Chapter 4 Exercise and H.W. Assignment

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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

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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

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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

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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.


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Problem 4.3 Plot the step responses for Problem 4.2 using MATLAB


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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 =


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Solution Problem 4.2 (b) H.W.


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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

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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

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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


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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

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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

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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

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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

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Computer response: Transfer function: M= 1 Transfer function: 1 ----s+8 M= 2 Transfer function: 0.5 ----s+4


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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

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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

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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

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Problem 4.9 Find the poles using MATLAB

s 2s 2 T ( s) ! 4 3 2 s 6s 4s 7 s 22


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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

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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


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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

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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

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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


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Problem 4.12Write the general form of the capacitor voltage for the electrical network shown below


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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

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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

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Problem 4.14Solve for x(t) in the system for a unit step f(t)


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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

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Problem 4.15Find the output response as a function of time. Assume the system is underdamped. The system has a unit step input


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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

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Problem 4.16 Derive the relationship for damping ratio as a function of percent overshoot


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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


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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


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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

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(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

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(d) H.W.

(e)

H.W.


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(f)

H.W.


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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.


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H.W.


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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.


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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.


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Problem 4.20 For each of the second-order systems that follow, find ^, [n, Ts,, Tp, Tr, and %OS


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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


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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.


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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

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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.


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H.W.


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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

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H.W.


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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

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Problem 4.29

Find the transfer function of the system


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Figure P4.9 (continued)


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Figure P4.10Steps in determining the transfer function relating output physical response to the input visual command


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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

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Figure P4.12


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Figure P4.13


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Figure P4.14


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Figure P4.15


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Figure P4.16


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Figure P4.17


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Figure P4.18


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Figure P4.19


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Figure P4.20 Pump diagram

1996 ASME. 2000, John Wiley & Sons, Inc. Nise/Control Systems Engineering, 3/e

Chapter 4 Exercise and H.W. Assignment

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Transcript of Chapter 4 Exercise and H.W. Assignment