# Stabilization of the Magnetic Levitation System
## 1 Introduction  
The magnetic levitation system represents a typical nonlinear unstable system. Since it presents several interesting problems for control, there exists numerous educational laboratory models that can be used for verifying control algorithms, demonstrating their properties. The most used are products from Quanser [1], Humusoft [2], Googol [3], LD Didactic Group [4], and Inteco [5]. For controlling the position of the ball in the magnetic field created by the solenoid, the system enables using a wide variety of methods, such as:  
* polynomial approach [6; 7; 8];
* LQR PID tuning [9];
* constrained control [10; 11];
* robust control [12];
* model predictive control; [11; 13; 14];
* state space control [15; 16];
* direct synthesis method for stable and unstable systems [17; 18];
* fuzzy control [19; 20]. The paper presents and compares two possible solutions to the magnetic levitation plant control problem. The first method is based on the knowledge of the transfer function of the linearized system (1) derived, considering parameters given by the manufacturer [21]. It starts withstabilizing the given unstable system. In the next step, the dynamics of compensation of the emerging disturbances are specified. By neglecting the fast stable mode of transients, the design procedure can be made simple and robust, similar to model-based PI control [22]. The controllers designed by the first approach make it possible to measure the input-output steady-state characteristic (IOSSCH) of the system, expressing the dependence of the output variable of the system on the input in steady states (Figure 2). While the analytically derived transfer function of the system results in a linear dependence of the output from the input at steady states, the measured dependence shows non-linear properties of the system. It indicates that, in an effort to accelerate transients by this approach, we would have to deal with the limited accuracy of the linear model. Although the proposed control meets most of the above requirements, it will be relatively slower in terms of transient speed. Therefore, to verify the plant dynamics, the second group of possible approaches begins with identifying the system. The authors of [10] have already pointed out several aspects that are important in the effort to achieve the fastest possible (and sufficiently smooth) transients at the input and output of the system. In view of the non-linearity of the system, in the second set of approaches, it is preferred to approximate the system's dynamics by means of the simplest second-order transfer function of a double integrator with a transport delay (DIPDT, double integrator plus dead time). Such an ultra-local model, typical of today's popular robust control approach with active disturbance rejection control (ADRC) [23; 24; 25; 26; 27], avoids the need to approximate nonlinear system feedback. A similar simplification is used by another method working with finite impulse response (FIR) filters. Due to omitting identification of internal plant feedback, it is also called model-free-control (MFC). Its application can lead to an intelligent proportional-derivative (iPD) controller [28; 29; 30; 31]. After determining the DIPDT plant model, simple proportional-derivative (PD) controllers can be used for the stabilizing controller design [31; 32]. When supplemented by low-pass filter design and combined with static feedforward (using inversion of the measured IOSSCH), it is possible to get control processes that are already very close to the performance limits given by the available hardware. To deal with possible disturbances (including the model imperfections), the proportional-integral-derivative (PID) controllers (supplemented again by the design of low-pass filters [33]), or internal model control (IMC) controllers may be added to the basic loop with a stabilizing PD controller [34].