![\"Steel](\"figures/fig_setup.jpg\")
Hybrid simulation relies on the transfer system to impose the computed displacements on the specimen. When the outer controller loop loses stability, the actuator drives the specimen beyond its target displacements. Such instability creates safety hazards for laboratory personnel and can also damage the specimen and expensive equipment such as the actuator and servo-valve. Instability then necessitates early termination of the test, and the experiment is lost before the planned loading is completed. Because the specimen may already be damaged, the test must be repeated with a new specimen. The dynamics of the experiment are not fixed, because the specimen itself can change during the test. Maghareh et al. [2] noted that a failure in part of the physical substructure can drastically change its dynamics. They also noted that such a change affects the safety and integrity of the experiment.
\nGomaa et al. [1] showed that specimen degradation can cause this instability even when the outer controller is properly tuned for the initial state. As the specimen degrades, its stiffness decreases and the same controller produces a larger response. This results in gain amplification in the outer controller loop, which reduces its gain margin. Conventional time-delay-based margins can miss this mechanism, so Gomaa et al. [1] proposed the NCF online stability indicator (NCF OSI) based on the NCF margin. This finding encourages the use of the NCF margin in this study. However, that finding was obtained from a virtual hybrid simulation, so this section examines whether the same mechanism appears in a physical experiment.
\nFigure 1 shows the experimental setup, in which the specimen is a steel column representing the physical substructure. Its base is anchored to a concrete block with four Hilti anchors, and its top is connected to the actuator. An MTS hydraulic actuator with a capacity of 10 kip imposes the lateral displacement at the top of the column. The internal LVDT of the actuator measures this displacement, which serves as the feedback signal of the outer controller loop. A Speedgoat real-time target runs the outer controller and sends the command to the actuator at 2048 Hz.
\nThe goal of the tests is to compare the outer controller loop before and after damage. The plant is first identified in the initial state, while the anchorage is undamaged and the specimen has its initial stiffness. The specimen is then pushed laterally until the concrete around the anchors cracks and the base loses part of its rotational stiffness. Finally, the plant of the degraded specimen is identified, and Section 2.3 compares the two states.
\nNo numerical substructure is involved in the tests of this section, and the measured force is therefore not fed back. The reference is a prescribed displacement, so only the outer controller loop is closed. As Figure 2 shows, the outer controller acts on the tracking error and sends the command to the plant . The measured displacement is then fed back to the controller, which closes the loop with unity feedback.
\nThe plant was experimentally identified before and after the anchorage damage, and the two identified plants are denoted and . In each state, band-limited white noise up to 80 Hz was applied through the loop of Figure 2 with the same PID controller . A parametric model was then fitted to the transfer function from the command to the measured displacement . In the initial state, the lateral stiffness of the specimen at the actuator control point was MN/m. After the anchorage was damaged, the lateral stiffness fell to 5.38 MN/m, or , which is a 25% loss.
\nIn the loop of Figure 2, the controller is the same in both states, and only the specimen changes. The open-loop transfer function is therefore in the initial state and in the damaged state. The chordal distance between and the critical point is
\nand its minimum over frequency is the NCF margin, presented by Vinnicombe [3] as the generalized stability margin. Figure 3 compares the experimentally identified frequency responses of the two states and their chordal distances.
\n![\"Open-loop](\"figures/fig_open_loop.svg\")
The 25% loss of lateral stiffness raised the loop gain by a factor of about 2.4. However, the phase of the loop changed only slightly, as Figure 3 shows. The loss of stiffness therefore mainly amplifies the loop gain rather than adding phase lag. Consequently, the NCF margin falls from 0.59 at 35 Hz in the initial state to 0.34 at 39 Hz in the damaged state, below the 0.4 guideline. These trends agree with the gain-amplification mechanism and the guideline proposed by Gomaa et al. [1], so the mechanism also appears in a physical experiment.
