This problem examines the steady$-$state error of a negative unity$-$feedback control system. Its forward path is comprised of a PID controller in series with a first$-$order plant described, respectively, by transfer functions K$($s$)=$ $+$ s $+$s and P$($s$)=1$ s $+$a $,$ where controller parameters $,$ and denote the $($real$-$valued$)$ gains of three independently adjustable amplifiers and plant parameter a is associated to an arbitrary $($real$-$valued$)$ pole location $($at s $=$a$).($a$)$ Determine the closed$-$loop transfer function T$($s$)($from reference input to controlled output, as usual$)$ in terms of all four parameters, expressing the result as a ratio of two polynomials in s$.($b$)$ Determine conditions on gains $,$ and $,$ expressed as a function of plant parameter a$,$ to ensure that the closed$-$loop system is stable. Note: You should see that your answer depends upon whether $0$ or $<0,$ but for simplicity consider only the case of $0.($c$)$ Again for only the case in which $0,$ what additional conditions on the gains $($if any$)$ to those in part $($b$)$ will yield less than $5\%$ error in steady state to a ramp input? $($d$)$ If power considerations limit the controller$$s three amplifiers to each have maximum magnitude$-$gain of $100,$ determine bounds on plant parameter a such that the specifications of part $($c$)$ can still be met. Hint: Answer after considering separately the cases of $>1$ and $<1.$