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If I were teaching I would use this example. I got it from a student that was posting this problem on LinkedIn. Most of the forum had different opinions as to whether the system was controllable. Even the student's instructor had doubts. The system consisted of two tanks and a pump. Each tank had a level feed back. The pump pumped water into tank 1. Tank 1 had a hole, orifice or fixed valve on the bottom that allowed water to flow into tank 2. Tank 2 had a similar hole, orifice or fixed valve on the bottom. The goal was to control the level in tank 2. To me it was obvious that the level in tank 2 could be controlled. One could look at the system as two low pass filters in series but there are a few "gotchas". Much of the forum suggested many ways of tweaking on PID gains. I chastised the forum because one doesn't go to college just to learn how to tweak gains. Monkeys could do that given enough time. People go to college to learn how to design systems. In this case one should go to college to learn how to be a designer, not simply one who commissions a system that is already built. After all, how does one chose the size of the pumps, orifices or valve, shape and height of the tanks? Nothing is learned by just tweaking gains and drinking coffee while watching the results.
The student got an 'A' but I wouldn't have given him one. I wouldn't have even given him a passing grade because he could not define his system. He did manage to control the level in the second tank by trial and error.
In my experience there are plant engineers that just keep things running. The next level is system integrators/OEMs. Finally there are the designers. Even among the designers I have 3 categories. There are those that kludge systems and hope they work, those that evolve systems, and then there are those that are golden that design and model their systems. These are the same engineers that design air craft, space craft or other things where learning by trial and error is not acceptable.
So below is a video for my solution to the problem. I do not assume anything is linear in the end. Gains are calculated on-the-fly. No system identification is required. From the differential equations it is possible to simulate the system with all its non-linearities. Too many so called engineers say the system is non-linear and throw up their hands and just tweak gains until they get an acceptable response but isn't what one should learn in college.
Here is my solution.
By now is should be obvious that I like to use differential equations. I do NOT see differential equations used in control books, just Laplace transforms and matrices and they are given. The students are paying too much money. The students are being cheated.
OK, I admit, I made a mountain out of a mole hill but it is one heck of a mole hill. In the end it is not the mole hill that matters but how one approaches problems.
I forgot to mention. NO Root Locus required.
Peter Nachtwey
Delta Computer Systems
The student got an 'A' but I wouldn't have given him one. I wouldn't have even given him a passing grade because he could not define his system. He did manage to control the level in the second tank by trial and error.
In my experience there are plant engineers that just keep things running. The next level is system integrators/OEMs. Finally there are the designers. Even among the designers I have 3 categories. There are those that kludge systems and hope they work, those that evolve systems, and then there are those that are golden that design and model their systems. These are the same engineers that design air craft, space craft or other things where learning by trial and error is not acceptable.
So below is a video for my solution to the problem. I do not assume anything is linear in the end. Gains are calculated on-the-fly. No system identification is required. From the differential equations it is possible to simulate the system with all its non-linearities. Too many so called engineers say the system is non-linear and throw up their hands and just tweak gains until they get an acceptable response but isn't what one should learn in college.
Here is my solution.
By now is should be obvious that I like to use differential equations. I do NOT see differential equations used in control books, just Laplace transforms and matrices and they are given. The students are paying too much money. The students are being cheated.
OK, I admit, I made a mountain out of a mole hill but it is one heck of a mole hill. In the end it is not the mole hill that matters but how one approaches problems.
I forgot to mention. NO Root Locus required.
Peter Nachtwey
Delta Computer Systems
