# Gain-Scheduled Model Predictive Control for
## 5 Conclusions  
A gain-scheduled model predictive control method for the CIPS with friction and disturbances is proposed to control the cart's movement to a new desired position. Meanwhile, during the motion process of the cart, the SIP returns to its upright position. Based on the Newton-Euler equation, the dynamic equation of the CIPS with friction and disturbances is given. According to the given dynamic equation of the CIPS, its motion characteristics and control process are analyzed. Then, the given dynamic equation of the CIPS is linearized to obtain a series of linearized models. Based on these linearized models, a series of MPCs are designed. In addition, a GSMPC is designed by proposing a gain-schedule algorithm to switch one of the designed MPCs to control the CIPS according to the measured output pendulum angle of the SIP. Finally, the effectiveness of the control strategy is verified by simulations.  
The friction and disturbances have great influences on the control of the CIPS. When the friction and disturbances are not considered, the stabilization control of the CIPS can be realized smoothly. When considering the friction and disturbances, the jitters occur. However, the designed GMPC can regulate the CIPS to overcome the influences of the jitters and achieve the stabilization control objectives. When applying pulse disturbances \(f_{m}\) and \(f_{M}\) simultaneously, two jitters occur. After overcoming these two jitters by using the GMPC, both the SIP and cart of the CIPS can smoothly reach the desired position. When applying Gaussian white noise disturbances \(f_{m}\) and \(f_{M}\) simultaneously, the SIP has a slight oscillation near the upright position and the cart also has a slight vibration around its desired position. However, the position errors of the SIP and the cart are small, demonstrating that the stabilization control objectives of the CIPS can be realized by using the proposed gain-scheduled model predictive control method. In addition, the proposed control method has a wide range of applications in aerospace and robotics, such as balance control in robot walking, verticality control in rocket launching, and attitude control in satellite flight.  
Conceptualization, J.H.; methodology, J.H. and Y.L.; software, Z.W. and Z.H.; validation, J.H., Y.L., Z.W. and Z.H.; formal analysis, J.H. and Z.W.; investigation, Y.L. and Z.H.; resources, Z.H.; writing--original draft preparation, J.H. and Y.L.; project administration, Z.W.; funding acquisition, Z.H. All authors have read and agreed to the published version of the manuscript.  
Figure 16: Control results of the CIPS with Gaussian white noise disturbance when \(x_{d}=40\).  
**Funding:** This work is supported by (1) Hubei Province Nature Science Foundation (no. 2023 AFB380); (2) Hubei Key Laboratory of Digital Textile Equipment (Wuhan Textile University) (no. KDTL202003); (3) Hubei Key Laboratory of Intelligent Robot (Wuhan Institute of Technology) (no. HBIRL202301).  
**Institutional Review Board Statement:** Not applicable.  
**Informed Consent Statement:** Not applicable.  
**Data Availability Statement:** The data presented in this study are available on request from the corresponding author. The data are not publicly available due to privacy issues.  
**Conflicts of Interest:** The authors declare no conflict of interest.