# Robust Control of Electric Tail Reduction System: Uncertainty and Performance Index
## 3 Robust Constraint Following Control
### Robust Constraint Following Control Design
The scalar \(\hat{\alpha}\) denotes the weighting factor and \(J_{2}(\kappa)\) reflects the control cost._  
For any \(t_{0}\), an optimal design problem is formulated as:  
\[\min_{\kappa}J(\kappa)\quad\text{subject to }\kappa>0 \tag{40}\]  
taking the first order derivative of \(J\) with respect to \(\kappa\)  
\[\frac{\partial J}{\partial\kappa}=\frac{1}{\kappa^{2}}(2\hat{\alpha}\kappa^{ 3}-\frac{\lambda_{M}(P)}{\lambda_{m}(P)}\frac{\varepsilon}{||\overline{\mu}|| ^{3}}) \tag{41}\]  
Let  
\[\frac{\partial J}{\partial\kappa}=0 \tag{42}\]  
That is,  
\[2\hat{\alpha}\kappa^{3}-\frac{\lambda_{M}(P)}{\lambda_{m}(P)}\frac{ \varepsilon}{||\overline{\mu}||^{3}}=0 \tag{43}\]  
or  
\[2\hat{\alpha}\kappa^{3}=\frac{\lambda_{M}(P)}{\lambda_{m}(P)}\frac{ \varepsilon}{||\overline{\mu}||^{3}} \tag{44}\]  
**Theorem 3**. : _For given \(\varepsilon>0\) and \(\overline{\mu}>0\), the solution \(\kappa>0\) to (43) always exists and is unique, which globally minimizes the performance index (39)._  
Proof. : Let \(\psi(\kappa):=2\hat{\alpha}\kappa^{3}\). Then, \(\psi(0)=0\) and \(\psi(\cdot)\) is continuous in \(\kappa\). In addition, since \(\hat{\alpha}>0\) and \(\frac{d\psi}{d\kappa}=6\hat{\alpha}\kappa^{2}>0\), \(\psi(\cdot)\) is strictly increasing in \(\kappa\). As \(\frac{\lambda_{M}(P)}{\lambda_{m}(P)}\frac{\varepsilon}{||\overline{\mu}||^{3} }>0\), the solution \(\kappa>0\) to (43) always exists and is unique. Furthermore, taking the second order derivative of \(J\) with respect to \(\kappa\), we have  
\[\frac{\partial^{2}J}{\partial\kappa^{2}}=\frac{1}{\kappa^{3}}(2\hat{\alpha} \kappa^{3}+\frac{\lambda_{M}(P)}{\lambda_{m}(P)}\frac{2\varepsilon}{|| \overline{\mu}||^{3}})>0 \tag{45}\]  
Therefore, the positive solution \(\kappa>0\) to (43) solves the constrained minimization problem (40). Q.E.D. The analytic solution of cubic Equation (43) is the optimal solution of the performance index. **Remark 9**. : _We established the performance index (39) based on the selection \(\kappa>0\) in (28). Then, for given \(\varepsilon\) and \(\overline{\eta}\), by minimizing the performance index \(J(\kappa)\), we can summarize the optimization design problem as (40). The optimization design problem is based on method, not on a numerical search algorithm. This method can be understood as: analyze the performance index of the minimum system and analyze the first-order and second-order conditions in (41)-(45). On the one hand, we explored all possible solutions of the optimal problem by taking the first derivative of the performance index. The necessary conditions are provided. On the other, we observed that the optimal solution is unique by taking the second derivative of the performance index. The sufficient conditions are given. The essence of this optimization method is to use a small control cost to keep a small error to achieve the purpose of the comprehensive optimization of control cost and error. For the specific engineering problems, the designer can seek the balance of control performance by adjusting the different weighting factors \(\hat{\alpha}\) including the choice of the control precision and the control cost._