import re, json, html, os, sys, datetime from urllib.parse import quote_plus from latex2mathml.converter import convert ARGMIN_BAD = 'argmin' def mml(tex, block=False): m = convert(tex, display="block" if block else "inline") return m.replace(ARGMIN_BAD, 'arg min') EQ = {} # key -> number WORD = {} # "eqN" -> UnicodeMath def eq(key, tex, word): n = len(EQ) + 1 EQ[key] = n WORD[f"eq{n}"] = word return (f'
{mml(tex, True)}
' f'({n})
') def ref(k): return f'Equation ({EQ[k]})' def inline(text): text = re.sub(r"\$(.+?)\$", lambda m: mml(m.group(1)), text) return text def eqs_range(a, b): return f'({EQ[a]})–({EQ[b]})' parts = [] P = lambda t: parts.append(f"

{inline(t)}

") H2 = lambda num, t, id_: parts.append(f'

{num}{t}

') E = lambda *a: parts.append(eq(*a)) RAW = parts.append H1 = lambda num, t, id_: parts.append(f'

{num}{t}

') def trow(cells, tag="td"): return "" + "".join(f"<{tag}>{inline(c)}" for c in cells) + "" # ---------------------------------------------------------------- content H1("2", "Problem formulation", "sec2") H2("2.1", "Motivation", "s21") P("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. {citet:maghareh} 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.") P("{citet:gomaa} 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 {citet:gomaa} 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.") H2("2.2", "Experimental setup", "s22") P("Figure 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.") RAW('
Steel column specimen anchored to a concrete block and loaded laterally by an MTS actuator
Figure 1. Experimental setup: steel column specimen anchored to a concrete block and loaded laterally by a hydraulic actuator.
') P("The 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.") P("No numerical substructure is involved in the tests of this section, and the measured force is therefore not fed " "back. The reference $u(t)$ is a prescribed displacement, so only the outer controller loop is closed. As Figure 2 " "shows, the outer controller $C(s)$ acts on the tracking error $u(t)-y(t)$ and sends the command $v(t)$ to the " "plant $H(s,k)$. The measured displacement $y(t)$ is then fed back to the controller, which closes the loop with " "unity feedback.") RAW('''
Outer controller loop with a prescribed reference u(t), used to identify the plant H(s,k) in the initial and damaged states The reference u enters a summing junction, where the measured displacement y is subtracted. The tracking error enters the controller C(s), which sends the command v to the transfer system and specimen H(s,k). The output y is fed back to the summing junction. C(s)Controller H(s, k)Transfer systemand specimen u(t)+− v(t)y(t)
Figure 2. Outer controller loop with a prescribed reference u(t), used to identify the plant H(s,k) in the initial and damaged states.
''') H2("2.3", "Effect of degradation on stability", "s23") P("The plant $H(s,k)$ was experimentally identified before and after the anchorage damage, and the two identified plants are denoted " "$H_0$ and $H_d$. In each state, band-limited white noise up to 80 Hz was applied through the loop of Figure 2 with " "the same PID controller $C(s)$. A parametric model was then fitted to the transfer function from the command $v(t)$ " "to the measured displacement $y(t)$. In the initial state, the lateral stiffness of the specimen at the actuator " "control point was $k_0=7.20$ MN/m. After the anchorage was damaged, the lateral stiffness fell to 5.38 MN/m, or " "$0.75\\,k_0$, which is a 25% loss.") P("In the loop of Figure 2, the controller $C(s)$ is the same in both states, and only the specimen " "changes. The open-loop transfer function is therefore $L_0=CH_0$ in the initial state and $L_d=CH_d$ in the " "damaged state. The chordal distance between $L(j\\omega)$ and the critical point $-1$ is") E("chi", r"\chi(\omega)=\frac{|1+L(j\omega)|}{\sqrt{2\big(1+|L(j\omega)|^{2}\big)}},", "χ(ω)=(|1+L(jω)|)/√(2(1+|L(jω)|^2)),") P("and its minimum over frequency is the NCF margin, presented by {citet:vinnicombe} as the generalized stability margin. Figure 3 compares the experimentally " "identified frequency responses of the two states and their chordal distances.") RAW('
Open-loop magnitude, phase and chordal distance for the initial and damaged systems
Figure 3. Experimentally identified open-loop frequency response of the outer controller loop and its chordal distance for the initial and damaged systems. Markers show the measured points, lines show the fitted models, blue dots show the NCF margins, and the dotted line shows the 0.4 guideline.
