PROBLEM #2 (40%) Consider a unity feedback control system having a nominal plant and controller transfer functions of 2 G(s)= C(s)= (s+4)(s+5) 60(s+1)(s+2)(s+4)(s+5) s(s²+1)(s+100) with an input disturbance of d,(t) = Asin (t+α) and reference input of r(t) a) Show that the closed-loop system is stable. b) Explain, using the inversion principle and sensitivity functions, why the controller is a sensible choice, i.e., show that the given controller leads to (i). the closed-loop transfer function from the reference input to the error will have an s in the numerator so that the steady-state error caused by any constant reference input would be zero when the closed-loop system is stable, and (ii). the closed-loop transfer function from the input disturbance to the output will have a factor of s² +1 in the numerator so that the steady-state error caused by any sinusoidal input disturbance of frequency of 1 rad/sec would be zero (the frequency response at c = 1 is zero). c) Assume that the true plant transfer function differs from the above nominal model by a MME which satisfies the following bound @ |G(jo) - √√²+10 Determine the robust stability of the above feedback loop.
PROBLEM #2 (40%) Consider a unity feedback control system having a nominal plant and controller transfer functions of 2 G(s)= C(s)= (s+4)(s+5) 60(s+1)(s+2)(s+4)(s+5) s(s²+1)(s+100) with an input disturbance of d,(t) = Asin (t+α) and reference input of r(t) a) Show that the closed-loop system is stable. b) Explain, using the inversion principle and sensitivity functions, why the controller is a sensible choice, i.e., show that the given controller leads to (i). the closed-loop transfer function from the reference input to the error will have an s in the numerator so that the steady-state error caused by any constant reference input would be zero when the closed-loop system is stable, and (ii). the closed-loop transfer function from the input disturbance to the output will have a factor of s² +1 in the numerator so that the steady-state error caused by any sinusoidal input disturbance of frequency of 1 rad/sec would be zero (the frequency response at c = 1 is zero). c) Assume that the true plant transfer function differs from the above nominal model by a MME which satisfies the following bound @ |G(jo) - √√²+10 Determine the robust stability of the above feedback loop.
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Transcribed Image Text:PROBLEM #2 (40%)
Consider a unity feedback control system having a nominal plant and controller transfer functions of
2
G(s)=
C(s)=
(s+4)(s+5)
60(s+1)(s+2)(s+4)(s+5)
s(s²+1)(s+100)
with an input disturbance of d,(t) = Asin (t+α) and reference input of r(t)
a) Show that the closed-loop system is stable.
b) Explain, using the inversion principle and sensitivity functions, why the controller is a sensible
choice, i.e., show that the given controller leads to
(i). the closed-loop transfer function from the reference input to the error will have an s in the
numerator so that the steady-state error caused by any constant reference input would be zero
when the closed-loop system is stable, and
(ii). the closed-loop transfer function from the input disturbance to the output will have a factor
of s² +1 in the numerator so that the steady-state error caused by any sinusoidal input
disturbance of frequency of 1 rad/sec would be zero (the frequency response at c = 1 is zero).
c) Assume that the true plant transfer function differs from the above nominal model by a MME
which satisfies the following bound
@
|G(jo) -
√√²+10
Determine the robust stability of the above feedback loop.
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