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3 shows that when a system has lhp slow zeros there

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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 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 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...
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