Wednesday, Due January 30 You may use Matlab/simulink wherever applicable 1. a)
EE/MFS 599
HW #3
Plot the step responses of the following first order systems with a transmission zeroes. Can you see the effect of the zero? How about on part ii)? i) H(s) = 10(s+1)/(.1s +1) ii) H(s) = 10(.1s+1)/(.1s +1) iii) H(s) = 10(.01s+1)/(.1s +1)
Soln:
EE/MFS 599: Plot of H(s) = 10(s+1)/(.1s +1 100
11 EE/MFS 599: Plot of H(s) = 10(0.1s+1)/(.1s +1
90
10.8
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70 60 50 40 30
10.4 10.2 10 9.8 9.6 9.4
20 10
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1
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EE/MFS 599: Plot of H(s) = 10(0.01s+1)/(.1s [phone]
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b)
Go back and finish problem 2e) from HW#2 (hint:Think delay)
Soln : See HW2 solution c) Make a Bode plot of the following 2nd order system with the transfer function of:
n Y(s) = 2 W(s) s 2 + 2 n s + n
2
When and n is 10 rad/sec and the value of the damping ratio is: i) 1 (critically damped) ii) 0.5 iii) 0.1 iv) 0 (no damping) Soln:
EE/MFS 599: zeta=1 0
EE/MFS 599: zeta=0.1 20 10 Magnitude (dB) Phase (deg)
-1 0 1 2 3
-20 Magnitude (dB)
-40
0 -10 -20 -30 -40 0
-60
-80 0 -45 Phase (deg)
-45
-90 -135 -180 10 10 10 10 10 Frequency (rad/sec)
-90 -135 -180 10
0
10
1
10
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Frequency (rad/sec)
EE/MFS 599: zeta=0 150
100 Magnitude (dB) Phase (deg)
50
0
-50 -180 -225 -270 -315 -360 10
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10
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10
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Frequency (rad/sec)
d)
Find the transfer function of the system with the following Bode plot (hint: compare the value of the resonant peak with your Bode plots above)
Soln: n is 100 rad/sec and the value of the damping ratio is about 0.. Therefore, H(s) = 1002(1s+1)/(10s+1)(.1s +1)(s2+20s+1002)
e)
What is the type number of the model you found above?
Soln: Type 0 system 2. a) For the critically damped system modeled in problem 1c), what is the bandwidth?
Soln: The bandwidth (-6 dB frequency) is at 10 rad/sec b) If we use the bandwidth as an estimate of the highest frequency present the in system, what is the Nyquist minimum sampling frequency we can use to recover all information? Soln: Must sample at twice the highest frequency. Thus, the minimum frequency is 20 rad/sec c) If we were going to control the system, what is a better choice of a sampling frequency (or periond Ts)? Soln: In controls, we should sample at 5 to 10 times the Nyquist rate (100 to 500 rad/sec) d) What are the three parts of the analog-to-digital conversion process learned in class today? Soln: sampling, quantizing, and encoding e) For the transfer function, H(s) = 10/(0.1s+1), pick a sampling time of 10 msec (is this reasonable?) and find an approximate discrete model H(z) using i) the bilinear transform ii) step invariant design
Ans. Using the bilinear transformation, H(z) = H(s) when s = (2(z-1)/[Ts(z+1)] = 25,600 [z-1]/[z+1] = [0.4762z-0.4762]/[z 0.9048]. Using step invariant (w(s) = 1/s), then Y(s) = H(s)/W(s)= 100/(s(s+10)) = 10/s -10/(s+10). Therefore, y(t) = 10(1-e-10t)u(t) and y(kTs) = 10u(kTs) 10e-0.1K u(kTs). Thus, Y(z) = 10z/(z-1) -10z/(z-e-0.1). W(z) = z/z-1. Thus, H(z) = Y(z)/W(z) = 10 10(z-1)/ /(z-e-0.1). While most of the processes we will encounter in industry are stable, we should at least be familia...