EE420 – Term project 

Report and files due electronically (using Canvas) by posted deadline on Canvas 

 

The purpose of this project is that you learn to tackle a fairly challenging/realistic control design problem. The project is to be an individual effort.

 

Problem description

 

The figure on the right^[1] shows a simplified   version            of         a             magnetic levitation system that can be built in a laboratory, where a ball of magnetic material is suspended by means of an electromagnet         whose current             is controlled    by        feedback          from    the optically measured ball position. To quote the excellent reference [1], “this system has the basic ingredients of systems constructed to levitate mass, used in gyroscopes, accelerometers, and fast trains”.

Several such examples are given in many technical references, some of which I have listed and are posted on Blackboard.

 

In any case, the equation of motion of the ball is m&y&ky&  mg  F(y,i)     (1)

where m is the mass of the ball, y  0 is the vertical (downward) position of the ball measured from a reference point (y = 0 when the ball is next to the coil), k is a viscous friction coefficient, g is the gravitational acceleration constant, and F(y,i) is the force generated by the electromagnet, i  0 being the coil current. The force F(y,i) is related to the energy E(y,i)  L(y)i² / 2 (where L(y) is the position-dependent coil inductance) stored in the coil by

F(y,i) E(y,i)           L0i2           2      (2)

y           2a(1 y / a)

where L₀,a are positive constants. This completes the system model if one controls the position by manipulating the coil current directly. The equations for controlling the ball d(Li) position by varying the coil voltage v include the additional equation v  Ri  dt

L⁰i y&  _(1L_y⁰i_/a)  _dtdi   (3), but this makes the problem harder (you can try this

Ri    2 _{a(1 y/a)} if you want!), and so we will stick to designing the current and not the voltage.

Control objective

 

1. It is desired to balance the ball at a certain position r > 0. Based on a linearized model of the nonlinear plant, design a compensator G_c(s), the input to which is the error e = r – y, and whose output is the coil current i, such that the specifications shown below are all satisfied:

 

• When the parameters of the system are exactly known, the steady-state error for a constant r must be ≤ 5%. You must not include an integrator as part of your controller. You are free to choose any configuration, lead, lag, lead-lag, feedforward, non-unity gain, whatever you wish, except that the system must not be type 1.
• Redesign your controller in the above part to meet the same specifications even when the parameters are perturbed from their nominal values. In this case, design the controller as an integral controller.

 

Numerical values for the parameters are as specified (taken from [1]):

m  0.1kg,k  0.001N /m/ s,g  9.81m/ s²,a  0.05,L₀  0.01H

 

Tasks to be completed

 

• Find the steady-state value I_ss of the current required to balance the ball at an arbitrary desired equilibrium point y = r > 0.
• Linearize the nonlinear equations specified in the “problem description” section about y^(s) the desired operating point to obtain the linearized model G_p(s)  where i_(s) y_(t)  y(t)  r,i_(t)  i(t)  I_ss denote the deviations of the position and current respectively about their desired equilibrium values. Since the point (y  r, y&  0,i  I_ss )

is an equilibrium point, if we start there, we will stay there for ever (that’s what it means to be an equilibrium point). Show that this equilibrium point in unstable, i.e., there are poles of G_p (s) in the right-half complex s-plane. What this physically means is that if the ball were moved by an infinitesimally small value from its equilibrium position (or equivalently, if the current changed slightly from its equilibrium value), the ball will move away from its equilibrium position. This is what necessitates the use of a compensator, we want to “stabilize” this equilibrium point, along with possibly meeting some nice (transient and steady-state) performance specifications.

• For your linearized plantG_p(s), design a linear compensator G_c (s) to regulate the value of the output to y = r = 0.05m.

A very important observation is that your linear compensator deals with the deviations from the equilibrium values !!  In other words, you are not designing the current i, but the current i_(t)  i  I_ss. Do not use integral control!

Perform simulations in simulink with your compensator design to verify that the specifications have been satisfied. Only your compensator should be linear, the plant should be the nonlinear, in other words, you must design for a linear system, but evaluate your controller’s performance on the nonlinear system. Plots should indicate that the responses of the various signals are satisfactory.

