This page is generated by the validation suite and checked by CI against what the code produces. It is not written by hand.

Verification and validation

What galata has been checked against, what agreement was measured, and what has not been checked at all.

The last column is the one to read. "Unvalidated" is not a placeholder here — it is a statement that no published reference value has been found for that case, and it is preferred to a number invented to fill the gap.

Summary

The table below is GENERATED from tools/validation/case_registry.cpp and is reconciled against the capability registry the CLI dispatches through. It used to be typed by hand and it drifted three times, most memorably by still saying "there is no aerodynamic model yet" in the commit that added one.

Four checks stand behind it, each a test rather than a convention:

• every case claiming a comparison names a reference and names its evidence;
• every piece of evidence names a test that is actually registered in the binary the case says it lives in, so a renamed or deleted test breaks the build rather than leaving a fictional citation;
• every capability declaring implemented and validated is backed by at least one validating case, which makes that declaration unfalsifiable by hand;
• a case marked not implemented may not name a capability that exists, so the row cannot outlive the thing being built.

Evidence

Every case above, and the tests that stand behind it. These names are checked against the tests actually registered in each binary, so they can be run:

ctest --preset dev -R '<test name>'

Capabilities, and the cases that validate them

Read straight out of the registry the CLI dispatches through, so a capability cannot appear here without existing, and cannot claim validation without a case backing it. A capability may legitimately be implemented, unvalidated — that is an honest state, and it means no published reference has been compared against.

U.S. Standard Atmosphere, 1976

Reference. U.S. Committee on Extension to the Standard Atmosphere (COESA), U.S. Standard Atmosphere, 1976, NOAA-S/T 76-1562 / NASA-TM-X-74335, U.S. Government Printing Office, October 1976. NTRS document 19770009539, https://ntrs.nasa.gov/citations/19770009539. Rights: Work of the US Gov. Public Use Permitted.

Transcription. The NTRS scan's OCR layer is unusable for numeric tables — roughly 95,800 extractable characters across 243 pages, with digits rendered as underscores and letters. No value came from it. The page images were extracted and read visually, with load-bearing digits re-cropped at native resolution, and a second independent pass re-read twelve values without reference to the first. The full method is in the header of tests/validation/reference/ussa1976.csv.

Agreement measured. Deviation is given in units of the last printed significant figure. Below 0.5 the computed value rounds to exactly what the document prints.

Gate. The validation suite requires every cell to agree within 1.0 units of the last printed place. That bound is chosen to be the tightest one the 1976 tables actually support, not the loosest one that passes: the measured disagreements are around one part in 10^5, while a wrong lapse-rate sign, a geopotential/geometric mix-up or the wrong Earth radius each move these values by one part in 10^2 or worse.

Cells that do not round to the printed value

There are 3 of them, out of 32 tabulated cells.

Known model-versus-table differences that are not errors

Molecular-scale versus kinetic temperature above 80 km. The seven-layer system is defined in terms of molecular-scale temperature T_M, which equals kinetic temperature T only where the mean molecular weight equals its sea-level value. Above roughly 80 km the ratio M/M_0 falls away from 1 — the document's Table 8 gives 0.9995788 at 86 km — so galata's temperature runs high relative to the tabulated kinetic temperature by 0.0759 K in the top 6 km. At 86 km the document tabulates T = 186.87 K and T_M = 186.95 K; galata computes 186.946 K, which is the T_M value.

The upper bound is stated twice and the two statements differ. The standard gives the ceiling as both 84.8520 km' geopotential and 86 km geometric. Equation (18) maps 86000 m to 84852.046 m', so the geopotential figure is the document's own rounding. galata bounds the envelope in geometric altitude, so that a query at exactly the standard's stated ceiling is accepted.

Ambiguities in the source document

These are inconsistencies in the 1976 publication itself, found during transcription. Each is recorded so that it is not later mistaken for a transcription error.

What is not validated here

Dynamic viscosity. galata implements equation (51), but no tabulated viscosity values were transcribed from the document, so there is nothing to compare against. The implementation is unvalidated. It additionally inherits the S = 110 versus S = 110.4 ambiguity above, which moves the result by 0.10%.

Altitudes between the tabulated points. Eight altitudes are checked against the tables. Continuity, monotonicity and the absence of steps at layer boundaries are checked on a 100 m grid across the whole envelope, which constrains the space between the sampled points but is not the same as checking it against published values.

Everything above 86 km. Out of scope: galata refuses the query rather than extrapolating.

