Online Stability Indicator (OSI) During RTHS

We call the experimentally identified plant $H(s,k)$. It maps the internal PID output $v$ to the measured response $y$, while $k$ identifies the current stiffness condition between 7k and 5k.

The PID and plant multiply to form the open-loop transfer function. The adaptive factor $\alpha\in[0.02,1]$ scales the controller’s influence; $\alpha=1.00$ means full PID gain.

Closing the negative-feedback loop gives the command → measurement response:

Let $u[n]$ be command displacement and $y[n]$ be measured displacement. A Hann window $w[n]$ reduces edge discontinuities before the FFT:

At time step $n$, the instantaneous input-power, output-power, and cross-power terms are

One window is noisy, so each spectrum is exponentially smoothed. An overbar denotes a smoothed power or cross-power estimate; $S$ is reserved for the feedback sensitivity function.

Magnitude-squared coherence asks: “At this frequency, how consistently does the input explain the output?”

Coherence alone is insufficient, so OSI gates it with input leverage for two reasons:

1. Feedback-created coherence. At frequencies with high loop gain, measurement noise can drive the feedback loop and create high coherence between feedback-related signals even when the intended external excitation is absent. The resulting relationship is driven mainly by noise and can produce a misleading transfer-function estimate.
2. Division by weak input power. Forming a transfer-function ratio requires division by the input spectrum. When $\bar p_{uu}$ is very small, numerical roundoff, measurement noise, spectral leakage, and ordinary estimation fluctuations can be greatly magnified.

The leverage factor therefore reduces confidence in frequency bins that do not contain meaningful input power. It cannot by itself distinguish strong feedback-correlated noise from intentional excitation.

First define the relative input power $p_{rel}(\omega)$: the power in the current frequency bin divided by the power in the strongest bin.

Next, $p_{floor}$ is a small positive floor ($10^{-3}$ here) that prevents tiny input values from receiving high trust. The input leverage $\ell(\omega)$ maps relative power to a number between 0 and 1:

Finally, $A$ is one activity score for the entire time window: $A=0$ means the command is quiet and $A=1$ means it is strongly driven. Stage 06 gives its exact RMS-based definition. The per-frequency confidence $c(\omega)$ combines activity, coherence, and leverage:

The spectra now provide the raw measured closed-loop estimate $\widehat T(\omega)$:

OSI also forms a model prior. It scales the healthy loop $L_0(\omega)$ by the currently applied adaptive gain $\alpha$, then converts that loop to its closed-loop response $T_{prior}(\omega)$. Here $\alpha$ is the applied a_cmd value: $\alpha=1.00$ means full PID gain, and it changes only if the separate adaptive-gain controller changes a_cmd. The estimator clamps the received value to $0.02\le\alpha\le1$:

The confidence $c(\omega)$ chooses how far the usable response $\widehat T_{use}(\omega)$ moves from the model prior toward the measured estimate. The final symbol $\widehat L(\omega)$ is the reconstructed open loop:

The closed-loop denominator is $1+L$. Instability becomes possible when $L(j\omega)$ approaches $-1$, because then $1+L\to0$.

Define $b(\omega,\alpha,k)$ as the normalized distance from that critical condition at frequency $\omega$, adaptive gain $\alpha$, and plant condition $k$:

Let $\Omega$ be the usable 2–100 Hz band after excluding the 58–62 Hz line-noise bins. The operator $\operatorname{median}_3$ is a three-bin median filter that suppresses isolated spikes. The identification branch keeps its margin as a curve over $\Omega$:

Let $r=1,2,3$ index the first three non-overlapping 0.5 s reference windows, which are assumed to represent the healthy loop. For window $r$, $y_r[n]\equiv d_{m,r}[n]$ is its measured-displacement segment and $Y_r(\omega)$ is the FFT of that segment after applying the estimator window $w[n]$. The median reference power is denoted by $p_{yy,\mathrm{ref}}(\omega)$:

For every later window, the observed ratio $q_{obs}(\omega)$ divides the current smoothed output power by that fixed healthy-reference power:

OSI represents damage-like loop growth with a frequency-dependent scale $g(\omega)$. For each of 25 candidate values between 0.5 and 6, the modeled output power $p_{yy,\mathrm{model}}(\omega,g)$ is normalized by the same healthy-reference power used in $q_{obs}(\omega)$. The resulting modeled ratio is $q_{model}(\omega,g)$:

The remembered fitted curve is updated in log space. The same recurrence is used for damage-evidence accumulation and recovery, with a different update gain for each case:

