# Robust Control of Electric Tail Reduction System: Uncertainty and Performance Index
## 3 Robust Constraint Following Control  
### Uncertain Mechanical System  
The dynamic equation of mechanical system with uncertainty can be obtained by the Udwadia-Kalaba theory [38]:  
\[M(q(t),\sigma(t),t)\ddot{q}(t)=\tau(t)+Q(q(t),\dot{q}(t),\sigma(t),t)+Q^{c}(q( t),\dot{q}(t),\sigma(t),t) \tag{16}\]  
where \(t\in R\) is the time independent variable; \(q\in R^{n}\) is the coordinate (not necessarily the generalized coordinate and based on the specific problem); \(\dot{q}\in R^{n}\) is the velocity; \(\ddot{q}\in R^{n}\) is the acceleration; \(\sigma\in\sum\subset R^{p}\) is the uncertainty parameter (possibly time-varying vector). In addition, \(M(q,\sigma,t)\) is the inertia matrix, \(\tau\in R^{n}\) is the control input, \(Q(q,\dot{q},\sigma,t)\) is the force imposed on the system (centrifugal force, gravitational force are included here, but control input is not here), \(Q^{c}(q,\dot{q},\sigma,t)\) is the constraint force. The functions \(M(\cdot),Q(\cdot),\) and \(Q^{c}(\cdot)\) are assumed to be continuous. The servo constraint is considered in the following form:  
\[\sum\limits_{i=1}^{n}A_{li}(q,t)\dot{q}_{i}+A_{l}(q,t)=0,\ l=1,2,\cdots,m \tag{17}\]  
where \(A_{li}(\cdot)\) and \(A_{l}\) are both \(C^{1},1\leq m\leq n\). The first order constraint (17) is mainly a limit of the velocity and displacement. The matrix form can be expressed as  
\[A(q,t)\dot{q}=c(q,t) \tag{18}\]  
where \(c=\left[c_{1},c_{2},\cdots,c_{m}\right]^{T}\), \(A=\left[A_{li}\right]_{m\times n}\). By Equation (17), the second order form of the constraint equation is shown as follows:  
\[\sum\limits_{i=1}^{n}\frac{d}{dt}A_{li}(q,t)\dot{q}_{i}+\sum\limits_{i=1}^{n}A _{li}(q,t)\ddot{q}_{i}+\frac{d}{dt}A_{l}(q,t)=0 \tag{19}\]  
where  
\[\frac{d}{dt}A_{li}(q,t)=\sum\limits_{k=1}^{n}\frac{\partial A_{li}(q,t)}{ \partial q_{k}}\dot{q}_{k}+\frac{\partial A_{li}(q,t)}{\partial t} \tag{20}\]  
\[\frac{d}{dt}A_{l}(q,t)=\sum\limits_{k=1}^{n}\frac{\partial A_{l}(q,t)}{ \partial q_{k}}\dot{q}_{k}+\frac{\partial A_{l}(q,t)}{\partial t} \tag{21}\]  
Equation (19) can be obtained  
\[ \begin{array}{c}\sum\limits_{i=1}^{n}A_{li}(q,t)\ddot{q}_{i}=-\sum\limits_{ i=1}^{n}\frac{d}{dt}A_{li}(q,t)\dot{q}-\frac{d}{dt}A_{l}(q,t)\\ =:b_{i}(\dot{q},q,t)\end{array} ,\ l=1,2,\cdots,m \tag{22}\]  
or  
\[A(q,t)\ddot{q}=b(\dot{q},q,t) \tag{23}\]  
where \(b=\left[b_{1},b_{2},\cdots,b_{m}\right]^{T}\) and \(A=\left[A_{li}\right]_{m\times n}\). **Remark 1**. : _Equations (17)-(23) were derived from [39]. In engineering applications, it is rational to constraint following approximately when initial conditions are uncertain and uncertainties exist. In other words, Equation (23) does not need to strictly follow performance requirements. Therefore, the servo constraint problem can be understood as: determine the servo control \(\tau\) so that the controlled system approximately approaches the constraint (18). If there is no uncertainty in the mechanical system (16), then \(\sigma\) is known._  
**Assumption 1**. : _The inertia matrix \(M(q,\sigma,t)\) is positive definite such as that for each \((q,t)\in R^{n}\times R\), \(\sigma\in\sum\)._  
**Assumption 2**. : _The constraint (23) is consistent. That is, for given \(A\) and \(b\), there exists at least one solution \(\ddot{q}\). This assumption ensures \(AA^{+}b=b\), which means the existence of the Moore-Penrose generalized inverse of \(A\). On the contrary, if the constraint Equation (23) is inconsistent, then \(\tau\) is no solution in the servo constraint problem._  
**Theorem 1**. : _Considering the mechanical system (16) and the constraint (23), suppose that rank \(A(q,t)\geq 1\) for each \(q\) and \(t\), according to Assumptions 1 and 2, the uncertain mechanical system's constraint force conforms to [PERSON]'s form of d'Alembert's principle, which can be expressed as follows [24; 40]:_  
\[ \begin{array}{c}Q_{S}(q,\dot{q},\sigma,t)=M^{1/2}(q,\sigma,t)(A(q,t)M^{-1/2 }(q,\sigma,t))^{+}\\ \left[b(q,\dot{q},t)-A(q,t)M^{-1}(q,\sigma,t)(Q(q,\dot{q},\sigma,t)+Q^{\zeta} (q,\dot{q},\sigma,t))\right]\end{array}