# Unstable systems

## Magnetic levitation system

- Details
- Last Updated on Thursday, 10 October 2013 10:42
- Published on Tuesday, 15 May 2012 18:24
- Written by Jaroslav Kolařík
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**System description**

The magnetic levitation system CE 152 depicted in the figure below represents a laboratory-scale model designed by TQ Education and Training Ltd for studying system dynamics and experimenting with control algorithms. It demonstrates control problems associated with nonlinear unstable systems.

The system consists of a coil levitating a steel ball in the magnetic field with the position sensed by an inductive linear sensor connected to an A/D converter. The coil is driven by a power amplifier connected to a D/A converter. A basic control task is to control the position of the ball freely levitating in the magnetic field of the coil. From the control theory point of view, the magnetic levitation system is a nonlinear unstable system with one input and one output.

**Parameters of the real system CE 152:**

The values of all important parameters are listed in the following table:

Parameter |
Symbol |
Value & unit |
Parameter |
Symbol |
Value & unit |

A/D converter gain | k_{AD} |
0.2 MU/V |
Coil constant | kc |
1.769 x 10-6 Nm2/A2 |

D/A converter gain | k_{DA} |
20 V/MU | Ball mass | mk |
8.27 x 10-3 kg |

Damping constant | kfv |
0.02 Ns/m | Coil offset | x_{0} |
7.6 x 10-3 m |

Position sensor gain | kx |
821 V/m | Gravity constant | g |
9.81 m/s2 |

Power amplifier gain | ki |
0.3 A/V | Position sensor offset | y_{0} |
0.0183 V |

MU ... voltage converted by the data acquisition card and scaled to ±1 machine unit (MU).

**Mathematical model**

A mathematical model of the system including both D/A and A/D converters can be derived in the following form of a second-order nonlinear differential equation (Humusoft, 2002; Gazdoš, Dostál & Pelikán, 2009):

where *y* denotes the controlled variable - ball position [MU] and *u* is the control input [MU], proportional to the voltage from the D/A converter. All constants and symbols are clearly defined in the previous table.

**Simulation files**

Downloading files is possible only for registered users.

**Information sources**

Humusoft. *CE 152 Magnetic levitation model – educational manual. *2002. Prague: Humusoft s.r.o.