and state space into several sections. Numerical methods, such as the linear matrix inequality (LMI) toolbox, offer a possibility to solve \(\sigma\). The precision of maximal triggering time depends on the number of sections. Higher precision of maximal triggering time requires more sections. In addition, if the state space dimension is high, the computation cost in the design process dramatically increases. This is not practical in power system control applications with a high state space dimension. Therefore, another algorithm is proposed. The \(\sigma(x)\) can be directly computed with a given \(x\) on-line by the proposed algorithm. Based on the result of Theorem 1, the following theorem is derived for calculating \(\sigma\). Theorem 2: _For the given parameters in Theorem 1, the dynamic system in Equation (5) has a minimal triggering interval for the global state space \(\tau_{min}=\min_{x\in R^{4n}}\sigma(x)\), and a maximal sampling interval \(\tau_{max}=max_{x\in R^{4n}}\sigma(x)\). \(\forall\tau\in[0,\sigma(x)],x^{T}\Phi_{P,\beta}(\tau)x\leq 0\), if the \(\sigma(x)\) is:_  
\[\sigma(x)=\arg_{\tau}\min_{\tau\in[\tau_{min},\tau_{max}]}\tau, \tag{12}\]  
where \(\tau_{min}\) and \(\tau_{max}\) are the lower bound and upper bound of self-triggering interval or sampling interval, respectively.