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This paper investigates the control problem of magnetic levitation system, in which velocity feedback signal is influenced by stochastic disturbance. Firstly, single-degree-freedom magnetic levitation is regarded as an energy-transform action device. From the view of energy-balance relation, the magnetic levitation system is transformed into port-controlled Hamiltonian system model. Next, based on the Hamiltonian structure, the control law of magnetic levitation system is designed by applying Lyapunov theory. Finally, the simulation verifies the correctness of the proposed results.

Magnetic levitation system is a class of typical nonlinear system, which is difficult to establish accurate mathematical model for the natural parameter of electromagnetic part dependent times [

In addition, the systems are always affected by stochastic disturbance in many practical control problems, which always lead to system instability [

Due to the highly nonlinear characteristics of the system, the controller design problem will be solved based on Hamiltonian energy theory. In fact, the energy-based Hamiltonian system method has been widely used in practical systems control [

As is known, the controllers and regulators of the systems are always unavoidably affected by stochastic disturbances, and the study of the controlled systems with stochastic disturbance is of practical significance. Different from what have been studied, this present paper deals with the controller design problem of the magnetic levitation system with stochastic disturbances. Current efforts have been made to dispose the control problem of the stochastic magnetic levitation system on the basis of Hamiltonian energy theory. We regard the magnetic suspension as the energy conversion device and then derive a mathematical model of the single degree of freedom stochastic magnetic levitation system from the point of energy balance, which is transformed into a port-controlled Hamiltonian system. Consequently, the controller of the stochastic magnetic levitation system is designed. Finally, a simulation example is given to verify the validity of the results.

The rest of this paper is organised as follows. Section

The physical model of the magnetic levitation train system, which includes the concentrated mass of train carriages (together with the supporting magnet) suspended on the rigid lead rail is shown in Figure

Physical model of magnetic suspension system.

By invoking Kirchoff’s voltage law and Newtons second law, the dynamic equations of the magnetic levitation system can be obtained by taking the vertical upward direction as the positive direction:

Here we regard the flux

To obtain a port-controlled Hamiltonian model, we take a suitable approximation for the inductance is

The objective of this paper is to find a feedback control law as

Obviously, system (

where

According to the equilibrium condition of the system, the speed of the rigid body reduced to zero when the system is stable. Meanwhile, the electromagnetic force of the rigid body is equal to the gravity that acting upon on it. Then, we can get

It is evident that

In order to design the controller of system (

If there exists a controller

Next we introduce some auxiliary lemmas which will be used in this paper.

For system

Let

For any given matrices

In this section, we will put forward the controller design scheme for stochastic magnetic levitation system (

Consider system (

According to It

If we set suitable scalars

Suppose the Hamiltonian function

Substituting (

Combining (

According to Lemma

Set

Since

system (

Since

Next, we consider the stochastic magnetic levitation system (

The stochastic magnetic levitation system (

Because of the stochastic magnetic levitation system (

In this section, a simulation example is given to verify the correctness of the results obtained in this paper. The relevant parameters are given as follows:

According to Theorem

The velocity curve of the rigid body is shown in Figure

Velocity response curve.

Displacement response curve.

This paper has investigated the control problem of stochastic magnetic levitation system. By regarding the magnetic levitation as the energy conversion device, we derived the mathematical model of single degree of freedom magnetic levitation system with stochastic disturbance from the point of view of the energy balance, and then the model can be transformed into a port-controlled Hamiltonian system. Then the controller of the stochastic magnetic levitation system has been designed based on the obtained Hamiltonian system model. Finally, the correctness of the conclusion has been verified by simulations. The main innovation of this paper is that we have fully taken into account the effect of random disturbances on the magnetic levitation system and solve the control problem under Hamiltonian systems framework by making full use of the dissipative structural properties of the Hamiltonian systems.

The authors declare that they have no conflicts of interest.