docs: add heat rate noise section to temp control calibration

Documents the stirring-induced rate noise problem, its root cause
(post-differentiation filtering), and three mitigation options: pre-filter
before differentiating, sliding-window regression, and model-based rate
from Pot.p_pot (Smith only).

Co-Authored-By: Claude Sonnet 4.6 <noreply@anthropic.com>
Claude-Session: https://claude.ai/code/session_01NSo8R6GjoBQdStB3j67zSs

docs/temp_control_calibration.md

@@ -117,3 +117,88 @@ A replay panel added to `utils/analyze_log.py` would show:
 - `temp_ist` (blue) vs `sim_temp` (red dashed) — model accuracy at a glance
 - Residual `temp_ist − sim_temp` in a separate subplot — reveals systematic
   parameter errors vs noise floor
+
+
+## 5. Heat rate noise from stirring
+
+Stirring creates low-frequency temperature fluctuations at the sensor. The heat
+rate is computed in both controllers (`temp_controller_smith.py:71-74`,
+identical in `temp_controller.py`) as:
+
+```python
+heatrate = (self.theta_ist - self.last_theta_ist) / self.dt * 60
+alpha = 0.1
+self.heatrate_ist = (1-alpha) * self.heatrate_ist + alpha * heatrate
+```
+
+Two problems compound:
+
+1. **Differentiation amplifies noise.** A temperature fluctuation δT at
+   frequency f becomes a rate fluctuation of `2π·f·δT·60` K/min. Low-frequency
+   stirrer noise (e.g. 0.05 Hz, δT = 0.05 °C) → ≈ 0.9 K/min rate noise, the
+   same order as a typical `rate_soll`.
+
+2. **The IIR filter comes after differentiation.** With α=0.1 and dt=1 s the
+   time constant is τ = 9 s (cutoff ≈ 0.018 Hz). Filtering *after*
+   differentiating is the least effective arrangement: noise has already been
+   amplified before the filter sees it.
+
+The noisy `heatrate_err` feeds into `pid_heat.process()` as the proportional
+error. With `kp=0.08`, ±1 K/min rate ripple produces ±8% power output noise.
+
+### Options
+
+#### Option A — Pre-filter theta_ist before differentiating (best return for effort)
+
+Apply a separate, heavier IIR to the raw temperature *before* the derivative:
+
+```python
+beta = 0.05  # smaller = more smoothing, more lag
+self.theta_ist_filtered = (1-beta)*self.theta_ist_filtered + beta*self.theta_ist
+heatrate = (self.theta_ist_filtered - self.last_theta_ist_filtered) / self.dt * 60
+```
+
+This is fundamentally better than post-filtering: the derivative of a smooth
+signal stays smooth. `beta` can be tuned independently of `alpha`.
+
+#### Option B — Sliding-window linear regression
+
+Fit a line through the last N samples of `theta_ist` and use the slope:
+
+```python
+slope, _ = np.polyfit(t_window, theta_window, 1)
+heatrate_ist = slope * 60  # K/s → K/min
+```
+
+Gives the minimum-variance rate estimate for additive white noise, with a clean
+constant lag of N/2 seconds. N ≈ 20–30 s is a reasonable starting point.
+
+#### Option C — Model-based rate (Smith predictor only)
+
+`Pot.process()` already computes `p_pot = delayed_power_in − L·M·(temp − theta_amb)`
+and integrates `d(temp)/dt = p_pot / (M·C)`. This can be used directly:
+
+```python
+rate_model = self.model.get_p_pot() / (self.model.M * self.model.C) * 60
+```
+
+(`get_p_pot()` is not yet public on `Pot` but trivially added.) The rate is
+completely noise-free because it is derived from the heater power signal, not
+from differentiating the sensor. Accuracy depends on model calibration — see
+sections 1–2 above.
+
+#### Option D — Reduce kp of the Heat/Cool PID
+
+Halving `kp` from 0.08 to 0.04 halves the power noise at the cost of slower
+rate convergence. Treats the symptom rather than the source.
+
+### Recommendation
+
+Start with **option A**: one extra IIR state, localised to the rate
+computation, no other structural changes. Set `beta` so that
+`τ = (1−beta)/beta · dt` is 2–3× the dominant stirrer-noise period (visible
+as the ripple frequency in `rate_ist` in the log).
+
+For the Smith predictor, **option C** is the most principled long-term
+solution — it replaces a noisy numerical derivative with the model's own
+analytical rate, which the Smith predictor is already tracking.