ECSE-4760 Real-Time Applications in Control & Communications Spring 2015

Exam #2Thursday 5/21 11:30 AM Troy-2018

Name _______________________ Section 1: MR 1:30

(Grades will be posted on SIS when they are completed.)

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. No softcopies are allowed!

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. It is more important to set up the equations for the solutions than to solve them completely. You may go back if you have time at the end to do so.

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 ______

10/0

01/0

10/0

11/0 00/1

01/1

01/100/011/0

A B C D 10/1

00/1

11/0

00/1

11/0

10/1

01/1

I Digital Logic 2015 Name _________________________

1. (6 pt) Find the reduced SOP expression for ( A+B C ) ( A D+B E ) B C E+C?

2. (4 pt) Reduce the expression F=BD+ A B C D+ ABC+AC D+ A B D+BC D+ A B C D using K- maps.

3. Given the state transition flow diagram below for a system with inputs and output:

a. (2 pt) Is it possible for a set of inputs to get to state D from state A only outputting 0s? If so, what set of inputs will do it?

b. (8 pt) Find the system’s next state and output tables.

II Voice Processing 2015 Name _________________________

1. (6 pt) Match each speech signal shown below (the sample frequency fs = 8.192kHz) with a choice below. i) Soft “S” sound ii) Voiced “TH” sound as in “this” iii) Long “O” sound iv) End “T” sound as in “left”

0 200 400 600 800 1000 1200 1400 1600 1800-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

0 50 100 150 200 250 300 350 400 450 500-0.025

-0.02

-0.015

-0.01

-0.005

0

0.005

0.01

0.015

0.02

0.025

0 100 200 300 400 500 600 700 800 900

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

2a. (2pt) Rate the following signals in order from lowest to highest crest factor: 2sin(2π4000t), a 1Vpp triangle wave at 500 Hz, a 2Vpp square wave at 100 Hz, a comb function (.1V spikes lasting 0.1ms and repeating every 2ms.?

b. (2pt) What is the lowest possible value for a crest factor that any signal can have?

II Voice Processing (cont.) 2015 Name _________________________

3. Two identical signals are illustrated below, that have been transmitted over two channels using two different delta modulators but both with a sample frequency of 320Hz. Assume there is no D.C. blocking or LPF.

a. (7 pt) Find the A, B, C, & D parameters for each delta modulator that will produce the corresponding output. Note that the y-axis scales are not the same in each plot.

b. (3 pt) For each modulator, draw the input/output characteristics on the axes below and include scale values.

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8In: 20Hz sin, Out: 2-bit delta modulation (fs=160Hz)

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5In: 20Hz sin, Out: 2-bit delta modulation (fs=160Hz)

VoutModulator 1

Vin

VoutModulator 2

Vin

III Binary Communications 2015 Name _________________________

1a. (3 pt).If the SNR = 10.46dB for a PCM digital signal with 100% Duty Cycle and 5V amplitude pulses, what is the RMS voltage of the noise?

b. (3 pt) What would the SNR be in 1a. if the Duty Cycle were reduced to 25%?

2. (2 pt) TRUE/FALSE: In-phase signaling alone (without any quadrature components) cannot use carrier modulated pulses.

3. A (7,4) Hamming code can be extended to (8,4) by adding a parity bit to the full 7-bit string:c1 c2 m1 c3 m2 m3 m4 p with c1 = m1 m2 m4,

c2 = m1 m3 m4, c3 = m2 m3 m4, and p = c1 c2 m1 c3 m2 m3 m4

a. (2 pt) What is the efficiency of this code?

b. (2 pt) What is the minimum hamming distance of the code? (Assume it can maximally detect one more bit error than the (7,4) code.)

c. (4 pt) What are the possible modes for error corrections and/or simultaneous error detection given the possibility of 1 bit error, 2 bit errors, 3 bit errors, etc.?

d. (4 pt) If the received codeword is 00110100 and it is known to have at least 1 bit error, determine how many bit errors it contains and if it is correctable, find the corrected codeword.

IV Digital Filter 2015 Name _________________________

1. A second order HPF is designed to have a cutoff frequency of 60Hz with a sampling frequency of 10kHz. The coefficients are entered into a DSP correctly, but instead of a setting the sampling frequency to 10kHz it is mistakenly set to 10krad/s. What is the end effect on the filter's operation?

a. (1 pt) What is its sample period?

b. (4 pt) What is its effective cutoff frequency in Hz?

c. (1 pt) Will it still behave as a HPF or will it be transformed into a LPF?

2. (8 pt) The impulse invariant transformation of H(s) is H ( z )=0.6 z−0.134z−0.67 . Assuming there is at least one pole

at s = -4, find H(s).

