PROJECT REPORT

Modeling and Control Design of

Magnetic Levitation System

Group Members- Project Mentor

Prashant Kapoor – IEC2013009 Dr. Arun Kant Singh

Yashaswa Jain – IEC2013037

Arushi Goel – IEC2013078

Vartul Sharma – IEC2013096

Introduction

The objective of this project is to design a controller that levitates the steel ball

from the post and keeps it stably levitating. The maglev system can be decomposed

into two subsystems, viz. a mechanical subsystem and an electrical subsystem

(current loop). The ball position in the mechanical subsystem can be controlled by

adjusting the current through the electromagnet whereas the current through the

electromagnet in the electrical subsystem can be controlled by applying controlled

voltage across the electromagnet terminals. Thus, the voltage applied across the

electromagnet provides an indirect control of the ball position.

Robust feedback control for magnetic levitation systems is considered problematic

due to the parametric uncertainties in mass, strong disturbance forces between the

magnets and noise effects inflowing from sensor and input channels. Therefore

robustness is a key issue in designing a control system for a magnetic levitation as

the models are never 100 percent accurate and the uncertainties in the model must

be accounted. In this project, closed loop PID control is investigated to bring the

magnetic levitation system in a stable region by keeping a magnetic ball suspended

in the air in the presence of uncertainties. The project report first presents the

complete non-linear and linear mathematical models and then it adopts the mixed

sensitivity design method for closed loop PID controller synthesis.

We have developed the governing differential equation and the Laplace domain

transfer function models of the electrical and mechanical subsystems. Finally we

will design and implement a Proportional-integral-derivative (PID) controller to

ensure that the mechanical subsystem ball position response tracks the desired

position command.

The Magnetic Levitation controller is simulated using SIMULINK in MATLAB.

MATLAB is a high performance software package for scientific and numeric

computation, signal processing and graphics in an environment where problems

and solutions are expressed just as they are written mathematically - without

traditional programming. The power of MATLAB environment is further extended

by Simulink - a block oriented environment for simulation of dynamic systems and

numerous toolboxes.

MAGNETIC LEVITATION MODEL

Levitation is the stable equilibrium of an object without contact and can be

achieved using electric or magnetic forces. In a magnetic levitation, or maglev

system a ferromagnetic object is suspended in air using electromagnetic forces.

These forces cancel the effect of gravity, effectively levitating the object and

achieving stable equilibrium. The model of magnetic levitation shown in Figure 1

consists of a coil levitating a steel ball in magnetic field. The position of the steel

ball is sensed by an inductive linear position sensor connected to A/D converter.

Mathematical Model

(I) Electromagnetic Subsystem:

Consider a schematic of maglev plant and its electromagnetic network model as

shown in Figure.

Apply Kirchhoff’s voltage law in electrical system network.

𝑉 = 𝑉𝑟 + 𝑉𝑙 ⇒ u(t) = iR + L(x) (di

dt)

where u, i, R and L are applied voltage input, current in the electromagnet coil,

coil’s resistance and coil’s inductance respectively.

(II) Mechanical Subsystem:

Energy stored in the inductor can be written as

𝑊𝑒 =1

2𝐿(𝑥)𝑖2

Since,

Power in electrical system (Pe) = Power in the mechanical system (Pm)

𝑃𝑒 = (𝑑𝑊𝑒

𝑑𝑡) and 𝑃𝑚 = −𝐹𝑚

𝑑𝑥

𝑑𝑡

(𝑑𝑊𝑒

𝑑𝑡) = −𝐹𝑚

𝑑𝑥

𝑑𝑡⇒ 𝐹𝑚 = −

𝑑

𝑑𝑥(1

2𝐿(𝑥)𝑖2)

= −1

2𝑖2 𝑑

𝑑𝑥𝐿(𝑥)

Since 𝐿(𝑥) =𝐾𝑐

𝑥 , therefore, we have

𝐹𝑚 = −1

2𝑖2

𝑑

𝑑𝑥(𝑘𝑐

𝑥)

𝐹𝑚 =𝐾𝑐

2 (

𝑖2

𝑥2)

Where, Kc=electromagnet force constant, x=actual air gap between core face and

ball surface.

If Fm is electromagnetic force produced by input current, Fg is the force due to

gravity and F is net force acting on the ball, the equation of force can be written as

𝐹𝑔 = 𝐹𝑚 − 𝐹

= 𝐹𝑚 − 𝑚 (𝑑2𝑥

𝑑𝑡2)

⇒ 𝑚 (𝑑𝑣

𝑑𝑡) = 𝐹𝑚 − 𝐹𝑔

Including the damping force, 𝐹𝑣 = 𝑘𝑓𝑣 (𝑑𝑥

𝑑𝑡)

we get,

𝑀𝑘 (𝑑2𝑥

𝑑𝑡2) + 𝑘𝑓𝑣 (𝑑𝑥

𝑑𝑡) = (

𝑖(𝑡)2𝐾𝑐

(𝑥(𝑡)−𝑥0)2) − 𝑀𝑘𝑔 (1)

Where,

i(t) - electric current [A]

x(t) - ball position [m]

Mk - mass of ball [kg]

Kc - coil constant [A/V]

x0 - coil offset [m]

g - gravity constant [m/s-2]

kfv - damping constant [N/m.s].

