First of all, sorry for my bad english.

Although it's a fluidic system, I put this topic here because it has an electrical equivalent (ON-TOPIC).

Consider the following system.

Using Matlab (simulink), observe the unitary step response to the interval [0, 5] seconds. Obtain the graph of the height (h1 and h2) of water in both tanks. Initial conditions are zero.

qi, qo = water flow (inbound and outbound)
R1, R2 = fluidic resistance
A1, A2 = area of the tanks

I have some doubts about one thing in this system but first, my resolution.

The areas of the tanks are constants.
General conditions:

Now, considering q1 as the water flow through R1:

Then, using Laplace transform:

Simplifying,

Now, I can get the following block diagram. (considering water density=1)

colours:
Yellow - Qi(s)
Violet - H1(s)
Blue - H2(s)
Red - Qo(s)

Well, my doubt is a bit silly. I don't know where or when I can use H=1...
One thing that I know is when h2(t)<1, qo(t) should be 0 (right?) but in my diagram this isn't true.
(h2(5)=0.20 and qo(5)=1 approximately)

well, any help? how to solve this?

 

First of all, you should show that in your equations. Instead of \(q_o=\frac{\rho g}{R_2}h_2(t)\), write \(q_o=\frac{\rho g}{R_2}(h_2(t)-H)\).

The fact that water won't flow out if it below H makes the system non-linear and you can't study that easily by math. Let Simulink do the work for you.
First, don't go as far as extracting the three trasnfer functions. Just make more simpler blocks to make the equeation set of Picture No3 (after you turn them into Laplace transformations). Then, select a Saturation block and use it as a "diode" to prevent water from flowing in, back to the tank. Set the upper limit to a very high value and the lower to zero. Place that block at the qo output position.

That should describe your system better. If I find some time I will try to build the systmem myself and see how it works.