I have slapped together few components to create a working prototype, STM32 Nucleo board with STM32L452 MCU, Texas Instruments development board with Bluetooth 5, some RTD temperature measurement development boards from MaximIntegrated, a bunch of relays, current sensors, water flow sensor from Digmesa (actually replacement part from one of Saeco machines), etc.

I am using a test board to detect existing espresso machine operation and perform readings using multichannel laboratory thermometer that measures external boiler case temperature, group head temperature, brewing water temperature (inside portafilter), tank water temperature and external temperature along with signals (when the brewing starts, stops, etc.) to provide experimental data to build the model.

I am using Matlab and Simulink to help me generate parts of the software using gathered experimental data. This lets me play with models and observe how it could behave without actually brewing any coffee:)

Predictors are often too hard in practice, but in this case seems simple as looks like you have the parameters you need to calculate the required energy as Q=ml (+losses) and just put exactly that amount of electrical joules into the heating element.

"Tune" factors like thermal losses by hand, and/or have a simple heuristic learning algo that tunes factors for you based on previous result errors.

I am assuming you have the level in the tank as well to calculate volume, as how else do you protect the boiler element from coming on when no water in the tank?

Predictor calculates future evolution of the system based on current state and system parameters. The horizon changes dynamically and typically is about 5 seconds when in steady state and up multiple minutes when cooling from steaming to brewing temperature (depends heavily on insulation, steaming temperature, how long it was steamed, etc.)

The parameters are not directly physical values like thermal mass or energy but are related to them. For example, I am calculating the amount of energy lost based on water temperature and outside temperature (outside is inside enclosure). This is in units used to drive the heater (the unit is half-cycle of AC power). I am calculating (predicting) cooling of water when it is pumped based on amount of water pumped and temperature of water in the tank (this is also being measured).

I calculate initial values based on step responses and then use moving horizon estimator to continually test past predictions vs actual readings and adjust parameters to match.

I haven't yet analyzed which parameters are truly important and which can be easily compensated for without having to sense them.

As to the level of water in the I could not find suitable sensor so I built one. It can only give low level alarm (water falling below set level). It is a plastic pipe with floating magnet attach to the inside of the tank. On the outside, not mounted to the tank so that I can freely remove the tank, but pressed with a flat spring are two reed switches. Using those reed switches I can detect when water falls when its being used or when it rises when it is replenished.

Also sorry, I should have said Q=mcdelta(t) in my post.