Figure P8.8 gives the simulation diagram of an automatic control system. The plant is th...

Figure P8.8 gives the simulation diagram of an automatic control system. The plant is the system that is controlled, and the compensator is an added system to give the closed-loop system certain specified characteristics.

(a) Write the stale equations for the plant only, with input m(t) and output y(t).

(b) Give the transfer function H_p(s) for the plant.

(c) Write the differential equation for the plant.

(d) Write the state equations for the compensator only, with input e(t) and output m(t).

(e) Give the transfer function H_c(s) for the compensator.

(f) Write the differential equation for the compensator.

(g) Write the state equations for the closed-loop system, with input u(t) and output y(t) choose as states those of parts (a) and (d). These equations can be written directly from the simulation diagram.

(h) Give the transfer function H(s) for the closed-loop system.

(i) Write the differential equation for the closed-loop system.

(j) Verify the results of part (h), using MATLAB.

(k) It can be shown that the closed-loop transfer function is given by

Verify your results in part (h) by showing that this equation is satisfied by the derived transfer functions in parts (b) and (e).