# Variable Structure PID Controller for Satellite Attitude Control Considering Actuator Failure
## 3 Results
### Integral Partition Design
\begin{array}{c}l_{2}=l_{1}k_{d}+k_{p}\\ l_{3}=l_{1}k_{I}+l_{4}k_{p}\end{array}  \tag{25}\]and the derivative of the Lyapunov function can be simplified as:  
\[ \begin{array}{rl}\dot{V}_{2}&=-\mathbf{\omega}^{T}(k_{d}\mathbf{I}_{3}-l_{1}\mathbf{I}\mathbf{F })\mathbf{\omega}-l_{1}k_{p}\mathbf{q}_{v}^{T}\mathbf{q}_{v}-l_{4}k_{1}\mathbf{v}^{T}\mathbf{v}-l_{ 4}\mathbf{q}_{v}^{T}\mathbf{l}\mathbf{\omega}-(k_{I}+l_{4}k_{d})\mathbf{\omega}^{T}\mathbf{v}\\ &\leq-\left[\begin{array}{c}\mathbf{\omega}\\ \mathbf{q}_{v}\\ \mathbf{v}\end{array}\right]^{T}\left[\begin{array}{ccc}k_{d}\mathbf{I}_{3}-l_{1} \mathbf{I}\mathbf{J}&\frac{1}{2}l_{4}\mathbf{J}&\frac{1}{2}(k_{I}+l_{4}k_{d})\mathbf{I}_{3}\\ \frac{1}{2}l_{4}\mathbf{J}&l_{1}k_{p}\mathbf{I}_{3}&0\\ \frac{1}{2}(k_{I}+l_{4}k_{d})\mathbf{I}_{3}&0&l_{4}k_{1}\mathbf{I}_{3}\end{array} \right]\left[\begin{array}{c}\mathbf{\omega}\\ \mathbf{q}_{v}\\ \mathbf{v}\end{array}\right]\end{array}  \tag{26}\]  
Hence, if the following inequality is satisfied, then \(\dot{V}_{2}\leq 0\), and the system is asymptotic stable:  
\[ \begin{array}{rl}4l_{1}k_{p}(k_{d}\mathbf{I}_{3}-l_{1}\mathbf{I})-l_{2}^{2}\mathbf{I}^{2 }>0\\ 4l_{1}l_{4}k_{p}k_{I}(k_{d}\mathbf{I}_{3}-l_{1}\mathbf{J})-l_{2}^{2}\mathbf{I}^{2}-(l_{2}+ l_{4}k_{d})\mathbf{I}_{3}>0\end{array}  \tag{27}\]  
Based on the discussion above, the constraints on control parameters could be concluded as follows:  
\[\left\{ \begin{array}{rl}2(l_{1}k_{d}+k_{p})\mathbf{I}_{3}-l_{1}^{2}\mathbf{I}\geq 0 \\ 2\big{(}l_{1}k_{d}+k_{p}\big{)}\big{(}l_{1}k_{I}+l_{4}k_{p}\big{)}\mathbf{I}_{3}-2 \big{(}l_{1}k_{d}+k_{p}\big{)}l_{4}^{2}\mathbf{J}-l_{1}^{2}\big{(}l_{1}k_{I}+l_{4} k_{p}\big{)}\mathbf{J}\geq 0\\ 4l_{1}k_{p}(k_{d}\mathbf{I}_{3}-l_{1}\mathbf{I})-l_{4}^{2}\mathbf{J}^{2}>0\\ 4l_{1}l_{4}k_{p}k_{I}(k_{d}\mathbf{I}_{3}-l_{1}\mathbf{J})-l_{4}^{2}\mathbf{J}^{2}-(l_{1}k _{d}+k_{p}+l_{4}k_{d})\mathbf{I}_{3}>0\end{array} \right. \tag{28}\]  
It could be found that, in constraint (28), there are some matrix calculations that are not convenient for manual calculation, and (28) could be further transformed to:  
\[\left\{ \begin{array}{rl}2\big{(}l_{1}k_{d}+k_{p}\big{)}-l_{1}^{2}\lambda_{M }(\mathbf{J})\geq 0\\ 2\big{(}l_{1}k_{d}+k_{p}\big{)}\big{(}l_{1}k_{I}+l_{4}k_{p}\big{)}-2\big{(}l_{ 1}k_{d}+k_{p}\big{)}l_{4}^{2}\lambda_{M}(\mathbf{J})-l_{1}^{2}\big{(}l_{1}k_{I}+l_ {4}k_{p}\big{)}\lambda_{M}(\mathbf{J})\geq 0\\ 4l_{1}k_{p}(k_{d}-l_{1}\lambda_{M}(\mathbf{J}))-l_{4}^{2}\lambda_{M}^{2}(\mathbf{J})> 0\\ 4l_{1}l_{4}k_{p}k_{I}(k_{d}-l_{1}\lambda_{M}(\mathbf{J}))-l_{4}^{2}\lambda_{M}^{2} (\mathbf{J})-\big{(}l_{1}k_{d}+k_{p}+l_{4}k_{d}\big{)}>0\end{array} \right. \tag{29}\]  
**Remark 4**. : _Both (28) and (29) are strict constraints on control parameters, which means that if any inequality is satisfied, the system asymptotic stability can be ensured. The difference between (28) and (29) is that (28) involves some matrix calculation and (29) is simply a scalar calculation. In addition, (29) is stricter than (28), which means that if (29) is satisfied, then (28) holds._  
In addition, noting that there exists unknown disturbance torque, the system state cannot stay at the equilibrium point but must pass through the equilibrium point continuously; hence, it is necessary to discuss the steady accuracy related to disturbance. Considering the disturbance torque, the derivative of the Lyapunov function can be re-calculated as:  
\[