ECSE-4760
Real-Time Applications in Control & Communications
Spring 2012
Exam #2
Wednesday 5/16 11:30am Ricketts-211
Name __________________________________________
Section: MR 1:30 ____
MR 6:30 ____
(Grades will be posted in SIS when they are available.)
This exam is open book, open notes, etc. with calculators, but NO laptops, Internet connections, cell phones, …,
and no sharing materials. You need your own copies of everything, including lab reports. If you didn’t bring
your own hardcopies you will be penalized.
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 (maximum) with the breakdown shown by each
questions.
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
2012
Name _________________________
Given the following logic circuit for a state machine, answer the following questions.
D0
D1
D2
D Q
D Q
D Q
C
C
C
Flip-flop D0 has output Q0, D1
has output Q1, and D2 has
output Q2.
1. (1pt) What is the maximum number of unique states?
2. (6pt) Draw the state diagram. Label the states as D2D1D0 for consistency with Q2Q1Q0. as the outputs of the
state on the transition arrows. The diagram should include all the possible states of the system.
3. (1pt) As a LFSR (Linear Feedback Shift Register), what is the number of states in its normal repeated cycle?
4. (3pt) Assuming all the flip-flops are initially reset, show the timing diagram for all 3 outputs and clock for 9
clock cycles. The flip-flops trigger on the positive clock edge.
I Digital Logic
2012
Name _________________________
5. (3pt) How does the answer to 4 change if all the flip-flops are initially set?
6. (4pt) Find the POS (product of sums) form of
7. (2pt) What is the dual of
.
?
II Voice Processing/Delta Modulation
2012
Name _________________________
1. (3pt) Which sound would have the most regular periodic wave shape?
i) Voiced “th” ii) unvoiced “th”
iii) “ah” iv) “sh”
2. (5pt) A 2Vpp sine wave with a 1V offset is the input signal to the 56030 DSP board used in the lab. The
frequency is 1kHz with the program setting: S – 13 – 48/3. Sketch 1 cycle of the output signal below. Include
scales on both axes.
3. (4pt) For the same input signal, sketch output for D0 – 13 – 48/3 and also sketch the ideal output if there were
no LPF on the reconstruction. Include scales on both axes.
II Voice Processing/Delta Modulation
2012
Name _________________________
4. (2pt) Draw the input error voltage vs. output voltage plot (input/output characteristics) for the settings in 3.
5. (3pt) For your answer in 4, what is the highest frequency (in Hz) sine wave with a 2Vpp amplitude the delta
modulator could track without slope overload?
6. (3pt) What are the bit rates for the two cases in questions 2 and 3 and how do they compare?
III Binary Communications
2012
Name _________________________
1. (6pt) For a channel with noise Vrms = 0.001V, what must the pulse amplitude be for a PCM signal with RTZ
(50% duty cycle) in order to insure a SNR greater than 80dB?
2. (6pt) The following signal is obtained from the Binary Communication program running on the PC.
It is noted that the reconstruction voltage changes approximately every 1ms. Determine the following:
Modulation: PAM or PCM
Format: NRZ or RZ
Pulse amplitude:
Noise voltage (RMS):
Bits per sample:
Sample frequency (measured):
III Binary Communications
2012
Name _________________________
3. (2pt) The signal in 2 was obtained somewhere from the first input to the final output, including the
transmission channel. Where in this sequence will this waveform most likely appear?
4. (2pt) What is the minimum Hamming distance between code words in a code that can correct up to 3 bit
errors and detect 2 more bit errors (detect 4 & 5 bit errors)?
5. (3pt) Give an example of a code with 4 unique code words separated by a minimum Hamming distance of 7.
6. (1pt) If the code words for the code in 5 are 16 bits long, what is the code’s efficiency?
IV Digital Filtering
2012
Name _________________________
1. (6pt) Given an analog filter H(s) = 10000/(s+10000), find an equivalent digital filter (same 3dB frequency)
that has a sampling frequency of 10,000Hz using the impulse invariant transformation.
2. Consider an analog signal x(t) = 3cos(2000πt) + 5sin(6000πt) + 10cos(12000πt).
a) (2pt) What is the Nyquist rate for this signal in Hz?
b) (5pt) If the signal is sampled with fsample = 5000samples/sec, what is the discrete time signal in terms of k
where k is the discrete time sample integer.
c) (3pt) What analog signal y(t) can be reconstructed with an ideal LPF whose fco is half the sample rate?
3. (3pt) Show why a digital filter obtained by using the scaled bilinear transformation has similar magnitude
characteristics for analog (a) and digital (d) frequencies when both values are close to 0.
4. (1pt) A digital elliptic filter designed with a sampling frequency of 24kHz has its first zero in the magnitude
of the output at 7kHz. If the filter is run with the same coefficients, but at a sampling frequency of 54kHz, at
what frequency will the first zero (null) in the output signal appear?
V Graphics Simulation
2012
Name _________________________
5000
K(s  z)
and Gc (s) 
using only negative feedback (K ≥ 0), both p and z being
2
s
s p
positive and p > z, what restrictions are there on where the zeros and poles of the controlled system may lie in
the s-plane?
1a. (1pt) With G(s) 

