Problem $3$

The equivalent discrete$-$time transfer function for a particular system with sampling time $T=25ms$ is

$G(z)=\frac{0\mathrm{.}0008183(z+0\mathrm{.}4968)}{z^3+2\mathrm{.}951z^2+2\mathrm{.}911z+0\mathrm{.}9604}$

The system's actuators can accept inputs $u[k]$ satisfying

$|u[k]|21$

A desirable response reduces the settling time to $20\%$ of the uncompensated system's settling time and follows a step$-$input with at most $1\%$ steady$-$state error. The gain margin must be $2,$ but it is preferred if the gain margin is at least $8.$ Similarly, the phase margin must by $45,$ but a phase margin of at least $60$ is desirable.

$($A$)$ Design an open$-$loop controller $F(z)$ for the system. Clearly write the transfer function $F(z)$ and provide plots of both the control input $u[k]$ and the output $y[k].[15$ points$]$

$($B$)$ Design a closed$-$loop compensator $D(z)$ for the system. Clearly write the transfer function $D(z)$ and provide plots of both the control input $u[k]$ and the output $y[k].[20$ points$]$

$($C$)$ Combine your designs from part $($A$)$ and part $($B$)$ into a bridged$-$T controller for the system. $[20$ points$]$

$(1)$ Draw a block diagram of the bridged$-$T controller with the transfer functions all written in the blocks.

$(2)$ Provide plots of both the control input $u[k]$ and the output $y[k].$

$(3)$ If the system plant is actually

$G_{\text{a}\text{c}\text{t}\text{u}\text{a}\text{l}}(z)=\frac{0\mathrm{.}001(z+0\mathrm{.}4968)}{z^2+1\mathrm{.}8635z+0\mathrm{.}9025}$

find the response of the actual system to the bridged$-$T controller designed and simulated in $($C$2).$ Note that the actual system plant is lower order than what was assumed.