CONTROL OF MECHATRONIC SYSTEMS
Homework Problems for MAE 461 and MAE 524
Note: Homework Problems with an X beside them are ONLY for MAE 524 students.
Grading Policy:

Points are taken off for poorly drawn diagrams and sloppy work.

Limited partial credit is given in these assignments.

No late homework is accepted.

Homework is to be done individually. Discussion between students
specifically about the homework assignments is not allowed. Violations of
this rule are enforced.
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CONTROL OF MECHATRONIC SYSTEMS
Homework No. 1
Consider the following single degree of freedom system:
This system consists of a disk immersed in a fluid and attached to a spring. The
unstretched length of the spring is H = 4 in, the radius of the disk is R = 3 in, the weight
of the disk is mg = 15 lb, the spring constant is k = 15 lb/in and the damping constant is c
= 0.01 lb·s/in.
Note: The disk will be acted on by an applied motor moment M (not shown here) in
another homework problem later.
(a) Draw a free body diagram of the system when it is rotated an arbitrary angle .
(b) Develop a general expression for the moment acting on the disk by the spring.
(c) Derive the general nonlinear differential equation governing the motion of the
system for an arbitrarily large angle.
(d) Derive the general nonlinear algebraic equation governing the static equilibrium
of the system for an arbitrarily large angle.
(e) Find all of the system's equilibrium positions.
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CONTROL OF MECHATRONIC SYSTEMS
Homework No. 2
Consider the system in Homework No. 1 and specifically the nonlinear
differential equation of motion that you derived in part (c).
(a)
Assume that the nonlinear differential equation of motion has the
f
general form   f ( ,). Find the stability derivatives
and

f
about each of the equilibrium positions.

