# Comparison of Nonlinear and Linear Controllers for Magnetic Levitation System
###### Abstract  
Nonlinear system control belongs to advanced control problems important for real plants control design. Various techniques have been developed in this field. In this paper we compare two different approaches to a nonlinear unstable Magnetic levitation system control. The first control design approach further develops our recent results on robust discrete-time pole-placement, based on convex \(D_{R}\)-regions. The second studied approach is based on feedback linearization and the simplified development of the corresponding nonlinear control law is provided. Both approaches are compared and evaluated. The efficiency of robust discrete-time pole-placement controller is shown as well as its competitiveness in comparison with nonlinear control for Magnetic levitation system. nonlinear control; robust control; pole placement; magnetic levitation; pole regions +
Footnote †: 2021: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. control design [7; 9; 10; 16], where \(D_{R}\) region is a convex domain in a complex plane, where the system poles should lie. The corresponding \(D_{R}\) stability condition in the form of LMI, provides computationally efficient way for the respective controller design. The existing results in literature have been mostly proposed for continuous-time systems [10; 14; 15; 17]. In our recent works [18; 19; 20] we developed specific \(D_{R}\) regions for discrete-time systems, motivated by the fact that in real world application, mostly the digital controllers are implemented. In this paper we aim at the comparison of our recent results on robust pole-placement discrete-time state feedback control design with a nonlinear control developed for a laboratory magnetic levitation (ML) plant. Magnetic levitation principle is widely used in applications, besides transport systems (as Maglev train) also magnetic bearings, frictionless contacts, microrobotics and others. Magnetic levitation plant, where a magnetic object is positioned in the air space, provides a challenging control task owing to strong nonlinearity and inherent instability. Various methods have been proposed to control the magnetic levitation, as for example feedback linearization approach in [21; 22], adaptive state feedback [23], frequency domain method [24] or CDM (Coefficient Diagram Method) based design in [4]. Most of them consider continuous-time control. In [18; 19; 20] we proposed \(D_{R}\) region based discrete-time robust pole-placement controller for ML system. This paper is devoted to a study and comparison of two principal approaches applied on a laboratory Magnetic levitation system. The first approach is based on feedback linearization [3; 4; 22], where an algebraic transformation is applied on nonlinear state space model together with a nonlinear control law to obtain the linearized closed-loop system. The second approach employs robust pole-placement controller design for a system linearized around the working point and described by an uncertain polytopic system [7; 9; 18; 19]. The main contribution of the paper is in simplified development of nonlinear control law with novel approximation of nonlinear terms, developed robust pole-placement controller for different levitating balls, comparison and evaluation of the obtained results received with nonlinear continuous-time control and discrete-time linear one. The advantages and disadvantages of both used approaches are summarized for the studied case. The paper is structured as follows. Section 2 describes the Magnetic levitation plant, its nonlinear model and parameters. Section 3 is devoted to state feedback control design, two approaches are presented, Section 3.1 recalls our recent results on a discrete-time \(D_{R}\) region based controller design, in Section 3.2, nonlinear control law is developed following the concept introduced in [22].