## 3 Controller Design
### Fixed-Time Disturbance Observer Design  
The HMW has strong nonlinearity and fast time variance, and there are strong aerodynamic uncertainties under hypersonic flight; at the same time, there are problems such as system non-modeling dynamics and internal and external perturbations during the flight of the HMW. Therefore, \(d_{i}(i=\alpha,\omega_{z})\) is a total disturbance considering various uncertainties. It should be noted that the disturbances are different in different environments, and the disturbances are also different at different moments in the same environment; thus, the disturbances have strong randomness. To ensure the robustness of the controller, in the control law shown in Equations (31) and (37), in order to eliminate the influence of disturbance \(d_{i}(i=\alpha,\omega_{z})\), the switching gain \(D_{i}\) is utilized. Under this circumstance, the design of this control law depends highly on the estimation of the upper bound of the disturbance since the upper bound of the disturbance is hard to obtain. In addition, due to the strong randomness of disturbance, its upper bound is possibly large, thus leading to a large switching gain \(D_{i}\), causing serious chattering in the control system, which is not applicable in practical engineering. In order to improve the robustness of the system and avoid chattering, the extended state observer [40] technique is used to estimate disturbances online and to compensate for the control laws accordingly. In this paper, the fixed-time extended disturbance observer for the AOA channel is designed as follows:  
\[\left\{ \begin{array}{l}\hat{\mathbf{a}}=\kappa_{1}\varphi_{1}\Big{(}\frac{ \alpha-\hat{\mathbf{a}}}{\varepsilon_{1}}\Big{)}+d_{\alpha}+f_{\alpha}+\omega _{z}\\ \hat{d}_{\alpha}=\frac{\kappa_{2}}{\varepsilon_{1}}\varphi_{2}\Big{(}\frac{ \alpha-\hat{\mathbf{a}}}{\varepsilon_{1}}\Big{)}\end{array} \right. \tag{38}\]Similarly, the fixed-time disturbance observer for the pitch rate channel can be designed as  
\[\left\{ \begin{array}{l}\dot{\omega}_{z}=\kappa_{3}\varphi_{3}\Big{(}\frac{ \omega_{z}-\hat{\omega}_{z}}{\varepsilon_{2}}\Big{)}+d_{\omega_{z}}+f_{\omega_{ z}}+g_{1}\delta\\ \dot{d}_{\omega_{z}}=\frac{\kappa_{4}}{\varepsilon_{2}}\varphi_{4}\Big{(}\frac {\omega_{z}-\hat{\omega}_{z}}{\varepsilon_{2}}\Big{)}\end{array} \right. \tag{39}\]  
where \(\hat{\lambda}\), \(\hat{d}_{\alpha}\), \(\hat{\omega}_{z}\), and \(\hat{d}_{\omega_{z}}\) are the estimations of \(\alpha\), \(d_{\alpha}\), \(\omega_{z}\), and \(d_{\omega_{z}}\), respectively; \(\kappa_{1}\), \(\kappa_{2}\), \(\kappa_{3}\), \(\kappa_{4}\) are positive constants, satisfying \(\kappa_{1},\kappa_{2}>0\), \(\kappa_{1}\geq 2\sqrt{\kappa_{2}}\) and \(\kappa_{3},\kappa_{4}>0\), \(\kappa_{3}\geq 2\sqrt{\kappa_{4}}\); \(\varepsilon_{1},\varepsilon_{2}\in(0,1)\) are amplification factors. The nonlinear functions \(\varphi_{1}\) and \(\varphi_{2}\) are defined as  
\[\left\{ \begin{array}{l}\varphi_{1}(x)=\text{sig}(x)^{\alpha_{1}}+\text{sig} (x)^{\beta_{1}}\\ \varphi_{2}(x)=\text{sig}(x)^{2\alpha_{1}-1}+\text{sig}(x)^{2\beta_{1}-1}\end{array} \right. \tag{40}\]  
And the nonlinear functions \(\varphi_{3}\) and \(\varphi_{4}\) are defined as  
\[\left\{ \begin{array}{l}\varphi_{3}(x)=\text{sig}(x)^{\alpha_{2}}+\text{sig} (x)^{\beta_{2}}\\ \varphi_{4}(x)=\text{sig}(x)^{2\alpha_{2}-1}+\text{sig}(x)^{2\beta_{2}-1}\end{array} \right. \tag{41}\]  
where \(\alpha_{j}\in(0.5,1)\), \(\beta_{j}\in(1,1.5)\). **Theorem 1**. : _Consider the system of the first equation of Equation (28), and design the fixed-time disturbance observer as Equations (38) and (39) with the appropriate parameters; then, the disturbance estimation errors \(\left|d_{\alpha}-\hat{d}_{\alpha}\right|\) and \(\left|d_{\omega_{z}}-\hat{d}_{\omega_{z}}\right|\) will converge to a neighborhood of the origin in fixed time._  
Proof of Theorem 1.: Firstly, define the observer estimation errors as  
\[\left\{ \begin{array}{l}e_{1}{=}\alpha-\hat{\lambda}\\ e_{2}=d_{\alpha}-\hat{d}_{\alpha}\end{array} \right.