') P("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 {citet:gomaa}, so the mechanism also appears in a physical experiment.") H2("2.4", "Problem statement", "s24") P("These 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.") H1("3", "Online NCF stability indicator framework for adaptive control", "sec3") RAW('

' + inline( "The 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, $b_{\\mathrm{id}}$, which " "estimates the closed-loop transfer function of the outer controller loop from the desired displacement $u(t)$ " "to the measured actuator displacement $y(t)$, with the $H_1$ cross-spectral estimator {citep:bendat}. The second is " "output-power inference, $b_{\\mathrm{pow}}$, 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, $b_{0}$, " "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 $\\lambda$, 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.") + '

') H2("3.1", "Framework Architecture", "s1") P("In the hybrid simulation of Figure 4, the numerical substructure computes the desired displacement $d(t)$ " "from the ground acceleration and measured force $f_m(t)$. To compensate for transfer-system lag and improve tracking, " "a phase-lead compensator receives $d(t)$ and produces the reference $u(t)$. The outer controller $C(s)$ takes the " "tracking error $u(t)-y(t)$, where $y(t)$ is the measured actuator displacement. It scales its output by the adaptive " "factor $\\lambda\\in(0,1]$ to produce the command $v(t)$ sent to the plant $H(s,k)$.") P("The plant comprises the specimen of stiffness $k$ and the transfer system, which includes the inner-loop " "controller, servo-valve, and actuator. The control–structure interaction discussed by " "{citet:dyke} is also accounted for in the transfer-system dynamics. The measured force $f_m(t)$ is fed back to the numerical substructure " "and closes the numerical substructure loop. Similarly, the measured displacement $y(t)$ is fed back to the outer " "controller and closes the outer controller loop.") P("The NCF OSI block runs alongside the outer-controller loop and does not alter the signals within it. It collects " "the reference $u(t)$ and the measured displacement $y(t)$ 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 $\\lambda$. A lower margin reduces $\\lambda$ to restore stability, while a sufficient margin " "lets $\\lambda$ recover toward one to retain tracking performance.") RAW('''
Hybrid simulation loop with the NCF OSI framework setup The numerical substructure computes the desired displacement d, which the phase-lead compensator turns into the reference u. The tracking error u minus d_m enters the controller C(s). The adaptive factor lambda scales its output to give the command v applied to the transfer system and specimen H(s,k), whose output is y. The measured displacement d_m returns to the summing junction. The measured force f_m returns to the numerical substructure. The NCF OSI collects the reference u and the measured displacement d_m and updates lambda. Numericalsubstructure Phase-leadcompensator C(s)Controller λAdaptivefactor H(s, k)Transfer systemand specimen NCF OSI d(t) u(t) +− v(t) y(t) dm fm update
Figure 4. Hybrid simulation loop with the NCF OSI framework setup.
''') P('The plant in Figure 4 is characterized by the transfer function $H(s,k)=Y(s)/V(s)$ from the command $v(t)$ ' 'to the measured actuator displacement $y(t)$. The open-loop transfer function of the outer controller loop and ' 'its closed-loop response from $u(t)$ to $y(t)$ are') E("L", r"L(s,\lambda,k)=\lambda\,C(s)\,H(s,k),", "L(s,λ,k)=λ C(s) H(s,k)") E("T", r"T(s,\lambda,k)=\frac{Y(s)}{U(s)}=\frac{L(s,\lambda,k)}{1+L(s,\lambda,k)},", "T(s,λ,k)=(Y(s))/(U(s))=(L(s,λ,k))/(1+L(s,λ,k))") P('Here $U(s)$, $Y(s)$ and $V(s)$ are the Laplace transforms of $u(t)$, $y(t)$ and $v(t)$, respectively. The NCF stability margin ' 'of the outer controller loop is evaluated for the loop $L$ under unity feedback {citep:glover}:') E("bncf", r"b_{\mathrm{NCF}}(L)=\inf_{\omega}\chi(\omega).", "b_(\"NCF\") (L)=\"inf\"┬ω χ(ω).") P('As the specimen changes during a hybrid simulation, the loop $L(j\\omega)$ changes with it, and so does its ' 'margin. {eq:bncf} ' 'could be applied directly only if $L(j\\omega)$ 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 $b_{\\mathrm{id}}$, output-power inference $b_{\\mathrm{pow}}$ ' 'and the prior identification of the initial state $b_0$.') H2("3.1.1", "Direct identification branch, " + inline("$b_{\\mathrm{id}}$"), "s2") P('Identifying the closed-loop transfer function requires the autospectrum of reference $u(t)$ and the cross ' 'spectrum of measured actuator displacement $y(t)$ with $u(t)$. The autospectrum of $y(t)$ 