# An Extended Proxy-Based Sliding Mode Control of Pneumatic Muscle Actuators
## 3 Controller Design and Stability Analysis
### Stability Analysis  
By substituting the controllers (11) into the model (6), the closed-loop system model is given by:  
\[\left\{ \begin{array}{l}\ddot{x}=K_{P}\left(p-x\right)+\ddot{x}_{d}+H\left( \dot{x}_{d}-\dot{x}\right)+I\left(x_{d}-x\right)+d(t)\\ m_{p}\ddot{p}=f_{SMC}-f_{c}=\Gamma\text{sgn}\left(\sigma_{p}\right)-K_{P}\left( p-x\right)+m_{p}\ddot{x}_{d}+m_{p}H\left(\dot{x}_{d}-\dot{p}\right)+m_{p}I \left(x_{d}-p\right)\end{array} \right. \tag{12}\]  
The stability of the closed-loop system can be analyzed as follows. First, we prove the stability when the lumped disturbance \(d=0\). **Theorem 1**. : _Consider the system described by (6) with \(d=0\), and the controllers are assumed as (11). The trajectory of system (6) can then be driven onto the sliding mode manifolds (10)._  
Proof. : Introduce the following quadratic function:  
\[V_{1}=\frac{1}{2}m_{p}\sigma_{p}^{\ 2}. \tag{13}\]  
By differentiating the above equation, we have  
Figure 2: The physical interpretation of proposed EPSMC. \[ \begin{array}{l}\dot{V}_{1}=\sigma_{p}m_{p}\phi_{p}\\ =\sigma_{p}\left(m_{p}\dot{x}_{d}-m_{p}\ddot{p}+m_{p}H\left(\dot{x}_{d}-\dot{p} \right)+m_{p}I\left(x_{d}-p\right)\right)\\ =\sigma_{p}\left(m_{p}\dot{x}_{d}-\left(\Gamma\text{sgn}\left(\sigma_{p}\right) -K_{P}\left(p-x\right)+m_{p}\dot{x}_{d}+m_{p}H\left(\dot{x}_{d}-\dot{p} \right)+m_{p}I\left(x_{d}-p\right)\right)\\ +\;m_{p}H\left(\dot{x}_{d}-\dot{p}\right)+m_{p}I\left(x_{d}-p\right)\right)\\ =\sigma_{p}\left(-\Gamma\text{sgn}\left(\sigma_{p}\right)+K_{P}\left(p-x \right)\right)\\ =-\Gamma\left|\sigma_{p}\right|+K_{P}\left(p-x\right)\left(\dot{x}_{d}-\dot{p }\right)+K_{P}H\left(p-x\right)\left(x_{d}-p\right)+K_{P}I\left(p-x\right) \int\left(x_{d}-p\right)dt\end{array}  \tag{14}\]  
Next, introduce another quadratic function:  
\[V_{2}=\frac{1}{2}\sigma_{x}^{2}. \tag{15}\]  
By differentiating the above equation, it follows from (12) that  
\[ \begin{array}{l}\dot{V}_{2}=\sigma_{x}\dot{\sigma}_{x}\\ =\sigma_{x}\left(\ddot{x}_{d}-\ddot{x}+H\left(\dot{x}_{d}-\dot{x}\right)+I \left(x_{d}-x\right)\right)\\ =\sigma_{x}\left(\ddot{x}_{d}-\left(K_{P}\left(p-x\right)+\ddot{x}_{d}+H\left( \dot{x}_{d}-\dot{x}\right)+I\left(x_{d}-x\right)\right)+H\left(\dot{x}_{d}- \dot{x}\right)+I\left(x_{d}-x\right)\right)\\ =\sigma_{x}\left(-K_{P}\left(p-x\right)\right)\\ =K_{P}\left(x-p\right)\left(\dot{x}_{d}-\dot{x}\right)+K_{P}H\left(x-p\right) \left(x_{d}-x\right)+K_{P}I\left(x-p\right)\int\left(x_{d}-x\right)dt\end{array}  \tag{16}\]  
Introducing the third quadratic function:  
\[V_{3}=\frac{1}{2}K_{P}(p-x)^{2}+\frac{1}{2}K_{P}I\bigg{(}\int\left(p-x\right) dt\bigg{)}^{2}. \tag{17}\]  
Similar to \(V_{1}\) and \(V_{2}\), we have  
\[\dot{V}_{3}=K_{P}\left(p-x\right)\left(\dot{p}-\dot{x}\right)+K_{P}I\left(p-x \right)\int\left(p-x\right)dt \tag{18}\]  
Choose the following Lyapunov function candidate:  
\[V=V_{1}+V_{2}+V_{3}. \tag{19}\]  
The derivative of \(V\) can then be calculated by combing (14), (16) and (18). It follows that:  
\[ \begin{array}{l}\dot{V}=\ddot{V}_{1}+\ddot{V}_{2}+\ddot{V}_{3}\\ =-\Gamma\left|\sigma_{p}\right|-K_{P}H\left(p-x\right)^{2}\leq 0\end{array}  \tag{20}\]  
This completes the proof. If there is no disturbance \(d(t)\), from Theorem 1 we know that the system trajectory will finally be driven onto the sliding mode manifolds \(\sigma_{p}=0\) and \(\sigma_{x}=0\), on which the tracking error \(x_{d}-x\) will converge to zero. On the other hand, when the lumped disturbance \(d\) is unavoidable, the system is found to be passive through the following theorem. **Theorem 2**. : _The second-order nonlinear system_  
\[ \begin{array}{l}\dot{x}=f(x,\dot{x})+b(x,\dot{x})u(t)+d(t)\\ y=-\sigma_{x}\end{array}  \tag{21}\]  
_mapping \(d(t)\) to \(y\) is passive._  
**Proof. ** Let us choose the storage function as (19). It can be found that the derivatives of \(V_{1}\) and \(V_{3}\) are the same even \(d\
eq 0\). For \(V_{2}\), we have  
\[