# Robust Control of Electric Tail Reduction System: Uncertainty and Performance Index
## 3 Robust Constraint Following Control
### Robust Constraint Following Control Design  
While designing the control \(\tau\), we considered the uncertainties of each part of the electric tail reduction system including the uncertainty of the equivalent rotational inertia \(J_{tr}\), the parameter uncertainty of the equivalent viscous damping coefficient \(B_{tr}\) and the external disturbance dynamic uncertainty of the equivalent load \(T_{tr}\). It is assumed that the matrix \(J_{tr}\), \(B_{tr}\), \(T_{tr}\) can be decomposed into the following forms [38]:  
\[ \begin{array}{c}J_{tr}(\theta_{m},\sigma,t)=\overline{J}_{tr}(\theta_{m},t)+ \Delta J_{tr}(\theta_{m},\sigma,t)\\ B_{tr}(\theta_{m},\hat{\theta}_{m},\sigma,t)=\overline{B}_{tr}(\theta_{m},\hat {\theta}_{m},t)+\Delta B_{tr}(\theta_{m},\hat{\theta}_{m},\sigma,t)\\ T_{tr}(\theta_{m},\hat{\theta}_{m},\sigma,t)=\overline{T}_{tr}(\theta_{m},\hat {\theta}_{m},t)+\Delta T_{tr}(\theta_{m},\hat{\theta}_{m},\sigma,t)\end{array}  \tag{25}\]  
where \(\overline{J}_{tr}\), \(\overline{B}_{tr}\), and \(\overline{T}_{tr}\) are the 'nominal' portions with \(\overline{J}_{tr}>0\), \(\Delta J_{tr}\), \(\Delta B_{tr}\), and \(\Delta T_{tr}\) are the uncertain portions. The functions \(\overline{J}_{tr}(\cdot)\), \(\overline{B}_{tr}(\cdot)\), \(\overline{T}_{tr}(\cdot)\), \(\Delta J_{tr}(\cdot)\), \(\Delta B_{tr}(\cdot)\), and \(\Delta T_{tr}  
Assumption 5 is to assure \(\lambda_{m}(\psi(\theta_{m},t))\) positively bounded from below. Therefore, we now propose the control  
\[\tau(t)=p_{1}(\theta_{m}(t),\dot{\theta}_{m}(t),t)+p_{2}(\theta_{m}(t),\dot{ \theta}_{m}(t),t)+p_{3}(\theta_{m}(t),\dot{\theta}_{m}(t),t) \tag{27}\]  
where  
\[ \begin{gathered} p_{1}(\theta_{m}(t),\dot{\theta}_{m}(t),t)=Q_{S }\\ p_{2}(\theta_{m}(t),\dot{\theta}_{m}(t),t)=-\kappa\overline{I}_{tr}^{-1}( \theta_{m},t)A^{T}(\theta_{m},t)P\beta\\ p_{3}(\theta_{m}(t),\dot{\theta}_{m}(t),t)=-\gamma(||\mu||)\mu( \theta_{m},\dot{\theta}_{m},t)\rho(\theta_{m},\dot{\theta}_{m},t)\end{gathered}  \tag{28}\]  
For the electric tail reduction system,  
\[ \begin{gathered} Q_{S}(\theta_{m},\dot{\theta}_{m},t)\\ =\overline{I}_{tr}^{1/2}(\theta_{m},t)(A(\theta_{m},t)\overline{I }_{tr}^{-1/2}(\theta_{m},t))^{+}\\ [b(\theta_{m},\dot{\theta}_{m},t)-A(\theta_{m},t)\overline{I}_{tr }^{-1}(\theta_{m},t)(-\overline{B}_{tr}(\theta_{m},\dot{\theta}_{m},t)\dot{ \theta}_{m}-\overline{T}_{tr}(\theta_{m},t))]\end{gathered} \]  
In (28), \(\kappa\in R\), \(\kappa>0\),  
\[\gamma(||\mu||)=\frac{\gamma_{1}(||\mu||)}{\gamma_{2}(||\mu||)} \tag{29}\]  
with  
\[\gamma_{1}(||\mu||)=\frac{(1+\dot{\rho}_{E})^{-1}}{||\overline{\mu}||}||\mu|| ^{n-2}\]  
\[\gamma_{2}(||\mu||)=||\mu||^{n-1}+||\mu||^{n-2}\varepsilon+\cdots+||\mu|| \varepsilon^{n-2}+\varepsilon^{n-1}\]  
where \(\varepsilon\in R\) and \(\varepsilon>0\), \(n\in\mathbb{N}^{*}\), and \(n\geq 2\). Here, \(\gamma(||\mu||)\) is generally based on system performance and fractional, which is denoted as the robust control gain. This kind of choice is not unique, which extends the application of robust constraint following the control scheme in engineering. In (28),  
\[\mu(\theta_{m},\dot{\theta}_{m},t)=\overline{\mu}(\theta_{m},\dot{\theta}_{m},t)\beta(\theta_{m},\dot{\theta}_{m},t)\rho \tag{30}\]  
where  
\[\overline{\mu}(\theta_{m},\dot{\theta}_{m},t)=\overline{I}_{tr}^{-1}(\theta_ {m},t)A^{T}(\theta_{m},t)P \tag{31}\]  
The bounding function \(\rho(\cdot):R^{n}\times R^{n}\times R\to R_{+}\) is chosen as  
\[ \begin{split}\rho\Big{(}\theta_{m},\dot{\theta}_{m},t\Big{)}& \geq\max_{\sigma\in\Sigma}||PA\Delta D(-B_{tr}\dot{\theta}_{m}-T_{tr}+p_{1} +p_{2})\\ &+PAD(-\Delta B_{tr}\dot{\theta}_{m}-\Delta T_{tr})||\end{split}  \tag{32}\]  
which defines the upper bound of the term of the right-hand side. **Remark 3**. : _The function \(\rho(\cdot)\) may be interpreted as the uncertainty bound. In a sense, the unknown upper bound vector function of uncertainty can be linearly parameterized with the bounding function \(\rho(\cdot)\).