magnetic levitation control system

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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

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Mathematical 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.

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  • 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 Kirchhoffs voltage law in electrical system network.

    = + u(t) = iR + L(x) (di

    dt)

    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

    =1

    2()2

  • Since,

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

    = (

    ) and =

    (

    ) =

    =

    (1

    2()2)

    = 1

    22

    ()

    Since () =

    , therefore, we have

    = 1

    22

    (

    )

    =

    2 (

    2

    2)

    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

    =

    = (2

    2)

    (

    ) =

    Including the damping force, = (

    )

    we get,

    (2

    2) + (

    ) = (

    ()2

    (()0)2) (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 :

    =()

    ()=

    + 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 : () = () A/D converter : () = ()

    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.

    (2

    2) + ( + ) (

    ) + = (

    ()2

    (()0)2) for x < 0

    (2

    2) + (

    ) = (

    ()2

    (()0)2) for 0 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:

    () = +

    + = (

    (1 + 1)(2 + 1)

    )

    Where,

    () = 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 is given by

    =(()())

    1 + ()()

  • Simulation Result:

    The proposed model is simulated using step input.