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Unformatted text preview: NAME: 16.060 QUIZ
October 24, 2002
All questions are worth 10 points each. 1. Answer TRUE or FALSE (you may include a sentence of explanation if you wish) a. A linear system with feedback may be characterized as stable for certain inputs and as
unstable for other inputs. FM. 93 I Siabluiy {S a F®F1°rtgf of IQ $753k"?! r1071 7th {apt/13 b. Increasing the rootlocus gain of a ﬁrstorder, linear feedback system always tends to
make the system more stable. {WW4 ‘7’”! LI”? 57(de (boo! {oft/.5 Mel‘s/J) 0. Good tracking and good disturbance rejection are typically the goal of control system .
design, but cannot be achieved simultaneously with the use of feedback control. ' HUB: we can 3:! Lo” 97 {74ny “130'" 44W“
More I? a”! 0 entn ﬁe 90f. d. The openloop zeroes of a linear system can be moved with feedback. FALSF' (Cm 0N; Molfc‘. 0L {point Lita“ ﬂea/£454 e. The rootlocus gain is important because it determines steadystate errors. FALIE; ch Siaw/a/r/l (yarn 4b denierrake €55 NAME: 2. Consider the system represented by the following polefzero diagram, where the pole at
the origin represents a step input. The system has unity gain. not to scale ﬁrm:
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residue magnitudes and angles, but deﬁne your notation. b. What is the root locus gain of the system?
c. Calculate the magnitude of each of the residues.
d. What is the decay time of each exponential mode? e. Is there a dominant mode or modes? Explain the rationale for your answer in a few
words.
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input D: a. If Gc is a proportional controller, GC= KC, what value of Kc will cause the steadystate
error for a unit ramp reference input to be 0.1? ' b. If we want a zero steadystate error for a unit ramp reference input, what form should
the controller Gc take?  ' . c. For a unit step disturbance input, what form of controller GE should we choose to make
the steadystate error zero?  (1. Assume that the reference input is a unit step, the disturbance input is zero and we are
using a proportional controller. What value (or values) of Kc will yield an underdaniped system with a settling time of 4 seconds? 6. Considering the same case as part d, what effect does increasing KC have on the
damping ratio? (a) CC:{(° ‘3" 71pr (1er .21} (/ “Jymﬁr 1,, 6(3))
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For the system shown below, sketch the root locus for K>0 and K<0 on separate plots. Where appropriate, calculate pg and the angles of departure of the complex poles. NAME: 5. The following plot shows the response of 5 different systems to a unit step input. The
dashed lines indicate the response of the baseline system. Step Response '1 Tiﬁne (secii 10 Consider the baseline polezero diagram shown below and the four modiﬁed polezero
diagrams shown on the next page. Each diagram corresponds to one of the responses
shown above. For each of the four modiﬁed systems, match the step response to the pole zero diagram. Justify your choice with a few words. Baseline: NAME: Polezero diagram 1: Pole diagram 2: )'(   —j 213
Which response? @ Which response? @
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applied at t=0. At t=0, the system was in its equilibrium position. Step Response
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0 1 2 3 4 5 6 7 8 9 10 ‘11 12 13 14 15
Two (sec) a. What is the 5% settling time (T5)? 2: 3 7 56604115 b. What is the percentage overshoot (P.O.)? RU, :5 9—1"? 3 (3 Z), c. What is the peak time (2;)? T!” :3, i. 6 Run!)
d. What is the riSe time (T)? fr 2 O“? 51¢“ ,1, e. What is the damped natural frequency (an)? :5 32. 51701); Mire: [79619 z) D 5.1:; 2i: ...,_ _
f. What are the locations of the poles? J 3'? N 2 {Gal/g T 2 T5 (5%} z (.2 seem! = i
3 J W” [M95 «J— 631)” :t. 0J6; : v0.57 1 2% Practice StateSpace Problem for Quiz 1
16.060: Principles of Automatic Control
October 8, 2003 For the following differential equation: 1 tﬁ+5w+6w=f+r+2r . Derive a state—space model in the form: a = AE+B€E
g“ = oe+oe Take r to he the input and w to be the output. . Draw a block diagram for the system, and clearly label the input, the state variables, and the output. _ (Hint: look at the diagram on page 4 of the Lecture 13 notes.) Using your state—space model, determine the transfer function 0(3) from the input to the output.
Calculate the state—transition matrix @(t).
. If the initial state is £(0) = E] , and there are no inputs, what is the state vector :E'(t) at time t > 0'?
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 Fall '03
 willcox
 Aeronautics

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