Determinism

ADR-0004 defines two tiers, and both are gated by .github/workflows/determinism.yml on Linux, macOS and Windows.

Tier 1 — same binary, same platform, byte-identical. tools/determinism emits 145 values at %.17g, which round-trips a double exactly, so byte-identical output means bit-identical values rather than values that merely print the same. Gated absolutely on every platform.

Tier 2 — cross-platform, bounded. Every platform PAIR is compared, not each against a nominated reference: with a reference, which platform holds that role is an arbitrary choice that then shows up in the published numbers. The gate is 1e-9 relative — far above the roughly 1e-16 that one math-library call costs, far below the roughly 1e-5 that any real divergence in the physics would produce, so it discriminates between "different libm" and "different answer".

98 of the 145 fingerprinted values are compared this way. The other 47 are downstream of a finite difference and are held byte-identical in tier 1 instead, for the reason given below.

The observed deviation is printed by every run rather than merely bounded; read it from the workflow log. It is not restated here, because a measured figure copied into a hand-maintained document is a figure that drifts.

Why tier 2 is not bit-identity. sqrt is required by IEEE 754 to be correctly rounded and agrees everywhere. sin, cos, tan, asin, atan2, exp, log and pow are not, and come from the platform's math library. galata cannot avoid them — angle of attack is an atan2, the atmosphere's pressure profile is a pow — so the honest claim is the two-tier one rather than a bit-identity claim that would be false.

A constraint on what may be fingerprinted. A cross-platform bound is only meaningful on a computation that does not amplify small differences; on a chaotic trajectory the gate would be measuring chaos rather than agreement. The fingerprint's rigid-body case was measured and amplifies a perturbation by a factor between 0.0563 and 1.01 over 60 s. A test asserts this, so a change that makes the battery chaotic fails there rather than as an intermittently red workflow.

Trim and linearisation

Reference. Heffley and Jewell, NASA CR-2144, Tables II-1, II-2 and II-7.

This is the end-to-end case, and it is a much stronger statement than the modal comparison above. There, the state matrix came from the report's own dimensional derivatives, so only the eigen-analysis was under test. Here the input is the NON-dimensional derivative set and the geometry; galata builds a nonlinear model, finds its trim, linearises about it, and both the dimensional derivatives and the modes fall out. Everything in between is under test: the atmosphere, every unit conversion, the coefficient buildup, the wind-to-body rotation, the equations of motion, the root-find and the finite differences.

Trim. Converges to a residual of 0, with quadratic convergence visible in the reported history. It is checked two ways: against the closed-form force balance L = mg - D tan(a) and T = D/cos(a), which it satisfies to a part in 10^6; and against the published flight condition.

The trimmed angle of attack sits 0.0519 deg below the published value, and the difference is understood rather than tolerated. The published pair (alpha and C_L) is related by the conventional level-flight relation C_L = W/(qS), which neglects the vertical component of drag in body axes. galata solves the exact balance, which needs a slightly smaller C_L. The shift is exactly D tan(a)/(qS)/C_L_alpha, and a test asserts it is that term and nothing else.

Linearised dimensional derivatives against Table II-7 — seven numbers the report computed from the same non-dimensional set by a different route:

The source prints its inputs and its outputs to three significant figures, so each carries up to about 0.5% of its own rounding and several combine in every one of these. The gate is 0.5%; the worst observed is 0.26%.

All five modes, from the same linearisation:

The worst is Dutch roll zeta at 1.05%.

An error this comparison caught. The first version of the model treated the report's lateral derivatives as body-axis when the report gives them in stability axes. At a trim angle of attack of about two degrees that looks like a 0.07% effect, since cos(alpha) = 0.9993. It is not: the rotation MIXES the rolling and yawing moments, and C_l_beta is 2.59 times C_n_beta, so the cross term dominates and C_n_beta moves by 10%. The Dutch roll damping came out 35% high. It was the derivative-by-derivative comparison above that localised it — the modes alone said only that something was wrong.

A second error was found the same way: the force assembly rotated the lift/drag/side triple through the full wind-axis rotation, which adds a -D sin(beta) term to the body y force that a body-axis C_Y_beta already contains. Double-counting it inflated the side-force derivative by 19%.

Both are the reason models/nt33a/nt33a-fc1.yaml must DECLARE which axes its lateral derivatives are in, rather than defaulting.