\nThese results expose a dilemma for a fixed outer controller, which must be tuned before the experiment begins. A controller tuned for the initial state becomes too aggressive once the specimen degrades, as Section 2.3 showed. Conversely, a controller tuned for the damaged state can remain stable but compromises tracking performance in the initial state. The specimen state is also highly uncertain during the experiment, so the appropriate tuning cannot be chosen in advance. The stability of the outer controller loop must therefore be quantified online based on the observed measurements. Updating the controller accordingly is then required so the experiment is completed safely and successfully. Section 3 presents a framework that meets both requirements by combining an online stability indicator with an adaptive controller.
\nThe NCF OSI introduced in Section 2.1 quantifies the robustness of the closed-loop system as the physical plant changes during the experiment. This study extends the analytical results to experimental implementation, discusses the challenges that arise, and proposes ways to address them. The proposed framework comprises two parts: an online estimator of the indicator and an adaptive controller that scales the outer controller output according to the estimated indicator. The indicator is estimated during the experiment from the desired displacement calculated by the numerical substructure and the measured actuator displacement. To this end, three estimator branches are combined at each frequency, each weighted by the confidence in its output. The first is direct identification, , which estimates the closed-loop transfer function of the outer controller loop from the desired displacement to the measured actuator displacement , with the cross-spectral estimator [4]. The second is output-power inference, , which can detect loop amplification at frequencies that cannot be reliably identified. This branch compares the output power measured during the experiment against the reference recorded before the excitation. The third is the prior identification of the initial state, , which serves as a fallback when insufficient information is available. The adaptive controller then uses the estimated indicator to scale the controller output by the adaptive factor , keeping the loop stable with an adequate margin as the specimen degrades while retaining as much tracking performance as possible. This section presents the framework first and then its calibration against cases with a known margin.
\nIn the hybrid simulation of Figure 4, the numerical substructure computes the desired displacement from the ground acceleration and measured force . To compensate for transfer-system lag and improve tracking, a phase-lead compensator receives and produces the reference . The outer controller takes the tracking error , where is the measured actuator displacement. It scales its output by the adaptive factor to produce the command sent to the plant .
\nThe plant comprises the specimen of stiffness and the transfer system, which includes the inner-loop controller, servo-valve, and actuator. The control–structure interaction discussed by Dyke et al. [5] is also accounted for in the transfer-system dynamics. The measured force is fed back to the numerical substructure and closes the numerical substructure loop. Similarly, the measured displacement is fed back to the outer controller and closes the outer controller loop.
\nThe NCF OSI block runs alongside the outer-controller loop and does not alter the signals within it. It collects the reference and the measured displacement over a moving window and estimates the NCF margin of the loop at each update. The adaptive law then compares this estimate with a target margin and updates the adaptive factor . A lower margin reduces to restore stability, while a sufficient margin lets recover toward one to retain tracking performance.
\nThe plant in Figure 4 is characterized by the transfer function from the command to the measured actuator displacement . The open-loop transfer function of the outer controller loop and its closed-loop response from to are
\nHere , and are the Laplace transforms of , and , respectively. The NCF stability margin of the outer controller loop is evaluated for the loop under unity feedback [6]:
\nAs the specimen changes during a hybrid simulation, the loop changes with it, and so does its margin. Equation (4) could be applied directly only if were identified with high confidence at every frequency, but online measurements rarely support this. The NCF OSI is therefore estimated from three complementary branches, which the following sections develop in turn. These are direct identification , output-power inference and the prior identification of the initial state .
\nIdentifying the closed-loop transfer function requires the autospectrum of reference and the cross spectrum of measured actuator displacement with . The autospectrum of is also needed to calculate coherence, which helps set the identification weight at each frequency. A moving window lets these spectra follow changes in the specimen during the experiment instead of averaging over the whole record.
