## 2 System Description and Preliminaries  
After linearizing the model of air-ground platform in reference [19], the state space equation is obtained, and the time-varying delay term is introduced to obtain the dynamic Equations (1) and (2). The model in reference [19] is shown in Appendix A. Considering the operation efficiency and the phenomenon of universal joint lock, the quaternion differential equation is Equation (3). \[\dot{x}(t)=Ax(t)+A_{m}x(t-d(t))+B(I+F(t))u(t)+Dv(t) \tag{1}\]  
\[y=Cx(t) \tag{2}\]  
\[\dot{q}=G(q)\big{[}x_{1}\quad x_{2}\quad x_{3}\big{]}^{T} \tag{3}\]  
where \(x(t)= \begin{bmatrix}x_{1}&x_{2}&x_{3}&x_{4}&x_{5}&x_{6}\end{bmatrix} ^{T}\in R ^{n}\) is roll angle, pitch angle, yaw angle, and corresponding angular rates; \(u(t)= \begin{bmatrix}F_{1}&F_{2}&F_{3}&F_{4}\end{bmatrix} ^{T}\in R^{m}\) is input for control; \(y(t)\in R^{q}\) is output vector for attitude angle. \(d(t)\), \(F(t)\) are system delay and time-varying term, respectively. \(A\), \(A_{m}\), \(B\), \(C\), and \(D\) are known matrices. \(v(t)\) is external interference. As shown in Figure 1, \(b\), \(d\) are, respectively, rotor lift coefficient and drag coefficient. \(l\) is the geometric distance from the center of the rotor to the center of rotation. Quaternion \(q= \begin{bmatrix}q_{1}&q_{2}&q_{3}&q_{0}\end{bmatrix} ^{T}= \begin{bmatrix} \overset{\rightarrow}{q}&q_{0}\end{bmatrix} ^{T}\) represents the rotation relationship of the body system relative to the inertial system, where \(\overset{\rightarrow}{q}^{\times}\) is skew symmetric matrix. The attitude quaternion satisfies Equations (4) and (5). \[\overset{\rightarrow}{q}^{T}\overset{\rightarrow}{q}+q_{0}^{2}=1 \tag{4}\]  
\[G(q)=\frac{1}{2} \begin{bmatrix}\overset{\rightarrow}{q}^{\times}+q_{0}I_{3} \\ \overset{\rightarrow}{q}\end{bmatrix}  \tag{5}\]  
For the attitude control system of air-ground platform, the time-varying term includes the current moment of inertia of the platform \(J\) changing \(\Delta J\) caused by the change of the total mass of the system and the change of the failure of the system actuator with time. In the paper, only additive bias fault and actuator efficiency loss (LOE) are considered for actuator time-varying fault. The additive fault is represented  
Figure 1: Mechanical model of air-ground platform. input of the rotor motor, and the multiplicative fault is represented by the loss of motor efficiency. Therefore, the actuator fault is modeled as Equation (6). \[u=(I-E)u_{c}+u_{a} \tag{6}\]  
In Equation (6), \(E\) is actuator efficiency matrix which satisfies \(E=diag\{e_{1},e_{2},e_{3},e_{4}\}\), \(e_{i}\in(0,1]\). \(e_{i}=0\) indicates that the actuator mechanism is completely damaged and \(e_{i}=1\) shows actuator intact. \(u_{a}\) is an additive offset input. The overall fault size of the system is expressed as \(f\), and Equation (1) is rewritten in combination with Equation (6) as:  
\[\dot{x}(t)=Ax(t)+A_{m}x(t-d(t))+B(I+F(t))u_{c}+f+Dv(t) \tag{7}\]  
\[f=-B(I+F(t))(Eu_{c}-u_{a}) \tag{8}\]  
The overall control structure of the attitude system is shown in Figure 2. The following assumptions are made for the system. **Assumption 1**. : \((A,B)\)_, \((A,C)\) are controllable and observable._  
**Assumption 2**. : _The time-varying term of the moment of inertia in the system \(F(t)\) is bounded and satisfies \(\overline{f}\leq F(t)\leq\underline{f}\). The time-delay term is continuously bounded and differentiable which satisfies \(0\leq d(t)\leq h\), \(\dot{d}(t)\leq\tau\leq 0\). The overall fault size of the system meets \(\|f\|\leq\mu\), \(\|\dot{f}\|\leq\delta\), where \(h\), \(\tau\), \(\mu\), \(\delta\) are constants._  
**Assumption 3**. : _The external disturbance is bounded and satisfies \(\|Dv(t)\|\leq\overline{d}\), where \(\overline{d}\) is constant._  
**Remark 1**. : _The controllability and observability of \((A,B)\), \((A,C)\) are the premise of system fault estimation and regulation. The state space model of air-ground platform in the paper conforms to controllability and observability.