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.') P('Each window contains $N=f_sT_w$ samples of $u(t)$ and $y(t)$, where $f_s$ is the sampling frequency and ' '$T_w$ is the window duration. Updates occur at intervals $\\Delta t$, so consecutive windows overlap when ' '$\\Delta t\eta_A\,\sigma_u,\\ 0, & \text{otherwise},\end{cases}", "A_n={■(1,&\"RMS\"(u_n)>η_A σ_u,@0,&\"otherwise\",)┤") P("where $\\operatorname{RMS}(u_n)$ is the root mean square of the command samples in the considered window $n$, " "$\\sigma_u$ is the standard deviation of the command noise measured in a quiet record, and $\\eta_A$ is a " "hyperparameter to be tuned. The second factor is the input leverage $\\ell_n$, which reduces " "the weight of bins that carry little command energy relative to the other bins. To measure this, the command power in each bin is first " "normalized by its maximum over $\\Omega$:") E("prel", r"\ell_{\mathrm{raw},n}(\omega)=\frac{\bar{p}_{uu,n}(\omega)}{\max_{\nu\in\Omega}\bar{p}_{uu,n}(\nu)}.", "ℓ_(\"raw\",n) (ω)=(p̄_(uu,n) (ω))/(\"max\"┬(ν∈Ω) p̄_(uu,n) (ν)).") P("The raw leverage then sets the weight of each bin through the threshold $\\ell_{\\mathrm{th}}$, expressed as a " "fraction of the peak command power. A bin with $\\ell_{\\mathrm{raw},n}$ well above $\\ell_{\\mathrm{th}}$ should " "receive a weight close to one, and a bin well below it a weight close to zero:") E("lev", r"\ell_n(\omega)=\frac{\ell_{\mathrm{raw},n}(\omega)}{\ell_{\mathrm{raw},n}(\omega)+\ell_{\mathrm{th}}},", "ℓ_n (ω)=(ℓ_(\"raw\",n) (ω))/(ℓ_(\"raw\",n) (ω)+ℓ_(\"th\") ),") P("which changes smoothly between these limits and equals one half when $\\ell_{\\mathrm{raw},n}=\\ell_{\\mathrm{th}}$, " "that is, when the bin carries $\\ell_{\\mathrm{th}}$ times the peak command power. The leverage is relative to the peak bin, so $A_n$ is still needed to reject " "a command that is weak at all frequencies. The third factor is the " "coherence $\\gamma_n^{2}$ from {eq:coh}, which multiplies the activity and the leverage to complete the weight") E("conf", r"w_{\mathrm{id},n}(\omega)=A_n\,\gamma_n^{2}(\omega)\,\ell_n(\omega).", "w_(\"id\",n) (ω)=A_n γ_n^2 (ω) ℓ_n (ω).") P("This weight sets the share of direct identification in the per-frequency fusion of the three branches in " "Section 3.1.3.") H2("3.1.2", "Output-power inference branch", "s3") P("At frequencies where the command carries little energy, the identification weight $w_{\\mathrm{id},n}$ falls toward " "zero, and direct identification provides little evidence of the margin. When the specimen degrades, direct " "identification therefore detects the change at the excited frequencies but not at the others. " "The margin can still fall at these frequencies, so another stability indication is needed there. " "In this case, the output power " "can be used because the outer controller loop shapes signals that pass through it, including sensor noise. " "{citet:wu} used the growth of the measured power error to detect the positive-feedback mode of a virtual " "synchronous generator. This idea inspired the output-power inference branch employed in this study, which treats growth in the measured " "output power as evidence of a reduced margin. As shown in Section 2.3, degradation raises the loop gain by a " "comparable factor at all frequencies. A loop change identified at the excited frequencies therefore shows that " "the specimen has degraded and that the prior loop is outdated. However, the identified loop alone does not reveal the " "amplification at the frequencies that the command leaves unexcited. The output power then measures how much the " "loop has changed at these frequencies, because the amplification varies with frequency. To this end, the averaged " "output power at each frequency is compared with its reference from the initial state:") E("qobs", r"q_{\mathrm{obs},n}(\omega)=\frac{\bar{p}_{yy,n}(\omega)}{p_{yy,0}(\omega)},", "q_(\"obs\",n) (ω)=(p̄_(yy,n) (ω))/(p_(yy,0) (ω)),") P("where $q_{\\mathrm{obs},n}$ is the observed power ratio at window $n$, which measures the change in output power at each frequency " "relative to the initial state. The averaged autospectrum of $y$ is $\\bar{p}_{yy,n}$ from {eq:ewyy}, and " "$p_{yy,0}$ is the reference autospectrum of $y$ recorded in the initial state.") P("The 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 $\\lambda=1$:") E("pyyref", r"p_{yy,0}(\omega)=\operatorname{med}\{p_{yy,n_1}(\omega),\,p_{yy,n_2}(\omega),\,p_{yy,n_3}(\omega)\},", "p_(yy,0) (ω)=\"med\"{p_(yy,n_1) (ω),p_(yy,n_2) (ω),p_(yy,n_3) (ω)},") P("where $n_1$, $n_2$ and $n_3$ are the corresponding updates, and the median discards one unusually loud or quiet