Chapter 4 Exercise and H.W. Assignment
-
Upload
muhammad-hassan -
Category
Documents
-
view
402 -
download
10
Transcript of Chapter 4 Exercise and H.W. Assignment
1
Chapter 2: Modeling in the Frequency Domain
ME 370 Chapter 4Homework and other Problems
2000, John Wiley & Sons, Inc. Nise/Control Systems Engineering, 3/e
2
Chapter 2: Modeling in the Frequency Domain
Homework 1. Solve Problem 4.2 (b). 2. Solve problem 4.8, also 4.17 &18 (d), (e), and (f) 3. Solve problem 4.19 4. Solve problem 4.24 5. Solve problem 4.252000, John Wiley & Sons, Inc. Nise/Control Systems Engineering, 3/e
3
Chapter 2: Modeling in the Frequency Domain
Problem 4.2 (a) and (b)Find the output response, c(t), for each of the system shown here. Also find the time constant, rise time, and settling time for each case2000, John Wiley & Sons, Inc. Nise/Control Systems Engineering, 3/e
4
Chapter 2: Modeling in the Frequency Domain
Solution Problem 4.2 a. C(s) = 5/{s(s+5)} = 1/s 1/(s+5) . Therefore, c(t) = 1 - e-5t. Also, T = 1/5 , Tr = 2.2/a = 2.2/5 = 0.44, Ts = 4/a = 4/5 = 0.8.
2000, John Wiley & Sons, Inc. Nise/Control Systems Engineering, 3/e
5
Chapter 2: Modeling in the Frequency Domain
Problem 4.3 Plot the step responses for Problem 4.2 using MATLAB
2000, John Wiley & Sons, Inc. Nise/Control Systems Engineering, 3/e
6
Chapter 2: Modeling in the Frequency Domain
Program: '(a)' num=5; den=[1 5]; Ga=tf(num,den) subplot(1,2,1) step(Ga) title('(a)')
Computer response: ans = (a) Transfer function: 5 s+5 ans =
2000, John Wiley & Sons, Inc. Nise/Control Systems Engineering, 3/e
7
Chapter 2: Modeling in the Frequency Domain
2000, John Wiley & Sons, Inc. Nise/Control Systems Engineering, 3/e
8
Chapter 2: Modeling in the Frequency Domain
Solution Problem 4.2 (b) H.W.
2000, John Wiley & Sons, Inc. Nise/Control Systems Engineering, 3/e
9
Chapter 2: Modeling in the Frequency Domain
Problem 4.4Find the capacitor voltage in the network shown here if the switch closes at t=0. Assume zero initial conditions. Also, find the time constant, rise time, and settling time for the capacitor voltage2000, John Wiley & Sons, Inc. Nise/Control Systems Engineering, 3/e
10
Chapter 2: Modeling in the Frequency Domain
Solution 4.4VC ( s ) 2 ! ! Vi ( s ) R 1 s2 Cs 5 Vi ( s ) ! s 10 5 5 VC ( s) ! ! s ( s 2) s s 2 VC (t ) ! 5 5e 2t 1 2.2 2.2 4 4 T ! , Tr ! ! ! 1.1, Ts ! ! ! 2 a a 2 2 22000, John Wiley & Sons, Inc. Nise/Control Systems Engineering, 3/e
1 Cs
By voltage division
11
Chapter 2: Modeling in the Frequency Domain
Problem 4.5 Plot the step responses for Problem 4.4 using MATLAB From the plots, find the time constant, rise time, and settling time
2000, John Wiley & Sons, Inc. Nise/Control Systems Engineering, 3/e
12
Chapter 2: Modeling in the Frequency Domain
Solution 4.5Program:clf num=2; den=[1 2]; G=tf(num,den) step(5*G)
Computer response:Transfer function: 2 s+22000, John Wiley & Sons, Inc. Nise/Control Systems Engineering, 3/e
13
Chapter 2: Modeling in the Frequency Domain
2000, John Wiley & Sons, Inc. Nise/Control Systems Engineering, 3/e
14
Chapter 2: Modeling in the Frequency Domain
Problem 4.6
M 8
(a) Find an equation that relates settling time of the velocity of the mass to M (b) Find an equation that relates rise time of the velocity of the mass to M2000, John Wiley & Sons, Inc. Nise/Control Systems Engineering, 3/e
15
Chapter 2: Modeling in the Frequency Domain
