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Unformatted text preview: 16.060 QUIZ 1
October 15, 2003
Each question is worth an equal number of points. 1 . Consider a pure secondorder, underdamped system (quadratic lag) with poles at (a ijb). Consider the effect on the system step response of the following changes.
\ You should discuss the qualitative effect on percentage overshoot, settling time,
' frequency of oscillation, peak time, and system stability. Maﬁa
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U55. Ii I“ (f) (r?) C(s) a) What is the closedloop transfer function? b) The nominal plant is K=1, 1:1. Calculate the static sensitivity of the closed
loop system to variations in K. Sketch a plot of this static sensitivity versus K;
for Kc>0. c) Consider the closed—loop system response to a unit step input with a controller
gain of K5]. i. What is egg, the steadystate output value? ii.  If the value ofK varies by 10% to K=l.1, to ﬁrst order what will be
the new value of css? iii. How could this steadystate output sensitivity to variation in K be
reduced? 9  IVE :Tfs)
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output response C(t). You do not have to calculate the residues, but you
should ﬁll in all other numerical values and deﬁne clearly any notation that you introduce. 0) Identify the dominant mode(s) and justify your answer. . _ .. 9.10.; a: .—, Cm” 5:225. 92: ,M
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4 4 50 [1:2 a) What is the steady state error (855) when r0) is a unit step input and d (t) = 0 ? 635:0 _
b) What is the steady state error (en) when r(r) is a unit ramp input and d(r)=0? 4ch 1 s) 955 s. 7)? s :1; c) What is the steady state output (CH) when do) is a unit step and r(t) = O ?
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go \[ur (CUT (I; 3 C556?” (“(C) I “O :m ' i/Uu (an also Solve (c) L), (mics/airy 1%: CW! UM“? (q; you (a LONG Work) Problem 5 5a) Find a state space model and A, B, C and D matrices for the system
represented by the following block diagram. 5.b) Given a state Space model with A, B, C and D matrices
A z 0 1 B = 0
—1 0 1
C = [2 _ 2] D=l ﬁnd the system transfer function. V  ’ 492 t. (W :. SZt‘£§_.:’5‘ (q) U, ‘” 51" 5+3 5' (unﬁt5) ' 594547
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where y is the system output and u is the system input y+y2a+3u 6.a. 1) Determine the system transfer function 6.32) Create a state space model of the system by determining A, B,
C and D matrices 6b) Given a system represented by the following matrix transfer function 5+3
s(s+l) 32+2
s(s+1) 6(3): 6b. 1) Find a state space model of the system by determining A, B, C
and D matrices Note: Be sure that the state vector for your model is of smallest
possible dimension. Hint: Your results from part 6.3.2) should be useful here 6.b.2) Check that your model is correct by applying your A, B, C and
D matrices to the formula 6(3) 2 Ch] — A]*‘B+D Proum 6
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 Fall '03
 willcox
 Aeronautics, state space model

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