window.") P("According to the experimental verification presented in " "Section 2.3, the change due to specimen degradation can be represented by an amplification $g$ of the open-loop gain. " "Consequently, a candidate loop $\\lambda_n g L_0$ with amplification $g$ predicts the power ratio relative to the " "initial state") E("qmod", r"q_{\mathrm{model},n}(\omega,g)=\left|\frac{1+L_0(j\omega)}{1+\lambda_n g\,L_0(j\omega)}\right|^{2},", "q_(\"model\",n) (ω,g)=〖|(1+L_0 (jω))/(1+λ_n g L_0 (jω))|〗^2,") P("where $q_{\\mathrm{model},n}$ is the predicted power ratio, and $L_0(j\\omega)=C(j\\omega)H_0(j\\omega)$ 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:") E("gfit", r"\hat{g}_n(\omega)=\underset{g}{\operatorname{arg\,min}}\;\left|\log q_{\mathrm{obs},n}(\omega)-\log q_{\mathrm{model},n}(\omega,g)\right|.", "ĝ_n (ω)=\"arg min\"┬g |log q_(\"obs\",n) (ω)−log q_(\"model\",n) (ω,g)|.") P("The amplification $\\hat{g}_n(\\omega)$ is reliable only when the observed power ratio reflects a change in the true system dynamics. At weakly excited frequencies, however, the output consists mainly of sensor noise, and its measured power fluctuates randomly. Consequently, $\\hat{g}_n(\\omega)$ can falsely indicate gain amplification even when the true system dynamics have not changed.") P("Such a false amplification is separated from a real one using the identified loop at the excited frequencies of the same window. Because degradation raises the loop gain at all frequencies, a real amplification is accompanied by an increase of $\\hat{L}_n$ there, whereas a false one is not. This increase is measured against $\\lambda_n L_0$, the prior loop scaled by the current adaptive factor, so a reduction of $\\lambda_n$ is not mistaken for degradation. The loop-increase factor $\\rho_n$ is the real scale that best fits $\\lambda_n L_0$ to the identified loop $\\hat{L}_n$ in the weighted least-squares sense:") E("rhols", r"\rho_n=\underset{\rho\in\mathbb{R}}{\operatorname{arg\,min}}\sum_{\omega\in\Omega}w_{\mathrm{id},n}^{2}(\omega)\,\big|\hat{L}_n(j\omega)-\rho\,\lambda_n L_0(j\omega)\big|^{2}.", "ρ_n=\"arg min\"┬(ρ∈ℝ) ∑_(ω∈Ω)▒〖w_(\"id\",n)^2 (ω) |L̂_n (jω)−ρ λ_n L_0 (jω)|^2〗.") P("Each frequency is weighted by $w_{\\mathrm{id},n}^{2}$ from {eq:conf}, so the well-identified excited frequencies dominate the fit and the weakly excited ones contribute little. The fit uses the identified loop $\\hat{L}_n$ from {eq:lhat}, taken before the median smoothing of {eq:lmed}. The scale is kept real, consistent with the real amplification $g$ in {eq:qmod}. Setting the derivative of the sum with respect to $\\rho$ to zero then gives the closed form") E("rho", r"\rho_n=\frac{\operatorname{Re}\{\sum_{\omega\in\Omega}w_{\mathrm{id},n}^{2}(\omega)\,\lambda_n L_0^{*}(j\omega)\,\hat{L}_n(j\omega)\}}{\sum_{\omega\in\Omega}w_{\mathrm{id},n}^{2}(\omega)\,|\lambda_n L_0(j\omega)|^{2}}.", "ρ_n=(Re{∑_(ω∈Ω)▒〖w_(\"id\",n)^2 (ω) λ_n L_0^* (jω) L̂_n (jω)〗})/(∑_(ω∈Ω)▒〖w_(\"id\",n)^2 (ω) |λ_n L_0 (jω)|^2〗).") P("For an unchanged specimen, $\\hat{L}_n$ agrees with $\\lambda_n L_0$ at the excited frequencies and $\\rho_n\\approx1$. A degraded specimen instead raises the identified loop at these frequencies and drives $\\rho_n$ above one. The confirmation $c_n$ converts this loop increase into the trust given to the amplification inferred from output power. It rises linearly with $\\rho_n$ and is clipped between a lower bound $\\epsilon$ and one:") E("confirm", r"c_n=\frac{\rho_n-1}{\rho_{\mathrm{th}}-1},\qquad c_n\in[\epsilon,\,1].", "c_n=(ρ_n−1)/(ρ_(\"th\")−1), c_n∈[ε,1].") P("Once $\\rho_n$ reaches the confirmation threshold $\\rho_{\\mathrm{th}}$, the loop increase confirms the amplification fully and $c_n=1$. The lower bound $\\epsilon=0.1$ 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 $\\rho_n$ is a single value at each update, the same confirmation applies at every frequency.") P("The confirmation acts by pulling the inferred amplification toward one, which represents an unchanged loop, and gives the accepted amplification") E("gaccept", r"\tilde{g}_n(\omega)=1+c_n\big(\hat{g}_n(\omega)-1\big).", "g̃_n (ω)=1+c_n (ĝ_n (ω)−1).") P("With full confirmation, $\\tilde{g}_n=\\hat{g}_n$, and the amplification inferred from output power is accepted unchanged. For an unchanged specimen, $c_n=\\epsilon$, so a false value of $\\hat{g}_n=2$ is accepted only as $\\tilde{g}_n=1.1$.") P("The accepted amplification still carries the scatter of $q_{\\mathrm{obs},n}$ 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 ({n:ewuu})–({n:ewyu}). The average is formed on a logarithmic scale, consistent with the logarithmic difference in {eq:gfit}:") E("gmem", r"\log\hat{g}_{\mathrm{s},n}(\omega)=(1-\eta_g)\log\hat{g}_{\mathrm{s},n-1}(\omega)+\eta_g\log\tilde{g}_n(\omega).", "log ĝ_(\"s\",n) (ω)=(1−η_g) log ĝ_(\"s\",n−1) (ω)+η_g log g̃_n (ω).") P("The smoothed amplification $\\hat{g}_{\\mathrm{s},n}$ acts as the memory of this branch and holds a detected amplification while the specimen remains degraded. The averaging factor $\\eta_g$ sets how strongly each new accepted amplification updates this memory.") P("The same memory, however, slows the release of an amplification that the observed output no longer supports. With {eq:gmem} alone, each update moves the remembered value toward the newest one by only the fraction $\\eta_g$. 