Unformatted text preview: ominant pole(s) with real part of –p, p > 0
pole(s)
A2: The zero is much slower than the dominant pole(s), i.e., η = c / p << 1
pole(s),
A3: Let K be a positive scalar satisfying v (t ) = 1 − y (t ) < Ke − pt , for all t ≥ t s
where y(t) is the unit step response
Then, the unit step response has an overshoot bounded below by Mp ≥ 1
e − cts ⎛
Kη ⎞
⎜1 − 1 − η ⎟
−1⎝
⎠ Proof: (see textbook)
Lemma 4.3 shows that, when a system has LHP slow zeros, there is a
tradeoff between having a fast step response and having small overshoot!
ME575 Session 6 – Dynamic Response School of Mechanical Engineering
Purdue University Slide 15 Transient and Steady State Responses
In general, the total response of a stable LTI system
total
stable
m
m−1
b s + bm−1s + L + b1s + b0 N (s) bm (s − z1 )(s − z2 )L(s − zm )
G( s ) ≡ m n
=
=
s + an−1s n−1 + L+ a1s + a0
D(s) (s − p1 )(s − p2 )L(s − pn )
to an input u(t) can be decomposed into two parts: y (t ) = yT (t ) + ySS (t )
{{
Transient where
Response
• Transient Response (yT(t)) Steady State
Response – Contains the free response yFree(t) of the system plus a portion of the forced
response.
– Will decay to zero at a rate that is determined by the poles of the system. • Steady State Response (ySS(t)) – will take the same form as the forcing input.
– Specifically, for a sinusoidal input, the steady state response is a sinusoidal
signal with the same frequency as the input but with different amplitude and
amplitude
phase.
School of Mechanical Engineering ME575 Session 6 – Dynamic Response Purdue University Slide 16 8 ME575 Handouts Steady State Response
• Final Value Theorem (FVT)
FVT) Given a signal’s LT F(s), if the poles of sF(s) all lie in the LHP (stable
signal’
the
all
region), then f(t) converges to a constant value f(∞). f(∞) can be obtained
without knowing f(t) by using the FVT: f ( ∞ ) = lim f (t ) = lim sF ( s )
t →∞ s→0 Ex: A model of a linear system is determined to be: && + 4 y + 12 y = 4u + 3u
&
&
y (1) if a constant input u = 5 is applied at t = 0, determine whether the output y(t) will
0,
converge to a constant value?
(2) If the output converges, what will be its steady state value?
value? ME575 Session 6 – Dynamic Response School of Mechanical Engineering
Purdue University Slide 17 9...
View
Full Document
 Fall '10
 Meckl
 Mechanical Engineering, Impulse response, Purdue University, School of Mechanical Engineering

Click to edit the document details