# Robust Control of Electric Tail Reduction System: Uncertainty and Performance Index
## 3 Robust Constraint Following Control
### Robust Constraint Following Control Design
\begin{array}{rl}&2\beta^{T}PA(D+\Delta D)p_{3}\\ =&2\beta^{T}PA(D+DE)(-\gamma\mu\rho)\\ =&2(DA^{T}P\beta\rho)^{T}(I+E)(-\gamma\mu)\\ =&2\mu^{T}(I+E)(-\gamma\mu)\\ =&-2\gamma\mu^{T}\mu-2\gamma\mu^{T}E\mu\\ \leq&-2\gamma||||^{2}-\gamma A_{m}(E+E^{T})||\mu||^{2}\\ \leq&-2\gamma(1+\beta_{E})||\mu||^{2}\end{array}  \tag{34}\]  
By Equations (28) and (31), we can obtain  
\[ \begin{array}{rl}&2\beta^{T}PADp_{2}\\ =&2\beta^{T}PAD(-x\overline{J}_{tr}^{-1}A^{T}P\beta)\\ =&-2x||\overline{\mu}||^{2}||\beta||^{2}\end{array}  \tag{35}\]  
Combining Equations (33)-(35), we have  
\[ \begin{array}{rl}&2\beta^{T}P\dot{\beta}\\ \leq&-2\kappa||\overline{\mu}||^{2}||\beta||^{2}\\ &-2\gamma(1+\beta_{E})||\mu||^{2}+2||\beta||\rho\end{array}  \tag{36}\]  
Based on that  
\[x^{n}-y^{n}=(x-y)(x^{n-1}-x^{n-2}y+\cdots xy^{n-2}+y^{n-1})\]  
Thus,  
\[ \begin{array}{rl}&||\mu||^{n}-\varepsilon^{n}\\ =&\big{[}||\mu||^{n-1}+||\mu||^{n-2}\varepsilon+\cdots+||\mu||\varepsilon^{n-2 }+\varepsilon^{n-1}][||\mu||-\varepsilon]\\ =&\gamma_{2}(||\mu||)[||\mu||-\varepsilon]\end{array}  \tag{37}\]  
By Equations (29) and (37), the second term on the RHS of (36) can be analyzed as  
\[ \begin{array}{rl}&-2\gamma(1+\beta_{E})||\mu||^{2}\\ =&-2\frac{(1+\beta_{E})^{-1}}{||\beta||}\frac{||\mu||^{n-2}}{2(||\mu||)}(1+ \beta_{E})||\mu||^{2}\\ =&-2\frac{1}{||\beta||}\frac{||\mu||^{n-2}}{2(||\mu||)}\\ \leq&-2\frac{1}{||\beta||}\frac{||\mu||^{n}-\varepsilon^{n}}{2(||\mu||)}\\ =&-2\frac{1}{||\beta||}(||\mu||-\varepsilon)\end{array}  \tag{38}\]  
Substituting (38) into (36), we can obtain  
\[ \begin{array}{rl}&2\beta^{T}P\dot{\beta}\\ \leq&-2\kappa||\overline{\mu}||^{2}||\beta||^{2}-2\gamma(1+\hat{\rho}_{E})||\mu ||^{2}\\ +2||\beta||\rho\\ \leq&-2\kappa||\overline{\mu}||^{2}||\beta||^{2}+2\frac{\varepsilon}{|| \beta||}\\ -2\frac{1}{||\beta||}||\mu||+2||\beta||\rho\end{array} \]  
By (30), hence,  
\[ \begin{array}{rl}\dot{V}&=2\beta^{T}P\dot{\beta}\\ \leq&-2\kappa||\overline{\mu}||^{2}||\beta||^{2}+2\frac{\varepsilon}{|| \beta||}-2\frac{||\mu||}{||\beta||}+2||\beta||\rho\\ =&-2\kappa||\overline{\mu}||^{2}||\beta||^{2}+2\frac{\varepsilon}{||\beta||} \end{array} \]  
**Definition 1**. : _We define \(\underline{d}\) as the infimum of the size of the uniform ultimate boundedness \(\overline{d}\), which can be written as_  
\[ \begin{array}{l}\underline{d}=\sqrt{\frac{\lambda_{M}(P)}{\lambda_{m}(P)}}R\\ =\sqrt{\frac{\lambda_{M}(P)}{\lambda_{m}(P)}}\sqrt{\frac{\varepsilon}{\kappa|| \overline{\mu}||^{3}}}\end{array} \]  
_where \(\underline{d}=\overline{d}-\varepsilon\); \(\overline{d}\) is the uniform ultimate boundedness; and \(\varepsilon\) can be a positive constant (arbitrarily small) [30]. For any \(t_{0}\), the system performance index \(J(\kappa)\) is expressed as_  
\[ \begin{array}{l}J(\kappa):\qquad=\underline{d}^{2}+\hat{\alpha}\kappa^{2}\\ \qquad=\frac{\lambda_{M}(P)}{\lambda_{m}(P)}\frac{\varepsilon}{\kappa|| \overline{\mu}||^{3}}+\hat{\alpha}\kappa^{2}\\ \qquad=J_{1}(\kappa)+\hat{\alpha}J_{2}(\kappa)\end{array}  \tag{39}\]  
_where \(\hat{\alpha}>0\) (scalar) is a weighting factor._  
**Remark 8**. : _The performance index reflects the consideration in the engineering application. It combines the steady-state performance and the control cost. In Equation (39), \(J_{1}(\kappa)\) is interpreted as the steady-state performance, that is, the value of the infimum of the size of uniform ultimate boundedness.