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I'm building a PI controller with inputs read using instrumentation amplifiers (for sensing resistors and floating grounds), rather than subtracting the two input signals with a third chip, is it possible to use the reference pin (as shown in the circuit) in series to accomplish this?

Circuit

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Your circuit is somehow weird. The purpose of the instrumentation amplifier is to have very high input impedance at the input stage and a good common mode rejection ratio. In your case you have some non symmetric components at the input stage, that will result in bad CMRR, extra you have a resistor placed in between both inputs, why?

The Vref needs to be connected to a low impedance, so connecting an opamp output to Vref input is a way to go. But there is one another strange thing: you have one measurement as Setpoint and other named Sense, in control theory you would need (Setpoint-Sense), what you'll get is (Setpoint+Sense), OK you can still swap the Sense inputs to get negative value.

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