A negative unity$-$feedback control system has as its forward element a variable gain K in cascade with a system P$($s$)$ having poles at$-1,-3$ and$-3$ but no $($finite$)$ zeros. $($a$)$ Given that its DC$-$gain is $1/9,$ determine the transfer function P$($s$)$ as a ratio of two polynomials in s$.($b$)$ Derive the characteristic equation of the closed$-$loop system as a function of parameter K$.($c$)$ Determine the values of K for which the closed$-$loop system is stable. $($d$)$ Determine the values of parameter K that render the closed$-$loop system marginally stable as well as the exact location in the s$-$plane of the associated imaginary$-$axis poles. $($e$)$ State whether the closed$-$loop system could track a step, ramp, or parabolic input with finite steady$-$state error. $($f$)$ For the input type stated in part $($e$),$ determine the condition on K such that the steady$-$state error could be below $10\%.($g$)$ With the gain fixed at K $=16,$ it can be verified $($you need not do so$!)$ that the step response of the closed$-$loop system for t $>0$ is c$($t$)=0\mathrm{.}640\mathrm{.}16$e$5$t $0\mathrm{.}48$e$$tcos$(2$t$)0\mathrm{.}64$e$$tsin$(2$t$).$