## 3 The Self-Triggered Controller Design
### Function \(\sigma\) for Self Triggering
Under the constraint of:  
\[x^{T}\Phi_{P,\beta}(\tau)x=\psi(\tau) \tag{13}\]  
then the origin system in Equation (2) is global \(\beta\)-stable with input-to-state stability from \(W\) to \(x\). Proof: From Theorem 1, it is known that the left-hand-side and the right-hand side of inequalities (9) are continuous, and \(\sigma(x)\in[\tau_{min},\tau_{max}]\), thus the maximal triggering interval \(\sigma(x)\) must be the minimal root of Equation (13). From Theorem 2, for a given triggering interval \(\tau=t_{k+1}-t_{k}\), the root of Equation (13) for triggering interval \(\sigma(x)\) should be the one which is closest to \(\tau_{min}\). Equation (13) can be written as a linear combination of \(e^{\lambda_{i}\tau}\) and \(e^{(\lambda_{i}+\lambda_{j})\tau}\), where \(\lambda_{i}\) and \(\lambda_{j}\) are the eigenvalues of \(A\). If we denote \(e^{\tau}\) by \(z\),  
\[\Phi_{P,\beta}(\tau)=\phi_{1}+\phi_{2}z^{v_{1}}+\phi_{3}z^{v_{2}}+\ldots+\phi_ {r}z^{v_{r-1}} \tag{14}\]  
where \(\phi_{k}\) is the coefficient matrix of \(z^{v_{k}}\), \(v_{k}\) is \(\lambda_{i}\) or \(\lambda_{i}+\lambda_{j}\), \(i\leq n,j\leq n\) and \(v_{i}\
eq v_{j}\). However, the computation cost of directly solving the equation \(x^{T}\Phi_{P,\beta}(\tau)x=0\) is very high, and it is not practical in online processing for power system control. Instead, the approximate root of this equation can be obtained by the two-point Taylor expansion method. The \(\Phi_{P,\beta}(\tau)\) can be expanded into two-point Taylor series in \(m\) orders at \(z_{1}=e^{\tau^{min}}\) and \(z_{2}=e^{\tau_{max}}\). The approximation of \(\Phi_{P,\beta}(\tau)\) is:  
\[ \begin{array}{l}H(z)=\sum_{k=0}^{m}\{[a_{k}(z_{1},z_{2})(z-z_{1})+\\ a_{k}(z_{2},z_{1})(z-z_{2})](z-z_{1})^{k}(z-z_{2})^{k}\},\end{array}  \tag{15}\]  
where \(z=e^{\tau}\), and \(a_{n}(z_{1},z_{2})\) is:  
\[ \begin{array}{l}a_{n}(z_{1},z_{2})=\\ \sum_{k=0}^{m}\{\frac{(n+k-1)!}{k!(n-k)}\frac{(-1)^{n+1}n\phi^{(n-k)}(z_{2})+ (-1)^{k}k\phi^{(n-k)}(z_{1})}{n!(z_{1}-z_{2})^{n+k+1}}\}.\end{array}  \tag{16}\]  
and the approximation of the right hand side of Equation (13), denoted \(h(z)\), can be obtained similarly as \(H(z)\). Above all, for a given state \(x\), we have the following theorem to calculate the \(\sigma(x)\). **Theorem 3**: _For the given parameters in Theorem 1, the dynamic system (5), the maximal and minimal triggering intervals \(\tau_{min}\), \(\tau_{max}\), and the maximal approximation error of \(\Phi_{P,\beta}(\tau)\):_  
\[\varepsilon_{1}=\max_{\tau\in[\tau_{min},\tau_{max}]}eig[\Phi_{P,\beta}(\tau)- H(z)]\]  
The maximal approximation error of \(h(z)\) is \(\varepsilon_{2}=\max_{\tau\in[\tau_{min},\tau_{max}]}\psi_{1}(\tau)-h_{1}(z)\), under \(\gamma-re^{\varepsilon\pi\tau}>0\), with the approximation expression (15). Then, under the constraint of \(\sigma(x)\in[\tau_{min},\tau_{max}]\), the maximal triggering interval function is chosen by:  
\[\sigma(x)=\ln z_{c} \tag{17}\]  
where,  
\[z_{c}=\arg_{z}\min_{x^{T}[H(z)+\varepsilon_{1}I]x=h_{1}(z)-\varepsilon_{2}}|z -z_{1}| \tag{18}\]  
Then, the origin system in Equation (2) is global \(\beta\)-stable with input-to-state stability from \(W\) to \(x\). **Proof. ** See Appendix B. \(\square\)  
For a given state \(x\), \(z_{c}\) can be easily obtained by solving the polynomial in Equation (18). Thus, under the constraint of \(\sigma(x)\in[\tau_{min},\tau_{max}]\), the maximal triggering interval can be obtained by Theorem 3. If the approximation error \(\varepsilon_{1}\) and \(\varepsilon_{2}\) are very small, the approximation maximal triggering interval obtained by applying Theorem 3 should be very close to the actual maximal triggering interval. It should be noted that \(\tau_{min}\) can be calculated by conventional discrete control theory. However, the maximal sampling interval \(\tau_{max}=max_{x\in R^{\infty}}\sigma(x)\) cannot be obtained without knowing \(\sigma\). Thus, \(\tau_{max}\) is set to be much larger than \(\tau_{min},\tau_{max}>\tau_{min}\) in the algorithm to guarantee \(\sigma(x)\leq\tau_{max}\).