Homework 2: EE 4314, Control Systems, Fall 2013 Posted 09/17/13, Due 10/1/13 Instructions:

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Homework 2: EE 4314, Control Systems, Fall 2013

Posted 09/17/13, Due 10/1/13



Instructions:

The purpose of this homework is to let you practice concepts learned in class


regarding system models, block diagrams, stability and response.
. To receive full credit, please
return handwritten or typed answers to me prior to the lecture on Sept. 17. Be as detailed in your
answers as possible, and show the steps you followed in arriving at your answers (simply listing
the correct answer does not

mean you will receive full credit). In addition, MATLAB problems
require you to print the code (*.m file), its ouput and associated plots. Finally, compile all such
code, output and plots into a
single
document in
Word format
. Name the document
'FA2013_EE
4314_Homework2
_Lastname.doc’.

If there is a need to send .m files or other
relevant files, include them and the above mentioned word document in a .zip file
'
FA2013
_EE4314_Homework2
_Lastname.zip'.
Please email this file to the TA,
suresh_sampathkumar@mavs.uta.edu
. You are encouraged to consult with your classmates while
you work on the homework, however, writing/coding and understanding final submissions must
be your own work.



Problem 1 (55 pts) (Dy
namic Models)


(i)

Consider the mechanical system in Figure 1, the
motorized
cart
-
pendulum system. The cart
has a moving mass

M, and is connected to a rotating

motor
that drives an inverted pendulum
.

The mass M linear motion has a damping coefficient of B and
a spring constant K
. The

pendulum of length l has
negligib
le inertia and

mass m
, and

is

a
ttached to the cart via a rotary
servo motor that has
motor constants Km

and Ke, armature resistance Ra and inductance La.
Assume that

the pendulum damping coefficient is b
, the linear actuator force is zero

and

the
rotar
y actuator torque is driven by the

dc servo motor

torque generated by armature voltage
ea(t
). Your tasks are as follows:



a)

Write the dynamical equations of motion

as a f
unction of the states (15
pts).

b)

Approximate the pendulum equations around
θ=0
, and find the transfer function
between input ea

and output θ (15

pts).

c)

Write a MATLAB simulation to animate the cart and pendulum response to a sine
wave input from an initial co
ndition
s

of x=0,
θ=0
.

Assume reasonable values for
simulation constants, and you m
ay refer to :

http://www.baldor.com/support/literature_load.asp?LitNumber=BR1202
-
F


for

the dc servo motor numerical specifications (indicate which motor you selected).
(25 pts
).




Figure 1: The cart
-
pendulum system



Problem 2 (15 pts) (Routh’s Criterion)

(a)

Using the Routh’s

table find out how many poles of the following function are in the
right half plane, in the left half plane, and on the


axis



(

)
























(b)

Using the Rout
h’s
-
Hurwitz criterion, find out how many closed loop poles of the system
shown

in the figure lie in the left half plane, in the left half plane, and on the


axis










Problem 3 (15 pts) (Stability)

(a)

A linearized model of a torque controlled crane hoisting a load with a fixed rope
length is


(

)





(

)


(

)












(







)



where










the rope length,



the mass of the car, a = combined rope
and car mass,


= the force input applied to the car, and




the resulting rope
displacement. If the system is controlled in a feedback configuratio
n by placing it in a loop as
shown in the figure below with K>0, where will the closed loop poles be located?



(b)

For the transfer function below, find the constraints on K1 and K2 such that the
function will have only two



poles





Problem 4 (15 pts
) (Transfer functions)


A virtual reality simulator with haptic(sense of touch) feedback was developed to simulate the
control of a submarine driven through a joystick input. Operator haptic feedback is provided
through joystick position constraints and si
mulator movement. Figure below shows the block
diagram of the haptic feedback system in which the input



is the force exerted by the muscle

of the human arm; and the outputs are


, the position of the simulator, and


,the position of the
joystick.




(a)

Find the transfer function


(

)


(

)


(b)

Find the transfer function


(

)


(

)