Rigid-body dynamics

Reference. These cases carry no transcribed numbers: the reference is an analytic solution of Euler's equations, with the derivation written out in tests/validation/test_rigid_body_dynamics.cpp so a reader can check it against the equations rather than against a table. That makes them the strongest validation in the suite — there is no transcription step to get wrong, and the expected values are exact rather than rounded.

H. Goldstein, C. P. Poole and J. L. Safko, Classical Mechanics, 3rd ed., Addison Wesley, 2002; L. D. Landau and E. M. Lifshitz, Mechanics, 3rd ed., Butterworth-Heinemann, 1976.

Conservation under torque-free motion. Asymmetric body with a non-zero product of inertia, 60 s of integration at a 1 ms fixed step — 60,000 RK4 steps.

The angular-momentum figure is the vector in the NAVIGATION frame, not its magnitude in body axes. That is deliberate and it is the stronger claim: the magnitude is conserved by the rotational dynamics alone, whereas the vector being fixed in NED requires the attitude kinematics and the rotational dynamics to agree with each other. A transposed direction-cosine matrix conserves the magnitude and fails this.

RK4 is not symplectic, so this drift is secular rather than oscillatory — it grows with integration length rather than staying bounded. Over the tens of seconds a flight simulation runs it is far below every other error in the model; over an orbit it would not be, and this is the wrong integrator for that.

Torque-free precession. For a body symmetric about its z-axis the transverse angular-velocity vector rotates in the body frame at lambda = (Ia - It) n / It with constant magnitude. Both an oblate case (lambda > 0) and a prolate case (lambda < 0, where the precession runs the other way round the body) are checked, and the check is that the error falls like h^4 as the step halves — not merely that it is small at one step. A solution converging to the wrong closed form would sit at a small constant error and pass an absolute check while failing this one outright.

Intermediate-axis instability. Rotation about the intermediate principal axis is unstable with growth rate sigma = W sqrt((I3 - I2)(I2 - I1) / (I1 I3)). Starting from a perturbation in e1 with e3 = 0 gives e1(t) = a cosh(sigma t) and e3(t) = a sigma I1 sinh(sigma t) / ((I2 - I3) W) — a cosh, not an exponential. Both are asserted pointwise to a relative tolerance of 1e-4 over four e-foldings. Rotation about the major and minor axes is checked to remain bounded.

Mass-property guard rails. MassProperties::validate() rejects a non-positive mass, an asymmetric inertia tensor, an indefinite one, and one whose principal moments violate the triangle inequality Ia + Ib >= Ic. The last of these is the one that catches a moment quoted about the wrong axis, which passes every other check.

Aircraft modal characteristics

Reference. R. K. Heffley and W. F. Jewell, Aircraft Handling Qualities Data, NASA CR-2144, Systems Technology Inc., December 1972. NTRS 19730003312, https://ntrs.nasa.gov/citations/19730003312. The report's own documentation page prints "Distribution Statement: Unclassified - Unlimited", with no copyright notice and no limited-rights legend.

Aircraft NT-33A at flight condition 1 — sea level, M = 0.204, power approach. The derivatives, the flight condition and the published modal results are transcribed in tests/validation/reference/nt33a_fc1.csv; the state matrices are assembled from the report's own equations (Appendix C pages C-1 and C-3), because that is the only way to compare against the report's own answers.

What is being validated. The eigenvalue decomposition, the modal metrics, the participation factors and the classification. Not a trim-and-linearise chain — neither capability exists yet.

How the tolerance is set. Both sides of the comparison are rounded: the report prints its derivatives to three significant figures and its modal results to three. So the test perturbs each input by half a unit in its own last printed digit, accumulates the resulting spread in each modal quantity, and adds the published value's own rounding. The gate is that the disagreement falls inside that band. The tolerance is therefore a property of the source document rather than of the author's patience.

One subtlety that is easy to get wrong and does change the answer: the step must be taken on the value AS PRINTED and then carried through whatever unit conversion the value went through. M_u* is printed per second-foot; taking half a unit in the last digit of its per-second-metre form understates its uncertainty by the conversion factor of 3.28.

Result. Six of the seven published modal quantities reproduce within that band: the spiral root, the roll-subsidence root, the Dutch-roll frequency and damping, the phugoid frequency, and the short-period frequency and damping. All five modes are also labelled correctly by participation factor alone, with scores between 0.717 and 0.993. The Dutch-roll period, published separately in Table II-10, agrees to within 1%.

The phugoid damping ratio: localised, and no longer open

Status changed. This was recorded as an unexplained discrepancy. It is now localised to the hand-assembled state matrix, and it does not appear in the full chain.

