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Lagoa Fall 2008

EE 380 - Homework #1Due Wednesday, September 17th

1. A system is described by the differential equation

y(4) + y(3) + 5y(2) + 7y(1) + y =  u(3) + 2u(2) + 3u(1) + 7u.

Find the transfer function  Y  (s)/U (s).

2. For the system in Problem 1, find the step response using  Matlab toolbox  Simulink .

(a) Plot your output for 30 seconds and comment. Please include descriptive titlesand axis labels.

(b) In view of the results in (a), a feedback gain   K   is introduced into the system.This leads to a systems with 7y(1) replaced by (7 + K )y(1). Using  Simulink , finda suitable value of  K  so that the unstable oscillations in (a) are eliminated. Plot30 seconds of the step response for the stabilized system. (Note: Try not to usean excessively large gain if avoidable.)

For this problem submit the two plots as well as a copy of the  Simulink  schematic fileused to generate the plots (only one schematic is needed). Please enlarge the necessaryblocks so that all the numbers inside are visible, e.g. the transfer function block.

3. Consider the following RLC circuit

R L

C+

-v (t)C

v (t)i

+

-i(t)

and assume that vc(0) = 0.

(a) Does the following integro-differential equation

vi(t) = Ri(t) + Ldi(t)

dt  +

  1

C 

   t

0i(τ )dτ 

describe the relationship between the input voltage   vi(t) and the current   i(t)?Justify.

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(b) Using the equation above, find the transfer function from the input voltage vi(t)and the current  i(t); i.e. find  I (s)/V i(s).

(c) Assume that vi(t) = 0 for t > 0 and i(0) = 0 . Does  i(t) tend to zero as  t →∞ ?Justify.

4. The block diagram of a control system is shown in the figure below.

)(1 sG

)(2 sG

)(1 s H

)(3 sG+

++ +

+

- -

)(s R)(s E

)(s N

)(sY

Draw the equivalent signal flow graph and find the transfer functions below

Y  (s)

R(s)

N =0

Y  (s)

N (s)

R=0

E (s)

R(s)

N =0

E (s)

N (s)

R=0

5. Signal flow graphs may be used to solve a variety of electric network problems. Shownin the figure in the next page is the equivalent circuit of an electronic circuit. Thevoltage source ed(t) represents a disturbance voltage. The objective is to find the valueof the constant k  so that the output voltage  e0(t) is not affected by ed(t). To solve thisproblem, it is best first to write a set of cause-and-effect equations for the network,This involves a combination of node and loop equations. Then, construct the signalflow graph using these equations., Finally, find the gain eo/ed  with all other inputs setto zero. For  ed  not to affect  e0  set e0/ed  to zero.

+

-

)(tes 1 R

2 R

3 R

4 R 5 R

S R

+

-

)(ted +

-

)(1 tke

+

-

)(0 te

+ -

)(1 te

)(2 ti )(3 ti

)(tis

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6. Find the following transfer functions for the signal flow graph shown below. Commenton why the results for parts (c) and (d) are not the same.

(a)  Y 7

Y 1

Y  8=0

(b)  Y 7

Y 8

Y  1=0

(c)  Y 7

Y 4

Y  8=0

(d)  Y 7

Y 4

Y  1=0

1G 2G3G 4G 5G

6G

2 H-

3 H-

1 1

1Y 2Y

3Y 4Y

5Y 6Y 7Y

7Y

8Y

1

1 H-