# Gain-Scheduled Model Predictive Control for
## 1 Introduction
Representative results include the proportional-integral differential (PID) control method [35], the sliding mode control method [36], the fuzzy control method [37], the model predictive control method [38], etc. More specifically, PID control algorithms are relatively simple, robust, and adaptable. They represent some of the earliest control strategies to gain popularity and remain the most widely used and versatile controllers to this day. Classical PID control applications continue to hold a considerable position in practical systems. One of the key issues in the design of PID controllers is the setting of parameters; the main methods of calibration include empirical and patchwork methods [39]. However, for PID controllers, it is difficult to solve the control problems of extremely complicated systems. For the CIPS, there tends to be a large error between the simulation (or experimental) results and the control objectives. So, the PID controller has certain limitations in the control of the CIPS. Model predictive control is a new type of computer control algorithm that began to develop in the early 1980s, which is a good state-of-the-art control strategy [40; 41]. The algorithm is directly generated from practical applications in industrial process control and continues to improve and mature through close collaboration with industrial applications [42]. Due to the use of multi-step prediction, rolling optimization, feedback correction, and other control strategies, the model predictive control algorithm has advantages such as good control effects and strong robustness [43; 44]. In [45], a single MPC is designed to stabilize the CIPS. However, in the design of this single MPC, the dynamic equation of the CIPS is overly simplified as a linearized system around the upright position of the SIP. Additionally, friction and external disturbances are not modeled. Therefore, establishing a dynamic equation of the CIPS that includes friction and external disturbances, and then realizing its stabilization control objective, remains a valuable topic. Based on the above analysis, a CIPS with friction and disturbances is studied in this paper and a gain-scheduled model predictive control method is proposed to realize the upright stabilization of its SIP when its cart is moved from one position to another desired position. First, a dynamic equation for the CIPS, including friction and disturbances, is formulated based on the Newton-Euler equation. Next, the given dynamic equation of the CIPS is linearized to obtain a series of linearized models at a series of pendulum angles. Then, a series of MPCs are designed based on the above-linearized models, respectively. Subsequently, by introducing the idea of the gain schedule, a gain-scheduled MPC (GSMPC) is designed to realize the control of the CIFS. Finally, several simulations are implemented to demonstrate the effectiveness of the proposed gain-scheduled model predictive control method. The main innovations are as follows:  
1. The dynamic equation of the CIPS with friction and disturbances is established based on the Newton-Euler equation. 2. The gain-scheduled model predictive control method is proposed to stabilize the SIP of the CIPS in the upright position while the cart moves to its desired position. 3. The proposed gain-scheduled model predictive control method is verified by several simulations. The remainder of this paper is organized as follows. Section 2 establishes the dynamic equation of the CIPS and analyzes its characteristics. Section 3 designs the GSMPC of the CIPS. Section 4 performs simulations to demonstrate the effectiveness of the proposed gain-scheduled model predictive control method. Section 5 concludes this paper.