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[answered] ME EN 3210 MECHATRONICS II HOMEWORK 9 Problem 1 Consider a


I need help with problem 3, part b and c is where i am stuck?


ME EN 3210 MECHATRONICS II

 

HOMEWORK 9

 

Problem 1

 

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.

 

Problem 2

 

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

 

each system.

 

s

 

a.

 

s5

 

s 3

 

b.

 

(s 2)(s 4)

 

1 c. 2

 

s

 

1

 

d. 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

 

HOMEWORK 9

 

Problem 3

 

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.

 

Problem 4

 

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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