I need help with problem 3, part b and c is where i am stuck?
ME EN 3210 MECHATRONICS II
Consider a mass-spring-damper system with the characteristic equation: ms2x + bsx + kx = 0. Assume a
mass m = 10 kg and a damping b = 100 Ns/m.
a. We want to find the spring constant k that makes this system critically damped. Calculate it
analytically, which is easy to do for this simple system.
b. Now that you know the correct answer, let's practice using a root-locus diagram to calculate the
same parameter. The root-locus method can be used with more complicated systems that are too
difficult to tackle analytically. First, reformulate the characteristic equation in a form that will let
you consider the characteristic equation's roots as k varies from zero to infinity. Then, use
MATLAB's rlocus function to plot the root locus (turn in your plot, and the m-file used to
generate it). Click on the location of the root-locus in the location that represents the critically
damped response, and find the value of k at that point. Turn in any additional calculations that
were required. Remember, since this is a numerical method, you might not get the exact answer
that you calculated in part (a), but it should be very close.
Now let's practice drawing and interpreting root-locus plots with MATLAB. For each of the plant
transfer functions given below, assume the plant is being controlled using negative-unity-feedback
control with a simple proportional controller with gain K, and plot the root-locus as K is varied from
zero to infinity. Turn in all of your root-locus plots, as well as a single m-file that plots them all. Finally,
use MATLAB's rlocus tools to determine the range of K for which the closed-loop system is stable for
(s 2)(s 4)
1 c. 2
(s 5)(s 4s 8) Problem 3 and 4 are on the following page. DEPARTMENT OF MECHANICAL ENGINEERING, UNIVERSITY OF UTAH
PAGE 1 OF 2 ME EN 3210 MECHATRONICS II
Consider the position control of a mass (for example, a free-floating body in space) with a transfer
function G(s) that maps input force F(s) to output position X(s). A common feedback controller is the
proportional-derivative (PD) controller C(s) = Kds + Kp which converts the error signal E(s) = Xdes(s) X(s) to the effort F(s). An alternative control method is proportional control with velocity feedback, in
which the control effort is set as F(s) = Kp(Xdes(s) - X(s)) - KvsX(s). The velocity feedback gain Kv acts
very much like the derivative gain Kd in a PD controller, but the two control methods are different in
some key ways.
a. Draw and label the block diagrams of the two systems described above.
b. Calculate the transfer function X(s)/Xdes(s) for each of the two systems. Comment on the
differences and similarities between the two control methodologies in terms of the form of the
resulting transfer functions.
c. Assuming a 1 kg mass, choose the three controller gains such that the characteristic equations of
the closed-loop systems are critically damped with a natural frequency of 1 rad/sec.
d. Use the MATLAB function step to generate unit-step-response plots (i.e., Xdes(s) = 1/s) of the
two transfer functions from part (b), using the values from part (c), on a single plot. Turn in the
labeled plot, as well the m-file used to generate it.
e. Use the MATLAB function bode to generate the Bode plots of the two transfer functions from
part (b), using the values from part (c), on a single plot. Turn in the labeled plot, as well as the
m-file used to generate it.
f. Comment on the potential pros and cons of using PD control vs. using P control with velocity
feedback. In doing so, consider your results from the step-response plot (in which the input was
constant), but also use the Bode plot to imaging how your system would be able to track
sinusoidal signals in which your input was constantly changing.
g. Often, our sensor is corrupted with noise N(s), which can affect our controller's performance. In
this case, our controller cannot utilize the true position X(s), but must rely on a noisy measured
position: Xmeas(s) = X(s) + N(s). Redraw the block diagrams for the PD-controlled system with
noise N(s) as a second input.
h. Calculate the transfer function X(s)/N(s) for the system from part (g). Comment on the results in
terms of the affect of sensor noise on the system's response.
We would like to design a finite state machine for a heating and air-conditioning system, so that the
furnace and air conditioner are not constantly turning on and off. The homeowner will set a desired
temperature Td. We are okay if the temperature in the room is 1 degree too warm or 1 degree too cold,
but we won?t tolerate anything more than that. The only actions that you can take are to turn the furnace
on and off, and to turn the air-conditioner on and off. (a) Draw the state transition diagram for the
heating/cooling system. Don?t worry about how to turn the whole system on/off. Just assume it?s always
running. (b) Write a pseudocode for this controller, using ?if? and ?else if? statements. DEPARTMENT OF MECHANICAL ENGINEERING, UNIVERSITY OF UTAH
PAGE 2 OF 2
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