\nEach window contains samples of and , where is the sampling frequency and is the window duration. Updates occur at intervals , so consecutive windows overlap when and share part of their samples. With update index and sample index within each window, the Hann-windowed discrete Fourier transforms are
\nwhere and are the samples of window , and is the angular frequency of bin . The samples of each window are weighted by the Hann window presented by Blackman and Tukey [7]:
\nWelch [8] proposed estimating the power spectrum by averaging, with equal weight, the squared Fourier magnitudes of overlapping windowed segments of a record. Cross spectra are estimated in the same way from products of paired Fourier transforms, with one transform conjugated. In the present study, the plant changes as the specimen degrades, so recent windows should carry more weight than earlier ones. Each window therefore provides one raw estimate of the autospectra of and and the cross spectrum of with :
\nwhere the superscript denotes the complex conjugate, and the subscript marks the raw estimate of window . These raw estimates are averaged with the geometric moving average proposed by Roberts [9], which updates an average from its previous value and the newest observation. Earlier spectra are retained with contributions that decay as new windows arrive, which reduces scatter between updates while still following changes in the plant. The first complete window initializes the average, and at each subsequent update weights the new spectrum and weights the previous average:
\nThe NCF OSI is evaluated over , the set of frequencies considered in the analysis.
\nAt each selected frequency, the loop response can be identified directly from the reference and the measured response . The averaged spectra give the estimate of the closed-loop frequency response [4] and the magnitude-squared coherence
\nInverting Equation (3) gives the identified loop, which is the open-loop transfer function estimated from the measured closed-loop response:
\nHere, the identified loop is smoothed with the running median over neighbouring frequencies, as proposed by Di Matteo et al. [10], so the median is taken over each frequency and its two neighbours. With three values, a single outlying bin cannot set the median, and the smoothed loop is given by
\nThe identified margin then follows from Equation (4), with the smoothed loop in place of the unknown loop :
\nAt each update, however, the estimate is uncertain at any frequency that the command does not excite sufficiently. The measured response at that frequency then stays close to the sensor noise and carries little information.
\nEach bin therefore receives an identification weight between zero and one, built from three factors. The first is command activity , which checks from the root mean square of the window that the command rises above the noise:
\nwhere is the root mean square of the command samples in the considered window , is the standard deviation of the command noise measured in a quiet record, and is a hyperparameter to be tuned. The second factor is the input leverage , which reduces the weight of bins that carry little command energy. To measure this, the command power in each bin is first normalized by its maximum over :
\nThe raw leverage then sets the weight of each bin through the threshold , expressed as a fraction of the peak command power. A bin with well above should receive a weight close to one, and a bin well below it a weight close to zero:
\nwhich changes smoothly between these limits and equals one half when , that is, when the bin carries times the peak command power. The leverage is relative to the peak bin, so is still needed to reject a command that is weak at all frequencies. The third factor is the coherence from Equation (15), which multiplies the activity and the leverage to complete the weight
\nThis weight sets the share of direct identification in the per-frequency fusion of the three branches in Section 3.1.3.
\nWhere the command carries little energy, the identification weight falls toward zero, and direct identification then provides little evidence of the margin. A degrading specimen can still reduce the margin of the outer controller loop at these frequencies, so a second source of evidence is needed. In this case, the output power still carries evidence, because the outer controller loop shapes every signal that enters it, including sensor noise. Wu and Wang [11] used the active-power deviation and its rate of change to detect the positive-feedback mode of a virtual synchronous generator. The output-power inference branch is inspired by this approach and uses the change in output power as evidence of a reduced margin. To quantify this change at each frequency, the averaged output power is compared with its reference from the initial state:
\nwhere is the observed power ratio at window , which measures the change in output power at each frequency relative to the initial state. The averaged autospectrum of is from Equation (12), and is the reference autospectrum of recorded in the initial state.
\nThe output-power branch detects a change by comparing the present output power with a reference from the initial state. A single window of output power fluctuates strongly, so a reference taken from one window could bias every later comparison. The reference autospectrum is therefore the median of three raw autospectra recorded in the initial state. These autospectra are taken from the first three non-overlapping windows of the experiment, with :
\nwhere , and are the corresponding updates, and the median discards one unusually loud or quiet window.
