The Magnetic levitation lab is used in the course SIE3055 Nonlinear Systems. The experimental system consists of a steel ball that is levitated by magnetic force. The force is generated with a coil through which the current-flow can be controlled. In order for the ball to levitate at a constant position, the magnetic force has to equal the force of gravitation. However, because of small disturbances this force can not be held constant but needs to be controlled via a feedback loop. The position of the ball is measured every 0.5ms by determining how much light reaches a sensor positioned inside a post under the ball. Based on the measurements, a new reference value for the current is generated.

The system is nonlinear because the magnetic force on the ball is inversely proportional to the square of the distance between the coil and the ball and proportional to the square of the current through the coil. The open-loop system is also highly unstable since if the current is held constant and the ball starts to move downwards, the magnetic force is decreased due to the increased distance between the coil and the ball. This again results in an increased acceleration in the downwards motion.

Control problems are based on constant reference positions as well as sine- and square waves. Linear and nonlinear control systems are compared. The system is also used to study limit cycles, which occur if for example a backlash element is included in the feedback. In this case the ball will oscillate similar to a sinewave, even though the reference position is a constant.

The Helicopter lab is used in the courses SIE3015 Linear System Theory and SIE3030 Optimalization and Control. The setup consists of two propellers mounted on each side of a helicopter frame. The propellers have parallell axes and the resulting thrust vector is normal to the frame. By varying the load on each motor, the frame can pitch around its centre, which is connected to another arm. This arm is gimbaled on a 2 DOF joint, connected to the base. This allows the helicopter to elevate and move in a rotating motion around the gimbal. The other end of the arm holds a weight which counteracts most of the weight of the helicopter, resulting in an equivalent mass of the helicopter of around 70grams.

If the total thrust force in the upwards direction is higher than the equivalent mass times the acceleration of gravity, then the helicopter will lift off and move along the elevation axis. If one of the motors receives more current than the other, then the helicopter will pitch. A sustained pitch gives rise to an acceleration around the axis of the base. Sliprings carry the electrical signals to and from the helicopter, thereby allowing the helicopter to turn an unlimited number of times around its base.

The system is nonlinear, but the linearization around an equilibrium is quite good for the practical operating range of the helicopter. Linear controllers therefore give good results.

Control problems that are investigated are monovariable control, multivariable control, state estimation, optimal open-loop control, and optimal closed-loop control (LQ) in one or two dimensions.

• 3-DOF Helicopter
• Magnetic Levitation
• ISA MultiQ

• Matlab/Simulink
• WinCon
• RTX