Addendum Handout for the ECE3510 Project The magnetic levitation system that is provided for this lab is a non-linear system. Because of this fact, it should be noted that the associated ideal linear responses will not be the response of the system. This design project will be an attempt at increasing the stability of the system using standard compensation techniques by treating the system within a small linear region of operation. By itself this wonderful maglev kit created by Guy Marsden (www.artec.net) is not very stable. You will notice once your kit is built that there is a bit of finesse required to get your object to remain stable in the magnetic field. This effect is due mostly to the fact that the closed-loop design has a pole on or very close to the jw-axis. So, a small disruption or jolt to the system can setup an oscillation in the system that will cause the object to fly off. Thus the system needs to be stabilized, but how do we stabilize a nonlinear system? First we pick a set point or a value around which the system will be operating. For this project the set-point you should use (as seen in the figure below) will be about 2 cm from the top of the magnets to the coil and the sensor at 1cm from the magnet. This will have the result of setting the current supplied of the system around 200 ma.
Figure 1 Set point Coil Sensor The system is basically a non-linear proportional controller, so the set point of the system controls how much gain is provided in the closed loop. For your design the first thing to do after getting the system working is break the feed-back loop by placing a noninverting summer circuit in the feed-back path as seen in the design below Figure 2 - Non-inverting summer Setting R1 = R2 =R3=R4 is the simplest design for this circuit
The Control input of the summer allows you to adjust the set point of the system by placing an offset into the sensor reading. As a result, the distance from the coil to the levitated object is adjusted to meet the requirements already mentioned. Transfer Function for the motion The next step would be to determine the LTI system model to use for the analysis. Page 70 of your text book (Feedback Control of Dynamic Systems, Gene Franklin et.la ) has a good example of the process to determine the linear system model for a levitator. Due to time constraints for this project, you do not need to make these measurements. The following figure shows the displacement versus the force. 0.9 0.8 0.7 Experimentally determined force curves I = 0.15 I = 0.2 I = 0.25 0.6 force (N) 0.5 0.4 0.3 0.2 0.1 0 1.6 1.7 1.8 1.9 2 2.1 2.2 2.3 2.4 2.5 2.6 displacement (cm) From this plot the values of K i and K x are determined to be: K i =.8 and K x = 33 With the given values and the mass of the object you are levitating the transfer function for the motion of the object can be obtained by taking the Laplace Transform of: m*x = K i * i + K x * x ( eqn 2.103 )
and solving for X(s)/I(s). The complete system model for the basic maglev is provided in Figure 4 Figure 4 Basic maglev model ref + v(t) PWM i(t) X(s)/I(s) x(t) Sensor Due to the non-linear nature of the PWM and the Hall Effect sensor another simplification is made to the system. This simplification is to assume the system is just the model X(s)/I(s) with a built-in proportional gain. The only thing left to determine is the loop gain (K) of the closed loop basic system. The final simplified system including a compensator is shown in figure 5 Figure 5 Simplified maglev model ref + v(t) C(s) K p i(t) X(s)/I(s ) x(t) Determining K p To be able to design the compensator of Figure 5, the value of the effective open-loop gain K p must be found. This value can be determined by measuring natural frequency of oscillation for your maglev system. Connect an oscilloscope probe to the output pin of
the hall effects sensor. Slightly disturb the floating object and capture the resulting waveform. Measure the frequency of the most dominant oscillation. Now you can perform a root-locus of the X(s)/I(s) transfer function. You should notice that the poles of the system move together and progress along the jw-axis. If you play around with the value of the gain of the root locus by using a specified gain range in the rlocus command of matlab you can find the amount of gain that corresponds to the poles on the jw-axis of your the basic system. With the all the values determined for the simplified model your can now use the compensation technique of section 5.5 of the book to design and implement your stabilized system. Overview of steps 1. Build the basic maglev system 2. insert the summer circuit of Figure 2 in the loop and set the operating-point using the control voltage input. 3. Find the transfer function of the system from equation 2.103. 4. Measure the natural frequency of oscillation for your system. 5. Estimate the value of K p using rlocus with the transfer function of motion and the natural frequency of oscillation 6. Design the Compensator Good luck and have fun!