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E near steady state since s 1 1 g sg s c p a

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Unformatted text preview: ies, at the low frequency end of the frequency response, i.e. near steady-state. Since ε ( s) = 1 [1 + G ( s)G ( s)] c p A plot of ε( jω ) for a closed-loop system where a P+I controller is being used to control a delay free first-order process is shown in Figure 2 below. We can see that the sensitivity function is bounded between 0 and 1. 3 of 7 Copyright 2002 Sensitivity Functions Figure 2. Sensitivity function of a feedback loop with a PI controller and a 1st-order system without delay Complementary Sensitivity Function The complementary sensitivity function is, as suggested by the name, defined as: η( s) ∆ 1 − ε ( s) Noise free system If there is no measurement noise, i.e. N(s) = 0, then since ε ( s) = 1 [1 + G ( s)G ( s)] c η( s) = 1 − 1 = G c ( s) G p ( s ) [1 + G ( s)G ( s)] [1 + G ( s)G ( s)] c p c p = p Y ( s) R ( s) In this case, the complementary sensitivity function simply relates the controlled variable Y(s) to the desired output, R(s). Thus, it is clear that η(s) should be as...
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