Solution 4.61 X (s) ! F ( s ) Ms 2 8s velocity V; V ( s ) ! sX ( s ) 1 sX ( s ) ! ! F ( s) Ms 8 s 8 M 4 1 2.2 Ts ! ! M , Tr ! ! 0.275M 8 8 2 M M2000, John Wiley & Sons, Inc. Nise/Control Systems Engineering, 3/e
1 M
16
Chapter 2: Modeling in the Frequency Domain
Problem 4.7 Program: Clf M=1 num=1/M; den=[1 8/M]; G=tf(num,den) step(G) pause M=2 num=1/M; den=[1 8/M]; G=tf(num,den) step(G)2000, John Wiley & Sons, Inc. Nise/Control Systems Engineering, 3/e
17
Chapter 2: Modeling in the Frequency Domain
Computer response: Transfer function: M= 1 Transfer function: 1 ----s+8 M= 2 Transfer function: 0.5 ----s+4
2000, John Wiley & Sons, Inc. Nise/Control Systems Engineering, 3/e
18
Chapter 2: Modeling in the Frequency Domain
2000, John Wiley & Sons, Inc. Nise/Control Systems Engineering, 3/e
19
Chapter 2: Modeling in the Frequency Domain
Problem 4.8 Find the locations of the poles and zeros, plot them on the s-plane, and then write an expression for the general form of the step response without solving for the inverse Laplace transform. State the nature of each response (overdamped, underdamped, and so on)2000, John Wiley & Sons, Inc. Nise/Control Systems Engineering, 3/e
20
Chapter 2: Modeling in the Frequency Domain
2 (a) T (s ) ! s2 (b ) T ( s ) ! (c ) 5
s 3s 6 10s 7 T (s) ! s 10s 20
20 (d ) T (s) ! 2 s 6 s 144 s2 (e) T ( s ) ! 2 s 9 s 7 ( f ) T (s) ! 2 s 102000, John Wiley & Sons, Inc. Nise/Control Systems Engineering, 3/e
21
Chapter 2: Modeling in the Frequency Domain
a. Pole: -2; c(t) = A + Be-2t ; first-order response. b. Poles: -3, -6; c(t) = A + Be-3t + Ce-6t; overdamped response. c. Poles: -10, -20; Zero: -7; c(t) = A+Be-10t +Ce-20t ; overdamped response. d. Poles: (-3+j11.6 ), (-3-j11.6 ) ; c(t) = A + Be-3t cos (11.6 t + ); underdamped. e. Poles: j3, -j3; Zero: -2; c(t) = A + B cos (3t + ); undamped. f. Poles: -10, -10; Zero: -5; c(t)=A+Be-10t +Cte-10t; critically damped.2000, John Wiley & Sons, Inc. Nise/Control Systems Engineering, 3/e
22
Chapter 2: Modeling in the Frequency Domain
Problem 4.9 Find the poles using MATLAB
s 2s 2 T ( s) ! 4 3 2 s 6s 4s 7 s 22
2000, John Wiley & Sons, Inc. Nise/Control Systems Engineering, 3/e
23
Chapter 2: Modeling in the Frequency Domain
Solution 4.9 Program: p=roots([1 6 4 7 2]) Computer response: p= -5.4917 -0.0955 + 1.0671i -0.0955 - 1.0671i -0.31732000, John Wiley & Sons, Inc. Nise/Control Systems Engineering, 3/e
24
Chapter 2: Modeling in the Frequency Domain
Problem 4.10 Find the transfer function and poles of the system x1 8 4 1 x1 1 x ! 3 2 0 x2 3u (t ) 2 x2 5 7 9 x3 7 0 y ! ?2 8 3Ax; x(0) ! 0 0
2000, John Wiley & Sons, Inc. Nise/Control Systems Engineering, 3/e
25
Chapter 2: Modeling in the Frequency Domain
Solution 4.10G ( s) ! C ( sI A) 1 B s 2 7 s 18 4s 29 s2 1 2 1 ( sI A) ! 3 2 3 3s 27 s s 77 s s 91s 67 2 5s 31 7 s 76 s 10 s 4 5s 2 136s 1777 G ( s) ! 3 2 s s 91s 67
Factoring the denominator, or using det(sI-A), we find the poles to be 9.683, 0.7347, -9.4179.2000, John Wiley & Sons, Inc. Nise/Control Systems Engineering, 3/e
26
Chapter 2: Modeling in the Frequency Domain
Solution 4.11 Program: A=[8 -4 1;-3 2 0;5 7 -9] B=[1;3;7] C=[2 8 -3] D=0 [numg,deng]=ss2tf(A,B,C,D,1); G=tf(numg,deng) poles=roots(deng) Computer response: A= 8 -4 1 -3 2 0 5 7 -9 B= 12000, John Wiley & Sons, Inc. Nise/Control Systems Engineering, 3/e
27
Chapter 2: Modeling in the Frequency Domain
3 7 C= 2 8 -3 D= 0 Transfer function: 5 s^2 + 136 s - 1777 --------------------s^3 - s^2 - 91 s + 67 poles = -9.4179 9.6832 0.7347
