Unformatted text preview: ominant pole(s) with real part of –p, p > 0
A2: The zero is much slower than the dominant pole(s), i.e., η = c / p << 1
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 ⎛
⎜1 − 1 − η ⎟
⎠ Proof: (see textbook)
Lemma 4.3 shows that, when a system has LHP slow zeros, there is a
trade-off 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
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 Response (yT(t)) Steady State
Response – Contains the free response yFree(t) of the system plus a portion of the forced
– 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
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
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
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...
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- Fall '10
- Mechanical Engineering, Impulse response, Purdue University, School of Mechanical Engineering