## 3 Disturbance Rejection Dynamic Inverse Control
### Anti-Windup Modification  
For the tailless aircraft model, the actuators will saturate for surface commands out of the deflection ranges listed in Table 2.  
In the presence of actuator saturation, the system of Equation (30) takes the form:  
\[\dot{x}_{1}=f(x_{1},x_{2})+g(x_{1},x_{2})sat(u)+d_{1} \tag{42}\]  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Surface** & **Notation** & **Position Bound** & **Rate Limit** \\ \hline Aileron/Elevator & \(\delta_{\alpha},\delta_{\varepsilon}\) & \(\pm 30^{\circ}\) & \(\pm 200^{\circ}\)/s \\ Rudder & \(\delta_{r}\) & \(\pm 40^{\circ}\) & \(\pm 200^{\circ}\)/s \\ \hline \hline \end{tabular}
\end{table}
Table 2: Deflection range of the surfaces on the tailless aircraft model.  
\[sat(u)=\left[\begin{array}{c}sat_{\delta_{a}}(u_{\delta_{a}})\\ sat_{\delta_{e}}(u_{\delta_{e}})\\ sat_{\delta_{e}}(u_{\delta_{e}})\end{array}\right] \tag{43}\]  
where \(sat_{j}\) is the saturation operator defined as following:  
\[sat_{j}(u_{j})=\left\{\begin{array}{c}u_{j}^{max},u_{j}\geq u_{j}^{max}\\ u_{j},u_{i}^{min}\leq u_{j}\leq u_{j}^{max}\\ u_{j}^{min},u_{j}\leq u_{j}^{min}\end{array}\right. \tag{44}\]  
When saturation is active, the controller of Equation (33) exhibits windup, which could lead to significant performance degradation and even instability of the closed-loop system. To attenuate the effect of windup, an anti-windup modification is made to the disturbance observer of Equation (32) referring to ref. [25].  
\[\left\{\begin{array}{c}\dot{z}_{d_{1}}=-l(x_{1})\big{[}\lambda(x_{1})+z_{d_{ 1}}+f(x_{1},x_{2})+g(x_{1},x_{2})u\big{]}\\ +K_{sat}[u-sat(u)]\\ \dot{d}_{1}=z_{d_{1}}+\lambda(x_{1})\end{array}\right. \tag{45}\]  
where the term of \(K_{sat}\) is an anti-windup gain matrix to be designed.  
Thus, using the modified observer of Equation (45) for estimation of the lumped disturbance in Equation (42), we have:  
\[\dot{e}_{d_{1}}=-Le_{d_{1}}+\dot{d}_{1}+[l(x_{1})g(x_{1},x_{2})-K_{sat}][u-sat (u)] \tag{46}\]  
The anti-windup gain matrix \(K_{sat}\) can be chosen as the following so that the effect of actuator saturation on the disturbance observer will be removed.  
\[K_{sat}=l(x_{1})g(x_{1},x_{2}) \tag{47}\]  
When Equation (47) is substituted into Equation (45), the modified disturbance observer yields  
\[\left\{\begin{array}{c}\dot{z}_{d_{1}}=-l(x_{1})\big{[}\lambda(x_{1})+z_{d_ {1}}+f(x_{1},x_{2})\\ +g(x_{1},x_{2})sat(u)]\\ \dot{d}_{1}=z_{d_{1}}+\lambda(x_{1})\end{array}\right. \tag{48}\]  
The architecture of the closed-loop system with the actuator saturation and modified observer (NDI-AW) is as presented in Figure 5. Note that the anti-windup modified disturbance observer is driven by the actual surface deflections rather than the surface commands. Combining the plant dynamics of Equation (42), the inner-loop NDI controller of Equation (33) and the anti-windup modified disturbance observer of Equation (48), it is readily reached that the closed-loop tracking error takes the same form as Equation (35). Hence, the effect of actuator saturation on the closed-loop tracking performance is also removed.  
Figure 5: Anti-windup modification of the disturbance observer-enhanced NDI controller.