2000, John Wiley & Sons, Inc. Nise/Control Systems Engineering, 3/e
28
Chapter 2: Modeling in the Frequency Domain
Problem 4.12Write the general form of the capacitor voltage for the electrical network shown below
2000, John Wiley & Sons, Inc. Nise/Control Systems Engineering, 3/e
29
Chapter 2: Modeling in the Frequency Domain
Solution 4.12 1 VC ( s ) V ( s ) 1 VC ( s ) !0 R Ls Cs R1 2 1 10s VC ( s ) R1 ! ! 2 1 1 1 V ( s) Cs s 20s 500 R1 R2 Ls 1 V ( s) ! s 10 VC ( s ) ! 2 s 20 s 500 VC (t ) ! Ae 10 t cos(20t J )2000, John Wiley & Sons, Inc. Nise/Control Systems Engineering, 3/e
Node equation at capacitor
30
Chapter 2: Modeling in the Frequency Domain
Solution 4.13 Program: num=[10 0]; den=[1 20 500]; G=tf(num,den) step(G) Computer response: Transfer function: 10 s ---------------s^2 + 20 s + 5002000, John Wiley & Sons, Inc. Nise/Control Systems Engineering, 3/e
31
Chapter 2: Modeling in the Frequency Domain
Problem 4.14Solve for x(t) in the system for a unit step f(t)
2000, John Wiley & Sons, Inc. Nise/Control Systems Engineering, 3/e
32
Chapter 2: Modeling in the Frequency Domain
Solution 4.141 1 X ( s) ! ! 2 2 F ( s ) Ms f v s k s s s 5 1 1 1 s 1 5 5 !5 X ( s) ! 2 s s2 s 5 s 1 19 s 2 4
19 1 1 0.5t 1 19 cos x (t ) ! e 2 t 19 sin 2 t 5 5 19 1 0.5t 0 cos x(t ) ! 1 e t 12.92 2 5 2000, John Wiley & Sons, Inc. Nise/Control Systems Engineering, 3/e
33
Chapter 2: Modeling in the Frequency Domain
Problem 4.15Find the output response as a function of time. Assume the system is underdamped. The system has a unit step input
2000, John Wiley & Sons, Inc. Nise/Control Systems Engineering, 3/e
34
Chapter 2: Modeling in the Frequency Domain
Solution 4.152 [n 1 s 2\[n C ( s) ! ! 2 2 2 2 s 2\[n s [n s s 2\[n s [ n s
s 2\[n s \[n \[n 1 1 C ( s) ! ! 2 2 2 2 s s \[n [ n \ [n s s \[ 2 [ 1 \ 2 n n
2
1 ! s
s \[n
\[n [n 1 \ 22
[n 1 \ 22
s \[n \[n t
[n 1 \
2
c (t ) ! 1 e
\ 2 2 cos [ n 1 \ t sin [ n 1 \ t 2 1 \ 2000, John Wiley & Sons, Inc. Nise/Control Systems Engineering, 3/e
35
Chapter 2: Modeling in the Frequency Domain
Problem 4.16 Derive the relationship for damping ratio as a function of percent overshoot
2000, John Wiley & Sons, Inc. Nise/Control Systems Engineering, 3/e
36
Chapter 2: Modeling in the Frequency Domain
Solution 4.16 \T
%OS ! e
1\ 2
*100
\T %OS ln ! 2 100 1 \ %OS ln 100 2 \ ! 2 2 %OS T ln 100 2
%OS ln 100 \ ! 2 2 %OS T ln 100 Negative solution gives a positive damping ratio
2000, John Wiley & Sons, Inc. Nise/Control Systems Engineering, 3/e
37
Chapter 2: Modeling in the Frequency Domain
Problem 4.17 Calculate the exact response of each system using Laplace transform techniques, and compare the results to those obtained from Problem 4.8
2000, John Wiley & Sons, Inc. Nise/Control Systems Engineering, 3/e
38
Chapter 2: Modeling in the Frequency Domain
2 (a) T (s ) ! s2 (b ) T ( s ) ! (c ) 5
s 3s 6 10s 7 T (s) ! s 10s 20
20 (d ) T (s) ! 2 s 6 s 144 s2 (e) T ( s ) ! 2 s 9 s 7 ( f ) T (s) ! 2 s 102000, John Wiley & Sons, Inc. Nise/Control Systems Engineering, 3/e
39
Chapter 2: Modeling in the Frequency Domain
(a)
1 1 1 ! C (s) ! s s 2 s s 2 c(t ) ! 1 e 2t
(b)
5 5 1 5 1 5 1 ! C (s) ! s s 3s 6 18 s 9 s 3 18 s 6 5 5 3t 5 6 t c(t ) ! e e 18 9 18
10s 7 7 1 3 1 13 1 (c) C ( s ) ! ! s s 10s 20 20 s 10 s 10 20 s 20 7 3 10t 13 20t e e c(t ) ! 20 10 2000, 20 Wiley & Sons, Inc. JohnNise/Control Systems Engineering, 3/e
40
Chapter 2: Modeling in the Frequency Domain
(d) H.W.