(b)
For each equilibrium position, let (t) = (t) - 0 represent an angle
measured from the equilibrium angle, and write equation the linear
approximation of f.
(c)
Write down the linear differential equation governing the motion of the
system in the neighborhood of each equilibrium position.
(d)
Find a general form of the solution of each linear differential equation.
(e)
Specify which equilibrium positions are stable and which are unstable.
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CONTROL OF MECHATRONIC SYSTEMS
Homework No. 3
Consider again the system in Homework No. 1. Assume that the system was
initially at rest, and that the initial angle was 10○, which is in the neighborhood of the first
equilibrium position. Determine the response of the system (t) in the following ways:
(a)
Use the linear differential equations you found in Homework No. 2 to find
the analytical solution of (t). Using MATLAB, plot (t) for about Tf = 3
periods.
(b)
Rewrite the nonlinear differential equation you found in Homework No. 1
in part (c) as two first-order nonlinear differential equations (state
equations).
(c)
Use the Euler Method to numerically integrate the state equations you
found in part (b) to obtain the numerical solution of (t). Using MATLAB,
write a short program to do this and plot (t) for about 3 periods. Use a
step size of about T = 0.005Tf seconds (The number of steps is then about
600.)
(d)
Describe what accounts for the differences between the responses obtained
in (a) and (c)?
(e)X
Linearize the nonlinear differential equation of motion in the state space
rather than in the configuration space. You may wish to review section
I.A-4.
(f)X
Find (t) numerically by solving the nonlinear state equations using the
Second-order Runga Kutta Method. Use the same step size as in (c). Write
a program in MATLAB to do this and plot your results.
(g)X
Find (t) numerically by solving the linear state equations using the
Second-order Runga Kutta Method. Use the same step size as in (c). Write
a program in MATLAB to do this and plot your results.
(h)X
Describe what accounts for the differences between the responses obtained
in (a), (c), (f), and (g)?
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CONTROL OF MECHATRONIC SYSTEMS
Homework No. 4
Consider again the system in Homework No. 1. The disk is now acted on by an
applied motor moment M = Rf. Two motor moments will be compared – a finite pulse f1
and an impulse f2. When the time of the pulse is “short” enough, the response of the
system acted on by the finite pulse should look like the response of the system acted on
by the impulse. The question arises how short a time is “short.”
The short-time pulse f1 and the impulse f2 are given by:
0  t  T0
A ,
f1   0
,
T0  t  
0,
f 2  A0T0 (t )
where T0 is the period of the pulse (not to be confused with step size of period). Let T0 =
1 sec and A0 = 200 lb. Notice for both functions that the integral over time of the force is
the same, that is
T0
0
T
f1 (t )dt  0 0 f 2 (t )dt  A0T0 .
(a)
Draw a free body diagram of the system: This time assume that the angle
of the disk is small. Remember that sin() =  and cos( ) = 1. (Recall
that the free body diagram that you drew in part (a) of Homework No. 1
assumed that the angle was arbitrarily large).
(b)
Using this free body diagram, find the linear differential equation that
describes the motion of the system. (It should be the same as the linear
differential equation you found in part (d) of Homework No. 1.)
(c)
Find the response (t) of the system subject to each of the forcing
functions assuming that the system is initially at rest. Using MATLAB,
plot each response for about 4 times the period of the pulse.
(d)
Look at the two responses and comment on the nature of the
approximation.
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CONTROL OF MECHATRONIC SYSTEMS
Homework No. 5
Consider again the system in Homework No. 4. Assume now that the system is
subject to a function f3 which is a square wave, as shown below.
Mathematically, the square wave can be written as
 A0 ,
f3  
  A0 ,
2( r  1)T0  t  (2r  1)T0
(2r  1)T0  t  2rT0
( r  1,2,...)
in which T0 denotes a half-period. (Use the same values of T0 and A0 here that you used in
Homework No. 4).
(a)
Find the response of the system by representing the forcing function f3 as a
Fourier Series. Plot the response for about 5T0 seconds.
(b)
Find the response of the system by representing the forcing function f3 as a
series of step functions. Again, plot the response for about 5T0 seconds.
(c)X
Find the response using the convolution integral.
(d)X
Find the response by numerical integration using the Second-Order RungaKutta Method.
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CONTROL OF MECHATRONIC SYSTEMS
Homework No. 6
Consider the 1 dof system shown below.
This homework problem is about the effectiveness of open-loop control. The interest lies
in designing an open-loop tracking controller to move the cart from rest position A (x(0) =
0) to rest position B in time Tf. Let m = 200 slug, Tf = 10 sec, L = 24 ft, and H = 10 ft.
(a)
Find the desired path of the cart.
(b)
Find the associated open-loop tracking force.
(c)
The next four parts evaluate the sensitivity of the response to parameter
errors: You’ll now assume that the actual parameters are slightly different
than the ones you postulated to get the expressions in (a) and (b).
Specifically you’ll change the mass and the initial conditions and see what
effect they have on the response of the system. First, plot the nominal case,
in which the postulated parameters are the same as the actual parameters.
(d)
Let m = m/5, x(0) = 0 and plot the response.
(e)