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 $q_{\\mathrm{obs},n}$ over $\\Omega$ 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 {eq:gmem}:") E("gdecay", r"\log\hat{g}_{\mathrm{s},n}(\omega)=r_g\,\log\hat{g}_{\mathrm{s},n-1}(\omega).", "log ĝ_(\"s\",n) (ω)=r_g log ĝ_(\"s\",n−1) (ω).") P("The recovery factor $r_g$, with $0 Structure of one estimator update Windowed displacements u and y feed a windowed FFT that forms the averaged autospectra and cross spectrum. These spectra feed the direct identification and output-power branches. A third branch is the initial-state prior. The adaptive factor lambda, shown in green, feeds the output-power and initial-state prior branches. The three margin curves are fused per frequency bin with the weights w_id, w_pow and w_0, and the minimum of the fused curve over the analysis band gives b_OSI, which updates the adaptive factor lambda. u y Windowed FFT p̄uu, p̄yy, p̄yu Equations ({n:dftu})–({n:ewyu}) Direct identification L̂s, wid → bid(ω) Equations ({n:that})–({n:conf}) Output-power inference qobs, ĝs → bpow(ω) Equations ({n:qobs})–({n:gdecay}) Initial-state prior L0 → b0(ω) Equation ({n:b0}) Per-bin fusion β(ω) Equations ({n:w0})–({n:beta}) Minimum over Ω bOSI Equation ({n:bosi}) Adaptive factor λ Equation ({n:lam}) wid wpow w0 bOSI
Figure 5. Flowchart of one NCF OSI update. The reference u and the measured displacement y provide the spectra for direct identification and output-power inference. The two measured margin curves are fused bin by bin with the prior identification of the initial state. The minimum of the fused curve, bOSI, then updates the adaptive factor λ.
''') T1 = [ ("$\\Delta t$", "0.125 s", "—", "Design", "Update interval"), ("$T_w$, $N$", "0.5 s, 1024", "—", "Design", "Window length at $f_s=2048$ Hz"), ("$\\Omega$", "2–80 Hz, 37 bins", "—", "Design", "Analysis set, 58–62 Hz rejected"), ("$\\eta_s$", "0.39", "0.25 s", "Calibrated", "Spectral averaging, Equations ({n:ewuu})–({n:ewyu})"), ("$\\eta_A$", "4.54", "—", "Calibrated", "Activity threshold relative to command noise, {eq:act}"), ("$\\ell_{\\mathrm{th}}$", "$10^{-3}$", "—", "Calibrated", "Input-leverage threshold, {eq:lev}"), ("$\\epsilon$", "0.1", "—", "Design", "Minimum confirmation, Equation ({n:confirm})"), ("$\\rho_{\\mathrm{th}}$", "1.5", "—", "Calibrated", "Confirmation threshold, Equation ({n:confirm})"), ("$\\eta_g$", "0.12", "1.0 s", "Calibrated", "Output-power memory, accumulation, {eq:gmem}"), ("$r_g$", "0.73", "0.4 s", "Calibrated", "Recovery factor per update, {eq:gdecay}"), ("$b_{\\mathrm{tar}}$", "0.40", "—", "Design", "Target NCF margin, Equation ({n:lerr})"), ("$\\kappa$", "0.625", "—", "Calibrated", "Adaptation gain per update, Equation ({n:lam})"), ("$\\mu$", "0.32", "—", "Calibrated", "Ratio of the recovery rate to the reduction rate of $\\lambda$"), ("$\\lambda_{\\min}$", "0.20", "—", "Design", "Smallest adaptive factor, Equation ({n:lam})"), ] H2("3.2", "Calibration", "s32") P("The 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 $\\eta_s$ of Section 3.1.1 illustrates. A larger $\\eta_s$ 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.") P("Table 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 $T_w$ of 0.5 s gives a frequency resolution of 2 Hz, which also sets the lower bound of $\\Omega$. The upper bound of 80 Hz matches the band over which the plants were identified in Section 2.3. The update interval $\\Delta t$ of 0.125 s sets how often the indicator and the adaptive factor are updated. The target $b_{\\mathrm{tar}}=0.4$ follows the guideline proposed by {citet:gomaa}, and $\\lambda_{\\min}=0.2$ keeps part of the controller output active. Finally, the minimum confirmation $\\epsilon=0.1$ is the safety minimum introduced in Section 3.1.2 for amplification that cannot be confirmed.") P("The 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.") RAW('
Table 1. Estimator parameters. Smoothing coefficients follow ' + inline("$\\eta=1-\\exp(-\\Delta t/\\tau)$") + ' from the listed time constant.