Two independent routes to the same published number:

The second route takes the report's NON-dimensional derivatives, builds a nonlinear aircraft, finds its trim, and perturbs it. It shares no arithmetic with the first beyond the source data, and it reproduces the published value.

So the discrepancy is in the hand assembly, not in the eigen-analysis and not in the published value. Substituting the chain's linearisation into the hand assembly ONE MATRIX ENTRY AT A TIME localises it the rest of the way: the single entry A(3,4) = M_wdot (-g sin theta)/(1 - Z_wdot) carries 98.8% of the gap, and zeroing that entry alone moves the hand-assembled damping ratio to 0.094829 — +0.03% from the published value, inside its own printing precision.

That entry is an artefact of closing Appendix C's descriptor form by substituting the WHOLE w_dot equation, gravity term included, into M_wdot w_dot. M_wdot multiplies the rate of change of AERODYNAMIC incidence; gravity acts at the c.g. and produces no moment about it, so no gravity term may reach q_dot at all. The chain gets exactly zero there because d(alphadot)/d(theta) = g sin(alpha - theta)/V, which vanishes identically in level flight.

The hand assembly therefore does not OMIT a term — it INCLUDES one that should not be there. What was ruled out on the way to that, and the two transcriptions that would settle it, are in the note on this discrepancy.

The regression lock on the hand-assembled route stays, and is now anchored to the trim-and-linearise result rather than to a bare measurement. It is kept DELIBERATELY: the hand assembly's worth is that it is a literal reading of Appendix C, independent of galata's own modelling opinions, and quietly correcting it would turn an independent check into a second implementation of the opinion it is supposed to be checking.

Detail of the hand-assembled discrepancy

The phugoid damping ratio does not reproduce within the source's precision. It disagrees by about 2% relative. Enumerating all 2^10 corners of the inputs' own printed rounding simultaneously gives zeta in [0.092521, 0.093219]: the published 0.0948 is not reachable from ANY corner, so the inputs' rounding does not explain it either.

The mode itself is in very nearly the right place. Published, the phugoid eigenvalue is -0.016306 ± 0.171225j; the hand assembly gives -0.015965 ± 0.171168j. The imaginary parts agree to a few parts in ten thousand. The disagreement is concentrated in a derived quantity: ζ = |Re| / |λ| divides a small real part by a small natural frequency, so a residual of a few times 1e-4 in the real part becomes 2.04% in the ratio — the entire discrepancy, accounted for but not explained.

Two things it is NOT, both checked rather than assumed:

1. The published value's own rounding. zeta = 0.0948 and omega_n = 0.172 are printed to three figures, so the published real part -zeta omega_n carries +/- 5.6e-5. The observed gap is 6.1 times that band, so the published number cannot be moved far enough to meet the assembly.
2. A units error in the per-foot scaling of the pitching-moment row. The obvious suspect, since that row is the one not invariant under a change of length unit — and eliminated by omega_n, which already agrees to 0.05%: omitting the scaling moves it to 0.13582 and applying it twice to 0.19001.

The earlier candidate — X_q left blank in the report and read as zero — is also numerically sufficient, at X_q = -0.217 m/s per rad/s. It is rejected as the primary explanation because it is FITTED to the one number, while the M_wdot term above predicts 98.8% of the gap with no free parameter. The two are separable by experiment, and the note says how.

The phugoid damping ratio stays listed as an open discrepancy rather than as validated, because acting on the finding is a decision about what the hand assembly is FOR and not an edit to a matrix. A regression lock — labelled as one, per charter rule 8 — holds the gap at its measured size so it cannot grow unnoticed, and fails if it shrinks.

Frequency response and stability margins

References. G. F. Franklin, J. D. Powell and A. Emami-Naeini, Feedback Control of Dynamic Systems, and K. J. Astrom and R. M. Murray, Feedback Systems, for the classical margins. A. J. Laub, "Efficient multivariable frequency response computations", IEEE TAC 26(2), 1981, for the Hessenberg method. P. Seiler, A. Packard and P. Gahinet, "An Introduction to Disk Margins", IEEE Control Systems Magazine 40(5), pp. 78-95, 2020, doi:10.1109/MCS.2020.3005277, for the disk margin — consulted as the authors' preprint https://arxiv.org/abs/2003.04771, since the published version is paywalled.

What is compared against what

Frequency response is gated against arithmetic, not a document. For a system whose transfer function can be written down, G(jw) is a ratio of polynomials evaluated at s = jw, and the state-space result must equal it to rounding. That is a stronger reference than any printed table, because it has no precision of its own to hide behind.

