This is a Chemical Process Control question from the Chemical Engineering course. Please do not respond if you are not sure.

A feedback control loop is represented by the block diagram in Figure 2:

Figure 2. Feedback control loop.

Where:

For the system in Figure 2 consider:

a) Write the closed-loop transfer function and the characteristic equation of the loop, if the Gc of the controller operates only with proportional action [Gc(s) = Kc].

b) Applying the R-H and Direct Substitution methods, determine the values of Kc for which the system will be stable.

c) For what controller gain values is the loop response to a step change in the set-point overdamped, critically damped, and underdamped? Can the mesh be unstable?

(c) Find the effective time constants, or second-order time constant and closed-loop damping factor for Kc = -2 and 2.

d) Determine the steady-state residual error for each of the gains in item (c) for a unit-step change in the set-point.

Gv=2s+10,8;Gp=1.5s+10,9;Gm=1;Gd=1.5s+10,85