ECSE-4760
Real-Time Applications in Control & Communications
Spring 2010
Exam #2
Wednesday 5/19 11:30 a.m. Lally-102
Name ______________________________
Section: MR 1:30___
MR 6:30___
(Grades will be posted on SIS when they are available.)
This exam is open book, open notes, etc. calculators, but NO laptops, Internet connections…
Answer questions in only 5 out of the 8 sets corresponding to each of the experiments in the lab. You do not
need to choose the experiments you performed but you will probably wish to do so. If you answer more than 5
sets, the first 5 will be graded. Each set is worth 20 points (max) with the breakdown shown by the questions.
If you are missing any grades on RPILMS, make sure you bring the reports to me (JEC-6028) before the end of
the day today to get your records corrected.
I Digital Logic ______
II Voice Processing ______
III Binary Communications ______
IV Digital Filter ______
V Interactive Graphics ______
VI Hybrid Control ______
VII DC Motor ______
VIII Optimal Control ______
TOTAL ______
I Digital Logic
2010
Name _________________________
001
1
0/0
0/0
010
2
1
000
0
0/1
0/1
1/0
0/0
011
3
1/0
1/0
1/0
100
4
1/0
1/1
1/1
110
6
101
5
0/0
0/0
1. (9pt) Given the state machine with Next State Table and Output Table below, find the expression for D2 the
MS bit in the D flip-flop implementation and simplify it using a Carnough map.
Input
0
1
P.S.
000 001 100
001 010 100
010 011 100
011 011 100
100 000 101
101 000 110
110 000 110
111
d
d
Next State Table
Input 0
1
P.S.
000
0
0
001
0
0
010
1
0
011
1
0
100
0
0
101
0
1
110
0
1
111
d
d
Output Table
I Digital Logic
2010
Name _________________________
2. (2pt) Find the logic expression for the output of the state machine and simplify.
3. (2pt) Draw the state diagram for a 4-bit counter that skips all the counts that are divisible by 5 without a
remainder. Represent the states in binary.
4. (7pt) Given the following logic expression for T, find the reduced inverse logic expression T .
T  CD  CD  ABC  ABD  ABCD
T  ?(use Carnough with 1 <=> 0)

II Voice Processing
2010
Name _________________________
1. (10pt) A low order antialiasing filter with too wide a pass band will result in aliasing of some frequencies. If
the antialiasing filter’s response is similar to the Sinc function below from 0 - 20Hz and then effectively zero
for all frequencies above 20kHz, what effect does this have on the spectrum in the 0 – 40kHz band of a 15kHz
input sine wave? (Use equations or appropriately scaled plots to explain results.)
Antialias
Filter
fco ≈=10kHz
20kHz
Ideal
Sampler
fs = 20kHz
Z.O.H.
Ideal LPF
Reconstruction
Reconstruction
fco = 10kHz
x-axis is frequency in kHz
Input
1.5
1
0.5
0
0
10
20
30
40
50
40
50
Antialias Filter
1.5
1
0.5
0
0
10
20
30
2. (2pt) If a 6-bit quantizer has a quantization noise SNRdB = 32dB, with the same technology using a 9-bit
quantizer, what will the new SNRdB approximately be by the “rule of thumb”?
II Voice Processing
2010
Name _________________________
3. (2pt) For the RTA DSP implementation (and all its limitations) with fs = 48kHz and a 1-bit deadband delta
modulator using a 0.317mV step size, how good would the reconstruction be on a 1Vp-p triangle wave with a
frequency of 5.0Hz? Explain your answer.
4. (4pt) For the following input triangle wave and ideal delta modulator output, determine a possible set of
values for A, B, C, and D.
5. (1pt) Is the modulator in 4. a deadband modulator? Circle one. YES
NO
CANNOT BE DETERMINED
6. (1pt) Which sound will have a larger amplitude and a more periodic (less random) waveform, a voiced “th”
or an unvoiced “th” sound?
III Binary Communications
2010
Name _________________________
1. (6pt) Given a Gaussian noise distribution on a voltage pulse such that A = 3 where A is the amplitude of the
pulse and  is the standard deviation of the noise, find the probability of a bit error from the pdf.
2a. (4pt) Find the minimum hamming distance between codewords for a code that will correct double bit errors
and simultaneously detect up to 4-bit errors.
2b. (3pt) Could this code be used to correct more than double errors while simultaneously detecting 4-bit
errors?
2c. (3pt) Find the minimum hamming distance between codewords for a code that will correct double bit errors
or detect up to 4-bit errors.
3. (2pt) What is the efficiency of a code with 5 message bits and a hamming distance of 4?
4. (2pt) TRUE or FALSE: A RZ (RTZ) modulation with a 50% duty cycle requires about twice the bandwidth
of a similar modulation with a 25% duty cycle.
IV Digital Filter
2010
Name _________________________
1. (4pt) A 6th order analog filter with fcutoff = 3kHz was transformed using a scaled bilinear with fs = 18kHz.
What will be the actual (digital) cutoff frequency of the filter when it is downloaded to the Hyperception DSP
board and executed?
2. Design a 5th order Butterworth high pass digital filter with fs = 36kHz, that is 3dB down @ 6kHz and >30dB
down @ 3kHz.
2a. (4pt) Sketch its approximate pole-zero diagram
2b. (3pt) Sketch |H(ej)| vs. for 0 ≤  ≤ 2 with scaled axes in Hz.
IV Digital Filter
2010
Name _________________________
3. (4pt) For a 6th order Butterworth & Chebyshev I HPF, what is the numerator of H(z)? (Don’t worry about
scaling for unity gain at peak output.) Give answer as a 6th order polynomial in z or as a product of second order
systems [az 6  bz 5  cz 4  dz 3  ez 2  fz  g or (az 2  bz  c)  (dz 2  ez  f )L ].