(e)
H.W.
2000, John Wiley & Sons, Inc. Nise/Control Systems Engineering, 3/e
41
Chapter 2: Modeling in the Frequency Domain
(f)
H.W.
2000, John Wiley & Sons, Inc. Nise/Control Systems Engineering, 3/e
42
Chapter 2: Modeling in the Frequency Domain
Problem 4.18 Find the damping ratio and natural frequency for each second-order system of Problem 8 and show that the value of the damping ratio conforms to the type of response (underdamped, overdamped, and so on) predicted in that problem.
2000, John Wiley & Sons, Inc. Nise/Control Systems Engineering, 3/e
43
Chapter 2: Modeling in the Frequency Domain
H.W.
2000, John Wiley & Sons, Inc. Nise/Control Systems Engineering, 3/e
44
Chapter 2: Modeling in the Frequency Domain
Problem 4.19 A system has a damping ratio of 0.5, a natural frequency of 100 rad/s, dc gain of 1. Find the response of the system to a unit step input.
2000, John Wiley & Sons, Inc. Nise/Control Systems Engineering, 3/e
45
Chapter 2: Modeling in the Frequency Domain
lim sX ( s ) ! 1s p0
k 2 lim 2 ! 1, k ! [n s p0 s 2:[ s [ 2 n n
dc gain=1
H.W.
2000, John Wiley & Sons, Inc. Nise/Control Systems Engineering, 3/e
46
Chapter 2: Modeling in the Frequency Domain
Problem 4.20 For each of the second-order systems that follow, find ^, [n, Ts,, Tp, Tr, and %OS
2000, John Wiley & Sons, Inc. Nise/Control Systems Engineering, 3/e
47
Chapter 2: Modeling in the Frequency Domain
2000, John Wiley & Sons, Inc. Nise/Control Systems Engineering, 3/e
48
Chapter 2: Modeling in the Frequency Domain
2000, John Wiley & Sons, Inc. Nise/Control Systems Engineering, 3/e
49
Chapter 2: Modeling in the Frequency Domain
Problem 4.21 Repeat Problem 20 using MATLAB. Have the computer program estimate the given specifications and plot the step responses. Estimate the rise time from the plots
2000, John Wiley & Sons, Inc. Nise/Control Systems Engineering, 3/e
50
Chapter 2: Modeling in the Frequency Domain
2000, John Wiley & Sons, Inc. Nise/Control Systems Engineering, 3/e
51
Chapter 2: Modeling in the Frequency Domain
2000, John Wiley & Sons, Inc. Nise/Control Systems Engineering, 3/e
52
Chapter 2: Modeling in the Frequency Domain
2000, John Wiley & Sons, Inc. Nise/Control Systems Engineering, 3/e
53
Chapter 2: Modeling in the Frequency Domain
Problem 4.22 Use MATLAB's LTI Viewer and obtain settling time, peak time, rise time, and percent overshoot for each of the systems in Problem 20.
