# Variable Structure PID Controller for Satellite Attitude Control Considering Actuator Failure
## 3 Results
### Integral Partition Design  
Although the PD controller for satellite attitude control has such advantages as discussed above, one of its main drawbacks is its low robustness to disturbances, especially high frequency random noise. Noting that this type of noise is quite common in satellite engineering practice (always caused by electric motors), and this type of noise is hard to measure and resist, it is necessary to take this perturbation into consideration when designing satellite attitude controllers. In engineering practice, a PID controller with an integral term is an effective method to resist this type of perturbation, but the integral term slow down system convergence rate and system is unstable when the integral term is not designed properly. Based on the discussion above, the idea to design a variable structure PID controller could be concluded as follows: (1) when the system state is away from its equilibrium point, the controller maintains a PD controller, as stated in the previous section, to improve the system convergence rate, and the drawback of an integral term such as a low convergence rate could be avoided; (2) noting that the influence of perturbations mainly exists when the system state approaches its equilibrium point, the integral term is added into the controller to resist the influence of disturbance torque; (3) the main difficulty of designing a PID controller is analyzing the system stability, and noting that the condition of the integral term is that the system nearly reaches its equilibrium point, some assumptions could be made when analyzing system stability. The variable structure PID controller proposed in this paper could be written as follows:  
\[\mathbf{u}= \begin{cases}-k_{d}\mathbf{\omega}-k_{p}\mathbf{q}_{\gamma}-\overline{d}sign( \mathbf{\omega})+\mathbf{\tau}&\|\mathbf{q}_{\gamma}\|\geq\alpha\\ -k_{d}\mathbf{\omega}-k_{p}\mathbf{q}_{\gamma}-\overline{d}sign(\mathbf{\omega})&\beta\leq \|\mathbf{q}_{\gamma}\|<\alpha\\ -k_{d}\mathbf{\omega}-k_{p}\mathbf{q}_{\gamma}-k_{1}\mathbf{v}&\|\mathbf{q}_{\gamma}\|<\beta \end{cases}  \tag{16}\]  
where \(k_{d},k_{p},k_{1}\), and \(\alpha,\beta\) are all positive scalars, the definition of control term \(\mathbf{\tau}\) is totally same as Equation (10), and the definition of integral term \(\mathbf{v}\) is given as follows:  
\[\dot{\mathbf{v}}=\mathbf{q}_{\gamma^{\prime}}\mathbf{v}(0)=0_{3\times 1} \tag{17}\]  
**Remark 3**. : _Based on the controller structure of (16), it could be found that this controller consists of three stages; the first two stages are totally the same as variable structure PD controller (8), to improve system performance when the system state is relatively large, and the third stage is the integral partition term to resist disturbance when the system state approaches its equilibrium point. It could also be found that there is no sign function term during this stage to avoid the high frequency vibration of system state. In addition, the parameters \(\alpha\) and \(\beta\) are designed to detect whether the system state is near zero or not, and the selection principle could be made depending on the system capability and control mission. The larger torque the actuator can generate and the faster a convergence rate is needed, a smaller \(\alpha\) and \(\beta\) should be selected, and vice versa._The next step is to discuss system stability under controller (16). Noting that the first two stages are totally the same as controller (8), and system stability during these stages has been proven in the previous section, this section only discusses the stability during the third stage. The Lyapunov function is selected as follows:  
\[V_{2}=\frac{1}{2}\mathbf{\omega}^{T}\mathbf{I}\mathbf{\omega}+l_{1}\mathbf{q}_{v}^{T}\mathbf{I}\mathbf{ \omega}+2l_{2}(1-q_{0})+\frac{1}{2}l_{3}\mathbf{v}^{T}\mathbf{v}+l_{4}\mathbf{v}^{T}\mathbf{J} \mathbf{\omega} \tag{18}\]  
where \(l_{1},l_{2},l_{3},l_{4}\) are all positive scalars.