4. (5pt) H(s) 

1
. Use the impulse invariant transformation with T = 0.1 seconds to find H(z).
(s  1)(s  2)
V Interactive Graphics
2010
Name _________________________
1a. (6pt) The transfer function of the DC motor was given as G1(s) 
5000
but the actual model is
s2
2250
. If both systems are configured with negative unity feedback gain, how will the impulse
s(s  6.25)
responses differ? Describe the forms of the response if you can’t find the actual expressions.

G2 (s) 
1b. (2pt) Classify the stability of the feedback systems with G1(s) and G2(s) in 1a.
2. (7pt) Given a Gaussian pdf f1 (x, y) 
1
exp12 (x 2  y 2 ), C1 = 10, and a second Gaussian pdf
2
1
exp12 [(x  0.2)2  (y  0.1)2 ], C2 = 1, with equal a priori probabilities, determine the decision
2
boundary between the two probability functions, looking down on the x-y plane.

f 2 (x, y) 
3. (5pt) Given 2 Gaussian pdfs with parameters below with equal a priori probabilities, what must the ratio
C1:C2 be for the decision boundary to be the line x =1.75.
f1(x,y)
f2(x,y)
x
1
1
y
3
3

0
0
x0
0
2
y0
0
0
Cost
C1
C2
VI Hybrid Control
2010
Name _________________________
ANSWER ANY COMBINATION OF QUESTIONS THAT ADD UP TO 20 POINTS.
1. (5pt) Which parameter or factor in a PID controller of a type 0 system will introduce a steady-state error
between the input and output when the feedback controller is applied?
i) KP
ii) KI
iii) KD
iv) the presence of a free integrator in series with the plant Gp(s)
2. (5pt) For any of the three digital approximations to PID controllers, which parameter, P, I, or D, causes the
controller value to exhibit quantization error for small sampling periods and why?
3. (5pt) TRUE or FALSE: Both an ideal ripple free (FST) and ideal minimal prototype controller on a 2nd order
system guarantee no continuous error after the second sampling interval.
4. (5pt) A PI Ziegler-Nichols controller for a process has Kp = 0.1 and Ki = 0.01. Find R and L for the plant’s
step response and find the value of Kp for a purely proportional Ziegler-Nichols P controller.
VI Hybrid Control
2010
Name _________________________
2
and a purely proportional gain constant in the feedback path, if
(s  5)(s  11)
the system has POSITIVE feedback with a gain of 26, is it stable?
5. (5pt) For a plant with G p (s) 

6. (10pt) Given the following system, find the pole placement gains that will place the state-controlled poles at
s = -2  j2.
xÝ1(t)  x 2 (t)
xÝ2 (t)  x1(t)  2x 2 (t)  3u(t)
y(t)  3x1(t)  4 x 2 (t)

7. (10pt) Set up the equations to find the PI gains for a controller that minimizes the integrated time absolute
s 1
error, given G p (s) 
.
(s  3) 2

VII DC Motor Control
1. (5pt) Show that H(s) 
2010
Name _________________________
 2 (s  z) 2
,  > 1, has Bode plot properties very similar to a phase lead compensator.
(s  z) 2

2. (5pt) Find the position error constant and the error velocity constant for the plant
compensator
Gc (s)  7
s2
with
s 10
G p (s) 
123s  11
s 2  3s 8
and
negative unity feedback when the input is a unit step and unit ramp


3a. (5pt) Find the digital controller Gc(z) obtained from the analog controller Gc (s) 
approximation with T = 0.1s.

3b. (2pt) Is Gc(s) a lead or lag compensator?
4. (3pt) What is the settling time for a controlled system with G(s) 

10
?
s  3s  9
2
10s  40
using the Tustin
s 12
VIII Optimal Control
2010
Name _________________________
0 1 
0
5 0 
1a. (4pt) For a 2nd order system with Ac  
, Bc   , Cc  1 2, Dc  0, Q  
, R  4 and whose
3 5
2
0 10
gain matrix for the LQR is G = [-0.5 -1], what must the p21 and p22 values be in the solution to the Riccati
equation?

1b. (1pt) Will the values of p11 and p12 have any effect on the feedback gains?
1c. (1pt) If all the same matrices were used in a discrete system, would the P matrix have the same values?
2. (8pt). Given x(k  1)   x(k)  u(k) and J(u(k), x(k), k)  H 
1
2
1
2

1
2
5x
2
(k)  3u 2 (k), find P used to
k 0
determine the feedback gain(s) but don’t find the gain(s). Assume there is no terminal state penalty.


3. (4pt) What are the required characteristics of the P matrix for the solution to the continuous Riccati equation?
4. (2pt) Given the continuous performance index J and the A, B, Q, R, and H matrices, what assumptions are
made about the limits of the integral term and the terminal state penalty matrix in finding a solution for P in the
differential Riccati equation?