1. For a certain ship steering system represented by the block diagram shown below, Y(S) is the ship's course, R(S) is the desired course, and A(S) is the rudder angle. Find the transfer function Y(s)/R(s) H($) A(s) R(s) K G(s) G(8) Y(s) H(s) H(s) 2. (i) Find the Laplace transform of f(t). f(1) = 5 + 2e2 sin (21+)-4e-2 cos(21) + 3e-4 If F(s) is the forward-path transfer function of unity-feedback control system, find the transfer function of the closed-loop system. (ii) Find the inverse Laplace transform of G(s). (s+1) G(s) = s(s+2) (s+25+2) 3. The following differential equations represent linear time-invariant systems, where r(t) denotes the input and y(t) the output. Find the transfer function Y(s)/R(s) for each of the systems. (Assume zero initial conditions.) (i) dy(t) d13 +2- dy(1) di +5 dy(f) di dr(t) +6y(t)=3- + r(t) dt dy(1) (ii) d13 +10- +2dy(1) di dt dy(1) + 2 dy(f) + y(t) +2. 25' dr(t) y(r)dr +2r(t) dt