2000, John Wiley & Sons, Inc. Nise/Control Systems Engineering, 3/e
54
Chapter 2: Modeling in the Frequency Domain
2000, John Wiley & Sons, Inc. Nise/Control Systems Engineering, 3/e
55
Chapter 2: Modeling in the Frequency Domain
2000, John Wiley & Sons, Inc. Nise/Control Systems Engineering, 3/e
56
Chapter 2: Modeling in the Frequency Domain
Problem 4.23 For each pair of second-order system specifications that follow, find the location of the second-order pair of poles. a. %OS = 12%; Ts = 0.6 second b. %OS = 10%; Tp = 5 seconds c. Ts = 7 seconds; Tp = 3 seconds2000, John Wiley & Sons, Inc. Nise/Control Systems Engineering, 3/e
57
Chapter 2: Modeling in the Frequency Domain
2000, John Wiley & Sons, Inc. Nise/Control Systems Engineering, 3/e
58
Chapter 2: Modeling in the Frequency Domain
Problem 4.24 Find the transfer function of a second-order system that yields a 12.3% overshoot and a settling time of 1 second.
2000, John Wiley & Sons, Inc. Nise/Control Systems Engineering, 3/e
59
Chapter 2: Modeling in the Frequency Domain
H.W.
2000, John Wiley & Sons, Inc. Nise/Control Systems Engineering, 3/e
60
Chapter 2: Modeling in the Frequency Domain
Problem 4.25
a. Find the transfer function G(s) = X(s)/F(s). b. Find ^, %OS, Ts, Tp, and Tr.2000, John Wiley & Sons, Inc. Nise/Control Systems Engineering, 3/e
61
Chapter 2: Modeling in the Frequency Domain
H.W.
2000, John Wiley & Sons, Inc. Nise/Control Systems Engineering, 3/e
62
Chapter 2: Modeling in the Frequency Domain
Problem 4.26
For the system shown in Figure P4.8, a step torque is applied at U(t), a. The transfer function, G(s) = U2(s)/T(s). b. The percent overshoot, settling time, and peak time for U2(s).2000, John Wiley & Sons, Inc. Nise/Control Systems Engineering, 3/e
63
Chapter 2: Modeling in the Frequency Domain
2000, John Wiley & Sons, Inc. Nise/Control Systems Engineering, 3/e
64
Chapter 2: Modeling in the Frequency Domain
Problem 4.29
Find the transfer function of the system
2000, John Wiley & Sons, Inc. Nise/Control Systems Engineering, 3/e
65
Chapter 2: Modeling in the Frequency Domain
Figure P4.9 (continued)
2000, John Wiley & Sons, Inc. Nise/Control Systems Engineering, 3/e
66
Chapter 2: Modeling in the Frequency Domain
Figure P4.10Steps in determining the transfer function relating output physical response to the input visual command
2000, John Wiley & Sons, Inc. Nise/Control Systems Engineering, 3/e
67
Chapter 2: Modeling in the Frequency Domain
Figure P4.11 Vacuum robot lifts two bags of salt
Courtesy of Pacific Robotics, Inc. 2000, John Wiley & Sons, Inc. Nise/Control Systems Engineering, 3/e
68
Chapter 2: Modeling in the Frequency Domain
Figure P4.12
2000, John Wiley & Sons, Inc. Nise/Control Systems Engineering, 3/e
69
Chapter 2: Modeling in the Frequency Domain
Figure P4.13
2000, John Wiley & Sons, Inc. Nise/Control Systems Engineering, 3/e
70
Chapter 2: Modeling in the Frequency Domain
Figure P4.14
2000, John Wiley & Sons, Inc. Nise/Control Systems Engineering, 3/e
71
Chapter 2: Modeling in the Frequency Domain
Figure P4.15
2000, John Wiley & Sons, Inc. Nise/Control Systems Engineering, 3/e
72
Chapter 2: Modeling in the Frequency Domain
Figure P4.16
2000, John Wiley & Sons, Inc. Nise/Control Systems Engineering, 3/e
73
Chapter 2: Modeling in the Frequency Domain
Figure P4.17
2000, John Wiley & Sons, Inc. Nise/Control Systems Engineering, 3/e
74
Chapter 2: Modeling in the Frequency Domain
Figure P4.18
2000, John Wiley & Sons, Inc. Nise/Control Systems Engineering, 3/e
75
Chapter 2: Modeling in the Frequency Domain
Figure P4.19
2000, John Wiley & Sons, Inc. Nise/Control Systems Engineering, 3/e
76
Chapter 2: Modeling in the Frequency Domain
Figure P4.20 Pump diagram
1996 ASME. 2000, John Wiley & Sons, Inc. Nise/Control Systems Engineering, 3/e