Let m = - m/5, x(0) = 0 and plot the response.
(f)
Let m = 0, x(0) = x(Tf)/10 and plot the response.
(g)
Let m = 0, v(0) = x(Tf)/(10Tf) and plot the response.
(h)
Describe the differences between the responses.
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CONTROL OF MECHATRONIC SYSTEMS
Homework No. 7
Consider again the system in Homework No. 4. Assume now that the system is
subject to a linear state (proportional-derivative) feedback control force f = fC . The
feedback controller is designed to reduce the peak-overshoot by a factor of 2 and to
dampen 90% of the motion in 4 vibration periods.
(a)
Determine the controller’s control gains.
(b)
Determine the response of the uncontrolled system and the control system
by the Euler method. Plot both over about 4 vibration periods. Let (0) = 0
and d (0)/dt = 0.1 rad.
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CONTROL OF MECHATRONIC SYSTEMS
Homework No. 8
Consider again the system in Homework No. 4. Use a PID (proportional-integralderivative) feedback control force fC to control the motion of the system. The feedback
controller is designed to reduce peak-overshoot by a factor of 2, and to dampen 90% of
the motion, including the errors associated with a “bias” acting on the system, in 4 natural
periods. A natural period is the period of the uncontrolled system. A bias is a constant
force acting on the system that causes its equilibrium position, if uncontrolled, to be nonzero. You’ll need to assume, in addition to the controller, that a bias force acts on the
system. Select a bias force that causes the equilibrium position of the uncontrolled system
to be 20○.
(a)
Determine the bias force fB.
(b)
Find the control gains g, h, and i.
(c)
Determine the response by the Euler method. Plot the uncontrolled and
controlled responses over about 4 natural periods. Let (0) = d (0)/dt = 0.
Note: In order to solve this problem, you’ll need to introduce the state
variables
t
x   ( s)ds, x   , x   .
1
0
2
3
The corresponding initial conditions are then
0
x1 (0)  0  ( s)ds  0, x2 (0)   (0),
(d)X
x3 (0)  (0).
Assume that there is a time delay Td associated with the electronics.
Examine the effect of the time delay on the response of the system and
comment on your observations.
Note: The time delay can be introduced in the computer program by letting
it equal to a multiple p of the step size T, i.e., let Td = pT. You will need to
store the values of the calculated force over the period of the time delay (p
values). The initial p values of the tracking force are zero.
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CONTROL OF MECHATRONIC SYSTEMS
Homework No. 9
Consider the two degree-of-freedom (2dof) mass-spring system shown below.
Let mg = 20 lb, L = 4 ft, H = 3 ft, and k = 10 lb/in.
(a)
Draw a free body diagram of the system assuming small angles.
(b)
Derive the linear equations governing the motion of the system.
(c)
Determine the system's mass and stiffness matrices.
(d)
Calculate the system natural frequencies of oscillation, natural modes of
vibration, and natural modes of force.
(e)
Assume that the system is initially at rest with the springs unstretched (in
the positions shown). Plot the response of the each degree of freedom as a
function of time.
10
CONTROL OF MECHATRONIC SYSTEMS
Homework No. 10
Consider again the system in Homework No. 9. The performance of the system
needs to be modified as follows: 90% of the motion needs to be damped out in 4
fundamental periods. Assume that a motor acting at point O on the bent beam produces a
control moment M and that an upward control force F acts on the block.(Neither M nor F
are shown in the figure.)
(a) Design a full-dimensional PD control system to control the system. Determine
the control gain matrices G and H.
(b) Write out the equations in the state space.
(c) Using Euler integration, find and plot both the uncontrolled response and the
controlled response (the uncontrolled response was found in Homework No.
9).
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CONTROL OF MECHATRONIC SYSTEMS
Homework No. 11
Consider again the system in Homework No. 9. The performance of the system
needs to be modified as follows: 90% of the motion needs to be damped out in 4
fundamental periods – including the steady-state error. Assume that a motor acting at
point O on the bent beam produces a control moment M and that an upward control force
F acts on the block.(Neither M nor F are shown in the figure.)
(a)
Determine the control gain matrices G, H, and I.
(b)
Write out the equations in the state space.
(c)
Using Euler integration, find and plot both the uncontrolled response and
the controlled response.
(d)X
Compare the forces (control effort) in Homework Nos. 10 and 11.
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CONTROL OF MECHATRONIC SYSTEMS
Homework No. 12
Consider again the system in Homework No. 9. The performance of the system
needs to be modified as follows: 90% of the motion needs to be damped out in 4
fundamental periods. Assume that a motor acting at point O on the bent beam produces a
control moment M and that an upward control force F acts on the block.(Neither M nor F
are shown in the figure.)
(a)
Assume that there is no sensor acting on the cart to measure the position or
velocity of the cart. Design an observer to estimate the position and
velocity of the cart.
(b)
Find the control gains of the observer.
(c)
Write out the state equations for the controlled system and the observer.
Use the control gains you found in Homework No. 10.
(d)
Using Euler integration, find and plot the controlled response.
(e)
Compare the control response in Homework Nos. 10 and 12.
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