# Gain-Scheduled Model Predictive Control for
## 4 Simulations
### Simulations I  
In the first set of simulations, the parameters of the CIPS are listed in Table 1. In addition, the pulse noise is used as the external disturbance. Im particular, the following four cases are considered:  
* **Case (I-A):** The CIPS is a nominal system, i.e., \(f_{m}=0\) and \(f_{M}=0\) in the simulation. * **Case (I-B):** The SIP of the CIPS is subjected to a pulse disturbance \(f_{m}\) with a magnitude of 2. As shown in Figure 7, for the ease of applying the pulse disturbance \(f_{m}\)  
Figure 9. The control inputs from Case (I-A) to Case (I-D) are shown in Figure 9e. For Case (I-A), the angle \(\theta\) of the SIP converges to the desired pendulum angle (i.e., \(\theta\rightarrow\theta_{d}=0\)) as the cart reaches the desired position (i.e., \(x\to x_{d}=1\)). Moreover, both the angular velocity of the SIP and the velocity of the cart converge to zero, i.e., \(\dot{\theta}\to 0\) and \(\dot{x}\to 0\). For Case (I-B), there is a jitter when the pulse disturbance \(f_{m}\) (see Figure 7) is applied to the SIP. However, the designed GMPC can control the cart to reach the desired position and stabilize the SIP in the upright position. Moreover, both the angular velocity of the SIP and the velocity of the cart can converge to zero. For Case (I-C), there is a jitter when the pulse disturbance \(f_{M}\) (see Figure 8) is applied to the cart. However, the designed GMPC can control the cart to reach the desired position and stabilize the SIP in the upright position and both the angular velocity of the SIP and the velocity of the cart can converge to zero. For Case (I-D), there are two jitters (see Figure 9). These two jitters occur when \(f_{m}\) is applied to the SIP (see Figure 7) and \(f_{M}\) applied to the cart (see Figure 8), respectively. However, even in this situation, the proposed GMPC can still control the CIPS to achieve the control objectives. When \(x_{d}\) is increased to \(x_{d}=20\) and the desired pendulum angle of the SIP is set at \(\theta_{d}=0\), the simulation results from Case (I-A) to Case (I-D) are shown in Figure 10. The control inputs from Case (I-A) to Case (I-D) are shown in Figure 10e. For Case (I-A), the angle \(\theta\) of the SIP converges to the desired pendulum angle (i.e., \(\theta\rightarrow\theta_{d}=0\)) as the cart reaches the desired position (i.e., \(x\to x_{d}=20\)). Near \(t_{11}=8\) (s), the above control objectives of the CIPS are reached. For Case (I-B), there is a jitter start from \(t_{11}=8\) (s), which corresponds to the start time of the pulse disturbance \(f_{m}\). For Case (I-C), there is a jitter start from \(t_{21}=15\) (s), which corresponds to the start time of the pulse disturbance \(f_{M}\). For Case (I-D), there are two jitters, which correspond to the application of the pulse disturbances \(f_{m}\) and \(f_{M}\), respectively. From Figure 10, it can be observed that even when \(f_{m}<f_{M}\), the jitter occurring upon applying the pulse disturbance \(f_{m}\) is larger than that resulting from applying the pulse disturbance \(f_{M}\). In addition, for Case (I-B), Case (I-C), and Case (I-D), the designed GMPC can regulate the CIPS to overcome the influences of the jitters and achieve the control objectives. These results indicate that using the proposed gain-scheduled model predictive control method, the SIP can be stabilized to the upright position while controlling the cart to reach a large desired position. Figure 9: Control results of the CIPS with pulse disturbances when \(x_{d}=1\). Further, even when \(x_{d}\) is increased to \(x_{d}=40\) and \(f_{m}\) and \(f_{M}\) are exerted, the SIP can still return to its upright position and the cart is able to reach the desired position (as shown in Figure 11). So, within a large motion range of the cart, the control objectives of the CIPS can be reached with or without considering the disturbances. Thus, the proposed gain-scheduled model predictive control method for the CIPS is effective and superior. Figure 11: Control results of the CIPS with pulse disturbances when \(x_{d}=40\).