## 2 The Attitude Control Problem Statement of the Prototype Ducted Fan MAV  
The prototype small-size ducted fan VTOL MAV, 0.8 m in height and 0.55 m in diameter, is shown in Figure 1. A duct is formed through the fuselage, and a ducted fan is mounted to the middle portion of the fuselage. Four deflecting vanes are mounted along the longitudinal and lateral axes of symmetry below the fan to provide pitch, roll, and yaw movements. Most of the anti-torque generated by the ducted fan is compensated by the stators configured inside the duct while the remaining anti-torque is balanced by the deflecting vanes. A cent-symmetric landing gear consisting of four legs, _i.e._, left, right, front, back, made from glass fiber reinforced plastics is installed on the MAV. Ducted fan aerodynamics has been discussed in more detail in listed references. A total of four servo motors are installed on the fuselage. The movements of the servo motors can change the direction of deflecting vanes, enabling the MAV to fly in all directions (forward, backward, left, right, _etc._). The movements of the left and right servo motors in the same directions can achieve pitch movement, while the movements of the front and back servo motors can achieve the roll movement. The movements of the opposite servo motors in the opposite direction can compensate the remaining anti-torque and further achieve yaw movement. Mathematically, the control allocation strategy is represented as:  
\[ \begin{bmatrix}\hat{\delta}_{t}\\ \hat{\delta}_{r}\\ \hat{\delta}_{f}\\ \hat{\delta}_{b}\end{bmatrix} = \begin{bmatrix}\frac{\hat{\delta}_{\mathit{p}it} }{2}+\frac{\hat{\delta}_{\mathit{y}aw}}{4}\\ \frac{\hat{\delta}_{\mathit{p}it}}{2}+\frac{\hat{\delta}_{\mathit{y}aw}}{4}\\ -\frac{\hat{\delta}_{\mathit{p}al}}{2}+\frac{\hat{\delta}_{\mathit{y}aw}}{4}\\ -\frac{\hat{\delta}_{\mathit{p}al}}{2}-\frac{\hat{\delta}_{\mathit{y}aw}}{4} \end{bmatrix}  \tag{1}\]  
The rotational speed of the ducted fan is maneuvered by one other servo motor to provide sufficient thrust force to lift the MAV. There is a sensor unit installed on the MAV to provide the flight controller with feedback signals containing measurement noises that increase the chatter in the input signal of actuators through feedback paths. We assume that the fuselage is a rigid body. The nonlinear kinematic equations can then be defined as follows:  
\[\dot{\hat{X}}\!\!=\!\!f\big{(}\tilde{X},\hat{U},\Theta\big{)} \tag{2}\]  
Figure 1: The prototype ducted fan MAV. where \(\hat{X}=\left[\hat{u},\hat{v},\hat{w},\hat{\theta},\hat{\varphi},\hat{q},\hat{p}, \hat{r}\right]^{\mathrm{T}}\) and \(\hat{U}=\left[\hat{\delta}_{\mathit{pr}},\hat{\delta}_{\mathit{rol}},\hat{ \delta}_{\mathit{yaw}},\hat{\delta}_{\mathit{rot}}\right]^{\mathrm{T}};\)\(\Theta\) represents the aerodynamic parameter set, which is not available in most cases. The nonlinear system can be linearized near the hover flight condition and the linearized model is given by:  
\[\hat{X}=\mathbf{AX}+\mathbf{BU} \tag{3}\]  
where \(X=\hat{X}-X_{e}=\left[u,v,w,\theta,\phi,\psi,q,p,r\right]^{\mathrm{T}}\) and \(\mathbf{U}=\hat{U}-\mathbf{U}_{e}=\left[\delta_{\mathit{pr}},\delta_{\mathit{rnd}}, \delta_{\mathit{yaw}},\delta_{\mathit{rot}}\right]^{\mathrm{T}}.\)\(X_{e}\) and \(\mathbf{U}_{e}\) are the trim state and input vectors, respectively. However the determination of the trim condition is quite complicated due to the uncertainties in \(\Theta.\) Therefore, the estimates of \(X_{e}\) and \(\mathbf{U}_{e}\) are, respectively, denoted by:  
\[\mathbf{X}_{0}=\mathbf{X}_{e}+\tilde{\mathbf{X}},\mathbf{U}_{0}=\mathbf{U}_{e}+\tilde{\mathbf{U}} \tag{4}\]  
Equation (3) should be rewritten as:  
\[\hat{X}=\mathbf{AX}+\mathbf{BU}+\mathbf{E} \tag{5}\]  
where \(X=\hat{X}-X_{0},\mathbf{U}=\hat{U}-\mathbf{U}_{0}\) and \(\mathbf{E}=\mathbf{A\tilde{X}}+\mathbf{BU}.\) Note that \(\mathbf{A},\mathbf{B}\) and \(\mathbf{E}\) are also unknown. The MAV is used mainly for low-speed flight missions.