\nAccording to the experimental verification presented in Section 2.3, the change due to specimen degradation can be represented by an amplification of the open-loop gain. Consequently, a candidate loop with amplification predicts the power ratio relative to the initial state
\nwhere is the predicted power ratio, and is the loop of the prior identification of the initial state. The amplification is then selected as the value that minimizes the logarithmic difference between the observed and predicted ratios:
\nThe amplification is reliable only when the observed power ratio reflects a change in the loop. At weakly excited frequencies, however, the output consists mainly of sensor noise, and its averaged power scatters because it is estimated from few windows. The reference is the median of only three windows, so it is equally uncertain at these frequencies. Consequently, can exceed one for an unchanged specimen, and the margin of this branch would then fall without any degradation.
\nSuch a false amplification must be separated from a real one before it enters the margin estimate. Direct identification cannot resolve the loop at these frequencies, but it remains reliable at the excited frequencies of the same window. Degradation changes the plant, so it also increases the magnitude of the identified loop at the excited frequencies. Section 2.3 observed such an increase by a factor of about 2.4 in the damaged specimen. An unchanged specimen and a degraded one therefore leave different traces in the measurements. For an unchanged specimen, a large at an unexcited frequency appears without any increase of at the excited frequencies. For a degraded specimen, the output power and the identified loop increase together. The amplification inferred from output power is therefore trusted only to the extent that the identified loop confirms it.
\nConfirmation requires a single measure of how much the identified loop exceeds the loop expected without degradation. This expected loop is , the prior loop scaled by the current adaptive factor. Including keeps a reduction of the adaptive factor from being mistaken for a change in the specimen. The loop-increase factor is the real scale that best fits to the identified loop in the weighted least-squares sense:
\nEach frequency is weighted by from Equation (22), so the well-identified excited frequencies dominate the fit and the weakly excited ones contribute little. The fit uses the identified loop from Equation (16), taken before the median smoothing of Equation (17). The scale is kept real, consistent with the real amplification in Equation (25). Setting the derivative of the sum with respect to to zero then gives the closed form
\nFor an unchanged specimen, agrees with at the excited frequencies and . A degraded specimen instead raises the identified loop at these frequencies and drives above one. The confirmation converts this loop increase into the trust given to the amplification inferred from output power. It rises linearly with and is clipped between a lower bound and one:
\nOnce reaches the confirmation threshold , the loop increase confirms the amplification fully and . The lower bound keeps some trust in this branch even when no loop increase is identified. It accounts for amplification that cannot be observed at any excited frequency, such as amplification from unmodeled dynamics or higher-mode effects. Because is a single value at each update, the same confirmation applies at every frequency.
\nThe confirmation acts by pulling the inferred amplification toward one, which represents an unchanged loop, and gives the accepted amplification
\nWith full confirmation, , and the amplification inferred from output power is accepted unchanged. For an unchanged specimen, , so a false value of is accepted only as .
\nThe accepted amplification still carries the scatter of and therefore varies between updates. To reduce this scatter while following a degrading specimen, successive accepted amplifications are averaged recursively, as the spectra were in Equations (11)–(13). The average is formed on a logarithmic scale, consistent with the logarithmic difference in Equation (26):
\nThe smoothed amplification acts as the memory of this branch and holds a detected amplification while the specimen remains degraded. The averaging factor sets how strongly each new accepted amplification updates this memory.
\nThe same memory, however, slows the release of an amplification that the observed output no longer supports. With Equation (31) alone, each update moves the remembered value toward the newest one by only the fraction . An outdated amplification would therefore keep the margin of this branch low after the loop has recovered, and the output power shows when this occurs. A median of over below one means that the output power has fallen below its initial reference at most frequencies. In that case, the remembered amplification is released by a recovery step in place of Equation (31):
\nThe recovery factor , with , moves the smoothed amplification gradually toward one. The release stops once the median ratio returns to one, and Equation (31) then resumes. As a result, the output power itself limits how far the amplification is released as the loop recovers.
\nThe smoothed amplification then scales the prior loop, and Equation (4) gives the margin of the output-power branch:
\nFor an unchanged specimen, stays close to one and stays close to the margin of the prior loop. A confirmed amplification instead lowers at the affected frequencies, including frequencies that direct identification cannot resolve.