The same is true of two of the margin cases. The loop 1/(s(s+1)(s+2)) has a phase crossover at exactly sqrt(2) rad/s — the phase reaches -180 degrees when 1 - w^2/2 vanishes — and a gain margin of exactly 6, since |L(j sqrt2)| = 1/(sqrt2 sqrt3 sqrt6) = 1/6. The loop 1/(s(s+1)^2) crosses at exactly 1 rad/s with a gain margin of exactly 2. Their gain crossovers are the roots of stated cubics, solved in the test by an independent bisection.

The delay margin is checked twice over, and the second check is the one that matters: applying exactly the reported delay must put the loop exactly on the critical point, 1 + L(jw) e^(-jw tau) = 0. A delay margin computed by the wrong formula can still satisfy the formula it was computed from; it cannot satisfy this.

The disk margin

Gated against the tutorial's worked example, the loop L(s) = 25 / (s^3 + 10 s^2 + 10 s + 10).

Every allowance is measured rather than chosen, in tests/validation/reference/seiler2020_disk_margin.csv, which also records the exact location of each value in the source and the rights position. ADR-0007 sets out why scalar results from a copyrighted paper may be committed at all and what the limits on that are.

The strongest evidence here is not a number. The theorem's proof constructs a perturbation on the boundary of the disk that destabilises the loop, placing a closed-loop pole on the imaginary axis at the critical frequency. galata computes that perturbation and the test closes the loop with it: the pole has to land where the theorem says. That check fails for any transcription error in alpha, delta_0 or f_0, individually or together, in a way that comparing printed digits does not.

A gap that had to be closed deliberately. At skew 0 the factors (1 - sigma) and (1 + sigma) are both 1, so the published example says nothing about which is which. Swapping them in gamma_min was verified to leave every skew-0 result identical. The intercepts are therefore additionally gated against the disk parameterisation itself at six skews, and against the paper's published closed forms for the two named one-sided margins.

The guaranteed phase variation is the one output with no peer-reviewed reference: the paper derives the formula for phi_m but prints no number for this example. It is gated against MathWorks' published diskmargin output for the same loop, which is vendor documentation and is labelled as such in the table above and in the reference file.

A discrepancy in the source

The paper prints a critical frequency of 1.94 rad/s. Its own printed delta_0 = 0.212 - 0.406j and f_0 = 1.128 - 0.483j are evaluated at that frequency, and neither is reproduced there: at 1.94 the construction gives Re delta_0 = 0.1955, out by 16.5 units in the last figure the paper prints. Both are reproduced near 1.955 rad/s, which is where |S - 1/2| actually peaks, and which is the value galata reports.

The printed critical frequency is therefore inconsistent with the rest of the example. Because |S - 1/2| is flat near its maximum — a peak's location is always more weakly determined than its height — this costs almost nothing in the margin itself, which is why the published alpha, gamma_min and gamma_max all agree. A test asserts the inconsistency directly, so that if a future reading of the paper resolves it, the test fails rather than passing quietly.

What these margins do not establish

Gain and phase margins are separately-varied margins: the tolerable gain change with no phase change, and the reverse. A loop can show a comfortable figure for each and still be fragile to a small simultaneous change in both. That is the whole reason the disk margin is computed alongside them, and examples/nt33a-bank-loop-margins is built to show it on a real aircraft — a loop whose gain margin is infinite and whose disk margin is not.

Applied one loop at a time to a multi-loop system, all four are optimistic. galata has no structured singular value, so it has no MIMO disk margin.

The disk margin's peak is found by searching a frequency grid and refining, not by the exact Hamiltonian-eigenvalue method. A grid maximum is a lower bound on the true peak, so the reported margin is an upper bound on the true one: the error is in the optimistic direction. The grid is refined around the closed-loop system's own lightly damped modes, which is where such peaks are, and every result records the band and point count that were searched.

Singular values and the sensitivity peaks

References. S. Skogestad and I. Postlethwaite, Multivariable Feedback Control: Analysis and Design, 2nd ed., Wiley, 2005 — the definitions, the peaks M_S and M_T, and equations (2.47), (2.48) and (2.50). The first author hosts the book: https://folk.ntnu.no/skoge/book/ps/bookall.pdf. Also P. Seiler, A. Packard and P. Gahinet, An Introduction to Disk Margins, IEEE CSM 40(5), 2020, for the identity linking these peaks to the disk margin.

