Description
1 Introduction and Purpose
The purpose of this experiment is to familiarize you with the simulation of a magnetic ball-suspension system using MATLAB. You will also become familiar with modelling a nonlinear system using SIMULINK
and performing linearization.
u
Figure 1: block diagram of a magnetic ball-suspension system
Consider the block diagram of a magnetic ball-suspension system (Figure.1). The position y of the
steel ball of mass M can be controlled by adjusting the current i in the electromagnet through the input
voltage u. The electromagnet is represented as a series RL circuit with winding resistance R and winding
inductance L. The force exerted by the electromagnet on the ball is given as
F =
i
2
y
2
.
The following physical parameters will be used throughout this lab:
• g = 9.8 m/s2
• M = 1 Kg
• R = 3 Ohm
1
• L = 1 H
Notice that this lab is made of two main parts: a ”preparation part” and an ”experimental part”. Each
part has different deadlines for the submission.
In the preparation part you will complete preliminary calculations that will be necessary for the second
part. You will first submit a pre-lab report where you report all the preliminary computations. You will
receive feedback for this pre-lab part.
For the second experimental part you will find a list of all the files required to submit at the end of this
lab report.
2 Preparation
In the preparation, you will first find a nonlinear state model of the magnetic ball-suspension system,
and then linearize it about an equilibrium point.
Carry out the following steps in your preparation and submit a pre-lab report (handwritten notes for
your pre-labs are acceptable as well).
1. Take the state vector x = [ y y i ˙ ]
T
. Write down the state model for the magnetic ballsuspension system with input the voltage u and output the position y:
x˙ = f(x, u)
y = h(x, u)
Hint: There are two subsystems to model. First, the steel ball is modelled using Newton’s second
law. It is subject to two forces, Mg pointing downward and F = i
2/y2 pointing upward. Second,
the electromagnet is modelled as an RL circuit with input voltage u and inductor current i. This is
the same current appearing in the expression of F above. Write the two equations arising from the
two subsystems, and then put them in state space form using the above definition of the state x.
2. Find all the equilibrium conditions of the system parametrized by the position of the steel ball.
Specifically, given y
?
, find ˙y
?
, i
?
, and u
? as functions of y
?
such that the pair (x, u) = ([y
? y˙
?
i
?
]
>, u?
)
is an equilibrium condition of the nonlinear system. You will find two solutions for the equilibrium
current i
?
. Pick the positive one.
Linearize the model about the equilibrium condition (x
?
, u?
), to obtain a system
d
dt(δx) = Aδx + Bδu
δy = Cδx + Dδu.
Find A, B, C, D, (the entries of these matrices will be functions of y
?
) and define δx, δu, δy. In your
pre-lab report, include both the computed matrices and the steps you followed to compute them.
3. Set y
? = 1, and derive the open-loop transfer function G(s) = δY (s)/δU(s) from the input δu to
the output δy of this linearized system. Find the poles and zeros of G(s).
4. Find the corresponding impulse-response function g(t) = L
−1
(G(s)) of this linearized system and
sketch a response of g(t) or include it with the pre-lab report.
3 Experiment
In the experiment, MATLAB will be used to build a SIMULINK model of the nonlinear system, and
then linearize it to get a linear approximation.
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3.1 Building the SIMULINK Model
1. Recall that if you take the state vector x = [ y y i ˙ ]
T
, the nonlinear differential equation for x
can be written in the following form
x˙ = f(x, u), y = h(x, u)
which has been obtained in your preparation.
Figure 2: The high-level Simulink model
2. You are now going to implement your state model in Simulink. You will begin by creating the block
diagram depicted in Figure 2. Open the Library browser of Simulink and look at the Commonly
Used Blocks. Using the following blocks:
• Subsystem
• Demux 1
• In1
• Out1
• Terminator
create a diagram identical to the one in Figure 2.
3. Next, you need to implement your mathematical model of the magnetic levitation system, which
will go inside the Magnetic Ball-suspension System block. Double click on this subsystem block.
You’ll see an input directly connected to an output. Erase the connection between input and output,
and replace it with the block diagram in Figure 3. In that diagram, you see three blocks 1/s. These
are integrators, you can find them in the Continuous library. There are two integrators in series.
Their input is ¨y, and their output is y. The input of these two integrators is computed using a
function block f(·), found in the User-Defined Functions library. This function takes as input
the output of the multiplexer, which is the vector [y y i u ˙ ]> = (x, u). Summarizing, the upper
part of the block diagram in Figure 3 implements an equation ¨y = f(x, u). You will have to insert
the correct function f according to your mathematical model. Now look at the single integrator
block in the lower part of the diagram. Its input is di/dt and its output is i. What feeds this
integrator is the output of another function block f(x, u) (this f is different than then one above).
Again, you will have to put the appropriate function in this block. Save your complete model in
SIMULINK as magball. Attach the file magball.slx as part of your submission.
1Double click on the Demux to choose the right amount of output needed
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Figure 3: Subsystem Block
3.2 Linearizing the Model in MATLAB
You will create and submit a unique script ex 3 2.m for all the computations in this section. Be sure to
add comment where necessary.
