Control System: DC Gain

Question

Mostly, whenever it is asked to find the DC gain of A system; in the solutions provided, they find the steady state error coefficient (corresponding to the Type of the system). But I figured out that finding the DC gain this way is equivalent to simply finding the open loop DC gain, and not the closed loop DC gain for the system. So is it something understood , that whenever we are asked find the DC gain we must find the DC gain of open loop transfer function?

For example for this question:

Question: A ramp input applied to a unity feedback system results in 5% steady state error. The Type number and zero frequency gain of the system are?

The solution describes the zero frequency gain of the system equal to the Velocity Error Constant, which is equal to 1/0.05=20. Isn't this the DC gain of open loop instead of the complete 'SYSTEM'?

Where am I going wrong?

Answer 1

The question dictates that a Ramp input has a steady state error, therefore it must be a Type 1 (1st order) System. I remember this from Uni. in '70's but found on web here.
p.5 on http://www.cs.mun.ca/av/old/teaching/cs/notes/steady_quad.pdf

From Steady state error, \$e_{ss}\$ using the final value theorem as follows;
E(s) is the Laplace transform of the error signal, e(t).

\$e_{ss}=e(t)_{t→∞}=sE(s)_{s→0}\$

Control System errors for unity negative feedback

Step response = \$\dfrac{1}{1+k_p}\$

• this is a zero frequency response normalized to 1 unit input

Ramp response = \$\dfrac{1}{k_v}\$

• this is a velocity dependent input rate

The overall steady state error is the sum from each input.