\nThe confirmation also sets how much this margin counts in the fusion of Section 3.1.3. Direct identification takes the share , and the output-power branch receives the remaining share in proportion to the confirmation:
\nConsequently, a confirmed amplification dominates the frequencies that direct identification cannot resolve, whereas an unconfirmed one receives only the small weight . The rest of that share is left for the third branch, the prior identification of the initial state.
\nSections 3.1.1 and 3.1.2 give two measured margins, and each is supported by evidence only under certain conditions. Direct identification needs an excited command, and output-power inference needs a loop increase that confirms the inferred amplification. At a bin where neither condition holds, a margin is still required, so the prior identification of the initial state supplies it. This third branch applies Equation (4) to the initial loop scaled by the current adaptive factor:
\nBecause is identified before the experiment, is available at every bin and at every update. However, this margin assumes that the specimen has not changed since the prior identification, so it serves only as the default estimate.
\nEach branch evaluates the chordal distance of Equation (1) for its own loop at every bin, before the infimum of Equation (4) is taken. These per-bin values, , and , form three margin curves over . The fusion combines the three curves bin by bin, and each weight should reflect the evidence available at that bin. The identification weight first sets the share of direct identification, and the confirmation then divides the remaining share between output-power inference and the prior. Output-power inference receives from Equation (34), and the prior receives the share that remains at each bin:
\nWith this split, the three weights add up to one at every bin. The fused margin curve is therefore a weighted average of the three branch margins:
\nThree limiting cases show how Equation (37) moves between the branches as the evidence at a bin changes. At a well-excited bin with high coherence, approaches one, and the fused margin follows the identified margin. At an unexcited bin, approaches zero, and the confirmation then decides between the two remaining branches. A loop increase that reaches gives , so the fused margin follows the output-power margin. Without an identified loop increase, stays at its lower bound , so the prior carries most of the weight.
\nThe same weights also handle an idle command, which excites no bin, without any special treatment. The activity from Equation (19) is then zero, so at every bin and no loop increase can be identified. The confirmation therefore falls to , and the fused margin returns mostly to the prior evaluated with the current adaptive factor. A remembered amplification in still lowers the fused margin, but only through the small weight .
\nFusing bin by bin, before any reduction over frequency, keeps each weight attached to the frequency where its evidence was obtained. Reducing each branch to its own minimum first would instead combine margins located at different frequencies, with weights taken from other frequencies. Because the adaptive controller needs a single value at each update, the fused curve is reduced to its minimum over , as in Equation (4):
\nThe NCF OSI margin is the only quantity that the adaptive law of Section 3.1.4 receives from the estimator. Figure 5 summarizes one complete update, from the measured signals to this margin and the adaptive factor that it sets.
\nThe NCF OSI margin indicates whether the outer controller loop keeps enough robustness, and the adaptive factor provides the means to restore it. Section 2.3 showed that specimen degradation mainly amplifies the loop gain, and scaling the controller output by offsets this amplification in Equation (2). For example, a loop amplified by a factor regains its initial gain when . Full cancellation is not required, however, because the loop only needs to regain the target margin. Every reduction beyond the target margin weakens the corrective control action and compromises the tracking performance. The adaptive law therefore reduces only as far as the margin requires, based on the margin error at update :
\nHere is the target NCF margin listed in Table 1, and a negative error indicates insufficient margin that calls for a smaller adaptive factor. Conversely, a positive error indicates spare margin, which allows to recover toward one and restore tracking performance.
\nReducing and restoring the adaptive factor carry different risks, which depend on how quickly each change is made. A slow reduction leaves the loop exposed to instability for as long as the margin remains insufficient. Conversely, a fast recovery can raise back toward instability before the margin estimate settles after a reduction. The adaptive factor is therefore reduced and restored at different rates, depending on the sign of the margin error:
\nHere is the adaptation gain, and is the ratio of the recovery rate to the reduction rate. Each update thus adds to , or when the margin is sufficient. The adaptive factor therefore keeps changing until the margin reaches the target or reaches a bound. With the values in Table 1, a margin 0.1 below the target reduces by about 6% per update. A margin 0.1 above the target raises by only about 2%, so recovery is about three times slower than reduction. The adaptive factor stays between the smallest permitted value and one, which corresponds to the full controller output.
