Determine the multivariable transfer function gs determine

Question

1. Given the following transfer function models:

y₁(s)/u₁(s) = g₁₁(s) = 2/(s+3)

y₁(s)/u₂(s) = g₁₂(s) = -1/(s+1)

y₂(s)/u₁(s) = g₂₁(s) = -1/(s+7)

y₂(s)/u₂(s) = g₂₂(s) = 0.5/(s+4)

a) Draw the 2x2 open-loop block diagram.

b) Determine the multivariable transfer function G(s).

c) Determine the multivariable gain matrix.

d) Determine the poles for the system.

e) Determine the zeros (if any) for the system.

f) Determine open-loop step response for a unit step in both u₁ and u₂ at time 0.

g) Determine the RGA for the system.

h) Suggest a controller pairing for the system.

i) Draw the 2x2 closed-loop block diagram with PI controllers.