1. For the real magnetic ball-suspension system, we have the following physical parameters:
• g = 9.8 m/s2
• M = 1 Kg
• R = 3 Ohm
• L = 1 H
Substituting in the values of the physical parameters, setting the equilibrium position y
?
to be
y
? = 1, and based on the linearized state equation obtained in your preparation, enter the linearized
system matrices A and B in the Matlab workspace.
2. Using the Matlab command eig, determine the numerical values of the eigenvalues of A. We will
see later in this course that if all eigenvalues of A have negative real part, then the linearized
system is stable. For the linearized maglev system, stability means that for any initial condition,
(y(t), y˙(t), i(t)) → (y
?
, y˙
?
, i?
) as t → ∞. Vice versa, if A has at least one eigenvalue with positive
real part, then the linearized maglev system is unstable, meaning that there are initial conditions
such that the triple (y(t), y˙(t), i(t)) does not converge to the equilibrium (y
?
, y˙
?
, i?
).
Look at the eigenvalues you just found: is the linearized system stable or unstable? Did you expect
the system to be stable or unstable? Add a comment about your considerations in the script you
are submitting.
Using the Matlab command ss2tf, use your matrices (A, B, C, D) to find the transfer function of
the linearized model. Compare it with the transfer function you found in your preparation and
check that they coincide.
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3. Now you will learn how to use Simulink to automatically linearize nonlinear models, and you
will compare the result to your own computations. Using the Matlab command linmod, you will
linearize your system at the equilibrium point (x
?
, u?
). First you need to understand the ordering
of states in Simulink. For that, issue the command
>> [sizes, x0, states]=magball
Look at the states array. The ordering of the strings in this array coincides with the ordering of
states used by Simulink. The string names are the labels you have used under each integrator block
(1/s). Now use a command of the form [A,B,C,D]=linmod(’magball’,xstar,ustar) to generate
the (A, B, C, D) matrices of the linearization at the equilibrium condition (x
?
, u?
) found in part 2 of
your prep, setting y
? = 1. Check that the matrices (A, B, C, D), other than a possible re-ordering
of the states, are the same as those obtained earlier in Step 1. 2
4. Use the MATLAB command impulse to obtain the impulse response and compare it with the result
in your preparation. Save your plot as a .png file with the name plot 3 1.png
5. Use the following two ways to obtain the step response (that is, the plot of the output signal when
the input fed to the system is the unit step).
(a) Use the MATLAB command step on your linearized system. Save your plot as a .png file
with the name step 3 1.png
(b) Create your linearized model in Simulink. A simple way to do this is by modifying the Fcn
blocks in Figure 3 to get the linearized model. The high level model will still look identical
as in Figure 2. The two equations in Fcn blocks are corresponding to the A and B matrices
you have obtained in Step 3. Save the new SIMULINK model as magball linear. Then use
Step block as the input source and place a Scope block to display the position y during the
simulation. Set the simulation stop time to be 2s and run the simulation. Save your plot as
a .png file with the name sim 3 1.png. Attach the file magball linear.slx as part of your
submission.
Check that the result is the same as the one obtained earlier in part (a).
4 Files to be submitted
Be sure to submit a unique folder with all the files listed below:
• A pdf file with all the pre-lab computations
• magball.slx
• ex 3 2.m
• plot 3 1.m
• step 3 1.m
• sim 3 1.m
• magball linear.slx
2Since you probably have already used the variables names A, B, C, D use some alternative naming so that you do not
subscribe the variables declared in Step 1.
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Appendix
The following is a list of MATLAB commands which may be useful for completing this lab. You may also
refer to the MATLAB Tutorial Handout on the course website (under the folder ”general handouts”) or
the reference library on the Mathworks website for more MATLAB functions and the SIMULINK usage.
• eig – Determines the eigenvalues of a matrix
• help – Access to help on MATLAB commands. e.g. help plot
• impulse – Simulates an impulse response for a LTI system. e.g. impulse(sys)
• legend – Creates a legend for a plot. e.g. legend(’Reference Step’,’Position’)
• linmod – Extract the continuous- or discrete-time linear state-space model of a system around an
operating point. e.g. linmod(’sys’, x0,u0), where x0 and u0 are the state and the input vectors.
If specified, they set the operating point at which the linear model is to be extracted.
• load – Loads mat files into the workspace.
• plot – Creates a plot on a figure. e.g. plot(time, output,’b’)
• sim – Runs a SIMULINK model file. e.g. sim(’sim position pv ip02’,10)
• ss – Creates a state-space system. e.g. sysss = ss(A,B,C,D)
• step – Simulates a step response for a LTI system. e.g. step(sys)
• tf – Creates a transfer function. e.g. systf = tf([1 -2],[1 7 2])
• tf2ss – Converts a transfer function to a state-space system. e.g. sysss=tf2ss([1],[1 -4 7])
• title – Creates a title for a plot. e.g. title(’Step Response for Linear System’)
• ss2tf – Converts a state-space system to a transfer function. e.g. systf=ss2tf(A,B,C,D)
• xlabel/ylabel – Creates a label for the axis of a plot. e.g. xlabel(’Time (seconds)’)
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