Consider the following generic feedback control system.

(a) Find the transfer function from the reference \(r(t)\) to the output \(y(t)\) i.e. \(T_{r}(s)\).

(b) Find the transfer function from the disturbance \(w(t)\) to the output \(y(t)\) i.e. \(T_{w}(s)\).

(c) For a generic reference, \(r(t)=\frac{t^{k}}{k !}\), show that the steady-state error with respect to the reference is given by \[
e_{s s}^{r}=\lim _{s ightarrow 0} \frac{\left(1-T_{r}ight)}{s^{k}}
\]

(d) For a generic disturbance, \(w(t)=\frac{t^{k}}{k !}\), show that the steady-state error with respect to the disturbance is given by \[
e_{s s}^{w}=\lim _{s ightarrow 0} \frac{-T_{w}}{s^{k}}
\]

(e) Explain the meaning of system type in terms of the steady-state errors \(e_{s s}^{r}\), and \(e_{s s}^{w}\).

Transcribed Image Text:

w(t) r(t) y(t) D(s) G(s) A General Control System