• How far up from the equilibrium point can you move the ball (i.e, you will have to specify an initial condition for the ball’s position, and its velocity, when you simulate the system in simulink) and still return to the equilibrium position ? How far down ? If the system were truly linear, what would these values be ?
• Using simulation, study the effect of perturbations of the mass m on your controller. Using simulation, determine by how much you can change the mass and still have the steady-state specs of SSE ≤ 5% met. Note that only the plant changes, the controller is still the same (i.e., the one you designed for m = 0.1 kg).
• The previous part shows that when the parameters are perturbed, there is a steady-state error. Redesign your control using integral control (PID) to get rid of the steady-state error. Now by simulation, study the effect of mass perturbations on your controller.
• In none of the previous simulations did we impose realistic constraints on the control magnitude. Suppose now that the control magnitude is limited to | i | I_max  15A. Place a limiter in your simulink model, and compare your results in steps (3) – (6) with what you get when the control is saturated, i.e., whether or not the range of values by which you can move the ball from the equilibrium position and/or the range of perturbations has decreased. This will be the case if the transient performance specs you chose were unrealistic, i.e., required a large control effort to meet the specs. Then, in the presence of constraints on the control magnitude, your performance, as expected, should suffer.
• Discretize your compensator G_c(s) in (7) to get a discrete compensator G_c (z). Choose the sampling period T sufficiently small. The input to the discrete compensator must be sampled, and the output of the compensator must be “held”, using a (zero-order hold) ZOH. Only the compensator must be discretized, the control must still be applied to the continuous-time nonlinear plant. Simulate the behavior of the closed-loop system, and comment on the performance. How does the choice of sampling rate affect the performance of the controller?
• (If time permits and for extra credit)

Look up papers/texts in the literature (for example [1,3]) on what nonlinear control methods have been used (these are all based on the state-space approach, because there is no transfer function if the system is nonlinear) in the control of magnetic levitation. Can you at least simulate such a controller in simulink ?

 

You are expected to submit a typed report summarizing your work. You can use any word processing system that you like. Typesetting systems such as LaTeX are especially recommended. The lengths of the reports may vary, but I am expecting that you will need at least 10 pages including simulink screenshots/plots and/or any matlab code. Some components that your report should contain are:

 

• Abstract:This is an overview of the report, a miniature version of 50 wordsor so. Someone reading the abstract should get a good idea of the problem tackled, what types of techniques were used to solve it, and what sort of solution was found.

 

• Problem description: Present the problem you are attempting to solve. Give some background. Explain why it is important or interesting.

 

• Solution of the problem: What techniques did you use to solve the problem? Did you use a technique that we haven’t discussed in class? Explain in detail.

 

• Conclusions: Are the results reasonable? Can you improve the solution technique so as to yield better results ? If so, explain.

 

• References: Please include a bibliography if you have used any references, e.g., books, journal articles, web pages.

 

References & suggested reading

 

1. K. Khalil. Nonlinear Systems, Third Edition. Prentice Hall, Upper Saddle River, New Jersey, 2002
2. Kent Lundberg, Katie A. Lilienkamp and Guy Marsden. Low-cost magnetic levitation project kits – IEEE Control System Magazine, October 2004 (on Blackboard as Reference1.pdf).
3. F. Al-Muthairi and M. Zribi. Sliding mode control of a magnetic levitation system – Mathematical Problems in Engineering, 2004, Vol 2, pg 93–107 (Reference2.pdf)

 

Besides being an important benchmark problem for the application of nonlinear control techniques, the magnetic levitation problem has also been widely used in control education, both in the classroom and in control laboratories, see for example :

 

4. Katie A. Lilienkamp and Kent Lundberg. Low-cost magnetic levitation project kits for teaching feedback system design – Proceedings of the 2004 American Control Conference (Reference3.pdf)
5. Magnetic levitation in the controls lab – Department of Electrical Engineering and Computer Science, Massachusetts Institute of Technology, Fall 2005 (Reference4.pdf)

 

 

[1] This image extracted from http://www.fbk.com/pdfs/control/matlab.pdf