# Variable Structure PID Controller for Satellite Attitude Control Considering Actuator Failure
## 2 Methodology  
The dynamic model of the flexible satellite using the [PERSON] theorem under the hypothesis of small elastic deformations could be written as follows.  
\[\mathbf{J}\dot{\mathbf{\omega}}=-\mathbf{\omega}^{\times}\mathbf{J}\mathbf{\omega}+\mathbf{u}+\mathbf{d} \tag{1}\]  
where \(\mathbf{J}\) is the inertia matrix of the satellite, which is a symmetric positive definite matrix. Generally, \(\mathbf{J}\) cannot be accurately known and could be written as \(\mathbf{J}=\mathbf{\hat{J}}+\mathbf{\hat{J}}\), where \(\mathbf{\hat{J}}\) is the inertia matrix best estimate and \(\mathbf{\hat{J}}\) is the error inertia matrix.  
In Equation (1), \(\mathbf{d}\) is the unknown disturbance torque with norm upper bound \(\|\mathbf{d}\|\leq\overline{d}\), and \(\mathbf{u}\) is the control torque to be designed. The product matrix \(\mathbf{r}^{\times}\) of the three-dimensional vector \(\mathbf{r}\) is defined as follows:  
\[\mathbf{r}^{\times}=\left[\begin{array}{ccc}0&-r_{3}&r_{2}\\ r_{3}&0&-r_{1}\\ -r_{2}&r_{1}&0\end{array}\right] \tag{2}\]  
The product matrix \(\mathbf{r}^{\times}\) has an important property, which will be used in a later part of this paper, that the eigenvalues of \(\mathbf{r}^{\times}\) satisfies  
\[\lambda(\mathbf{r}^{\times})=0,\left\|\mathbf{r}\right\|_{2} \tag{3}\] \[\lambda_{\text{max}}(\mathbf{r}^{\times})=\left\|\mathbf{r}\right\|_{2}\]  
In order to simplify the text, the maximum and minimum eigenvalue of matrix \(\mathbf{A}\) is described as \(\lambda_{M}(\mathbf{A}),\lambda_{m}(\mathbf{A})\).  
The kinetic model of the satellite could be written as follows:  
\[\dot{\mathbf{q}}=\left[\begin{array}{c}\dot{q}_{0}\\ \dot{\mathbf{q}}_{v}\end{array}\right]=\left[\begin{array}{c}-\frac{1}{2}q_{v}^{T} \mathbf{\omega}\\ \frac{1}{2}(q_{0}I_{3}+\mathbf{q}_{v}^{\times})\mathbf{\omega}\end{array}\right]=\frac {1}{2}\left[\begin{array}{c}-\mathbf{q}_{v}^{T}\\ \mathbf{F}\end{array}\right]\mathbf{\omega} \tag{4}\]  
where \(\mathbf{F}=q_{0}\mathbf{I}_{3}+\mathbf{q}_{v}^{\times}\), and the eigenvalue of \(\mathbf{F}\) satisfies  
\[\begin{array}{c}\lambda(\mathbf{F})=|q_{0}|,1\\ \lambda_{M}(\mathbf{F})=1\end{array} \tag{5}\]  
In addition, the Euler angle/axis description for satellite attitude is widely used, and its kinetic model is given as follows:  
\[\left\{\begin{array}{c}\dot{\mathbf{e}}=\frac{1}{2}\mathbf{e}^{\times}\big{(}\mathbf{I} _{3}-\cot\frac{\varphi}{2}\mathbf{e}^{\times}\big{)}\mathbf{\omega}\\ \dot{\varphi}=\mathbf{e}^{T}\mathbf{\omega}\end{array}\right. \tag{6}\]  
where \(\mathbf{e}\) is defined as the Euler axis, and \(\varphi\) is defined as the Euler angle.  
The next two assumptions used in this paper are introduced as follows.  
**Assumption 1**.: _It is assumed that the disturbance torque \(\mathbf{d}\) is the norm upper bounded, i.e.,_  
\[\|\mathbf{d}\|\leq\overline{d} \tag{7}\]  
_where \(\overline{d}\) is a positive scalar._  
**Assumption 2**.: _It is assumed that, in this paper, if matrix \(\mathbf{A}\) satisfies \(\mathbf{A}\geq 0\), it means that matrix \(\mathbf{A}\) is a positive definite matrix, i.e., \(\mathbf{x}^{T}\mathbf{Ax}\geq 0\) holds for any non-zero vector \(\mathbf{x}\)._