A tie is a gap smaller than its neighbours

Where a ringing mode stops being a pair found one failure of balanced truncation that was not gradual. On a six-mode oscillator at dampings 0.07 and 0.1, the reduction to six states cut between two Hankel singular values that agreed to under one per cent, and came back with a pole four to eight hundred times closer to the imaginary axis than any pole of the system it came from. Its error was a spike at zero frequency, twice what the dropped σ read. The essay left a warning and a test for it. Cutting at five or seven instead keeps or drops both members of the tie; “whether either of those reductions keeps its poles near the full model’s at every damping — and what the extra or missing state costs in error — is one more pair of orders on the same sweep, and would turn the warning into a rule: never cut inside a tie.”

The pair of orders answers that case at once, and the figure at the top is what the rule needs before it can be a rule: a census. One system is one place where a cut went wrong. A rule has to say what a tie is, and a census is what says whether the definition catches the failures and how much it costs where there was nothing to catch.

The census #

The systems are the earlier essay’s construction — six modes with natural frequencies spaced evenly in logarithm from 1 to 30, each a lightly damped pair of poles $-\zeta\omega \pm i\omega\sqrt{1-\zeta^2}$, with input and output weights drawn at random — at twelve seeds, of which seed 3 is the earlier essay’s system. Each is taken at the same eleven dampings from 0.005 to 0.5 and reduced to every order from 1 to 11. That is 1,452 balanced truncations, odd orders included, because an odd order cuts inside a mode’s own pair and the question is about cuts between any two singular values.

For each reduction three things are recorded. The H∞ error, over twice the first dropped singular value $2\sigma_{r+1}$, which the earlier essays established as the reading: a reduction whose error is about one reading has done what its σ promised. A reduction whose error is more than one and a half readings is called doubled, because every one of them turned out to be close to two. The slowest pole of the reduced model, over the damping, since every pole of the full model is at least $\zeta$ from the axis. And the gap at the cut, $\sigma_r/\sigma_{r+1} - 1$, with the same gap measured relative to the smaller of the two gaps beside it.

Ten reductions are doubled. They come from five of the twelve systems, at dampings from 0.03 to 0.3, at orders four, six and eight, and their errors are 1.69 to 1.93 readings. The other 1,442 run from 0.56 to 1.42, and 1,276 of them are within a fifth of one — mostly just below it, which is the earlier essay’s sum-of-the-pair effect at even orders. Doubling is not the tail of a distribution here. It is a separate population.

One system, every order #

On the earlier system at damping 0.1 the error tracks the reading at every order but six. At six it jumps to 7.13 against a reading of 3.87; at five it is 3.59 and at seven 3.54. Both sides of the tie are clean. Neither has a slow pole — the slowest are at 0.58 and 0.57 of the damping — and the extra state of order seven buys 1.5 per cent of error over order five. At damping 0.07 the numbers are 5.67 at five, 11.3 at six and 5.67 at seven: two more states for nothing.

That is the first answer to the question as posed, and it has a shape worth stating before the census generalises it. The two members of the tie belong to two different modes; the earlier essay identified them as the third and fourth modes, whose σ nearly coincide. Keeping one member keeps half of a mode, which buys nothing a reading can see. Keeping both keeps half of each of two modes, which buys almost nothing either. The tie’s own states are worth about what the singular values say — the same — and so a cut that keeps one of them and drops the other is a cut that the σ cannot rank.

Turn the dial down to 0.005 and the staircase appears that a mode that rings is counted twice explained: the error falls only at even orders, because each mode’s two singular values agree to about $2\zeta$ — one per cent at this damping — and every odd order cuts inside a pair. Those cuts are ties too, by any definition that looks at the gap, and they are harmless: the error at order five is 96.5 against 97.4 at four. A tie inside one mode wastes a state; a tie between two modes can double the error. The census has to tell them apart.

What a tie is #

The warning as written was a gap under one per cent. In the census 65 cuts satisfy it and seven of the ten doubled reductions are among them. The other three have gaps of 1.85, 2.13 and 2.18 per cent, at dampings 0.2 and 0.3, and a one-per-cent rule would cut there without complaint.

