I wondered how airliners do it, because you can’t really feel anything, while they may be descending at -2000fpm plus.
Obviously, they cannot start repressurising the cabin until the top of descent – assuming they flew with the max permitted differential.
Is there an automatic system which simply runs a linear pressure profile in the cabin, from the cruise altitude (say 8000ft, typically) all the way down to the airport altitude?
The pressurization controller needs to know the destination airfield elevation (pressure). During climb it will increase the pressure gradually to control the stress on the fuselage until its max differential pressure is reached. On descent it will do the reverse, arriving at the programmed airfield pressure at the right time. You can feel the pressure change during descent, at least I can 
OK, but what is the setpoint (i.e. the target) for this controller?
Is it maintaining the max allowed pressure differential? If it was that, you would see a rapid cabin pressurisation once the aircraft descends below about 8000ft (and continues to go down fast in some cases) yet this is definitely not the case.
The FMS knows when and where the aircraft will land and the controller adjusts the pressure accordingly so that it will have outside pressure upon landing but there is no unpleasant rapid pressure change. Not that difficult to implement I believe.
And Peter in the Meridian at top of descent it works to smoothly bring the pressure down. So it doesn’t wait until 8000ft and then use ambient. It is the same on the way up.There is a pressure setting (set for usually 1000ft above arrival airport elevation and a rate setting for how fast you want the cabin to climb or descend. It will use that rate if possible otherwise maintain Max differential if it isn’t.
OK, I can see that a rate (in effect a vertical speed for the cabin) would work, but that means somebody has to calculate the rate, to achieve a linear repressurisation. Maybe airliners have an automatic calculator?
You set the destination field elevation in the pressurisation controller and it adjusts the differential smoothly from max differential pressure in the cruise to zero differential pressure as you land. Wheels down/weight on wheels usually dumps pressurisation anyway so it needs to be zero diff just before landing to avoid this being noticeable.
OK, let me rephrase it differently
Does the controller (on any aircraft) use the EET (or the distance to run) so as to achieve a linear ramp on the cabin pressure, from the TOD to the touchdown?
I mine the aircraft starts to descend the cabin when you leave cruise altitude but it is simplistic in the sense it just uses rate bounded by max diff pressure. So no linear calculation other than rate.
Does the controller (on any aircraft) use the EET (or the distance to run) so as to achieve a linear ramp on the cabin pressure, from the TOD to the touchdown?
Not that I know. It is connected to the air data computer which supplies it with altitude and rate of descent/climb and, as others have said, with manually selected cruising level and destination field elevation. When it senses altitude and power changes above certain threshold values it will enter climb and descent mode and initially assume constant rate all the way and adjust the rate of differential pressure buildup/reduction accordingly. This is then fine tuned with the actual ROC/ROD, again observing certain limitations. It it actually not a very complex problem to solve in terms of control, on older aircraft (eg. Cessna piston twins) this is an entirely pneumatic-mechanical device with it’s own altimeters, one for inside, one for outside. The electronic ones have several backup modes in case it loses all or part of the connection with the air data computer. The pressurisation controller has six pages in the manual with diagrams of the pressurisation schedule. If you are intersted, I can scan them and send them to you by email one of these days.