5000
K(s  z)
and Gc (s) 
using only negative feedback (K ≥ 0), both p and z being
2
s
s p
positive but removing the condition that p > z, what restrictions are there on where the zeros and poles of the
controlled system may lie in the s-plane?
1b. (1pt) With G(s) 

5000
K(s  3)
, what is the smallest nonzero positive gain K that will result in
and Gc (s) 
2
s
s  50
the controlled system having only real poles? You do not need to find the exact solution.
2. (5pt) For G(s) 

2. (2pt) TRUE or FALSE: by properly adjusting the cost parameters and the a priori probabilities of both 2-D
Gaussian distributions it is possible to change the decision boundary from a straight line to a parabola or vice
versa for some PDFs.
3. (12pt) Classify the decision boundary for each of the pairs of PDFs (line, circle, ellipse, parabola, hyperbola,
…). Assume equal a priory probabilities and  = 0 in each case.
f1(x,y)
f2(x,y)
x
y
C1
x0
y0
C2
x
y
0
0
x
y
1
2
3
0
0
1
2
3
2
4
5
2
2
3
0
0
2
2
2
4
4
1
3
1
2
1
1
2
2
3
2
2
1
4
3
3
2
4
1
1
1
-2
0
3
5
2
3
0
0
1
2
2
0
4
2
6
1
2
0
0
0.1
2
4
1
1
10
1.
2.
3.
4.
5.
6.
VI Hybrid Control (Must answer whole questions) 2012
Name _________________________
1. (5pt) Given a plant model Gp(s) = A/(s + a) + B/(s + b), find an equivalent cascade model with states suitable
for a pole-placement state feedback controller.
2. (5pt) In s Ziegler-Nichols PID controller, Kp is small and yields a rise time that is longer than desired (too
slow). If Kp is increased by 15%, by how much should Kd be adjusted to maintain stability?
3. (10pt)
0
0 
1 
a. (5pt) A plant in its standard form model has Ac  
, Bc    and Cc  1 2. Find its
8 15
4 
characteristic equation and poles.



0
0 
1 
b. (5pt) A plant in its standard form model has Ac  
, Bc    and Cc  1 2. Find the state
8 15
4 
feedback gains –kc1 and –kc2 that place the poles at -4±j4.



VI Hybrid Control
2012
Name _________________________
0
0 
1 
4. (10pt) A plant in its standard form model has Ac  
, Bc    and Cc  1 2, Dc = 0. Find the
8 15
4 
expression whose inverse Laplace transform is used to find Ad. Write the expression with the actual matrices
and their element values. Also find Cd and Dd.



5. (10pt) An FST controller of a 1st order system with a sampling period of 1s reaches the desired output in one
period when the control input is 1V. What must the control input be for the same system to reach the same
desired output still in one period if the sampling period is shortened to 0.5s and lengthened to 2s? {HINT: the
form of the system’s response is y(t) = A[1 – exp(-t/)] }.
6. (5pt) Given that the roots of a 2nd order system can be found from a standard form equation s2  2 n s   n2 ,
what is the value of  for the closed loop roots when a Graham-Lathrop PID controller is used. Assume the
controlled system is 2nd order.


VII DC Motor Control
2012
Name _________________________
1. (4pt) For a pure proportional analog controller, where are the roots of the closed loop system when the gain
yields a dead zone of 5°?
2. (3pt) Which plant(s) will have no band of gain values that will yield a stable output with proportional closed
loop feedback?
1
1
1
1
1
1
i)
ii)
iii) 2
iv)
v) 3
vi)
s
s2
s
s
s(s  2)
s(s  2) 2
4.04z  3.96
3. (7pt) The Tustin approximation yields a digital plant G p (z) 
. The original analog plant model
2.06z
1.94





had a zero at s = -2. Find Gp(s) and fs, the sampling frequency.

4. (3pt) On an ideal DC Motor setup, what would be the difference between the response to an FST controller
and a minimal prototype controller?
5. (3pt) Why is it necessary to use the additional amp between pins 7 and 13 on the DC Motor servo system
along with the analog computer amps?
VII Optimal Control
2012
Name _________________________
1. (3pt) In a Linear Quadratic Regulator (LQR), R is chosen to be of the form r (r is a scalar, r > 0) . If the value
of r is increased while Q is kept fixed, what is likely to happen?
i) It will increase the overall energy consumption
ii) It will decrease the overall energy consumption
iii) It will slow down the system response
iv) None of the above
2. (3pt) If Q = 0 (R is still the same as the previous problem), for this particular optimal control problem (called
minimum energy control), what is the most likely K (u = -Kx) in the case where there are no constraints on the
final state?
3. (4pt) For the system x(t)   xÝ(t)  u(t), Q = 2 and R = 3. Find P.

r1 0 
4. (3pt) If R is a 2-D matrix of the form 
, where r1  r2  0 , what is the purpose of having a weighting
0 r2 
matrix like this?

5. (3pt) What is the purpose of having the matrix Q?
6. (4pt) For a 1st order system with A, B, C, D, Q, R, G and H, and the know integral values


0
x 2 (t)dt  m,

 u (t)dt  n , find an expression for the performance index in terms of these constants and the
2
0
terminal state penalty. For this controlled system the steady state value of x is 0.