The residual margin $b_{res}(\omega)$ is the chordal margin predicted by the frequency-dependent loop-growth hypothesis, kept as a curve just as $b_{st}(\omega)$ is. A three-bin moving median is applied across adjacent frequency bins:

The 0.5 s command RMS defines the binary activity $A$ directly. The approved threshold is $\eta_A=0.0575$, approximately $4.54\sigma_{\mathrm{noise},u}$ based on the clean quiet-record estimate:

Activity is no longer a regime switch that selects one scalar margin or the other. It enters only where it belongs: inside the confidence $c(\omega)$ of the identification branch, and in the fast drive-ratio detector of the residual branch.

Two independent kinds of evidence exist at every bin, and each one has to pay for the claim it makes. The first is the confidence $c(\omega)$ already built in Section 03: command energy and coherence say the loop is directly visible at this frequency, so the identified margin $b_{st}(\omega)$ is worth believing.

The second is a detection weight $d(\omega)$. The fitted loop growth $g(\omega)$ only means something once it clears the resolution of the $g$ fit itself; below the threshold $g_{det}=1.75$ the deviation carries little residual evidence. Where $g$ does clear it, the residual model $b_{res}(\omega)$ is worth believing to the extent the loop is not directly visible:

The two evidence weights never sum to more than one. Whatever is left over is the prior weight $w_{prior}$: the share of the bin that neither branch can account for, and which therefore falls back on the current authority-scaled commissioning prior $b_{prior}(\omega)$.

Carrying the unsupported share on the prior, rather than on the ceiling, is what makes the reported value an estimate of the loop margin instead of treating “unknown” as the largest possible margin.

After forming $b(\omega)$ at every retained frequency, the estimator trims the single smallest value and selects the second-smallest bin at $\omega_*$. That selected value is the scalar instantaneous margin $b_i[n]$:

For the configured sampling rate $f_s=2048\,\mathrm{Hz}$ and 0.5 s moving window, each FFT uses 1024 samples and has frequency spacing $\Delta f=f_s/1024=2\,\mathrm{Hz}$. The 2–100 Hz band therefore contains 50 bins; rejecting 58, 60, and 62 Hz leaves 47 retained bins. Thus $b_{(2)}$ is the second of 47 sorted margins, with empirical cumulative rank $2/47\approx4.3\%$—approximately the 4th percentile—and only the single lowest margin across the retained frequency bins is discarded.

The reported scalar margin is a first-order lag on $b_i[n]$, with $\eta_b$ the fraction of the new reading admitted at each update:

Recovery

Recovery uses the median observed power ratio over $\Omega$:

When $r_{\Omega}<1$, the remembered growth $g$ decays toward the healthy value $g=1$ using the recovery gain:

Each qualifying update applies the log-domain recurrence of Section 05 with the recovery correction toward $g=1$; otherwise the remembered value remains unchanged.

The adaptive factor uniformly scales the inner open loop:

Because a positive scale changes magnitude but not phase, the $-180^\circ$ crossover frequency stays approximately fixed. The gain margin changes in closed form:

Let $GM_{target}$ be the desired gain margin. The scale required to meet that target is $\alpha_{required}(k)$:

The adaptive law keeps $\alpha$ between the permitted minimum $\alpha_{min}$ and 1. When damage raises loop gain and $b$ falls, reducing $\alpha$ can restore separation from the critical condition. Recovery should be slower and evidence-based so one optimistic window cannot rush the adaptive factor upward.

The measured displacement is drawn once per implementation, because in every closed-loop test the authority factor scales the command that drives the plant: where the two traces separate, the implementations have driven the specimen differently. Dashed vertical marks give the plant blend used by the synthetic scenarios, where $\lambda=0$ is the identified 7k model and $\lambda=1$ the 5k model. Where a test provides ground truth, the true chordal margin is drawn dashed in each implementation's own colour, since it depends on the authority that implementation applied. The diagnostic gain, where the record carries one, is drawn against the right-hand axis of the authority panel. Click any legend entry to hide or restore that trace; the vertical scales follow what is left visible, so a single implementation can be read out of a crowded plot.

Scorecard

Read directly from the saved Fabby Bench 2 results, so this table cannot disagree with the benchmark that produced it. A raw composite is reported only when all seven gates pass; the failing implemented method is shown for reference and marked as gated out. Hover or tab onto any metric name for what it measures, whether high or low is the good direction, and which test in the dropdown above to look at.