# Gain-Scheduled Model Predictive Control for
## 1 Introduction  
Mechanical systems have greatly contributed to the development of human society. From the aspect of the relationship between the number of control inputs and degrees of freedom, mechanical systems can be classified into fully actuated mechanical systems and underactuated mechanical systems. In fully actuated mechanical systems, the number of their control inputs is equal to their degrees of freedom. Thus, a fully actuated mechanical system can be easily actuated to a desired state once appropriate control inputs are applied [1]. For underactuated mechanical systems, the number of their control inputs is less than their degrees of freedom. Such systems are not completely controllable [2; 3; 4; 5]. To promote social progress, various underactuated mechanical systems are widely applied, such as underactuated manipulators [6; 7; 8], cranes [9; 10; 11], helicopters [12; 13; 14], quadrotors [15; 16; 17], unmanned ships [18; 19], and soft robots [20; 21; 22]. The cart-inverted pendulum system (CIPS) belongs to a class of typical underactuated mechanical systems, characterized by having only one control input but multiple outputs [23]. In general, a CIPS consists of a cart and a single inverted pendulum (SIP), where the control input is the driving force applied to the cart and the outputs are the displacement of the cart and the pendulum angle of the SIP [24; 25]. The underactuated nature of the CIPS gives it complex characteristics, such as high state coupling, open-loop instability, and nonlinear dynamic properties [26]. Therefore, controlling the CIPS is a challenging problem. In addition, this topic continues to attract significant attention in the field of control and systems [27; 28]. This is mainly due to two reasons: (1) as an experimentalplatform, it is low-cost and simple in structure, and control experiments can be conducted using both analog and digital methods; (2) as a controlled object, the CIPS is a complex system in itself, being nonlinear, high-order, unstable, multivariate, and strongly coupled, only taking the appropriate control methods to make it stable [29]. The exploration of inverted pendulum systems is not only of strong theoretical significance but also of important practical engineering significance. Since the CIPS is an abstract model of many engineering control problems and its control methods can be generalized in practical control systems, the studies on the control methods of the CIPS have significant practical applications. The theoretical study of any system depends on a correct mathematical description of that system. In order to study the control problems of the CIPS, it is necessary to model their dynamics. There are two methods to establish the dynamic model of a system to analyze its complex characteristics [30]: classical Newtonian mechanics [31] and Lagrange equations [32]. Studies on the dynamic modeling of underactuated mechanical systems have been widely performed using both classical Newtonian mechanics and Lagrange equations. However, in most of these studies, only the nominal model of the underactuated mechanical systems is established, while the influences of external disturbances on each component are not modeled. Due to the complex nonlinear characteristics of high-order, instability, multivariate, and strong coupling in underactuated mechanical systems, external disturbances and other uncertainties significantly impact the response characteristics of these systems [33]. Considering that the CIPS is also an underactuated mechanical system, a comprehensive dynamic model of the CIPS should be established, taking into account the input force and external disturbances, thus providing a basis for achieving control of the CIPS. Due to the typicality and complexity of the inverted pendulum system, its control problem is currently considered to be one of the most challenging problems in the field of nonlinear control, and many control methods can be used for its stabilization control study [34].