# Robust Control of Electric Tail Reduction System: Uncertainty and Performance Index
## 3 Robust Constraint Following Control
### Robust Constraint Following Control Design
In the application, the suitable \(\rho(\cdot)\) is selected to ensure the effective handling of uncertainty. The bounding function \(\rho(\cdot)\) is guaranteed to exist as long as \(\sum\) is a compact set. Therefore, the proposed control method is suitable for dealing with common uncertainties such as random uncertainty, fuzzy uncertainty, and polytope uncertainty. However, there is a possibility that, for a given set \(\sum\) finding the analytical expression such a \(\rho(\cdot)\) may be difficult if the dimension of the system is high and the function \(I_{tr}(\cdot)\), \(B_{tr}(\cdot)\), \(T_{tr}(\cdot)\) is highly nonlinear. Then, we may slightly enlarge the set \(\sum\) for the convenience of finding such a \(\rho(\cdot)\)._  
**Remark 4**. : _For the uncertain and nonlinear characteristics of the electric tail reduction system, the proposed control method can deal with parameter uncertainty, external disturbance dynamic uncertainty, etc. In particular, our control design is able to deal with nonlinear time-varyingsystems. In addition, the proposed control was based on second-order form constraints, which can be holonomic and nonholonomic._  
**Theorem 2. **_Considering the mechanical system with uncertainty (16). Subject to Assumptions 1-5, the control (27) renders the following system performances._  
_(i) uniformly bounded: For any \(r>0\), there is a \(d(r)<\infty\) so that if \(||\beta(\theta_{m}(t_{0}),\dot{\theta}_{m}(t_{0}),t_{0})||\leq r\), then \(||\beta(\theta_{m}(t),\dot{\theta}_{m}(t),t)||\leq d(r)\) for all \(t\geq t_{0}\)._  
_(ii) uniformly ultimately bounded: For any \(r>0\) with \(||\beta(\theta_{m}(t_{0}),\dot{\theta}_{m}(t_{0}),t_{0})||\leq r\), there exists a \(\underline{d}>0\) such that \(||\underline{\beta}(\theta_{m}(t),\dot{\theta}_{m}(t),t)||\leq\overline{d}\) for any \(\overline{d}>\underline{d}\) as \(t\geq t_{0}+T(\overline{d},r)\), where \(T(\overline{d},r)<\infty\). Furthermore, \(\overline{d}\to 0\) as \(\varepsilon\to 0\)._  
**Proof. ** Let \(V=\beta^{T}P\beta\) be a legitimate Lyapunov function candidate. The derivative of \(V\) is evaluated as  
\[ \begin{array}{cc}\dot{V}&=2\beta^{T}P(A\ddot{\theta}_{m}-b)\\ &=2\beta^{T}P\Big{\{}A[J_{tr}^{-1}(-B_{tr}\ddot{\theta}_{m}-T_{tr})+J_{tr}^{- 1}\tau]-b\Big{\}}\end{array} \]  
Decomposing \(J_{tr}^{-1}\), \(B_{tr}\), and \(T_{tr}\), then we have  
\[ \begin{array}{cc}A[J_{tr}^{-1}(-B_{tr}\dot{\theta}_{m}-T_{tr})+J_{tr}^{-1} \tau]-b\\ =&A[(D+\Delta D)(-\overline{B}_{tr}\dot{\theta}_{m}-\overline{T}_{tr}-\Delta B _{tr}\dot{\theta}_{m}-\Delta T_{tr})\\ &\qquad\qquad\qquad\qquad+(D+\Delta D)\tau]-b\\ =&A[D(-\overline{B}_{tr}\dot{\theta}_{m}-\overline{T}_{tr}+p_{1}+p_{2})+D(- \Delta B_{tr}\dot{\theta}_{m}-\Delta T_{tr})\\ &\quad+\Delta D(-B_{tr}\dot{\theta}_{m}-T_{tr}+p_{1}+p_{2})+(D+\Delta D)p_{3}] -b\end{array} \]  
We know, by Equation (24) with the special case that \(\sigma\equiv 0\),  
\[A\ddot{\theta}_{m}-b=0\]  
and  
\[\overline{J}_{tr}\ddot{\theta}_{m}+\overline{B}_{tr}\dot{\theta}_{m}+ \overline{T}_{tr}=p_{1}\]  
Thus,  
\[\overline{J}_{tr}^{-1}(-\overline{B}_{tr}\dot{\theta}_{m}-\overline{T}_{tr}+p _{1})=\ddot{\theta}_{m}\]  
\[A[\overline{J}_{tr}^{-1}(-\overline{B}_{tr}\dot{\theta}_{m}-\overline{T}_{tr}+ p_{1})]-b=0\]  
By \(D(\theta_{m},t):=\overline{J}_{tr}(\theta_{m},t)^{-1}\),  
\[A[D(-\overline{B}_{tr}\dot{\theta}_{m}-\overline{T}_{tr})+Dp_{1}]-b=0\]  
Next, by Equation (32),  
\[ \begin{array}{cc}&2\beta^{T}PA[D(-\Delta B_{tr}\dot{\theta}_{m}-\Delta T_{ tr})+\Delta D(-B_{tr}\dot{\theta}_{m}-T_{tr}+p_{1}+p_{2})\\ \leq&2||\beta||\
u PA[D(-\Delta B_{tr}\dot{\theta}_{m}-\Delta T_{tr})+\Delta D (-B_{tr}\dot{\theta}_{m}-T_{tr}+p_{1}+p_{2})\|\\ \leq&2||\beta||\rho\end{array}  \tag{33}\]By Equations (26) and (28), \(\Delta D=DE\) and \(\overline{J}_{tr}^{-1}=D\), we can obtain  
\[