$3.(40$ pts$)$

Figure $2$

It is intended to automatically maintain the liquid level in the gravity drain double water tank pictured above. Water first flows into Tank $1$ thanks to the pump $($actuator$).$ When the first tank is full, the water in Tank $1$ flows to Tank $2$ by gravity, and after this tank is filled, the water returns to the reservoir. This cycle continues like this. The valve in the figure can be used as a disturbance, and thanks to this valve, water can be filled directly to Tank $2.$ Here LT $($Level Transmitter$)$ is a sensor$,$ LC $($Level Controller$)$ sends a signal to the pump.

$1.(10$p$)$ Obtain the block diagram of the system $($including the controller$)$

$2.(10$p$)$ Obtain FOPDT$-$First order plus dead time model of the system

$($Fitting technique with graphical method using step test can be used. See Figure $3$ and Figure $4.$ Height $2$ will be taken as output. The graph of this situation is also given in Figure $4.$ Data is also given to obtain the figure. See "data.txt$")$

$3.(20$p$)$ Design your own PI or PID controller for acceptable performance. Explain the design details clearly.

Figure $3$ Step response of the system Figure $4.$ Input $($pump$)$ and Output $($height $2)$