Perfect is the enemy of good.  I can’t perfectly control some of the “overshoot” in house temperature .. ie due to sunshine and / or burning too much firewood / and/or rapid ambient temperature changes from chinooks … and I’m not worried about that either.  A few tics in house temp higher or lower isn’t a big deal .. a first world problem really. 

How true, and an argument I am having elsewhere with those who want every ASHP installation to be totally bespoke, even at the end of the market where you are fitting a 5kW come what may because its the lowest rated one most manufacturers do.

The tricky one is the hydronic slab due to its much slower reaction time as you point out.  My strategy with it is to run it on the cool side during the heating season (about 21C) as at that temp it would act as a base load heat source for the house without any overshoot possible.  

That should work, although there is still the possibility of overshoot depending on the reaction time of the other components

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Our house is actually really stable regardless of ambient temperature due to how well insulated and tight it is .. where some houses drift materially within an hour, ours drifts over about 4 hrs if that makes sense .. even at -30C!

Ours is like that as well, despite being a 1930s originally solid wall (albeit partly upgraded) house.  The excursions due to delay will likely be slow, but nonetheless annoying.  As I say its house dependent.  Ours now runs (except at the very end of the season) on pure weather compensation with no thermostats or other room influence, much better than it ever ran with loads of TRVs etc.  Basically the only variable taken into account by any part of the control system is OAT.  Its not absolutely perfect, but its a whole lot better than it was previously

The tricky one is the hydronic slab due to its much slower reaction time as you point out.  My strategy with it is to run it on the cool side during the heating season (about 21C) as at that temp it would act as a base load heat source for the house without any overshoot possible.  I would then rely on the fireplace and air handler loop (gas or HP) to raise the last few degrees to keep the space comfortable.  Sweaters also work great to even out some lumps in heating ;) Our house is actually really stable regardless of ambient temperature due to how well insulated and tight it is .. where some houses drift materially within an hour, ours drifts over about 4 hrs if that makes sense .. even at -30C!

My design vision was to allow the gas furnace to run on its own thermostat set a couple of degrees cooler (ie 18C) than the HP target (20C) and simply allow the furnace to cut in whenever the HP couldn’t keep up.  I also planned to set the HP to cut out below -15 or -20C as COP becomes pretty poor below that anyway.  The fireplace can also heat the whole house and even the slab on its own (it is an interesting set up) .. so my HP design needs to accommodate that as well.  Fun eh ?

Definitely.

If I were you I would read up a bit on the engineering topic of 'control theory'.  This deals with how you control real world systems which have inbuilt delays (in the case if a house these are enormous), noisy interference, nonlinearities, the possibility to control only in one direction (in this case up only) and other embuggerations.  The maths is horrendous and well beyond me even when I was at uni, but some understanding of the basic principles, even if you don't go beyond the effects of delay (what @bobflux calls phase shift) and the basic principles of pid, will definitely help. 

I think you may have a simplistic view that measuring and controlling based on the desired output (in this case room temperature) is guaranteed to work.  Unfortunately, even though intuitively this the true, it is not true in fact.

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Some heat pumps have a "bivalent" dry contact output that means "It's too cold, I give up, turn on the boiler." You're supposed to wire that to the boiler, probably to its thermostat input. 

Note if it's too cold for the heat pump to run, something has to be done about the water freezing in the pipes.

My design vision was to allow the gas furnace to run on its own thermostat set a couple of degrees cooler (ie 18C) than the HP target (20C) and simply allow the furnace to cut in whenever the HP couldn’t keep up.  I also planned to set the HP to cut out below -15 or -20C as COP becomes pretty poor below that anyway.  The fireplace can also heat the whole house and even the slab on its own (it is an interesting set up) .. so my HP design needs to accommodate that as well.  Fun eh :) ?

Re thermostat control.  I planned to use a slab temperature sensor wired to a thermostat to control that loop at about 24C in winter (lower floor) and a separate regulator thermostat for the air exchanger loop (both floors).  Both would be wired to signal the HP to run on demand.  

As @bobflux says this is unlikely to work.

 I get how the HP should modulate using just the temperature curve alone .. but don’t you still need a way to turn it on/off once you get outside of the demand zone? (

In principle no because, by adjusting the flow temp according to the outside temperature you are matching the heat input to the heat loss from the house.  I know its a bit counterintuitive, but the physics is sound and it works.  The variation in heat gain from windows and life is generally small compared to the house loss, other than at the ends of the season, so not a problem and more to the point generally much less of a problem that the phase shift in a control loop involving room temperature.

To check the temperature inside the slab, I simply put a temperature probe on the return pipe from the slab.

Surely your heat pump does this anyway.

It needs to handle enough flow and suck enough thermal power from the water to allow the heat pump to run above its minimum power level and not cycle. The fan in the air handler can't be allowed to turn off, if it does the heat pump will cycle like crazy. In other words you can use a thermostat to control both heat pump and heat exchanger fan. If the heat pump has remote control or indoor temperature probe you can set the flow temperature according to your cooling needs.

I got a bunch of fan coils dirt cheap, I'm going to install them. So I'll need motorized valves and a bit of automation to send 18°C water to the slabs, and 7°C water to the fan coils, and set the heat pump flow temperature via Modbus. Not possible to do both at the same time, but slab cooling is good during the night, outside air is not hot so it works at fantastic efficiency, whereas air cooling is good during the day where photovoltaic electricity is free.

Note well water is oxygenated (so it'll corrode steel) and sometimes a bit acidic (so it'll corrode copper) so I hope it doesn't mix in your circuit...

To check the temperature inside the slab, I simply put a temperature probe on the return pipe from the slab. When the circulator is running, it gives good information about how much heat is stored inside the concrete, ie how much heat is going to come out in the next hours, so the control system (currently an ESP32) uses that. Basically it does weather compensation on the slab return temperature.

Thermostat is useful to stop the heat pump and avoid overheating if sunlight enters through the windows and warms the room. It depends on room solar gain so it depends on the building. If you have a lot of solar gain it's useful, otherwise meh. Thermostat as regulation for a slab... in control terms the slab has a ton of phase shift so if there's too much gain it will oscillate. Gain is simply deltaT between flow temperature and air. So if you heat the slab with 23°C water a thermostat can work. With 30-35°C water you get massive overshoot.

Re thermostat control.  I planned to use a slab temperature sensor wired to a thermostat to control that loop at about 24C in winter (lower floor) and a separate regulator thermostat for the air exchanger loop (both floors).  Both would be wired to signal the HP to run on demand.  Heat rises and my air always circulates .. so the slab would provide most of the houses heat.  I get how the HP should modulate using just the temperature curve alone .. but don’t you still need a way to turn it on/off once you get outside of the demand zone? (Heat gain from windows and “life” often increases winter house temperature above 20C).  The HP manufacturer appears to have designed their system using buffer temperature for this purpose (turning the HP on/off) .. my experiment would use the “manual” on/off terminals on the HP for this purpose and I’d leave the buffer terminals open. 

 That said, the system is designed to turn on/off based on buffer temperature .. so I’m not sure if the system will be ok with me using the home thermostats to turn it on/off.  I may have to just try it to find out.  It does seem to have inputs that appear to enable external on/off control. 

I doubt you will want to use (or be successful with using) the home thermostats, it will probably lead to oscillation because the slab will take so long to respond. 

Also remember that turning off the thermostat won't stop the slab emitting stored heat (you essentially have a storage heater but without adjustable output flaps, so you need to get the amount stored right in advance!)

More likely just use weather compensation.  Something perhaps to experiment with but I would definitely start with weather compensation only and no thermostatic control