Tank level

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TANK LEVEL
Practice
ALEJANDRO VICENTE GONZALEZ

TANK LEVEL. Alejandro Vicente González. “Universitatea Dunarea de Jos” González.

Our experiment is the following:

Water flows into the first tank through pump 1 a rate of fi which obviously affects the height of the water in tank 1 (denoted by h1). Water flows out of ). tank 1 into tank 2 at a rate of f12 , affecting both h1 and h2 .Water then flows out of tank 2 at a rate of fe controlled by pump 2. Given this information, the challenge was then issued to the students of the course: control the heights of the tanks. However, a problem arose when the sensor for the height of tank 1 failed. Graham then suggested that it would be possible to build a virtual sensor (or observer/soft sensor) to estimate the observer/soft height oftank 1 based on measurements of the height of tank 2 and the flows f1 and f2.

TANK LEVEL - PART 1

Before make the study of this simulation we have to establish a model with which we can analyse the all the points that are asked. The height of tank 1 can be calculated by the equation

The height of tank 2, h2 is given by

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TANK LEVEL. Alejandro Vicente González. “Universitatea Dunareade Jos” González.

The flow between the two tanks can be approximated by the free fall velocity for the difference in height between the two tanks.

Now, if we measure the heights of the tanks in % (where 0% is empty and 100% is full), we can convert the flow rates into equivalent values in % per second (where f1 is the equivalent flow into tank 1 and f2 is the equivalent flow out of tank 2).The model for the system is:

Where

We can then choose to linearise this model for a nominal steady state height difference (or operating point). Let:

Then

since

then

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TANK LEVEL. Alejandro Vicente González. “Universitatea Dunarea de Jos” González.

This yields the following linear model

Where

Since we are assuming that h2 can be measured and h1 cannot, we set C = [ 01 ] and D = [ 0 0 ]. The resulting system is both controllable and observable (which you can easily verify). Now we wish to design an observer

to estimate the value of h2. The characteristic polynomial of the observer is

so we can choose the observer poles which gives us values for J1 and J2. If we assume that the operating point is H = 10%, then k = 0.0411. If we wanted poles at s =-0.9822 and s = -0.0531, then we would calculate that J1 = 0.3 0.9822 0.0531, and J2 = 0.9. If we wanted two poles at s = -2, then J1 = 3.9178 and J2 = 93.41. The equation for the observed system is then

Where

is the measured value of h2. Here we model

as follows:

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TANK LEVEL. Alejandro Vicente González. “Universitatea Dunarea de Jos” González.
where v represents the noise.Alternatively, we could use the non linear model in our observer to attempt to non-linear get better performance:

We are going to start with the simulation in Java.

In this simulation we have some things to try, we are going to describe them in the following paragraph. g We’ll make two kinds of simulation, one of them with the linear observer and the other with a non-linear observer. We’ll notice thatthe non linear observer linear non-linear has no error in its output ( we’ll show this with some graphics obtained from the simulation ). The linear observer uses a nominal height difference of 10%. lation We’ll notice also that with the linear observer, the system will perform worse when the system is far from his operating point. We are going to move too the observer poles out towards negativeinfinity. We see the difference that makes the observer response speed and accuracy. If there’s noise in the experiment this is added to the actual heigh h2, to give height, m the measured height, h 2. The difference between the set point and the tween measured value, hm2 is the input to the controller. We are going to see the s controller. graphics in the following page.

The graph has a...
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