# Passivity-Based Control Design for Magnetic Levitation System
## 1 Introduction  
Due to many advantages of magnetic levitation system, including absence of contact, low noise, low energy consumption, it has been widely used, such as magnetic levitation bearings [1], magnetic levitation trains [2], magnetic levitation wind turbines [3], magnetic levitation vehicle operation lines [4]. Magnetic levitation system is highly nonlinear unstable system, it is a strong need to control the suspension gap. In recent years, various research methods have been proposed, such as Proportional-Integral-Derivative (PID) control [5; 6], Linear Quadratic Regulator (LQR) control [7], and sliding mode controller [8], fuzzy control is presented in [9]. When PID controller is designed, the system has to be linearized, sliding mode control has chattering effect. As for fuzzy control, the membership function is selected entirely by experience.  
In recent years, [PERSON] proposed a passivity-based design method for nonlinear system control based on the concept of energy variation in control systems [10; 11]. The superiority of this design method has been recognized and gradually applied to various fields. [PERSON] et al. [12; 13] presented passivity-based controller to solve low-frequency voltage oscillation. [PERSON] et al. [14] applied sensorless passivity-based control to solve the trajectory tracking task associated with the bidirectional DC/DC Buck power converter-inverter-DC motor system. [PERSON] et al. [15] addressed the problem of stabilization of port-Hamiltonian systems via the ubiquitous PID controller. [PERSON] et al. [16] designed a passivity-based robust switching controller for the posture stabilization of wheeled mobile robots with model uncertainty. Passivity-based theory and [PERSON]'s stability theory can be used to study system stability. The passivity-based theory has unique advantages. For example, the storage function in passivity-based theory is related to the physical characteristics of the system, which is easy to obtain according to the structure of the system. [PERSON] et al.'s method [16] is based on [PERSON]'s method, has a complicated design process and needs to switch between two controllers. The passivity-based design idea is different from the traditional idea, it is based on (error) signal and (frequency) response. It retains more nonlinear characteristics. The physical meaning of the design process is clear, especially the structure of the system, Moreover, the structure of the controller is simple and easy to implement.  
Passivity-based control needs to solve a partial differential equation. In order to solve the problem easily, the mathematical model of the system needs to be transformed into the Hamilton model. However, passivity-based control is generally more focused on qualitative considerations and is more difficult to apply to specific designs. In this paper, passivity-based control is combined with the bias-based feedback control and the response characteristics of the system, combines qualitative and quantitative methods. The magnetic levitation system gap control (suspended at the expected value) is presented as an example to illustrate the detailed design process, and a practical design method for passivity-based control of nonlinear systems is provided.  
The rest of paper is organized as follows. Section 2 briefly introduces passivity-based control. In Section 3, we briefly introduce sensitivity function. Passivity-based control for magnetic levitation systems is presented in Section 4. In Section 5, we analysis the role of the interconnection and damping assignment. The concluding remarks are contained in Section 6.