Estudio de una bola suspendida

CONTROL OF A MAGNETIC LEVITATION SYSTEM
By José Guamán , Geovanny Guaño
National Polytechnic School "EPN"
Electrical and Electronic Engineering
Campus José Rubén Orellana R.

OBJECTIVE:
Study the functioning of the levitation of a sphere through state-space model, analyze controlability, stability, observability and make the design of a controller and observer applying the conceptsof modern control
INTRODUCTION:
This paper addresses the problem of controlling the height of a steel sphere with respect to a reference level, by suspending (or levitation) against the force of gravity through the use of an electromagnet.
This system has been studied previously because of its usefulness in demonstrating many basic principles of control engineering but has also been used toimplement more complex control techniques or innovative.
SUMMARY:
This paper proposes the control of a magnetic levitation system which aims to maintain a metal sphere suspended by adjusting the current in an electromagnetic with input voltaje.
First, we present a detailed description of the mathematical model of the magnetic levitation system.
Since this is an unstable non-linear model, we presentthe same linearization procedure is performed and then design a compensator for a stable system showing its response to a step input.
Below we describe the design procedure in the state space model of a regulator and observer, presented the results graphically.
Finally, we present the results of the analysis total levitation system, offset and the respective designs made, besides the discrete model.DESCRIPTION: MODELLING
Figure 1 shows a system of magnetic levitation suspension or suspended keeps a small metal sphere of mass m against the force of gravity. The system regulates the value of the current i (t) of the circuit of the electromagnet, so that the area is so suspended at a constant distance x (t) = X, the electromagnet.
The voltage applied to the circuit or voltage is V (t) andacts as a control variable.

Figure 1. Simplified diagram of magnetic levitation system
The equations describing the dynamic behavior of the system:
Lditdt=-Rit+Vt
md2xtdt2=mg-fmx,t
fmx,t=Ci2(t)x2(t)
Where i (t) is the circuit current and x (t) is the displacement of the sphere measured fromthe solenoid, L is the inductance of the coil of the electromagnet, C is a constant known andfm (x, t)is the force attraction of the magnet on the sphere.
Then we have the following system of first order differential equations:
x1=x2
x2=g-Ci2(t)x2(t)
x3=-RLx3+1Lu
y=x1
To linearize the system using the Lyapunov stability theory the first step is to calculate the equilibrium point:
x1=0, x2=0, x3=0.
Then: x2=0
g-Cx32mx12=0
-RLx3+1Lu=0
The matrices A, B y C of the linearized systemaround the equilibrium point the Jacobian is calculated using non-linear system:
∂f1∂x1∂f1∂x2∂f1∂x3∂f2∂x1∂f2∂x2∂f2∂x3∂f3∂x1∂f3∂x2∂f3∂x3=0102gX0-2CgmX200-RL
∂f1∂u∂f2∂u∂f3∂u=001L
∂h∂x1∂h∂x2∂h∂x3=∂x1∂x100=100
Substituting the parameters of the system of Figure 2 we have the linear system:

Figure 2. Parameters of the system

A=01010000-2236.100-50;b=005
c=100 ;d=0

STABILITY ANALYSISObtain the transfer function using MATLB (tf) command and then see the response to a step input using the tool ltiview.

Then we see that the system is unstable and make a design of a compensator with a zero at s = -25 and a pole at s = -300 in order to move the poles of the closed loop system. For this location of the poles of the levitation system in closed loop the value of the compensator gain is Kc= -89.

obtaining the time response of the compensated system:

OBSERVABILITY
We find the matrix watch it with the following equation:
Mo=CCA⋮CAn-1
As we have a 3x3 system we have:
∴Mo=CCACA2
Replacing and calculating the respective values ​​to the following matrix:
Mo=10001010000-2236.1
Observability to determine where the determinant must be different from 0

detMo=-2236.1≠0...