What those three share with the seven is visible in the figure at the top. Plotted against the gap at the cut divided by the smaller of the gaps on either side, every doubled reduction lies to the left of a fifth: from 0.007 to 0.168. A gap of two per cent at damping 0.3 sits between gaps of thirteen per cent and more, where the modes’ own pairs have split by about $2\zeta$; it is a tie because its neighbours are not. A gap of one per cent at damping 0.005 sits between gaps of close to a hundred per cent, and is a tie for the same reason, but of the harmless kind.

So the relative definition catches all ten, and it pays for that by catching a great deal else: 431 of the 1,452 cuts are ties by it — 68 at each of the two lightest dampings, where every odd order is one, falling to 17 at 0.1. A rule that refuses to cut at any of them refuses thirty per cent of all orders. Whether that matters depends on what refusing costs, which is the next measurement.

The near-integrator was a symptom #

The earlier essay’s failure came with a pole near the axis, and it is natural to make the pole the thing to test for: balanced truncation guarantees a stable reduced model and nothing about how stable, and a pole at $10^{-4}$ of the damping is a model that rings for ten thousand of the full model’s time constants. Thirty-two reductions in the census have a pole closer to the axis than a tenth of the damping. All ten doubled reductions are among them.

The other twenty-two are not doubled. Their errors run from 0.75 to 1.42 readings, and one of them has its slowest pole at $4.4\cdot 10^{-4}$ of the damping — slower than seven of the ten that doubled. Most sit at damping 0.5, where the modes are barely oscillatory and the σ no longer come in pairs, or at order eleven, where the cut drops a single σ and the reduced model can afford a slow pole whose residue is small. A pole near the axis is necessary for the failure in this census and nowhere near sufficient for it. A slow pole with a small residue contributes a narrow spike of small height; a slow pole with a large one is the failure. The pole locates the failure without measuring it, which is the same distinction a model that cannot be run drew for a reduced model whose construction had done exactly what it promised: the property a construction guarantees is not the property a user needed.

Which side to cut #

Moving every tied cut one order down — dropping both members of the tie — is never doubled in the census and never leaves a near-integrator. Each of the ten doubled errors falls to between 0.46 and 0.50 of what it was, which is the reading: the doubled cuts were paying twice and the cut below pays once. On the 340 ties at dampings up to 0.3 the median cost of the move is one per cent and the worst eleven, because one order fewer is one singular value more dropped and at a tie that singular value is nearly equal to the one that was already being paid for.

At damping 0.5 the same move costs up to 2.47 times, on four of the twenty-three ties there by more than twelve per cent. At that damping the modes have stopped pairing, the σ fall without structure, and a gap that is small against its neighbours is no longer two modes’ values coinciding; it is one mode’s single value next to another’s, and dropping it drops a mode. The relative definition was built from lightly damped systems and it describes them. On heavily damped ones it flags cuts that need no flagging and the remedy is not free.

The cut above keeps both members of the tie, and it is safe too: never doubled, and two harmless near-integrators. What it buys over the cut below, at the price of two more states, depends on what kind of tie it is. Where the tie is a mode’s own pair — a gap within thirty per cent of $2\zeta$ — keeping both members keeps the whole mode, and the error halves: a median of 0.495 over 198 such ties. Where the tie is between two modes’ values the two extra states are half of each of two modes, and the median buys 0.68, anywhere from five times better to slightly worse.

So the rule the earlier essay hoped for has a direction. At a tie between two modes, cut below it: one state fewer than the tied cut, at most eleven per cent worse to damping 0.3, never doubled. If two more states are affordable, cut above it, and expect about a third off. Never cut inside it. That rule reads nothing but the singular values, which are in hand before any reduced model is built.

A check that needs no threshold #

The relative definition has a threshold, a fifth, chosen by looking at where the doubled cuts fell, and a threshold chosen that way is a fit to this census. There is a test that has no threshold to fit. Every one of the ten doubled reductions has its largest error at zero frequency, $\omega = 0$ — the spike the earlier essay found, in every case — and so every one of them sits on the diagonal of this figure. And no other reduction has an error at zero frequency above 1.42 readings.