' + "" + trow(["Symbol", "Value", "$\\tau$", "Type", "Role"], "th") + "" + "".join(trow(r) for r in T1) + "
") P("A known margin requires a plant that is known at every instant, including while the specimen degrades. The two plants $H_0$ and $H_d$ 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 $v(t)$, and their outputs are mixed:") E("mix", r"y(t)=\big[1-\delta(t)\big]\,y_0(t)+\delta(t)\,y_d(t),", "y(t)=[1−δ(t)] y_0 (t)+δ(t) y_d (t),") P("where $y_0$ and $y_d$ are the outputs of $H_0$ and $H_d$, and $\\delta(t)\\in[0,1]$ is the damage index. Thus $\\delta=0$ represents the initial state, and $\\delta=1$ represents the damaged state identified in Section 2.3. While $\\delta$ is constant, superposition makes this mixture equivalent to the single linear plant $(1-\\delta)H_0+\\delta H_d$ driven by $v(t)$. The true margin at each update therefore follows from {eq:bncf} applied to the loop $\\lambda C\\big[(1-\\delta)H_0+\\delta H_d\\big]$. This true margin is computed independently of the estimator, so it serves as the benchmark for judging $b_{\\mathrm{OSI}}$. The mixture interpolates the responses of the two states and does not represent an intermediate stiffness.") P("The 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 $\\delta$ from zero to one at 40 s and back to zero at 80 s. Gradual damage ramps $\\delta$ 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 $\\delta=0$ reproduces a loop that regains its margin. This return therefore tests the recovery step of Section 3.1.2, which releases the remembered amplification.") P("The second feature is the excitation band of the reference $u(t)$, 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 $\\eta_s$, activity threshold $\\eta_A$ and leverage threshold $\\ell_{\\mathrm{th}}$. 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 $\\rho_{\\mathrm{th}}$ and the averaging factor $\\eta_g$, and their return sets the recovery factor $r_g$.") P("The third feature is the adaptive factor, which is either fixed at one or active during the case. With $\\lambda$ fixed, the cases test whether the indicator alone estimates the margin correctly as the damage changes. With $\\lambda$ active, the same cases test whether the loop is held near the target while tracking is retained. These active cases set the adaptation gain $\\kappa$ and the ratio $\\mu$ 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 $\\lambda$ alone. Table 2 lists the resulting eight cases, which cover every combination of the three features.") RAW('
Table 2. Calibration cases.
' '' + "" + trow(["Case", "Damage history", "Excitation", "Adaptive factor"], "th") + "" + "".join(trow(r) for r in [ ("1", "Sudden", "Wideband, 2–80 Hz", "Fixed, $\\lambda=1$"), ("2", "Gradual", "Wideband, 2–80 Hz", "Fixed, $\\lambda=1$"), ("3", "Sudden", "Narrowband, 2–10 Hz", "Fixed, $\\lambda=1$"), ("4", "Gradual", "Narrowband, 2–10 Hz", "Fixed, $\\lambda=1$"), ("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"), ]) + "
") P("Figures 6 and 7 show the cases with $\\lambda$ fixed and active, respectively, and each column shows one case in three panels. The first panel shows the command $v(t)$ and the measured displacement $y(t)$, and the second shows the damage index $\\delta$. The third panel compares the estimated $b_{\\mathrm{OSI}}$ with the true margin and the target $b_{\\mathrm{tar}}$, and it also shows $\\lambda$ in the adaptive cases.") RAW('
Calibration cases 1 to 4 with the adaptive factor fixed
Figure 6. Calibration cases 1–4 with the adaptive factor fixed at one.
