# Performance-Constraint Fault Tolerant Control to Aircraft in Presence of Actuator Deviation
## 4 FTC-FT Controller Applied on the Unmanned Helicopter  
The yaw motor is highly sensitive to the actuator adjustment of the main rotor for an unmanned helicopter,so this section built a FTCFT controller according to the proposed Theorem 1. Firstly, the yaw-vertical channel coupling dynamics model is established as follows [35]  
Figure 1: Diagram of FTC-FT control scheme. \[\dot{\psi}(t) =\frac{S_{\phi(t)}}{C_{\phi(t)}}q(t)+\frac{C_{\phi(t)}}{C_{\phi(t)}}r(t) \tag{27}\] \[\dot{r}(t) =p(t)q(t)\frac{\left(I_{xx}-I_{yy}\right)}{I_{zz}}+\frac{\left(N_{ mr}+N_{of}+N_{tr}\right)}{I_{zz}}\] \[\dot{h}(t) =C_{\phi(t)}C_{\psi(t)}u(t)+\left(S_{\phi(t)}S_{\phi(t)}S_{\psi(t )}-C_{\phi}S_{\psi}\right)v(t)\] \[\quad+\left(C_{\phi(t)}S_{\phi(t)}C_{\psi(t)}+S_{\phi(t)}S_{\psi(t )}\right)w(t)\] \[\dot{w}(t) =u(t)q(t)-v(t)p(t)+gC_{\phi(t)}C_{\theta(t)}+\frac{1}{m}\left(Z_{ mr}+Z_{fus}+Z_{hf}\right)\]  
where \(\mathbf{\Omega}=\{\phi,\theta,\psi\}\) represents Euler angle,\(V_{b}=\{u,v,w\}\) is the body translation velocity along \(X,Y,Z\) axis, \(\mathbf{\omega}_{b}=\{p,q,r\}\) is the body angular rate, the \(h\) is the vertical position of the center of gravity described in the ground coordinate. \(S_{\cdot(\cdot)},C_{\cdot(\cdot)}\) denote the sine and cosine operator. \(N_{mr},N_{vf},N_{tr}\), represent the aerodynamic torques generated by main rotor, vertical fin, and tail rotor in yaw channel respectively. \(Z_{mr},Z_{hf},N_{fus}\), denote the vertical force from main rotor, fuselage and horizon fin. \(I_{(\cdot)}\) is the rotational inertia. Consider the trim in hover state, thus the aerodynamic force can be linearized, and keep the nonlinear parts of state coupling, the model is simplified as  
\[\left[ \begin{array}{c}h(t)\\ \dot{\psi}(t)\end{array} \right] =f_{1}(\mathbf{\Omega}(t),V_{b}(t),\mathbf{\omega}_{b}(t))+g_{1}( \mathbf{\Omega}(t))\left[ \begin{array}{c}w(t)\\ r(t)\end{array} \right] \tag{28}\] \[\left[ \begin{array}{c}\dot{w}(t)\\ \dot{r}(t)\end{array} \right] =f_{2}(\mathbf{\Omega}(t),V_{b}(t),\mathbf{\omega}_{b}(t))+(g_{2}(t)+ \Delta g_{2}(t))\delta+\mathbf{\xi}_{c}(t)\]  
The control input \(\delta=[ \begin{array}{c}\delta_{col},\delta_{tr}\end{array} ]^{T}\), \(\Delta g(t)\) represents the control coefficient matrix perturbation results from actuator deviation, and satisfies the upper bound condition \(|\Delta g_{2}(t)|\leq\varpi\). \(\mathbf{\xi}_{c}(t)=[ \begin{array}{c}\xi_{mr},\,\xi_{tr}\end{array} ]^{T}\) is external disturbance, \(\mathbf{f_{1}}\), \(\mathbf{f_{2}}\), \(\mathbf{g_{1}}\), \(\mathbf{g_{2}}\) are expressed as  
\[\mathbf{f_{1}} =\left( \begin{array}{cc}C_{\theta(t)}C_{\psi(t)}u(t)+\left(S_{ \phi(t)}S_{\phi(t)}S_{\psi(t)}-C_{\phi}S_{\psi}\right)v(t)\\ S_{\phi(t)}q(t)/C_{\phi(t)}\end{array} \right)\] \[\mathbf{g_{1}} =\left( \begin{array}{cc}C_{\phi(t)}S_{\theta(t)}C_{\psi(t)}+S_{ \phi(t)}S_{\psi(t)}&0\\ 0&C_{\phi(t)}/C_{\theta(t)}\end{array} \right)\] \[\mathbf{f_{2}} =\left( \begin{array}{c}u(t)q(t)-v(t)p(t)+Z_{w}w\\ p(t)q(t)\left(I_{xx}-I_{yy}\right)/I_{zz}+N_{r}r(t)\end{array} \right)\mathbf{g_{2}} =\left( \begin{array}{cc}Z_{col}&0\\ N_{col}&N_{tr}\end{array} \right)\]  
the \(Z_{w},Z_{col},N_{r},N_{col},N_{tr}\) represent the ratio of the aerodynamic force to the inertia. Generally, unmanned helicopters fly at low altitudes, often cruising below a certain value, vertical speed should also be limited in order to prevent the fuselage stall, which needs to be further reduction once occurring of collective pitch actuator fluctuation, similarly, yaw rate should restricted in an appropriate range. We have already built the all state-constraint FTC-FT controller design method in above section, this part will establish a partial state constraint controller for yaw and vertical motion in a simple way of modifying intermediate matrix. The restriction of altitude, vertical velocity and yaw rate of aircraft are defined as: \(\left|h(t)\right|<\bar{h},\left|w(t)\right|<\varpi,\left|r(t)\right|<\bar{r}\), giving the expected outputs signal \(y_{d}=\left[h_{d}(t),\psi_{d}(t)\right]^{T}\).