## 2 The Control Problem for the Prototype UH Stability Augmentation System  
The prototype UH, with net weight 180 Kg and height 1.9 m, is a vertical takeoff and landing aircraft which includes two coaxial rotors and a fuselage with toroidal portion, as shown in Figure 1. A duct is formed through the fuselage. A propeller assembly is mounted to the top portion of the fuselage with a main rotor, 4.4 m in diameter, above the fuselage. A ducted rotor assembly in fuselage is used to compensate the propeller antitorque as well as providing some fraction of lift. The coaxial rotors, main and ducted, rotate at 800 rpm in the opposite directions with the main rotor providing about 70% of lift, drag, and pitch and roll movements of UH and the ducted rotor providing about 30% of lift and yaw movement. The stability augmentation system design for the prototype UH is a challenging task because the UH dynamics are highly nonlinear and subject to parametric uncertainties. In addition, the plant parameters change with the flight environments (e.g., the aerodynamic constants) or the system conditions (e.g., lift curve slopes). The adaptive control strategy is adopted to solve the above problems in this study. By assuming that the fuselage is a rigid body and the main rotor speed is constant, we can describe the nonlinear kinematic equations associated with the six degrees of freedom (6-DOF) as an equivalent block in the following manner:  
\[\dot{\hat{\mathbf{X}}}=f(\hat{\mathbf{X}},\hat{\mathbf{U}},\Theta) \tag{1}\]  
where \(\hat{\mathbf{X}}\)=\(\left[\hat{a},\hat{v},\hat{w},\hat{\theta},\hat{\phi},\hat{\psi},\hat{a}, \hat{p},\hat{r}\right]^{\mathrm{r}}\) and \(\hat{U}=\left[\hat{\delta},\hat{\delta}_{\ast},\hat{\delta},\hat{\delta}_{ \ast}\right]^{\mathrm{r}}\) represent the state vector and the control vector, respectively; \(\Theta\) represents the unsteady aerodynamic parameters set, which is difficult and expensive to measure and is, therefore, not available in most cases. Of the state variables in \(\hat{x},\hat{U}\), here \(\hat{u},\hat{v},\hat{w}\) are forward velocity, lateral velocity and vertical velocity, respectively; \(\hat{\theta},\hat{\phi},\hat{\psi}\)  are pitch angle, roll angle and yaw angle, respectively; \(\hat{q},\hat{p},\hat{r}\) are pitch rate, roll rate and yaw rate, respectively; \(\hat{\delta}_{\ast},\hat{\delta}_{\ast},\hat{\delta}_{\ast},\hat{\delta}_{ \ast}\) are pitch cyclic, roll cyclic, ducted rotor collective and main rotor collective, respectively. One common method to solve the control problem for the nonlinear system in Equation (1) is through linearization. The linearized model about a trim condition of the nonlinear dynamics is then represented as  
Figure 1: The prototype unmanned helicopter. \[\dot{\mathbf{X}}=\mathbf{A}\mathbf{X}+\mathbf{B}\mathbf{U} \tag{2}\]  
where \(\mathbf{X}=\bar{\mathbf{X}}-\mathbf{X}_{e}=[u,v,w,\theta,\phi,\psi,q,p,r]^{\text{T}}\) and \(\mathbf{U}=\bar{\mathbf{U}}-\mathbf{U}_{e}=[\delta_{e},\delta_{a},\delta_{r},\delta_{c}]^{ \text{T}}\) are the increments at a specified trim condition; \(\mathbf{A}\) and \(\mathbf{B}\) are the system matrix and the control-input matrix, respectively. Accordingly, \(\mathbf{X}_{e}\) and \(\mathbf{U}_{e}\) are respectively the trim state and the trim input with respect to hovering, lifting or forward flight, which must satisfy the equation  
\[f(\mathbf{X}_{e},\mathbf{U}_{e},\Theta)=0 \tag{3}\]  
Note that \(\mathbf{X}_{e}\) and \(\mathbf{U}_{e}\), obtained by solving Equation (3), are unknown as well. Therefore, we define  
\[\mathbf{X}_{0}=\mathbf{X}_{e}+\mathbf{E}_{x},\mathbf{U}_{0}=\mathbf{U}_{e}+\mathbf{E}_{u} \tag{4}\]  
\(\mathbf{X}_{0}\) and \(\mathbf{U}_{0}\) can also be regarded as the estimates of \(\mathbf{X}_{e}\) and \(\mathbf{U}_{e}\), respectively.