The error at zero frequency is one evaluation: $H(0) = -CA^{-1}B$ for the full model, the same for the reduced one, and their difference. A reduction that compares that number with $2\sigma_{r+1}$ and moves its cut down when the ratio is above one and a half catches all ten doubled reductions in the census and touches nothing else. It costs a solve with the full model’s state matrix, which a code that built the Gramians has already had to decompose, and it needs no definition of a tie at all.

It is narrower than the singular-value rule in one way that matters. It tests for the failure this census contains, which always lands at zero frequency because a near-integrator is a pole near the origin and its spike is centred there. A tie could misbehave elsewhere — a reduced pole pulled close to the axis at a resonance rather than at zero — and the check would not see it. In 1,452 reductions it did not happen, which is a measurement of these systems rather than a theorem about balanced truncation.

What a reduction code would do #

A balanced truncation code chooses its order one of two ways: the user names it, or the solver takes the smallest order whose bound, twice the dropped σ summed, is under a tolerance. Both choose a cut without looking at the gap there, and the second is the more exposed, because a tolerance is crossed wherever the sum happens to cross it and the σ near a cluster are the slowly falling ones a sum crosses inside. On the earlier system at damping 0.1 a tolerance between 12.5 and 16.4 lands on order six, the doubled cut.

The census suggests three lines on top of either choice. After the order is chosen, compute the gap at the cut and the gaps beside it; if the cut is a tie and the damping is light, move it down one order. Then evaluate the error at zero frequency and compare it with twice the first dropped σ; if it is more than one and a half times that, move down again. And report the order actually used, since a user who named six and received five should know why. None of the three needs the reduced model’s frequency response, which is the expensive part of measuring a reduction, and the second needs no threshold that this census chose.

Two things the procedure does not do. It does not certify the reduced model’s stability margin, which half the conditions and a certificate showed is a property some reductions can prove and balanced truncation does not try to. And it does not make the order a statement about how reducible the system is: a condition number that is not the model’s found that the quantities a code has to hand while balancing say little about that, and a tie is one more place where the order a reduction picks reflects its own arithmetic rather than the system’s structure.

What the bound was saying all along #

The bound that is known in advance introduced balanced truncation through its a priori bound, twice the sum of the dropped σ, and the doubled reductions do not violate it: at order six on the earlier system the bound is 12.5 and the error 7.13. What they violate is the reading the later essays found inside the bound, and the census says the reading’s failures are not scattered. They are one mechanism at one kind of cut. The state that is removed is not a mode warned that a balanced state is not a physical mode and its σ is not a mode’s size; a tie is where that warning bites, because the balancing transformation is not unique inside a tied pair’s span, and which combination the truncation keeps is decided by rounding rather than by the system. The product nobody had to form computed those σ to full relative accuracy; it is precisely their accuracy that makes a gap of 0.4 per cent visible and so makes a rule possible.

What twelve systems do not show #

One family: six modes, logarithmically spaced, single input and output, random positive weights. The doubled cuts come from five of the twelve systems, so the census is enough to say what they share and too small to give a rate. Systems with modes clustered on purpose, multi-input systems whose σ come in larger groups, and dampings between 0.3 and 0.5 where the definition stops describing the systems are all unmeasured.

Still open: clusters on purpose, and where the definition stops #

A cluster of modes. A periodic structure has several modes at nearly the same frequency, and every cut near the cluster is a cut near a tie. The prediction with a sign is that the doubled cuts become common there — more than one system in three, against five in twelve here with at most four doubled cuts each — and that the zero-frequency check still catches every one, because the mechanism is still a near-integrator.

Where the definition stops. The relative definition costs nothing to damping 0.3 and up to two and a half times at 0.5. Between the two the σ stop pairing, and somewhere there a gap small against its neighbours stops meaning two modes coinciding. Whether the crossover can be read from the σ themselves — from whether the values beside the cut still come in pairs split by about $2\zeta$ — is the measurement that would make the singular-value rule as safe as the zero-frequency check.