# Gain-Scheduled Model Predictive Control for
## 2 Dynamic Equation and Characteristics Analysis
### Analyses of Motion Characteristics and Control Process  
During the process of dynamic modeling, the friction force, \(f\), the disturbance, \(f_{M}\), acting on the SIP, and the disturbance, \(f_{M}\), acting on the cart are considered, resulting in the dynamic Equation (14) of the CIPS complex. The complex dynamic Equation (14) brings great difficulty when it comes to analyzing the motion characteristics of the CIPS.  
For the sake of simplicity, we employ the nominal dynamic equation of the CIPS to analyze its motion characteristics. When setting \(f_{m}=0\) and \(f_{M}=0\) in the dynamic Equation (14) of the CIPS, its nominal dynamic equation is obtained, i.e.,  
\[\begin{cases}\dot{X}_{1}=X_{2}\\ \dot{X}_{2}=\varpi\\ \dot{X}_{3}=X_{4}\\ \dot{X}_{4}=\frac{g\sin X_{3}}{l}+\frac{\varpi\cos X_{3}}{l}\end{cases} \tag{15}\]  
where \(\varpi=\frac{u-\eta X_{2}-m\sin X_{3}\left(1X_{4}^{2}+g\cos X_{3}\right)}{M+m \sin^{2}X_{3}}\).  
Let \(u=0\) and \(\dot{X}_{i}=0\)\((i=1,2,3,4)\); under these conditions, the equilibrium points of the CIPS are obtained, i.e.,  
\[\begin{cases}(x_{\epsilon_{1}},\theta_{\epsilon_{1}})=(x,0)\\ (x_{\epsilon_{2}},\theta_{\epsilon_{2}})=(x,\pi)\end{cases} \tag{16}\]  
where the equilibrium point \((x_{\epsilon_{1}},\theta_{\epsilon_{1}})\) represents the state where the cart is stationary at the \(x\) position of the \(x\)-axis and the SIP holds upright. The equilibrium point \((x_{\epsilon_{2}},\theta_{\epsilon_{2}})\) indicates that the cart is stationary at the \(x\) position of the \(x\)-axis, and the SIP is in a vertically downward position. In addition, \((x_{\epsilon_{1}},\theta_{\epsilon_{1}})\) is a stable equilibrium point of the CIPS, while \((x_{\epsilon_{2}},\theta_{\epsilon_{2}})\) is an unstable equilibrium point of the CIPS.  
In this paper, the initial states of the CIPS are \(X_{0}=\left[\begin{array}{cccc}0&0&0&0\end{array}\right]^{T}\), i.e., the cart is stationary and the SIP is located in the upright position. The control objective of the CIPS is to control its states to reach the desired states \(X_{2}=\left[\begin{array}{cccc}x_{2}&0&0&0\end{array}\right]^{T}\), where \(x_{2}\) denotes the desired position of the cart and \(X_{2}\) denotes the desired states of the CIPS. In more detail, the control objective can be categorized as follows:  
1. When the desired position \(x_{2}\in\left[-x_{\max},x_{\max}\right]\) is changed as a step setpoint, the cart can be controlled to reach this position, where \(x_{\max}\) is the maximum controllable motion range of the cart.
2. During the process of controlling the cart to track the desired position, \(x_{2}\), the SIP oscillates. When the cart reaches the desired position, \(x_{2}\), the pendulum angle of the SIP converges to zero, i.e., the SIP returns to the upright equilibrium point.
3. When considering the friction exerted to the cart, an impulse disturbance, \(f_{m}\), applied to the SIP, and an impulse disturbance applied to the cart, the above two control objectives can still be realized.  
The above control objectives require stabilizing the SIP at the upright position, i.e., the CIPS is stabilized at one of its unstable equilibrium points. These control objectives are challenging. In addition, the dynamic Equation (14) of the CIPS has strong nonlinearity, which increases the difficulty of this control task.