Position of the ball in the magnetic field is controlled by electric current i(t), which

is generated from the power amplified. The power amplified is designed as a

source of constant current and is described by transfer function Fz :

𝐹𝑧 =𝐼(𝑠)

𝑈(𝑠)=

𝐾𝑖

𝑇𝑎𝑆 + 1

Where,

I(s) - Laplace of electric current i(t)

U(s) - Laplace of input voltage u(t)

Ki - coil and amplified gain [A/V]

Ta - coil and amplified time constant [s]

The signal incoming from inductive sensor for communication with surrounding is

necessary adjusted for further processing and therefore D/A resp. A/D converter is

added into mathematical model. The converters can be described by linear

equations:

D/A converter : 𝑈(𝑡) = 𝑘𝐷𝐴𝑈𝑚𝑢(𝑡) − 𝑈𝑜

A/D converter : 𝑌𝑚𝑢(𝑡) = 𝑘𝐴𝐷𝑌(𝑡) − 𝑌𝑚𝑢𝑜

Where,

u(t) - converter output voltage [V]

uMU(t) - converter input voltage [MU]

kDA - converter gain [V/MU]

u0 - converter offset [V]

yMU(t) - converter output voltage [MU]

y(t) - converter input voltage [V]

kAD - converter gain [MU/V]

yMU0 - converter offset [MU]

By, taking Laplace transform of equation (1), the following model for magnetic

levitation is designed.

The detailed model is based on the simple model, but in addition to this the

influence of input amplifier dynamics, limits of the ball movements and ball

damping is taken into account. To model the limits, model constants have to vary

according to the ball position.

𝑀𝑘 (𝑑2𝑥

𝑑𝑡2) + (𝑘𝑓𝑣 + 𝑘𝑓𝑙) (𝑑𝑥

𝑑𝑡) + 𝑘𝑑𝑥 = (

𝑖(𝑡)2𝐾𝑐

(𝑥(𝑡)−𝑥0)2) − 𝑀𝑘𝑔 for x < 0

𝑀𝑘 (𝑑2𝑥

𝑑𝑡2) + 𝑘𝑓𝑣 (𝑑𝑥

𝑑𝑡) = (

𝑖(𝑡)2𝐾𝑐

(𝑥(𝑡)−𝑥0)2) − 𝑀𝑘𝑔 for 0 <x < 1

𝑀𝑘 (𝑑2𝑥

𝑑𝑡2) + (𝑘𝑓𝑣 + 𝑘𝑓𝑙) (𝑑𝑥

𝑑𝑡) + 𝑘𝑑(𝑥 − 𝑙) = (

𝑖(𝑡)2𝐾𝑐

(𝑥(𝑡)−𝑥0)2) − 𝑀𝑘𝑔 for x > l

Where,

kfl - limit constant - elasticity

kdl - limit constant – damping

Applying the limits to our originally designed model, we get the Magnetic

levitation model as shown in the figure.

PID Controller Design:

Consider the closed-loop feedback system shown in Figure, where GR(s) is

transfer function of a PID controller; GP(s) is a transfer function of the real plant;

w, e, u and y are the reference, control error, manipulated variable and output of

the plant signals, respectively.

Standard PID controller can be described by a 2-nd order transfer function:

𝐺𝑅(𝑠) = 𝐾𝑝 +𝐾𝑖

𝑠+ 𝑠𝐾𝑑 = 𝐾 (

(𝑇𝑠1𝑠 + 1)(𝑇𝑠2𝑠 + 1)

𝑠)

Where,

𝐺𝑅(𝑠) = controller transfer function

Kp= controller proportional constant

Ki =controller integral constant

Kd=controller derivative constant

K= controller gain

Ts1, Ts2= time constants corresponding with controller zeros.

Each of the PID controller blocks (P, I and D) plays an important role. However

for some applications, the integral and derivative part has to be excluded to give

satisfactory results. The Proportional block is mostly responsible for the speed of

the system reaction. However for oscillator plants it might increase the oscillations

if the value of P is set to be too large.

The Integral part is very important and assures zero error value in the steady state,

which means that the output will be exactly what we want it to be. Nevertheless the

integral action of the controller causes the system to respond slower to the desired

value changes.

The Derivative part has been omitted in this project to avoid further slowing of the

system.

For the above shown closed loop control system, transfer function 𝑇𝑐𝑠is given by

𝑇𝑐𝑠 =(𝐺𝑅(𝑠)𝐺𝑝(𝑠))

1 + 𝐺𝑅(𝑠)𝐺𝑝(𝑠)

Simulation Result:

The proposed model is simulated using step input.