IV Digital Filter (cont.) 2015 Name _________________________

3. A digital filter with fs = 18kHz has the following pole-zero diagram (top half only).

a. (1 pt) What type is it? i) LPF ii) BPF ii)BSF iv) HPF v) All Pass Filter (only changes phase)

b. (1 pt) What is the order of the filter? (None of the poles shown are repeated poles.)

4. (4 pt) What is/are the cutoff frequency(ies)?

V Interactive Graphics 2015 Name _________________________

1. (3 pt) How far apart must the pole and zero be in a lead compensator for the maximum phase to approach 20?

2. (5 pt) For Gp (s )= 3 .986

s(0 .4839 s+1)= 8 .2369

s( s+2 . 0664) (the model of the new D.C. motor system) what purely proportional feedback gain will result in repeated poles?

V Interactive Graphics (cont.) 2015 Name _________________________

3. For the given signals whose statistical parameters are (assuming the noise is zero mean Gaussian):Signal #1 Signal #2

Mean x0 1 0 1 2

Probability 3/4 1/4Cost C 1

a. (8 pt) Find the linear or quadratic equation that can be solved to give the decision boundary in terms of Signal #1’s cost C.

b. (4 pt) If the a priori probabilities above are changed to ½ and ½ and both functions have the same standard deviation, while increasing C, the width of Signal #1’s region would (select all that apply):i) decrease ii) remain unchanged iii) increase iv) move to the left v) move to the right iv) can’t be determined

VI Hybrid Control 2015 Name _________________________

ANSWER ANY COMBINATION OF QUESTIONS THAT ADD UP TO 20 POINTS. 10-POINT QUESTIONS MUST BE FULLY ANSWERED.

1. Assuming the 2.5V step input is applied at t = 0, the following step response for a system is observed.

a. (7 pt) Find the Ziegler-Nichols L and R values and the PID feedback control gains.

b. (3 pt) If L doubles but R remains the same, find the new values for the PID gains.

2.5

0 1 2 3 4 5 time (s)

VI Hybrid Control (cont.) 2015 Name _________________________

2a. (7 pt) Use the Graham-Lathrop method to find the PID gains for plant GP ( s)= 2(s+3)2 with ω0 = 1.5.

b. (3 pt) Find the PD controller gains for the above plant?

VI Hybrid Control (cont.) 2015 Name _________________________

3. (10 pt) For a system x’(t) = Ax(t) + Bu(t), y(t) = Cx(t) + Du(t), if the pole placement gains are kc1 = 4.5 and kc2 = 1.5 for a response with ζ = 0.707 and ωn = 4, determine the A and B matrices (assume canonic form and that a1/a2 = 3.29).

VI Hybrid Control (cont.) 2015 Name _________________________

4. (10 pt) For GP=e−0.064 s

(s+1)(s+5) find the Gallier-Otto PID controller gains.

VII DC Motor Control 2015 Name _________________________

1a. (6 pt) Find D(z), the minimal prototype compensator for the lab D.C. motor system when the sampling period is 150ms.

b. (3 pt) If the D.C. motor system in a. is hardwired with an amplifier whose gain is 10 and considered to be part of the new system, how would D(z) change?

2. (3 pt) Given GC (s )=7(s+10)s+20

∧GP ( s )= 50s(s+0.5), find the velocity error constant and the acceleration error

constant for the system.

VII DC Motor Control (cont.) 2015 Name _________________________

3. (8 pt) An FST controller for a 1st order system requires a control output of 100V at t = 0 when the sample period T is set to 1.0ms What must the sample period be increased to in order for the control output to be reduced to 10V at t = 0. Remember, the form of the output for a 1 st order system is A(1-e-t), assuming a unity time constant.

VIII Optimal Control 2015 Name _________________________

1. (1 pt) How many positive solutions exist to the Riccati equation for a 1st order digital system.

2. (3 pt) TRUE/FALSE: The state feedback gains determined by the Riccati solution are completely independent of the output equation coefficients but not the initial state of the system.

3. (3 pt) TRUE/FALSE: The calculations of the feedback gain vector G for the continuous optimal LQR does not depend on the system’s A matrix.

4. (6 pt) Given the gain is calculated by G=−( 34 )( 3

2 ) (2 )( 23 √3−1), find the matrix equation for the original

system of x as a function the states and input u and the values of Q and R.

5. (7 pt) For the system: x (t )=[ 0 0 10 1 0

−2 −5 −8] x (t )+[054 ]u (t ) y (t )=[−1.5 √2 9 ] x (t )+√3u ( t )

J ( t )=12∫0

∞ {x ' ( t )[1 0 00 3 00 0 1 ]x (t )+u ' (t )8 u (t )}dt with Riccati solutions (rounded) :

P=[−1.7 −0.4 0.2−0.4 1.6 −0.10.2 −0.1 −0.1] , [ 1.7 −0.4 0.2

−0.4 1.6 −0.10.2 −0.1 0.1 ], find the optimal feedback gain vector G.