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Unformatted text preview: In the
following discussion, we shall be considering the conventional feedback loop shown in the
figure below: Figure 1. Schematic of conventional feedback control loop. 1 of 7 Copyright 2002 Sensitivity Functions Sensitivity Function
The sensitivity function that we will use is defined in the Laplace domain as:
ε ( s) ∆ E ( s)
R ( s) − d ( s) where the symbol ‘ ∆ ’ is used to denote ‘definition’. Thus the sensitivity function, ε(s),
relates the external inputs, R(s) and d(s), to the feedback error E(s). Notice, however, that it
does not take into account the effects caused by the noise, N(s).
From the block diagram in Figure 1, we can see that [ E ( s) = R( s) − Y ( s) = R( s) − G p ( s).U ( s) + d ( s)
But, U ( s) = G c ( s). E ( s) i.e. E ( s) = R ( s) − G c ( s)G p ( s). E ( s) − d ( s) Rearranging, E ( s) 1 + Gc ( s) G p ( s) = R( s) − d ( s) Hence, 1
E ( s)
R ( s) − d ( s) 1 + G c ( s ) G p ( s ) Since 1
d ( s ) 1 + Gc ( s )G p ( s ) [ [ [ it follows that
ε ( s) = E ( s)
Y ( s)
R ( s) − d ( s) d ( s) Thus, the sensitivity function has an important role to play in judging the performance of the
controller because it also describes the effects of the disturbance, d(s), on the controlled
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This note was uploaded on 02/11/2014 for the course EECS 320 taught by Professor Philips during the Spring '06 term at University of Michigan.
- Spring '06