magnetic levitation control system
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DESCRIPTIONMathematical model of a magnetic levitation system and design of a feedback controller for the system along with matlab simulink design for the designed controller and maglev system.
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
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
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
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.
(I) Electromagnetic Subsystem:
Consider a schematic of maglev plant and its electromagnetic network model as
shown in Figure.
Apply Kirchhoffs voltage law in electrical system network.
= + u(t) = iR + L(x) (di
where u, i, R and L are applied voltage input, current in the electromagnet coil,
coils resistance and coils inductance respectively.
(II) Mechanical Subsystem:
Energy stored in the inductor can be written as
Power in electrical system (Pe) = Power in the mechanical system (Pm)
) and =
Since () =
, therefore, we have
Where, Kc=electromagnet force constant, x=actual air gap between core face and
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
Including the damping force, = (
2) + (
) = (
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 :
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
D/A converter : () = () A/D converter : () = ()
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.
2) + ( + ) (
) + = (
(()0)2) for x < 0
2) + (
) = (
(()0)2) for 0 l
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:
() = +
+ = (
(1 + 1)(2 + 1)
() = 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
The Derivative part has been omitted in this project to avoid further slowing of the
For the above shown closed loop control system, transfer function is given by
1 + ()()
The proposed model is simulated using step input.