\nThe updated factor scales the controller output during the next update interval and also enters Equation (25), Equation (28), Equation (33) and Equation (35). Consequently, the next margin estimate already accounts for the controller output reduced or restored at this update.
\nThe equations of Section 3.1 fix the structure of the estimator and the adaptive law, but they leave several numerical values open. These values include the averaging factors, the evidence thresholds and the rates of the adaptive law. Each value balances two competing needs, as the spectral averaging factor of Section 3.1.1 illustrates. A larger lets the spectra follow a degrading specimen sooner, but it also leaves more scatter between updates. Calibration selects these values so that the indicator follows the true margin and the adaptive factor holds the loop near the target margin.
\nTable 1 lists the 14 parameters of the framework with their values and roles, and six of them are design values. These design values follow from the test setup and the analysis, so they are set directly rather than calibrated. The window duration of 0.5 s gives a frequency resolution of 2 Hz, which also sets the lower bound of . The upper bound of 80 Hz matches the band over which the plants were identified in Section 2.3. The update interval of 0.125 s sets how often the indicator and the adaptive factor are updated. The target follows the guideline proposed by Gomaa et al. [1], and keeps part of the controller output active. Finally, the minimum confirmation is the safety minimum introduced in Section 3.1.2 for amplification that cannot be confirmed.
\nThe remaining eight parameters are hyperparameters, which are marked as calibrated in Table 1 and are tuned in this section. They set how quickly the indicator follows a change, how much evidence each branch requires and how strongly the adaptive factor responds. Their suitable values depend on the laboratory, the equipment, the specimen and its material, which differ between experiments. They therefore cannot be fixed from theory alone and must be calibrated against cases in which the true margin is known.
\n| Symbol | Value | Type | Role | |
|---|---|---|---|---|
| 0.125 s | — | Design | Update interval | |
| , | 0.5 s, 1024 | — | Design | Window length at Hz |
| 2–80 Hz, 37 bins | — | Design | Analysis set, 58–62 Hz rejected | |
| 0.39 | 0.25 s | Calibrated | Spectral averaging, Equations (11)–(13) | |
| 4.54 | — | Calibrated | Activity threshold relative to command noise, Equation (19) | |
| — | Calibrated | Input-leverage threshold, Equation (21) | ||
| 0.1 | — | Design | Minimum confirmation, Equation (29) | |
| 1.5 | — | Calibrated | Confirmation threshold, Equation (29) | |
| 0.12 | 1.0 s | Calibrated | Output-power memory, accumulation, Equation (31) | |
| 0.73 | 0.4 s | Calibrated | Recovery factor per update, Equation (32) | |
| 0.40 | — | Design | Target NCF margin, Equation (39) | |
| 0.625 | — | Calibrated | Adaptation gain per update, Equation (40) | |
| 0.32 | — | Calibrated | Ratio of the recovery rate to the reduction rate of | |
| 0.20 | — | Design | Smallest adaptive factor, Equation (40) |
A known margin requires a plant that is known at every instant, including while the specimen degrades. The two plants and identified in Section 2.3 meet this requirement for the column specimen of this study. To impose damage numerically, both plants receive the same command , and their outputs are mixed:
\nwhere and are the outputs of and , and is the damage index. Thus represents the initial state, and represents the damaged state identified in Section 2.3. While is constant, superposition makes this mixture equivalent to the single linear plant driven by . The true margin at each update therefore follows from Equation (4) applied to the loop . This true margin is computed independently of the estimator, so it serves as the benchmark for judging . The mixture interpolates the responses of the two states and does not represent an intermediate stiffness.
\nThe calibration cases are built from three features, and each feature targets one part of the framework. The first feature is the damage history, which tests how quickly the indicator follows a change over a case duration of 120 s. Sudden damage steps from zero to one at 40 s and back to zero at 80 s. Gradual damage ramps up between 20 and 50 s and back down between 70 and 100 s. A damaged specimen does not regain its stiffness, but the return to reproduces a loop that regains its margin. This return therefore tests the recovery step of Section 3.1.2, which releases the remembered amplification.
