Problem $1.[50$ pts$]$

Consider a causal system with input u$($t$)$ and output y$($t$).$ The output of the system to unit

step input, u$($t$),$ is given by

y$($t$)=(1)/(2)(1-2$e$^(-$t$)+$e$^(-2$t$)),$t$>=0$

Determine if the system is bounded input bounded output stable.

Determine the transfer function of the system and the impulse response. Plot the poles and

zeros of the transfer function in the s$-$plane.

With initial conditions given by $($dy$)/($dt$)(0)=1,$y$(0)=2$ and input x$($t$)$ to be the step input. Find

the unforced and the forced response of the system.

Write down the state space model of the system with following definition of the states z$\_(1)=$y

and z$\_(2)=($dy$)/($dt$).$

Let the transfer function of the system be denoted by H$($s$)$ and consider the feedback con$-$

figuration of the system H$($s$)$ and controller C$($s$)=(\backslash$alpha s$+\backslash$beta $)$ as shown in Fig. $1.$ Determine

the value of $\backslash$alpha and $\backslash$beta so that the closed loop system has performance specification given by

damping ratio $\backslash$zeta $=0\mathrm{.}6$ and natural undamped frequency $\backslash$omega $\_($n$)=2.$