Question

The following is a schematic diagram of an experimental 4-tank system. Develop a linear state space model for the system for steady state conditions h0=[20 20 10 10] cm. The tank model and dimensions of the system are given below, where $A_i$ stands for the tank cross sectional area, $a_i$ denotes pipe cross sectional area, $f_i$s are the flowrates to the tanks. Assume that there are two sensors measuring the heights of Tank 1 and Tank 2. Parameters unit x-sectional area of tank, cm$^2$ $A_1, A_2$ x-sectional area of tank, cm$^2$ $A_3, A_4$ x-sectional area of outlet, cm$^2$ $o_1, o_2$ x-sectional area of outlet, cm$^2$ $o_3, o_4$ $\frac{dh_1}{dt} = -\frac{a_1}{A_1}\sqrt{2gh_1} + \frac{a_3}{A_1}\sqrt{2gh_3} + \frac{f_1}{A_1}$ $\frac{dh_2}{dt} = -\frac{a_2}{A_2}\sqrt{2gh_2} + \frac{a_4}{A_2}\sqrt{2gh_4} + \frac{f_2}{A_2}$ $\frac{dh_3}{dt} = -\frac{a_3}{A_3}\sqrt{2gh_3} + \frac{f_3}{A_3}$ $\frac{dh_4}{dt} = -\frac{a_4}{A_4}\sqrt{2gh_4} + \frac{f_4}{A_4}$

          The following is a schematic diagram of an experimental 4-tank system. Develop a linear state space model for the system for steady state conditions h0=[20 20 10 10] cm. The tank model and dimensions of the system are given below, where $A_i$ stands for the tank cross sectional area, $a_i$ denotes pipe cross sectional area, $f_i$s are the flowrates to the tanks. Assume that there are two sensors measuring the heights of Tank 1 and Tank 2.

Parameters
unit
x-sectional area of tank,
cm$^2$
$A_1, A_2$
x-sectional area of tank,
cm$^2$
$A_3, A_4$
x-sectional area of outlet,
cm$^2$
$o_1, o_2$
x-sectional area of outlet,
cm$^2$
$o_3, o_4$
$\frac{dh_1}{dt} = -\frac{a_1}{A_1}\sqrt{2gh_1} + \frac{a_3}{A_1}\sqrt{2gh_3} + \frac{f_1}{A_1}$
$\frac{dh_2}{dt} = -\frac{a_2}{A_2}\sqrt{2gh_2} + \frac{a_4}{A_2}\sqrt{2gh_4} + \frac{f_2}{A_2}$
$\frac{dh_3}{dt} = -\frac{a_3}{A_3}\sqrt{2gh_3} + \frac{f_3}{A_3}$
$\frac{dh_4}{dt} = -\frac{a_4}{A_4}\sqrt{2gh_4} + \frac{f_4}{A_4}$

The following is a schematic diagram of an experimental 4-tank system. Develop a linear state space model for the system for steady state conditions h0=[20 20 10 10] cm. The tank model and dimensions of the system are given below, where Ai stands for the tank cross sectional area, ai denotes pipe cross sectional area, fis are the flowrates to the tanks. Assume that there are two sensors measuring the heights of Tank 1 and Tank 2.

Parameters
unit
x-sectional area of tank,
cm^2
A1, A2
x-sectional area of tank,
cm^2
A3, A4
x-sectional area of outlet,
cm^2
o1, o2
x-sectional area of outlet,
cm^2
o3, o4
(dh1)/(dt) = -(a1)/(A1)√(2gh1) + (a3)/(A1)√(2gh3) + (f1)/(A1)
(dh2)/(dt) = -(a2)/(A2)√(2gh2) + (a4)/(A2)√(2gh4) + (f2)/(A2)
(dh3)/(dt) = -(a3)/(A3)√(2gh3) + (f3)/(A3)
(dh4)/(dt) = -(a4)/(A4)√(2gh4) + (f4)/(A4)

Added by Mar-A R.

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