Consider the control system with negative and unity feedback The open-loop transfer function is given by Gc(s)G(s), wher
Posted: Thu May 05, 2022 5:29 pm
Consider the control system with negative and unity feedback
The open-loop transfer function is given by Gc(s)G(s), where Gc(s) is a phase-advance compensator
A phase-advance compensator is defined as
for 0 < α < 1, in which α and T are determined by the angular condition of the locus of the roots and Kc ´e
determined by the modulus condition of the root locus.
Based on the transient specifications for the closed-loop compensated system, determine:
a) The pair of dominant closed-loop conjugate complex poles of the compensated system, given by
considering a maximum overshoot of Mp = 25% and an accommodation time of ts = 2.5s, for a unit-step input. The damping factor ξ and the natural frequency ωn are determined based on the expressions
b) The angular contribution of the pole-zero pair of the phase-advance compensator, necessary so that the dominant closed-loop pair of complex conjugate poles of the compensated system belonging to the place of roots;
c) The pole and zero of the phase-advance compensator, based on the geometric methodology
from the place of the roots;
d) The tuning gain Kc of the phase-advance compensator;
e) The closed loop transfer function of the compensated system
R(s) + E(s) Ge(s) G(s) C(s)
G(s) = = 1 s(s+1)
Ge(s) Kea = Ts+1 aTs +1 Ke s+ S+ QT
8 = = {wn = jwn √/1-²
Mp =e {TT ·² ⇒ E= [In (Mp)]² [In(M₂)]²+7² e t 4 = ⇒Wn Ew → W = 1 {Wn
C(s) Ge(s)G(s) R(s) 1+ Ge(s)G(s)
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determined by the modulus condition of the root locus.
Based on the transient specifications for the closed-loop compensated system, determine:
a) The pair of dominant closed-loop conjugate complex poles of the compensated system, given by
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c) The pole and zero of the phase-advance compensator, based on the geometric methodology
from the place of the roots;
d) The tuning gain Kc of the phase-advance compensator;
e) The closed loop transfer function of the compensated system
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G(s) = = 1 s(s+1)
Ge(s) Kea = Ts+1 aTs +1 Ke s+ S+ QT
8 = = {wn = jwn √/1-²
Mp =e {TT ·² ⇒ E= [In (Mp)]² [In(M₂)]²+7² e t 4 = ⇒Wn Ew → W = 1 {Wn
C(s) Ge(s)G(s) R(s) 1+ Ge(s)G(s)