page 121
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dtX = AX + BU |
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Y = CX + D |
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7.3 DESIGN OF CONTINUOUS CONTROLLERS
7.4 PRACTICE PROBLEMS
1.Short answer questions.
a)What is a Setpoint, and what is it used for?
b)What does feedback do in control systems?
c)What is Aliasing and what does it have to do with the Nyquist Criterion?
d)What can A/D and D/A converters be used for in computer control of processes?
2.Simplify the block diagram below.
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3.Given the transfer function below, and the input ‘x(s)’, find the output ‘y(t)’ as a function of time.
page 122
y---------( s) |
= |
-----------5 |
2 |
x( t) = 5 |
t ≥ 0 sec |
x( s) |
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4.Find the steady state error when the input is a ramp with the function r(t) = 0.5t. Sketch the system error as a function of time.
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1 + 0.2s
5. Simplify the following block diagram.
G1
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6.Given the block diagram below, select a system gain K that will give the overall system a damping ratio of 0.7 (for a step input). What is the resulting undamped natural frequency of the system?
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7.What is the transfer function for a second order system that responds to a step input with an overshoot of 20%, with a delay of 0.4 seconds to the first peak?
page 123
8.Draw a detailed root locus diagram for the transfer function below. Be careful to specify angles of departure, ranges for breakout/breakin points, and gains and frequency at stability limits.
( ) 2K( s + 0.5) ( s2 + 2s + 2) G s = -----------------------------------------------------------
s3( s + 1) ( s + 2)
9. Draw the root locus diagram for the system below. specify all points and values.
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10. Draw the root locus diagram for the transfer function below,
( ) K( s + 4) 2
G s = ----------------------
s2( s + 1)
11. Draw the root locus diagram for the transfer function below,
( ) K( s + 1) ( s + 2)
G s = -------------------------------------
s3
12. The block diagram below is for a motor position control system. The system has a proportional controller with a variable gain K.
page 124
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s + 2 |
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2
a) Simplify the block diagram to a single transfer function.
ans. 200K
-----------------------------------
s2 + 2s + 200K
b) Draw the Root-Locus diagram for the system (as K varies). Use either the approximate or exact techniques.
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– 2 ± 4 – 4( 200K) |
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1 – 200K |
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c) Select a K value that will result in an overall damping coefficient of 1. State if the Root-Locus diagram shows that the system is stable for the chosen K.
ans. s2 + 2s + 200K = s2 + 2ζω ns + ω n2 |
ω n = 1 |
K = 0.005 |
From the root locus graph this value is critically stable.
13. Draw a Bode Plot for either one of the two transfer functions below.
page 125
( s + 1) ( s + 1000) |
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5 |
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OR |
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( s + 100) 2 |
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s2 |
14. The block diagram below is for a servo motor position control system. The system uses a proportional controller.
a) Convert the system to a transfer function.
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b) draw a sketch of what the actual system might look like. Identify components.
15.Given the system transfer function below.
θo 20K
=-----------------------------
θd s2 + s + 20K
a)Draw the root locus diagram and state what values of K are acceptable.
b)Select a gain value for K that has either a damping factor of 0.707 or a natural frequency of 3 rad/sec.
c)Given a gain of K=10 find the steady state response to an input step of 1 rad.
d)Given a gain of K=10 find the response of the system as