Something I have recently become enamored with is the idea of modding Espresso machines with microcontrollers and digital measurements. Namely for the Gaggia Classic E24, which is an established boiler-based machine, a distinct ecosystem from my thermoblock-based Bambino Plus. Currently, there are two competing modifications that have captured my interest: the Gaggiuino and the GaggiMate.
Both projects utilize “fly-by-wire” control, using microcontrollers to close the loop on temperature, pressure, and flow. However, where this led into a parallel of my math side is the algorithms utilized for state estimation. Namely, the Gaggiuino implementation (and another experiment surrounding “predictive puck wetting”) leans heavily on a Kalman Filter.
The challenge they face is non-trivial, in that modeling the hydraulic resistance of a coffee puck is highly non-linear, especially when you account for the compression of gas bubbles in the boiler (acting as a capacitor in an RC circuit). Because of this, a standard linear filter often introduces phase lag. To solve this, we can apply an Extended Kalman Filter, utilizing a Jacobian matrix to linearize the system at each step. This allows the machine to use noisy pressure and heat values to predict flow rate before the sensor confirms it, effectively “hallucinating” the correct state to maintain consistency.
While I will most likely choose the GaggiMate for its integrations with Home Assistant and support for a proper scale like the BOOKOO Themis mini coffee scale, the mathematical rabbit hole of the Gaggiuino sent me back into control theory.
Revisiting the subject, I found some really interesting things. For one, Kalman filters parallel Predictive Coding and Free Energy Principle implementations exceptionally well. We can think of them as a Recursive Bayesian Estimator.
In terms of neuroscience, a Kalman filter is essentially a mechanistic implementation of the Bayesian Brain We have our Internal Model(Prior), which is the brain projecting the state forward based on physics (also known as process model). It predicts where the ball should be, ignoring the noise of the moment. Next is the Sensory Input (Likelihood), which is like a human eye; providing discrete, noisy measurements. Last is the Update (Posterior), which integrates these two based on their relative uncertainty.
My big parallel came with understanding the Kalman Gain . In the world of neuro, functions almost as the Synaptic Precision or sensory weighting. If sensory noise () is high, drops, and the brain relies on its Internal Model (top-down processing). If the Internal Model is uncertain (high Process Noise ), increases, and the brain relies on the immediate sensory data (bottom-up processing).
The filter works by minimizing innovation (the difference between the measurement and the prediction: ). This is mathematically synonymous with minimizing Surprise or Free Energy. I hadn’t considered the Kalman filter to be of particular importance to my Computational Neuroscience self-study, but it provides a rigid, mathematical framework for how biological agents might balance Process Noise against Measurement Noise.
For now though, I am curious to see if this recursive Bayesian estimation can help me pull a slightly better shot of Espresso.