\nThe second feature is the excitation band of the reference , which determines which branch can observe the change. In both bands, the reference is band-limited white noise with an RMS of 0.15 mm. Sensor noise with a standard deviation of 0.05 mm is added to the measured displacement in every case. Wideband excitation covers 2 to 80 Hz and therefore excites the frequencies of the minimum margin, between 35 and 39 Hz in Section 2.3. The wideband cases therefore test direct identification, and they set its spectral averaging , activity threshold and leverage threshold . Narrowband excitation covers only 2 to 10 Hz, so direct identification cannot observe the change near the minimum margin. Only output-power inference can then detect the reduced margin, and the narrowband cases set its hyperparameters. Their damage onset sets the confirmation threshold and the averaging factor , and their return sets the recovery factor .
\nThe third feature is the adaptive factor, which is either fixed at one or active during the case. With fixed, the cases test whether the indicator alone estimates the margin correctly as the damage changes. With active, the same cases test whether the loop is held near the target while tracking is retained. These active cases set the adaptation gain and the ratio of the recovery rate to the reduction rate. The fixed and active versions of each case use the same reference and sensor-noise records, so their differences come from alone. Table 2 lists the resulting eight cases, which cover every combination of the three features.
\n| Case | Damage history | Excitation | Adaptive factor |
|---|---|---|---|
| 1 | Sudden | Wideband, 2–80 Hz | Fixed, |
| 2 | Gradual | Wideband, 2–80 Hz | Fixed, |
| 3 | Sudden | Narrowband, 2–10 Hz | Fixed, |
| 4 | Gradual | Narrowband, 2–10 Hz | Fixed, |
| 5 | Sudden | Wideband, 2–80 Hz | Active |
| 6 | Gradual | Wideband, 2–80 Hz | Active |
| 7 | Sudden | Narrowband, 2–10 Hz | Active |
| 8 | Gradual | Narrowband, 2–10 Hz | Active |
Figures 6 and 7 show the cases with fixed and active, respectively, and each column shows one case in three panels. The first panel shows the command and the measured displacement , and the second shows the damage index . The third panel compares the estimated with the true margin and the target , and it also shows in the adaptive cases.
\n![\"Calibration](\"figures/fig_calibration_fixed.png\")
![\"Calibration](\"figures/fig_calibration_adaptive.png\")
Four metrics in Table 3 summarize each case, and they are evaluated after the first 5 s of the record. The estimation error is the mean of , which measures how closely the indicator follows the true margin. The roughness is the median absolute second difference of , multiplied by 1000, so steady changes are not penalized. The travel is the total variation of , which measures how much the controller output is adjusted. The unsafe time is the percentage of time with , during which the true margin lies clearly below the target.
\n| Case | Estimation error | Roughness | Travel of | Unsafe time (%) |
|---|---|---|---|---|
| 1 | 0.014 | 10.5 | — | 34.8 |
| 2 | 0.014 | 11.8 | — | 25.1 |
| 3 | 0.017 | 0.010 | — | 34.8 |
| 4 | 0.032 | 0.52 | — | 25.1 |
| 5 | 0.014 | 10.1 | 1.44 | 1.5 |
| 6 | 0.015 | 11.1 | 1.38 | 0.0 |
| 7 | 0.022 | 0.023 | 2.44 | 3.6 |
| 8 | 0.037 | 0.53 | 2.64 | 1.0 |
With fixed, the estimation error stays at or below 0.032, so the indicator follows the true margin in both bands. The unsafe time of 34.8% or 25.1% then measures how long the damage holds the true margin below the target, because stays at one. Activating with the same signals reduces the unsafe time to at most 3.6%, and the estimation error stays at or below 0.037. The narrowband cases adjust more, with a travel of 2.44 and 2.64 against 1.44 and 1.38 in the wideband cases.
\nThe estimator and its calibration rely on the following assumptions, and each one limits how the reported indicator should be interpreted.
\n