The Nyquist plot of an all-pole second order open-loop system is shown in Figure. Obtain the transfer function of the system.

A unity feedback system has the plant transfer function G_p(s)=$${1 \over {\left( {s + 1} \right)\left( {2s + 1} \right)}}$$
(a) Determine the frequency at which the plant has a phase lah of 90^o.
(b) An intergral controller with transfer function G_c(s)=$${k \over s}$$ , isplaced in the forwardpath the value of k such that the compensated system has an open loop gain margin of 2.5.
(c) Determine the steady state errors of the compensated system to unit-step and unit-ramp inputs.

The transfer function Y(s)/U(s) of a system described by the state equations
$$\mathop x\limits^ \bullet $$(t) = -2x(t)+2u(t)
y(t) = 0.5x(t) is

The block diagram of a linear time invariant system is given in Figure is (a) Write down the state variable equations for the system in matrix form assuming the state vector to be $${\left[ {{x_1}\left( t \right)\,\,{x_2}\left( t \right)} \right]^T}$$
(b) Find out the state transition matrix.
(c) Determine y(t), t ≥ 0, when the initial values of the state at time t = 0 are $${x_1}$$(0) = 1, and $${x_2}$$(0) = 1.