Why a MIMO system needs singular values

A SISO system has one gain at each frequency. A MIMO system has a range of them, because the gain depends on the DIRECTION of the input, and the singular values are exactly that range. The shipped example examples/nt33a-lateral-mimo makes the point on a real aeroplane: at 0.01 rad/s its principal gains are 43.8 and 0.458, a spread of about 96 to one at a single frequency.

Reading a MIMO system through its individual element transfer functions instead is not a weaker analysis, it is a misleading one — every element can be small while the largest gain is not. The unit test uses the smallest example that shows this: for [[1,1],[0,1]] every element has magnitude at most 1, and the singular values are the golden ratio 1.618 and its reciprocal.

What the singular values are gated against

Algebra, not a document. For a 2x2 matrix the singular values are the square roots of the eigenvalues of A^T A, which for the test cases come out in closed form: the golden ratio above, the element magnitudes of a diagonal system, and exactly one singular value equal to |G| for a SISO system — which is the check that ties this back to the already-validated frequency response.

The sensitivity peaks, and an identity between two of galata's own routes

M_S and M_T are computed WITHOUT inverting a matrix:

sigma_max(S) = sigma_max((I+L)^-1) = 1 / sigma_min(I + L)

so the sensitivity is read straight off the singular values of I + L. T comes from SOLVING (I + L) T = L. This matters because near the peak I + L is close to singular by definition — that is what a peak in S IS — so it is the one place where forming an inverse would destroy the digits being reported.

The strongest evidence for these numbers is not a comparison against a document at all. Seiler, Packard and Gahinet prove that the disk margin at skew +1 is 1/M_S and at skew -1 is 1/M_T. galata computes those two quantities by routes that share nothing below the frequency response — one takes the peak of a scalar sensitivity, the other inverts the smallest singular value of a matrix — and they agree to below 1e-12 relative. For the tutorial loop:

The shortest distance from that loop's Nyquist curve to the critical point is 0.402146, which is 1/M_S — the geometric statement the book makes on its page 36.

The margins M_S guarantees, and the scope of that guarantee

Skogestad and Postlethwaite, equations (2.47) and (2.48), printed page 36:

GM >= M_S/(M_S - 1)    PM >= 2 arcsin(1/(2 M_S)) >= 1/M_S   [rad]
GM >= 1 + 1/M_T        PM >= 2 arcsin(1/(2 M_T)) >= 1/M_T   [rad]

The [rad] is printed on the equations themselves. Note the two gain-margin bounds have DIFFERENT functional forms, which is easy to blur from memory: at M_S = M_T = 2 they give 2 and 1.5 respectively, and the book prints both.

Reproducing those two sentences is the weaker half of the validation. The stronger half is that the inequalities are checked to BOUND real loops: across four loop gains, the margins galata measures independently are at least what the peaks promise. Equation (2.50) — that |S| and |T| are equal at the gain crossover and both equal 1/(2 sin(PM/2)) — is an exact identity, and it ties three separate parts of galata together at a single frequency.

These bounds are SISO only, and that is the source's own scope. Equations (2.47) and (2.48) sit in a chapter whose stated remit is SISO, and the book never restates them for MIMO. It goes further: its spinning-satellite example shows a plant with excellent margins "when considering one loop at a time" that is destabilised by small SIMULTANEOUS input gain errors. galata therefore refuses to compute these bounds for a loop that is not 1x1, and says so in the report rather than omitting them silently.

What these do not establish

Singular values carry no phase, and there is no MIMO equivalent of a Bode phase plot here. They are also not eigenvalues: a system whose eigenvalues are all small can have a large sigma_max, and reading a singular value plot as a pole plot is a mistake the two have no relationship to prevent.

Every peak in this section is a GRID MAXIMUM and therefore a LOWER bound on the true H-infinity norm — the exact computation is the Hamiltonian-eigenvalue method (Boyd & Balakrishnan; Bruinsma & Steinbuch), which galata does not have. For a robustness margin that error is optimistic. The tests demonstrate the shortfall rather than hiding it: 1/(s+1) has an H-infinity norm of exactly 1 attained at zero frequency, which no logarithmic grid contains, and the reported peak climbs towards it from below as the sweep is widened.

Finally, the condition number depends on how inputs and outputs are SCALED. Comparing it between two models scaled differently compares the scalings, and strict SI does not make a metre and a radian commensurate.

Generated from galata 0.2.0 by tools/validation/report_main.cpp.