QUESTION $1$

A position control system is shown in Figure Q$1.$

rigure $41$

a$)$ Show that its open loop transfer function is$,\frac{K}{s(s+1)(s+5)}.$

$(1$ marks$)$

b$)$ Sketch the locus of roots $($in color$)$ of the closed loop system as K varies from zero to infinity, in the S$-$plane. Indicate the asymptotes, break points on the real axis, cross$-$over points at the imaginary axis, angle of departure or arrival, where appropriate.

$(9$ marks$)$

c$)$ The system is operating at $22\%$ overshoot with a natural frequency of $0\mathrm{.}9$ radians per second. It is desired for the steady state error to be less than $1,$ for a unit ramp signal, without changing the transient response appreciably.

Prove that the design cannot be achieved with gain alone.

$(5$ marks$)$

d$)$ Design a compensator using root locus so that the transient response will behave according to a second order system.

The objective is to achieve a steady state error of $0\mathrm{.}234$ radians or less. Determine the appropriate compensator gain and zero, given that the compensator pole is at $-0\mathrm{.}005.$

Determine the open loop transfer function of the compensated system with appropriate gain, $K{K}_C.$

Show that the desired condition is satisfied.

$(5$ marks$)$

e$)$ Prove that the dominant pole lies on $($or close to$)$ the root locus of the compensated system.

$(5$ marks$)$