課程名稱︰ 數位控制系統
課程性質︰ 選修
課程教師︰ 陳明新
開課學院： 工學院
開課系所︰ 機械系
考試日期（年月日）︰ 2008/05/01
考試時限（分鐘）： 180
是否需發放獎勵金： 是
試題 :
Digital Control, Midterm Term, 2008
[1]
(a) For a continuous time system G(s) with satisfactory performance,
the poles of G(s) should be in the good region as shown in Figure 1.
What is the corresponding good region in the unit circle for a
discrete time system H(z) ?
(b) What is the discrete time PID controller ?
(c) What should be the corner frequency of an anti_alising filter if the
sampling time is T seconds.
[2]
(a) Find the closed-loop characteristic polynomials of the block diagram in
Figure 2.
(b) Use the bilinear transformation and the Routh stability test to find out
if the closed-loop system can be stabilized by a positive constant K. If
the answer is yes, find out the range of positive constant K > 0 so that
the closed-loop system is stable.
[3]
(a) If the frequency spectrum of a continuous signal r(t) is R(ω), what is
the relationship between the sampled spectrum R*(ω) and the original
spectrum R(ω)?
(b) Let the sampling time be T. What is the condition on the spectrum R(ω)
of a continuous signal so that after sampling, the sampling spectrum
R*(ω) remains un-distorted ?
(c) What is the frequency observed when a signal f(t) = sin(10t) is sampled
with a sampling time T = 0.3π second.
[4]
(a) What is the discretized transfer function y(k) = H(z)u(k) of continuous
plant y(t) = G(s)u(t) = (1/s)u(t) when using a zero-order hold with a
sampling time T = 0.1 second, where u(t) = u(k), t 屬於 [kT,(k + 1)T).
(b) Repeat problem (a) using a predicative first order hold, where u(t) = u(k)
+ (t-kT)×[u(k+1) - u(k)]/T, t 屬於 [kT,(k+1)T).
Hint: substitute the control u(t) into the following equation
kT+T
y(k+1) = y(k) + ∫ u(t)dt = …
kT
(c) Use the root locus technique to design an implementable digital feedback
controller C(z) to stabilize the discretized system in (b).
[5]
(a) Let the Z-transform of a sequence f(k) be F(z) = Σf(k)z^-k. Please prove
that -z(dF(z)/dz) is the Z-transform of sequence kf(k).
(b) What is the Z-transform of the sequence (k^2)f(k) in terms of F(z)?
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