## 2 Problem Formulation
### Nonlinear Attitude Model of HSV with Unexpected Centroid Shift
#### 2.1.1 Attitude Angles Affine Nonlinear Equations of HSV with Centroid Shift  
According to [8; 25], the attitude angle subsystem of HSV with unexpected centroid shift can be developed as:  
\[\dot{\alpha} =q-\big{(}p\cos\alpha+r\sin\alpha\big{)}\tan\beta+\big{(}c_{31} \Delta x+c_{32}\Delta y+c_{33}\Delta z\big{)}\big{/}(mV\cos\beta)\] \[\quad+\big{(}-L+mg\cos\gamma\cos\mu\big{/}(mV\cos\beta) \tag{1}\] \[\dot{\beta} =p\sin\alpha-r\cos\alpha+\big{(}Y+mg\cos\gamma\sin\mu\big{)} \big{/}mV+\big{(}c_{21}\Delta x+c_{22}\Delta y+c_{23}\Delta z\big{)}\big{/}mV\] (2) \[\dot{\mu} =\sec\beta(p\cos\alpha+r\sin\alpha)-\tan\beta(-L+mg\cos\gamma \cos\mu)\big{/}mV\] \[\quad+\tan\gamma\Big{\{}-(\sigma_{1}\sigma_{2}+\dot{\sigma}_{3}) \bar{p}_{x}^{p}+(\sigma_{1}^{2}+\sigma_{3}^{2})\bar{p}_{y}^{p}-(\sigma_{2} \sigma_{3}-\dot{\sigma}_{1})\bar{p}_{z}^{p}\Big{\}}\Big{/}V\] \[\quad-\tan\beta(c_{31}\Delta x+c_{32}\Delta y+c_{33}\Delta z) \big{/}mV+\tan\gamma(Y\cos\mu+L\sin\mu)\big{/}mV \tag{3}\]  
where \(V\) is the velocity vector of the system and \(\gamma\) represents the flight path angle. \(\alpha,\ \beta\) and \(\mu\) are the attack angle, sideslip angle and roll angle, respectively, and \(\omega=[p,q,r]^{T}\) stands for the attitude angular rate vector that consists of rolling rate (\(p\)), pitching rate (\(q\)) and yawing rate (\(r\)). \(L\) and \(Y\) are the lift and side force of HSV. As for the influences of unexpected centroid shift, \(\bar{\rho}=[\Delta x,\Delta y,\Delta z]^{T}\) denotes the variation of the center-of-mass location (\(O_{b}^{t}\)) with respect to the original centroid (\(O_{b}\), as shown in Figure 1). Moreover, \(C_{i,j},i,j=1,2,3\) are intermediate variables, shown in (A1), and \(\sigma_{1},\sigma_{2}\) and \(\sigma_{3}\) are the angular velocities in view of flight-path coordinate frame with respect to the earth-surface inertial coordinate frame, provided in (A2). \(\bar{p}_{x}^{p},\bar{p}_{y}^{p}\) and \(\bar{p}_{z}^{p}\) are the functions of \(\Delta x,\Delta y\) and \(\Delta z\), standing for the influence of centroid shift on the system of HSV, seen in (A3). Finally, \(m\) is the mass of HSV. In what follows, in order to facilitate the subsequent theoretical analysis, based on (1)-(3), the affine attitude angle subsystem of HSV is obtained as  
\[\dot{\Omega} =f_{s}+\Delta f_{s}+g_{s}(I-F)\omega_{c}+d_{s} \tag{4}\]  
where \(\Omega=[\alpha,\beta,\mu]^{T}\) is defined to describe the attitude angle vector. \(f_{s}=[f_{\alpha}\ f_{\beta}\ f_{\mu}]^{T}\) denotes the system matrix, and \(\Delta f_{s}=[\Delta f_{\alpha}\ \Delta f_{\beta}\ \Delta f_{\mu}]^{T}\) represents the uncertainty of \(f_{s}\). In addition, \(\omega_{c}\) is the tracking index vector of attitude angular rate based on attitude angle tracking error. \(F\) stands for the fault factor of the slow-loop controller and the details are seen in (5), which represents a kind of signal gain fault [16]. \(D_{s}=\Delta f_{s}+g_{s}F\omega_{c}+d_{s}\) denotes the influence of centroid shift, external disturbance and virtual controller gain fault of slow-loop of HSV, satisfying that \(D_{s}\) is bounded. The details are described as follows:  
\[f_{\alpha} =\big{(}-L+mg\cos\gamma\cos\mu\big{)}\big{/}(mV\cos\beta),f_{ \beta}=\big{(}Y+mg\cos\gamma\sin\mu\big{)}\big{/}mV\] \[f_{\mu} =\big{(}Y\cos\mu+L\sin\mu\big{)}\tan\gamma\big{/}mV_{p}-\big{(}-L+ mg\cos\gamma\cos\mu\big{)}\tan\beta/mV_{p}\] \[\Delta f_{\alpha} =\big{(}c_{31}\Delta x+c_{32}\Delta y+c_{33}\Delta z\big{)}\big{/} (mV\cos\beta),\Delta f_{\beta}=\big{(}c_{21}\Delta x+c_{22}\Delta y+c_{23} \Delta z\big{)}\big{/}mV\] \[\quad+\big{(}-L+mg\cos\gamma\cos\mu\big{/}(mV\cos\beta) \tag{1}\]0 \[\quad+\big{(}-L+mg\cos\gamma\cos\mu\big{/}(mV\cos\beta) \tag{1}\]1 \[g_{s} =\left[ \begin{array}{ccc}-\tan\beta\cos\alpha&1&-\tan\beta\sin \alpha\\ \sin\alpha&0&-\cos\alpha\\ \sec\beta\cos\alpha&0&\sec\beta\sin\alpha\end{array} \right],F_{i}(t-T)=\left\{ \begin{array}{ccc}F_{i}(t-T)=0,&t\leq T\\ F_{i}(t-T)=1,&t\geq T\end{array} \right.