## 3 The Self-Triggered Controller Design
### The Selection of Feedback Gain for Controller  
The system performance to some extent depends on the feedback gain. If the selected feedback gain is not proper for the control system, the \(\tau_{min}\) in Theorem 1 may not exist, so that \(\beta\)-exponential stability and input-to-state stability cannot be satisfied even under the continuous control. Therefore, the feedback gain should be selected for satisfying the exponential stability and input-to-state stability in continuous control (\(\sigma(x)=0\)) first. Otherwise, the cost function should be considered when we select the feedback gain. In dealing with nonlinear control problems, many optimal control theories are proposed, such as Hamilton-Jacobi-Bellman equations, Euler-Lagrange equations and [PERSON]'s formula [44; 48]. However, it is difficult to apply those methods to solve this optimal control problem with the cost function described in Section 2. The reason is that solving the Hamilton-Jacobi-Bellman equations and Euler-Lagrange equations is extremely difficult. In addition, [PERSON]'s formula requires a fixed standard form of cost function, which does not coincide with our cost function. It is difficult to obtain the value of cost function directly. Therefore, instead of directly calculating the value of the cost function, inequalities for estimation are derived to select a proper feedback gain for guaranteeing the cost function within an upper bound. **Theorem 4**: _For given feedback gain \(K\), \(\gamma\), \(\beta\), and \(\tau_{min}=\min_{x\in R^{N}}\sigma(x)\) for global state space, which satisfies the condition of Proposition 1 and Assumption 1. Then the cost function (8) can be estimated by:_  
\[J<\frac{\lambda_{max}(Q)}{\beta^{2}\lambda_{min}(P)}\{\gamma n\eta T^{\prime}_ {f}\beta-[\gamma n\eta+V_{0}\beta](1-e^{-\beta T^{\prime}_{f}})\}+\frac{ \varepsilon T^{\prime}_{f}}{\tau_{min}}, \tag{19}\]  
_where,_  
\[T^{\prime}_{f}=\frac{1}{\beta}ln[(V_{0}\beta-\gamma n\eta)/\
u]\geq T_{f} \tag{20}\]Proof. : From the proof of Theorem 1, the value of the Lyapunov function can be estimated by \(V(t)\leq e^{-\beta t}V_{0}+\gamma n\eta/\beta(1-e^{-\beta t})\). To integrate both sides of the above inequality during 0 and \(T_{f}\), the cost of state can be estimated by: \(\int_{0}^{T_{f}}x^{T}Pxdt\leq\frac{1}{\beta^{T}}\{\gamma n\eta T_{f}\beta-( \gamma n\eta+V_{0}\beta)(1-e^{-\beta T_{f}})\}\). The estimated terminal time \(T_{f}^{\prime}\) for \(T_{f}\) can be obtained by solving the following inequality: \(V(T_{f})\leq e^{-\beta T_{f}}V_{0}+\gamma n\eta/\beta(1-e^{-\beta T_{f}})=V_{T}\), and the time derivatives of \(V\) have the inequality \(dV(T_{f})/dt\leq(\gamma n\eta-V_{0}\beta)e^{-\beta T_{f}}\). Therefore, the result is: \(T_{f}\leq\frac{1}{\beta}ln[(V_{0}\beta-\gamma n\eta)/\
u]=T_{f}^{\prime}\). The communication cost can be estimated by \(c\leq\varrho T_{f}/\tau_{min}\leq\varrho T_{f}^{\prime}/\tau_{min}\). Above all, the value of the cost function at terminal time \(T_{f}\) can be estimated by summing the estimation of state cost and communication cost. For a given \(P\) for Lyapunov function, the decay rate \(\beta\), and \(\gamma\) for the inhibition of disturbance effect, the minimal sampling interval \(\tau_{min}\) and the feedback gain \(K\) can be figured out by conventional robust discrete control technique. Then, the upper bound of cost function \(J\) can be estimated by a given Lyapunov function value \(V_{0}\) and \(V_{T}\) in initial time and terminal time by Theorem 4. The objective is to minimize the upper bound of J, so that the value of the cost function \(J\) can be guaranteed on a required level. Therefore, the feedback gain \(K\), decay rate \(\beta\), parameter \(\gamma\) for input-to-state stability, minimal sampling interval \(\tau_{min}\), and Lyapunov function parameter \(P\), can be estimated by minimizing the upper bound of J.