Time t is in seconds; angles are in radians

Time \(t\) is in seconds; angles are in radians. The unit step satisfies \(u(t)=1\) for \(t\ge 0\) and \(u(t)=0\) for \(t<0\). A type-1 plant with one free parameter, \[ G(s)=\frac{K}{s(s+a)},\qquad K>0,\quad a>0, \] is placed in unity-gain negative feedback with \(H=1\). The closed-loop characteristic polynomial is \(s^{2}+a s+K\), so \[ \omega_{n}=\sqrt{K},\qquad \zeta=\frac{a}{2\sqrt{K}},\qquad K_{v}=\lim_{s\to 0}sG(s)=\frac{K}{a}. \] Assume \(K\) and \(a\) are such that \(0<\zeta<1\) and the closed loop is asymptotically stable (automatic for \(K>0\), \(a>0\)). (a) First freeze \(a=10.0\,\mathrm{s}^{-1}\) and allow only \(K>0\) to vary. Determine whether there exists \(K\) such that both \(\zeta=0.600\) and \(K_{v}\ge 8.00\,\mathrm{s}^{-1}\) hold. If not, compute the \(K\) that meets \(\zeta=0.600\) and the resulting \(K_{v}\), and compute the \(K\) that meets \(K_{v}=8.00\,\mathrm{s}^{-1}\) and the resulting \(\zeta\). (b) Now allow both \(K>0\) and \(a>0\) to vary. Determine the set of pairs \((K,a)\) that satisfy \(\zeta=0.600\) and \(K_{v}\ge 8.00\,\mathrm{s}^{-1}\) simultaneously. Report the infimum of admissible \(K\) and the corresponding \(a\). Write \(T(s)=G/(1+G)\) at that infimum point. (c) At the infimum design of (b), compute the prototype percent overshoot \(100M_{p}\) with \[ M_{p}=\exp\Bigl(-\frac{\pi\zeta}{\sqrt{1-\zeta^{2}}}\Bigr), \] the envelope \(2\%\) settling time \(4/(\zeta\omega_{n})\), and the steady-state error to the ramp \(r(t)=0.50\,t\,u(t)\,\mathrm{rad}\). Compute the closed-loop poles.

** With the plant pole \(a\) frozen, express both \(\zeta\) and \(K_v\) in terms of \(K\) and compare the two one-parameter constraints.