Consider three identical tanks in a series subject to an input function
. The heights of the liquid in the three tanks (i.e.
) obey the following equations:
is the cross-sectional area of a tank and
is related to the discharge coefficient for the exit pipes.
Suppose the height of tank 3 is sampled for a given input function
to give the following data list:
Then the constants
can be estimated using a least-squares optimization method. That is, we define the following objective function
is the height in tank 3 predicted by the model at time
is the value of
measured at time
. The goal then is to determine
such that sum of squares
is minimized for
spanning the duration of the experiment.
One finds as shown in the second snapshot
. It is possible then to solve the governing equations shown above and determine the height of tanks 1 and 2. The second snapshot presents the height versus time for tanks 1, 2, and 3 in blue, magenta, and brown, respectively.
have been determined, one can run simulations for various forms of the input function: impulse input, triangle input, square input, and staircase input. The subsequent snapshots show the responses for all the above mentioned special input functions, which are shown in red in a separate plot.