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I need some tips to approach a complex challenge.

The system

I have a system that consists of four connected units. Every unit has its own light bulb and temperature. First, the brightness of the bulb and temperature are same in each unit.

I do not know how a change in, for instance, unit 1 bulb's brightness affect the other units' temperature. I did some measurements and collected the data. I created a neural network to learn the relationships between the units. Now I have a model that predicts from the units' brightness level the temperatures in each units. We assume that the system is closed.

Let's say we disturb the system. Now, the aim is to create a mechanism that returns temperature back to it original value in all units. In other words, to what level should I set the level of brightness in each units in order to keep the temperature constant everywhere in the system.

I have tried my best, but haven't come up with anything smart yet. Do you have some ideas? I would really appreciate it.

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  • $\begingroup$ Not my area, but look up 'control theory'. $\endgroup$
    – mkt
    Jun 6 '18 at 11:31
  • $\begingroup$ Hi. Thank you for your reply. I tried to simulate the system with simulink, but I guess my control theory skills are not good enough.Do you know any good sources to learn the basics of control theory? Don't you think there would exist some simple feedback mechanism to model this? $\endgroup$
    – Avocado
    Jun 6 '18 at 12:43
  • $\begingroup$ Afraid that it's totally outside my domain, so I can't really help. You might consider whether other tags would help get your question more attention - or check if there are other SE sites that may be a better fit for this question. $\endgroup$
    – mkt
    Jun 6 '18 at 13:00
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First, this is a regulating problem in control theory. As you want to maintain constant temperature (output).

I'm assuming that if your system is of this form: $$\dot{\textbf{x}} = f(x) + g(x) u$$ where x is the state of the system and you have already approximated $f(x)$ and $g(x)$ it with data from actual dynamics. Now you have to use that approximated model to design a controller $u$, which can be of following form: $$u = (\hat{g}(x))^{-1}(\hat{f}(x) - K_v * x)$$ where $K_v$ is gain, which you have to select on the basis of your system. So if you assume that $$g(x) \approx \hat{g}(x) \quad \& \quad f(x) \approx \hat{f}(x)$$ , Then we obtain $$\dot{\textbf{x}} \approx -K_v * x$$ which is stable system, you can easily prove it.

This become danger, if $g(x) \approx 0$ and you have to add some small constant to avoid it, that may not be stable. But in your case it will perfectly work.

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  • $\begingroup$ Thank you for your reply. What do you mean by "x is the state of the system". How do I quantify that? You also said that I have already approximated f(x) and g(x) from the data. It is true that I have a neural network that predicts my temperature based on the brightness. But how are these related to the functions you mentioned? $\endgroup$
    – Avocado
    Jun 18 '18 at 6:41
  • $\begingroup$ Any linear system can be written in the above equation form. So in your system, it seemed that u is brightness and x is temperature. Let's see this how i defined these parameter. Your objective in this system is to maintain a constant temprature for any changes in the brightness. So our x_d is constant and we designed a controller u, which will force the system's state x to remain constant by counter affecting the changes in brightness. For more information, you can look into "Indirect Adaptive Control" in control theory. $\endgroup$ Jun 19 '18 at 3:24

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