# Gain-Scheduled Model Predictive Control for
## 4 Simulations
### Simulations II  
In the second set of simulations, the parameters of the CIPS are also listed in Table 1. In addition, the Gaussian white noise is used as the external disturbance. Moreover, according to the applications of the disturbances \(f_{m}\) and \(f_{M}\), the following four cases are considered.  
* **Case (II-A):** The CIPS is a nominal system, i.e., \(f_{m}=0\) and \(f_{M}=0\) in the simulation.
* **Case (II-B):** The SIP of the CIPS is subjected to the disturbance, \(f_{M}\), in the form of Gaussian white noise. \(f_{m}\) is shown in Figure 12.
* **Case (II-C):** The cart of the CIPS is subjected to the disturbance, \(f_{M}\), in the form of Gaussian white noise. \(f_{m}\) is shown in Figure 13.
* **Case (II-D):** The SIP of the CIPS is subjected to the disturbance, \(f_{M}\), in the form of Gaussian white noise (see Figure 12). Additionally, the cart of the CIPS is subjected to the disturbance, \(f_{M}\), in the form of Gaussian white noise (see Figure 13).  
First, the desired position of the cart is set at \(x_{d}=1\) and the desired pendulum angle of the SIP is set at \(\theta_{d}=0\). The simulation results from Case (II-A) to Case (II-D) are shown in Figure 14. The control inputs from Case (II-A) to Case (II-D) are shown in Figure 14. For Case (II-A), the angle \(\theta\) of the SIP is stabilized at the desired pendulum angle (i.e., \(\theta\rightarrow\theta_{d}=0\)) while the cart reaches the desired position (i.e., \(x\to x_{d}=1\)). Moreover, both the angular velocity \(\dot{\theta}\) of the SIP and the velocity \(\dot{x}\) of the cart converge to zero, i.e., \(\dot{\theta}\to 0\) and \(\dot{x}\to 0\). From Case (II-B) to Case (II-D), the angle \(\theta\) and angular velocity \(\dot{\theta}\) of the SIP, as well as the position \(x\) and the velocity \(\dot{x}\) of the cart exist disturbances. However, under the regulation of the proposed gain-scheduled model predictive control method, the angle \(\theta\) of the SIP approaches zero and the position \(x\) of the cart approaches the desired position \(x_{d}\) when reaching the stabilization phase of the CIPS. In addition, both the angular velocity of the SIP and the velocity of the cart approach zero.  
Then, the desired position of the cart is increased to \(x_{d}=20\), and the external disturbances \(f_{m}\) and \(f_{M}\) are set as Gaussian white noise. In addition, the desired pendulum angle of the SIP is set at \(\theta_{d}=0\). The simulation results from Case (II-A) to Case (II-D) are shown in Figure 15. The control inputs from Case (II-A) to Case (II-D) are shown in Figure 15. From Figure 15, the angle \(\theta\) of the SIP and the position \(x\) of the cart can be controlled to approach their desired values when applying the Gaussian white noise disturbance \(f_{m}\), or Gaussian white noise disturbance \(f_{M}\), or even both \(f_{m}\) and \(f_{M}\). In addition, after entering the stabilization phase, both the angular velocity \(\dot{\theta}\) of the SIP and the velocity \(\dot{x}\) of the cart can be gradually  
Figure 12: The Gaussian white noise, \(f_{m}\), applied to the SIP of the CIPS.  
Figure 13: The white disturbance, \(f_{M}\), applied to the cart of the CIPS.  
controlled to approach zero. So, the proposed gain-scheduled model predictive control method works well when \(x_{d}=20\). Finally, even the desired position of the cart is increased to \(x_{d}=40\), the SIP can still return to the neighborhood of the upright position and the cart is able to reach the neighborhood of the desired position (as shown in Figure 16). So, within a large motion range of the cart, the control objectives of the CIPS can be reached with or without considering Gaussian white noise disturbances. Thus, the proposed gain-scheduled model predictive control method for the CIPS is efficacious and applicable.  
Figure 14: Control results of the CIPS with Gaussian white noise disturbance when \(x_{d}=1\).  
Figure 15: Control results of the CIPS with Gaussian white noise disturbance when \(x_{d}=20\).