1. For the linear control system with input r(t) and output y(t) which is described by the simulation diagram shown below:
a. Define the first order differential equations using the state variables i.e.,
x.(t) = A(t) x(t) + B(t) r(t)
b. Define a state variable model for the system in a matrix notation.
c. Use the state space mode to find the transfer function using the following formula:
Y(s)/R(s) = C(SI - A)-1B + D
2. For the feedback control system shown in the block diagram below, find the transfer function
3. For the feedback control system shown below, sketch the Root-Locus for 0 < k < ∞ on the S-Plane.
where: G(s) = ((S + 2)(S + 4)(S + 6))/((S+ 1)(S + 3)(S+ 5)(S +7)(S +9)(S + 11))