# Identification of conical tank for level control using Hammerstein model

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#### P. Venkatesan

I have a conical tank of height 49cms with upper dia 18cms and bottom dia 2cms. I need to control the level of the liquid in the tank. I have to identify the parameters associated with the tank using Hammerstein model. How do you go about in modelling the tank?

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#### Hakan ozevin

I was expecting for some days that a mathematician would answer such a question and we could benefit from his knowledge, but noone did, therefore as an engineer I will try to give my comments about it. An engineer must be partly a mathematician, but may have some mistakes in his analysis in such a pure mathematical subject, so please correct me if I did a mistake.

First of all, the question has a lacking point: What is the variable of such a system? The author wants to make a level control system, but he has to have a variable. I assume that there is a flow to this conical tank with a discreet function f(k).

Secondly, he wants to make a Hammerstein model of it, but does not explain it why. A Hammerstein model is suitable where the system to be analysed is *non linear* and *in some cases* of this nonlinearity. In this question, however, the system is linear (a conical geometry gives linear output for level to an input in flow). Therefore, I assume that this is a theoretical question and asks if "Is there a practical Hammerstein model for such a system, or are there better models to analyse it?". In the following lines I will try to show a way to analyse such a system and if it does not fit to Hammerstein model I can do nothing about it. I can only say that models are there to explain and analyse physical realities, but not the other way round. If the model does not explain or fit to be the easiest way for the truth, you have to change/choose another model.

I will take volume inside the cone as the key point for this problem (a**b means a to the power of b and L is the level of the cone at any instant).

1. At any instant k, volume of the cone filled with the flow equals to Sum( f(L)) as L=0 to L=k
2. The cone can be assumed to be a perfect cone and not cut, since we can always calculate the fixed volume below the cut (non existing) cone
3. At any instant k, there is a level L(k) and there is a radius of the filled cone r(k). By simple geometry we can calculate r(k) in terms of L (k) and L(k)=c1*r(k)+c2, where c1 and c2 are both constants.
4. Think about thin slices of the cone. Each will have an area of 2*pi*(r**2).
5. The volume of a such thin slice will be 2*pi*(r**2)*delta L, where delta L is the thickness of such a slice.
6. If L is taken small enough, volume=sum(2*pi*(r**2)*L) as L=0 to L=k
7. If we combine (3) and (6) we get volume=c3*sum(L**3)+c4 as L=0 to L=k. This is very logical since for a cone, volume is (1/pi)*r**3
8. If we combine (1) and (7) we get sum(f(L))=c3*sum(L**3)+c4 as L=0 to L=k
9. Since both sides are definite summations with the same limits, we can write as L(k)=c5*f(k)**1/3 +c6, where c6 is the last calculated value of L before instant k (dependent of L) and c5 is a contant where the author should find with geometry. This is written in PLC terms, because it is easy to read the preceeding value and add it to the actual term when using a PLC (or any microP).
10. In mathematical terms (9) can be written as L(k)=c7*sum(f(m)**1/3)+c8 as m=0 to m=k, where c8 is the level between the cut cone and c7 is a contant (can it be equal to c5?).

I think (10) gives a model (but a linear one) that the author is looking for.

There are some practical conclusions of this analysis:
1. You can calculate the level in a tank with known geometry if you have a flowmeter (provided that you know the initial conditions and have a system to reset the errors that will occur from the sensor)
2. You can calculate the volume in a tank with known geometry if you have a level sensor ( no need to reset). I already made an application for this and it really works.

Share with me such interesting applications of integral or derivative in real life.

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Hakan Ozevin

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#### Venkatesan

I want to control the liquid level in the conical tank by varying the inlet flow rate using a control valve. The outlet flow rate is constant. For a certain percentage of valve opening the level in the tank varies non-linearly with respect to time. I would like model the conical tank using Hammerstein modelling i.e., I need to find the parameters (a1,a2,.., b1,b2,... and c1,c2,...) that describe the non-linear block and the linear block of a Hammerstein model. I have a DPT to continuously monitor the Level inside the tank. My manipulated variable is inlet flow rate and the controlled variable is the liquid level inside the tank. The height of the tank is 49 cms and the diameter at the top of the tank is 17 cms and at the bottom is 0.5 cms.
How do I model the given conical tank to a Hammerstein Model?

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#### Hakan ozevin

Correction: Such a system is not linear. I had a mistake of thinking that the output of such a system is volume, instead of level. It is obvious that if we increase the flow, say two times, level will not increase two times at any instant k. However, the system is continuous, therefore the solution can be easier than a complex tank (for example a tank with conical lower part and a cylindirical upper part).

Hakan Ozevin

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#### Bob Hogg

Dear Sir/Madam, We build continuous level controls - our simple dial indicates the visual readings. Your situation seems straight forward. Send us an email for more info. Bob Hogg Almeg Controls [email protected]