') RAW('
Calibration cases 5 to 8 with the adaptive factor active
Figure 7. Calibration cases 5–8 with the adaptive factor active.
') P("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 $|b_{\\mathrm{OSI}}-b_{\\mathrm{NCF}}|$, which measures how closely the indicator follows the true margin. The roughness is the median absolute second difference of $b_{\\mathrm{OSI}}$, multiplied by 1000, so steady changes are not penalized. The travel is the total variation of $\\lambda$, which measures how much the controller output is adjusted. The unsafe time is the percentage of time with $b_{\\mathrm{NCF}}
Table 3. Calibration metrics with ' + inline("$b_{\\mathrm{tar}}=0.4$") + '.
' + "" + trow(["Case", "Estimation error", "Roughness", "Travel of $\\lambda$", "Unsafe time (%)"], "th") + "" + "".join(trow(r) for r in [ ("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"), ]) + "
") P("With $\\lambda$ 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 $\\lambda$ stays at one. Activating $\\lambda$ 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 $\\lambda$ more, with a travel of 2.44 and 2.64 against 1.44 and 1.38 in the wideband cases.") H1("", "Appendix A. Assumptions and limitations", "appA") P("The estimator and its calibration rely on the following assumptions, and each one limits how the reported indicator should be interpreted.") RAW('
    ' + "".join(f"
  1. {inline(t)}
    2. " for t in [ "The identification branch assumes that the output disturbance is uncorrelated with the reference $u$, as the $H_1$ estimator requires. Feedback through the numerical substructure returns part of this disturbance to $u$ through the measured force $f_m$, which can bias $\\hat{T}_n$ even at high coherence.", "The output-power branch assumes that the disturbance spectrum remains as it was during the reference period formed by the first three windows. It also assumes that the loop during this period is the initial loop at full controller output, with $\\lambda=1$. A change in disturbance intensity or in the command-driven response can therefore alter $q_{\\mathrm{obs}}$ without any change in stability.", "The amplification $g$ is real, so the output-power branch captures changes in loop magnitude but not in loop phase.", "Frequencies outside $\\Omega$ are not evaluated, so $b_{\\mathrm{OSI}}$ is not a guaranteed lower bound on the full-frequency margin $b_{\\mathrm{NCF}}$.", "The calibration cases use the two identified linear plants and their output mixture, which do not include nonlinear specimen behavior. The calibrated values can therefore require adjustment when the framework is applied to another specimen or laboratory.", ]) + "
") REFS = { # key -> (in-text name, year, entry). Cite as {citet:key} "Name [n]" or {citep:key} "[n]". Numbered by first appearance. "glover": ("Glover and McFarlane", "1989", 'Glover, K., and McFarlane, D. (1989), “Robust stabilization of normalized coprime factor plant descriptions with H∞-bounded uncertainty,” IEEE Transactions on Automatic Control, 34(8), 821–830. DOI: 10.1109/9.29424'), "vinnicombe": ("Vinnicombe", "1993", 'Vinnicombe, G. (1993), “Frequency domain uncertainty and the graph topology,” IEEE Transactions on Automatic Control, 38(9), 1371–1383. DOI: 10.1109/9.237648'), "dyke": ("Dyke et al.", "1995", 'Dyke, S. J., Spencer, B. F., Quast, P., and Sain, M. K. (1995), “Role of control-structure interaction in protective system design,” Journal of Engineering Mechanics, 121(2), 322–338. DOI: 10.1061/(ASCE)0733-9399(1995)121:2(322)'), "gomaa": ("Gomaa et al.", "in press", 'Gomaa, M., Patino, E. D., Villamizar, J. N., Condori Uribe, J. W., and Dyke, S. J. (in press), “An online stability indicator for real-time hybrid simulation emphasizing physical substructure degradation,” author manuscript.'), "welch": ("Welch", "1967", 'Welch, P. D. (1967), “The use of fast Fourier transform for the estimation of power spectra: A method based on time averaging over short, modified periodograms,” IEEE Transactions on Audio and Electroacoustics, 15(2), 70–73. DOI: 10.1109/TAU.1967.1161901'), "roberts": ("Roberts", "1959", 'Roberts, S. W. (1959), “Control chart tests based on geometric moving averages,” Technometrics, 1(3), 239–250. DOI: 10.1080/00401706.1959.10489860'), "bendat": ("Bendat and Piersol", "2010", 'Bendat, J. S., and Piersol, A. G. (2010), Random Data: Analysis and Measurement Procedures, 4th ed., Wiley, Hoboken, NJ.'), "blackman": ("Blackman and Tukey", "1958", 'Blackman, R. B., and Tukey, J. W. (1958), “The measurement of power spectra from the point of view of communications engineering — Part I,” The Bell System Technical Journal, 37(1), 185–282. DOI: 10.1002/j.1538-7305.1958.tb03874.x'), "dimatteo": ("Di Matteo et al.", "2021", 'Di Matteo, S., Viall, N. M., and Kepko, L. (2021), “Power spectral density background estimate and signal detection via the multitaper method,” Journal of Geophysical Research: Space Physics, 126(2), e2020JA028748. DOI: 10.1029/2020JA028748'), "maghareh": ("Maghareh et al.", "2020", 'Maghareh, A., Fu, Y., Montoya, H., Condori, J., Wang, Z., Dyke, S. J., and Montoya, A. (2020), “A reflective framework for performance management (REFORM) of real-time hybrid simulation,” Frontiers in Built Environment, 6, 568742. DOI: 10.3389/fbuil.2020.568742'), "wu": ("Wu and Wang", "2020", 'Wu, H., and Wang, X. (2020), “A mode-adaptive power-angle control method for transient stability enhancement of virtual synchronous generators,” IEEE Journal of Emerging and Selected Topics in Power Electronics, 8(2), 1034–1049. DOI: 10.1109/JESTPE.2020.2976791'), } body = "\n".join(parts) # ---------------------------------------------------------------- page ARTICLE = f"""

Manuscript draft. Readers are assumed to be familiar with {{citet:gomaa}}.

