a$.(10$ pts$)$ Obtain the closed$-$loop transfer functions $\frac{Y(s)}{Y_{sp}(s)}$ and $\frac{Y(s)}{D(s)}$ for a proportional$-$integral $($PI$)$

controller with the proportional gain $K_c$ and integral time ${}_{\text{I}}.$

b$.(20$ pts$)$ Approximate the time delay $e^{-t_{d}s}$ in the transfer function using first$-$order Pade

approximation and find the characteristic equation for the closed$-$loop system with a proportional $($P$)$

controller $(K_c$ : proportional gain$).$

c$.(20$ pts$)$ Estimate the ultimate value of proportional gain $(K_{cu})$ for a stable controlled system.