Problem 1 Consider the closed-loop LTI system in Figure 1 as well as its ability to reject additional errors due to disturbance and noise inputs, where • the forward-path transfer function G(s) = K(s)P(s) denotes the loop gain; • the signals r(t), d(t) and n(t) denote the reference input, the disturbance input and the noise input, respectively; • the signal c(t) denotes the controlled output; and • the error signal is defined as e(t) = r(t) – c(t), which we note is not necessarily the same signal as a(t) due to noise input n(t). D(s) A(s): R(s) K(s) P(s) C(s) waigri dd eseacto sal dolq elag N(s) Figure 1: Unity-Feedback Configuration with Loop Gain G(s) = K(s)P(s) (a) Express the output C(s) as a function of inputs R(s), D(s) and N(s). You should observe the form C(s) = T,(s)R(s) +Ta(s)D(s) + T,(s)N(s), so answer by providing each of the three closed-loop transfer functions in terms of the loop gain G(s) = K(s)P(s). "b) Given transfer functions P(s) = 1/(s+ 1) and K(s) = K, + K1/s with (proportional-integral) control parameters Kp and KI denoting two adjustable real-valued gains, for what ranges of Kp and K1 is the closed-loop system stable? c) Suppose for all t > 0 that reference input r(t) = t+1 and inputs d(t) = n(t) = 0. What additional conditions (if any) to those in part (b) will yield less than 2% error in steady state?
Problem 1 Consider the closed-loop LTI system in Figure 1 as well as its ability to reject additional errors due to disturbance and noise inputs, where • the forward-path transfer function G(s) = K(s)P(s) denotes the loop gain; • the signals r(t), d(t) and n(t) denote the reference input, the disturbance input and the noise input, respectively; • the signal c(t) denotes the controlled output; and • the error signal is defined as e(t) = r(t) – c(t), which we note is not necessarily the same signal as a(t) due to noise input n(t). D(s) A(s): R(s) K(s) P(s) C(s) waigri dd eseacto sal dolq elag N(s) Figure 1: Unity-Feedback Configuration with Loop Gain G(s) = K(s)P(s) (a) Express the output C(s) as a function of inputs R(s), D(s) and N(s). You should observe the form C(s) = T,(s)R(s) +Ta(s)D(s) + T,(s)N(s), so answer by providing each of the three closed-loop transfer functions in terms of the loop gain G(s) = K(s)P(s). "b) Given transfer functions P(s) = 1/(s+ 1) and K(s) = K, + K1/s with (proportional-integral) control parameters Kp and KI denoting two adjustable real-valued gains, for what ranges of Kp and K1 is the closed-loop system stable? c) Suppose for all t > 0 that reference input r(t) = t+1 and inputs d(t) = n(t) = 0. What additional conditions (if any) to those in part (b) will yield less than 2% error in steady state?
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Please answer parts A - C. Thank you!
Transcribed Image Text:Problem 1 Consider the closed-loop LTI system in Figure 1 as well as its ability to reject additional errors due to disturbance
and noise inputs, where
• the forward-path transfer function G(s) = K(s)P(s) denotes the loop gain;
• the signals r(t), d(t) and n(t) denote the reference input, the disturbance input and the noise input, respectively;
• the signal c(t) denotes the controlled output; and
• the error signal is defined as e(t) = r(t) – c(t), which we note is not necessarily the same signal as a(t) due to noise
input n(t).
D(s)
A(s):
R(s)
K(s)
P(s)
> C(s)
abaigorsi ab eseecto al dolq olag
fstoyten di aasor okr telon oif
N(s)
Bow as e
Figure 1: Unity-Feedback Configuration with Loop Gain G(s) = K(s)P(s)
(a) Express the output C(s) as a function of inputs R(s), D(s) and N(s). You should observe the form
C(s) = T,(s)R(s) + Ta(s)D(s) +Tn(s)N(s),
so answer by providing each of the three closed-loop transfer functions in terms of the loop gain G(s) = K(s)P(s).
%3D
(b) Given transfer functions P(s)% = 1/(s+ 1) and K(s) = Kp + K1/s with (proportional-integral) control parameters Kp
and Ki denoting two adjustable real-valued gains, for what ranges of Kp and K1 is the closed-loop system stable?
%3D
%3D
(c) Suppose for all t >0 that reference input r(t) = t+1 and inputs d(t) =
to those in part (b) will yield less than 2% error in steady state?
n(t) = 0. What additional conditions (if any)
(d) Assume all conditions in part (c) hold and that the value of Kp is fixed at zero. Sketch on the axes below the magnitude
response for closed-loop transfer functions Ta(s) and Tn(s) as a function of radian frequency w, also indicating their
dependence on gain K1. (Hint: Recall how a transfer function's pole/zero diagram relates to its frequency response.)
(e) Explain conditions on disturbance input d(t) and noise input n(t) such that the steady-state error specification of part (c)
can be met even when both d(t) and n(t) may be nonzero for t >0.
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