{body}

References

    {{REFS}}
""" CITED = [] def cite(m): # numbered by first appearance: narrative "Name [n]" or bare "[n]" keys = m.group(2).split(",") for k in keys: if k not in CITED: CITED.append(k) nums = "[" + ", ".join(f'{CITED.index(k) + 1}' for k in keys) + "]" return (" and ".join(REFS[k][0] for k in keys) + " " + nums) if m.group(1) == "t" else nums ARTICLE = re.sub(r"\{eq:([A-Za-z0-9_]+)\}", lambda m: ref(m.group(1)), ARTICLE) ARTICLE = re.sub(r"\{n:(\w+)\}", lambda m: str(EQ[m.group(1)]), ARTICLE) ARTICLE = re.sub(r"\{cite([tp]):([a-z0-9_,]+)\}", cite, ARTICLE) def scholar(t): # title links to a Google Scholar search, DOI stays plain text m = re.search(r"“(.+?)(,?)”|(.+?)", t) title = m.group(1) or m.group(3) a = (f']+>", "", title))}" ' f'target="_blank" rel="noopener">{title}') return t.replace(m.group(0), f"“{a}{m.group(2)}”" if m.group(1) else f"{a}", 1) ARTICLE = ARTICLE.replace("{REFS}", "\n".join(f'
  • {scholar(REFS[k][2])}
    • ' for k in CITED)) if "--preview" in sys.argv: # comment suggestions: write the article only, no new version open(sys.argv[sys.argv.index("--preview") + 1], "w", encoding="utf-8").write(ARTICLE) sys.exit() # ---- content store read by the review app (writing/review-app builds manuscript2_journal.html) # usage: python build_manuscript2_journal.py "short note about this revision" # A new version is appended whenever the article changes. HERE = os.path.dirname(os.path.abspath(__file__)) VERS = os.path.join(HERE, "manuscript2_journal_versions.js") HEAD = "window.MANUSCRIPT_VERSIONS = " if os.path.exists(VERS): versions = json.loads(open(VERS, encoding="utf-8").readline()[len(HEAD):].rstrip().rstrip(";")) else: old = open(os.path.join(HERE, "manuscript2_html.html"), encoding="utf-8").read() stamp = datetime.datetime.fromtimestamp(os.path.getmtime(os.path.join(HERE, "manuscript2_html.html"))) versions = [{"v": 0, "date": stamp.strftime("%Y-%m-%d %H:%M"), "note": "original manuscript2_html", "html": old[old.index("
    ") + 9:old.index("
    ")]}] args = [a for a in sys.argv[1:] if not a.startswith("--")] if versions[-1].get("author") == "user" and versions[-1]["html"] != ARTICLE and "--merged" not in sys.argv: sys.exit(f"v{versions[-1]['v']} holds edits made in the review page. Port them into this script, " "then rerun with --merged.") if versions[-1]["html"] != ARTICLE: versions.append({"v": versions[-1]["v"] + 1, "date": datetime.datetime.now().strftime("%Y-%m-%d %H:%M"), "note": args[0] if args else "", "html": ARTICLE}) print("saved version", versions[-1]["v"]) versions = versions[-10:] # keep the last 10 versions open(VERS, "w", encoding="utf-8").write(HEAD + json.dumps(versions, ensure_ascii=False) + ";\n" + "window.MANUSCRIPT_WORD = " + json.dumps(WORD, ensure_ascii=False) + ";\n") print(len(EQ), "equations,", len(versions), "versions")