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transfer function of time invariant system

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bobobano

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I need some help with my controls class project.

I'm designing a control system to keep the light level in my room constant across the day by adjusting the blinds and the ceiling lamp. To do this I made a system of 4 LDR sensors, 1 per corner. Unfortunately since I don't have access to a light meter and I'm not confident in the formula I derived to change from the LUX measured by the LDRs is correct.

Because of this I'm using the units of the LDR measured by the micro controller (integer 0-1024) and I have derived the formula to calculate what my PWM output to the lamp should be to get my desired LDR reading

My questions are these:
1) Since my formula is time invariant, I'm having some issues finding the transfer function of it, as far as I know you can only find the laplace of a time dependent system, and I need the transfer function so I can simulate it in matlab.

2) Am I correct in assuming that if I multiply the measured LUX of the LDR (by measuring the voltage across it and converting to resistance and then to lux) by the area of the LDR head I can calculate the lumens?
 

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