## 3 Controller Design
### Fixed-Time Backstepping Controller Design  
Generally, to achieve the desired flight profile of the HMW, the purpose of controller design is to track the AOA command precisely. And the backstepping control is a feasible method with the step-by-step recursive design process for this nonlinear system [35; 36]. Therefore, in order to apply the backstepping technique, the longitudinal short-period dynamic equations with a strict feedback form are considered as  
\[\left\{\begin{array}{l}\dot{\alpha}=\omega_{z}+f_{\alpha}+d_{\alpha}\\ \dot{\omega}_{z}=g_{1}\dot{\delta}+f_{\omega_{z}}+d_{\omega_{z}}\end{array}\right. \tag{28}\]  
where \(g_{1}\)= \(QS_{0}L_{0}\overline{m}_{z,\lambda}^{\alpha}/J_{z}\); \(f_{\alpha}=\left[-L+mg\cos(\dot{\theta}-\alpha)-F_{Sx}\sin\alpha-F_{Sy}\cos \alpha\right]/(mV)\); \(f_{\omega_{z}}=\left[QS_{0}L_{0}\left(\overline{m}_{z,\lambda}^{\alpha}+ \overline{m}_{z,\lambda}^{\beta z}\delta^{2}+\overline{m}_{z,\lambda}^{\beta 3}\delta^{3}\right)+M_{SG}+M_{SD}\right]/J_{z}\); and \(d_{\alpha}\) and \(d_{\omega_{z}}\) are disturbances in the AOA channel and pitch rate channel, respectively.  
**Assumption 1** ([37; 38]).: _The unknown disturbance \(d_{i}(i=\alpha,\omega_{z})\) and its first-time derivatives are bounded, satisfying \(|d_{i}|\leq D_{i}\) and \(\left|\dot{d}_{i}\right|\leq\overline{D}_{i}\), where \(D_{i}\) and \(\overline{D}_{i}\) are the upper bound._  
According to the backstepping control methodology, there are two steps for the attitude controller design.  
Step 1: Define the AOA tracking error as  
\[s_{1}=\alpha-\alpha_{c} \tag{29}\]  
where \(\alpha_{c}\) is the reference AOA command.  
Taking the time derivative and substituting (28), the AOA tracking error dynamic equations can be obtained as  
\[\dot{s}_{1}=\omega_{z}+f_{\alpha}+d_{\alpha}-\dot{\alpha}_{c} \tag{30}\]  
The virtual control input \(\omega_{zc}\) is designed as  
\[\omega_{zc}=-\big{(}f_{\alpha}-\dot{\alpha}_{c}+a_{1}\text{sig}(s_{1})^{r_{1}} +c_{1}\text{sig}(s_{1})^{r_{2}}+D_{\alpha}\text{sign}(s_{1})\big{)} \tag{31}\]  
where \(a_{1}\), \(c_{1}\), and \(D_{\alpha}\) are positive constants, \(0<\)\(r_{1}<1\), \(r_{2}>1\).  
To avoid the problem of the \"explosion of terms\" caused by directly differentiating the virtual control \(\omega_{zc}\), the idea of dynamic surface control [39] (DSC) is used. Different from the traditional DSC which uses a first-order filter, in this article, a nonlinear first-orderfilter is designed, to guarantee the fixed-time stability of the overall system. The nonlinear first-order filter is given as  
\[\dot{\omega}_{zd}=-\frac{1}{\tau}\text{sig}(\omega_{zd}-\omega_{zc})^{r_{1}}- \frac{1}{\tau}\text{sig}(\omega_{zd}-\omega_{zc})^{r_{2}} \tag{32}\]  
where \(\tau\) is the time constant of the filter and \(\omega_{zd}\) is the output of \(\omega_{zc}\) after being filtered.  
Define the filtering error \(y_{1}\) and pitch rate tracking error \(s_{2}\) as follows:  
\[\left\{\begin{array}{l}y_{1}=\omega_{zd}-\omega_{zc}\\ s_{2}=\omega_{z}-\omega_{zd}\end{array}\right. \tag{33}\]  
Then, the AOA tracking error dynamic equation becomes  
\[\dot{s}_{1}=s_{2}+y_{1}+\omega_{zc}+f_{\alpha}+d_{\alpha}-\dot{\alpha}_{c} \tag{34}\]  
Step 2: The pitch rate tracking error dynamic equation can be expressed as  
\[\dot{s}_{2}=\dot{\omega}_{z}-\dot{\omega}_{zd} \tag{35}\]  
Substituting (28) into (35) gives  
\[\dot{s}_{2}=g_{1}\delta+f_{\omega_{z}}+d_{\omega_{z}}-\dot{\omega}_{zd} \tag{36}\]  
To achieve fixed-time convergence, the control input \(\delta\) is designed as  
\[\delta=-1/g_{1}\cdot\big{(}f_{\omega_{z}}-\dot{\omega}_{zd}+a_{2}\text{sig}(s _{2})^{r_{1}}+c_{2}\text{sig}(s_{2})^{r_{2}}+s_{1}+D_{\omega_{z}}\text{sign}( s_{2})\big{)} \tag{37}\]  
where \(a_{2},c_{2},\) and \(D_{\omega_{z}}\) are positive constants.