The LAG you may understand, but you missed the turn off at zero voltage and zero current which may not be co-incident.
Rectifying the mains won;t necessarily work. You would need 240 VDC with no losses. Your vacuum motors are what HP?
Not sure if regulating voltage squared or voltage would be better.
I can walk you through writing a PID controller on any processor that supports floating point. I just haven't done autotune, The PID loop is nothing more than numerical integration and differentiation.
A auto-tuning process controller, I think is your best option. 1st job find one. Next interface it to read pressure in engineering units. You may or may not need an isolated output.
My original thought was to rectify the mains hence the PWM on the circuit I proposed with the ? next to it, I thought I would just use suitable Mosfets to deliver the DC power to the motors, DC will have Tighter control.
Well, probably a transformer would be involved to get 240 ish VDC. The good part; is it's easy. I've used this
https://www.jameco.com/webapp/wcs/stores/servlet/Product_10001_10001_120539_-1 guy before, so most of the work is done for you.
Now, the triac thing, probably best done with a microprocessor. But would you want to do it with a look-up table without a processor. A/D creates a index to a look-up table for the phase angle of some average voltage, 0-5 is say 0-100% (0-511). Look up value of phase angle or maybe it's a digital divider. That's where you would start firing and you would continue this until the voltage went to zero again.
Even though we might look at it as 0-100% to a phase angle, the "phase angle" could be a time base divider.