[試題] 101上 王富正 自動控制 第2次期中考
課程名稱︰自動控制
課程性質︰必修
課程教師︰王富正
開課學院：工學院
開課系所︰機械系
考試日期（年月日）︰2012/12/4
考試時限（分鐘）：160分鐘
是否需發放獎勵金：是
（如未明確表示，則不予發放）
試題 :
1.(25%)Considering Figure 1 with G(s)=2/(s+2) and C(s)=K/(s+1)
(1)(5%)Determine the closed-loop transfer function.
(2)(5%)Find the steady-state errors of the system to a step input R. How
to make the steady-state error to zero?
(3)(5%)Decide the value of K such that the damping ratio of the closed-
loop system is ζ=1/2.
(4)(5%)Continued from (3), find the settling time(for 2% settling) of the
system's step response.
(5)(5%)Continued from (3), find the peak time and percentage overshoot of
the system's step response.
(Note: For a standard 2-order system T(s)=ω^2/(s^2 + 2ζωs + ω^2) = Y/R
, the response to step input R(s)=1/s is
y(t)=1-(1/β)*exp(-ζωt)*sin(ωβt+θ)
where β=√(1-ζ^2) and θ=arccosζ.)
┌───┐ │d ┌───┐
R │ │ ↓ │ │ Y
——→○———→│C(s) │─→○──→│G(s) │────→
+ ↑ └───┘ └───┘ │
-│ │
│ ↓ n
└──────────────────────○←─
Fiqure 1
2.(25%)Consider Fiqure 1, draw the root-loci of the closed-loop system
as α:0→∞
(1)(10%)with G(s)=(s+1)/s(s+2) and C(s)=α/s(s+5)
(2)(15%)with G(s)=1/(s^2 +2s +α) and C(s)=1/(s+2)
Please specify the breakaway points, the departure/arrival angles, and the
ranges of α for closed-loop stability.
3.(50%)Consider the closed-loop system of Fiqure 1, where G(s)=1/(s+2)
(1)(5%)Using an I-control C(s)=Ki/s, determine the nature frequency ω
of the closed-loop system in terms of Ki.
(2)(5%)Continued from (1), determine the damping ratio ζ in terms of Ki.
(3)(5%)Continued from (1), decide the settling time. (for 2% settling)
(4)(5%)In order to reduce the settling time, we can apply a PI-control
C(s)=Kp + Ki/s, decide the settling time in terms of Kp and Ki.
(for 2% settling)
(5)(5%)Continued from (4), decide the minimum value of Kp such that
the settling time is equal to or less than 1 second.
(6)(5%)Using the Kp obtained in (5), draw the root-loci of the closed-loop
system as Ki:0→∞.
(7)(5%)Continued from (4)and(5), decide the maximum value of Ki such that
the damping ratio is equal to or greater than 1/√2.
(8)(5%)Decide the steady-state error due to step input R and step
disturbance d, simultaneously. (From 4.5.6.7)
(9)(10%)Using the values obtained from (5)and(7), derive the output
response y(t) to a step input r.
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12/07 19:24, , 1F
12/07 19:24, 1F