# Why the controller block diagram is not working (while tracking the reference and water level in tank 2 at specific level)?

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Sana Mohamed on 27 Nov 2022
Commented: Sam Chak on 6 Dec 2022
Suvansh Arora on 30 Nov 2022
In order to understand this better, please send the following information:
• Your solution approach to the above problem.
• Details about the issue you are encountering.

Sam Chak on 30 Nov 2022
Edited: Sam Chak on 30 Nov 2022
It seems that your equation for u causes the level of Tank #2 to dip below 0, thus term sqrt(x(2)) returns an error message related to the complex number. In the attached Simulink model, your original u is disabled and two lines are added in the "Level Controller" Function Block:
where and are the reference levels of Tank #1 and Tank #2, respectively.
Note: I didn't know how you derived your original equation for u. But from your scripts, the Dual Tank Liquid Level System is given by
with the initial condition
and you want to regulate the level of Tank #2 to . Since can only be affected by , I designed the reference level of Tank #1, so that
.
Then, I backstepped the process and designed the control equation for u so that
.
Sam Chak on 6 Dec 2022
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Back to your query, I actually used basic algebra manipulation and the exponential decay principle. No difficult math!
I started from your original error definition, where you want to track the tank #2 level
.
Taking the time derivative of
, because due to , a constant.
Making substitution to obtain the error dynamics
.
Following the principle that guarantees exponential decay
where , is chosen as the indirect manipulated variable because we can directly manipulate in dynamics through the direct manipulated variable u.
Thus, we can equate both equations where
and solve for
.
Making substitution for to obtain the reference where must track
.
Similarly, repeat the design steps to obtain for u:
.
Here, because I find it a little mathematical tedious to obtain the time derivative for , I let , and thus it is reduced to
.
By the way, how did you derive the original equation for u? Can you show me?
% -------------------------------------------
Afterthoughts: I think you can possibly design based on
and

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