A PI controller is used to control a third$-$order process system that is defined by the

following third$-$order transfer function

$G(s)=\frac{K}{({}_{1}s+1)({}_{2}s+1)({}_{3}s+1)}$

Note that the above transfer function represents the product of three first$-$order transfer

functions that may describe the transfer functions of various process sub$-$systems.

If ${}_1=1,{}_2=\frac{1}{2},{}_3=\frac{1}{3}$ and ${}_{\text{I}}=\frac{1}{3},$ derive the conditions for closed$-$loop stability in

terms of the product of the controller gain $K{c}_c$ and the process gain $K.$ It can be assumed

that the product of these gains is such that $K_{c}K>0.$