The error transfer function relates an output of error (which
is just the output, e(s), from the feedback summation block) to
your input commanded value, which we'll refer to as 'e_ref(s)'.
ERTF(s) = e(s)/e_ref(s)
To evaluate this for our generic model, where:
C(s) = the Controller TF
P(s) = the TF of the Plant (or Physical system) you are controlling
M(s) = the Measurement TF
simply write out the algebraic relationship between desired output
(e(s)) and input (e_ref(s)):
e(s) = e_ref(s) - e(s) * C(s)*P(s)*M(s)
Then, move all terms in 'e(s)' [output] to one side and
all term in 'e_ref(s)' [input] to the other:
Use the ERTF(s) to evaluate the steady-state response of a
closed-loop system.
Refer to the lecture notes entitled
"Time-domain performance of feedback-controlled systems"
(distributed Oct-24-00) for more details.
The response to a STEP input is found by evaluating the following for
's=0' (i.e., steady state):
e_ss[step input] = 1 / (1 + P(s=0)*C(s=0)*M(s=0))
The response to a RAMP input is:
e_ss[ramp input] = 1 / (s + s*P(s=0)*C(s=0)*M(s=0))
(Note that the difference is in a multiplied by '1/s'.)