A mode that rings is counted twice

The bound that is known in advance found something unusual for a bound. Balanced truncation promises that the reduced model’s error, measured as the largest gain of the difference over all frequencies, is at most twice the sum of the Hankel singular values it throws away. On the collection’s model of heat flowing along a rod, that bound over the measured error was 1.0000 at every order, to four decimal places in every sweep — “not a bound being attained by luck”, the essay said, “a formula for the error”. It traced the equality to two facts: removing exactly one state costs exactly twice its σ, and on a spectrum falling that fast the removals do not interact. And it noted one place where the sum was loose — by 1.52, on a model with two nearly equal σ — and the state that is removed is not a mode went on to show that the looseness followed the signs of the residues rather than the spacing.

Every system in those essays had real poles. A heat mode decays without oscillating, and each carries one Hankel singular value. The most common systems a reduction is asked to handle — structures, circuits, anything with inertia — have lightly damped modes, poles in complex pairs close to the imaginary axis, and a mode like that rings. This essay asks what the bound does on them.

Six oscillators #

The system has twelve states: six modes, each a 2 × 2 block with poles $-\zeta\omega \pm i\omega\sqrt{1-\zeta^2}$, frequencies ω spread geometrically from 1 to 30, one input and one output coupling to both coordinates of every mode with seeded weights of mixed sign. The damping ratio ζ is the same for every mode and is set to 0.7, 0.3, 0.1, 0.03 and 0.01 — from a mode that barely overshoots to one that rings for a hundred cycles. The Gramians are solved as the Kronecker system $(I \otimes A + A \otimes I)\operatorname{vec}P = -\operatorname{vec}BB^{\mathsf T}$, since the collection’s Lyapunov solver works in the real Schur form and refuses its 2 × 2 blocks, and the Hankel singular values come from the square-root route the product nobody had to form recommended. Each order from 1 to 11 is reduced by balanced truncation, and the error’s largest gain is found on a grid of six thousand frequencies between 0.1 and 300, refined by golden section at the best — fine enough to put several points inside the narrowest resonance, which at ζ = 0.01 and ω = 1 is 0.02 wide.

Loose at every order but the last #

At ζ = 0.1 the bound over the error is 3.23, 4.53, 3.52, 4.47, 4.55, 1.75, 2.43, 3.16, 2.01 and 2.13 at orders 1 to 10, and 1.000 at order 11. The last number is the one-state result, and it holds exactly at every damping: removing only the smallest σ costs twice that σ, as it did on every system the earlier essays drew. Every other order over-states. Turn the dial: at ζ = 0.7, where the modes barely ring, the ratios run from 1.26 to 3.43; at 0.3 from 2.02 to 5.05; at 0.03 from 1.89 to 4.84; at 0.01 from 1.86 to 5.00. Nowhere below order 11 is the bound within a quarter of the error, and at its worst it is five times it.

So the equality was not a property of balanced truncation on “ordinary problems”. It was a property of a problem whose modes do not ring, and the moment they do — even at a damping ratio of 0.7 — the sum over-states.

Two singular values for every mode #

The singular values show where the extra comes from. At ζ = 0.7 they fall steadily, 3.56, 0.33, 0.23, 0.087 and so on, with no structure a reader would notice. As the damping falls they arrive in pairs. At ζ = 0.01 they are 120 and 118, then 63.9 and 62.7, then 24.4, 23.9, 23.2 and 22.9, then 10.1 and 9.88, then 4.28 and 4.19: one pair for each mode, whose two members agree to within a few per cent, and two modes whose pairs happen to sit close together. A lightly damped mode is a two-dimensional state space that the input drives and the output sees almost equally in every direction, because the state rotates through the plane at the mode’s frequency; balancing finds two directions of nearly equal importance, and the Hankel singular values say so.

The bound adds both. Removing a mode removes both of its σ, and the bound charges twice their sum — four times one of them. The error, measured, is not four times one of them.

Removing a mode costs one σ #

Take the two smallest σ, the last mode, and remove both. At ζ = 0.7 the bound over the error is 1.40; at 0.3, 2.02; at 0.1, 2.13; at 0.03, 2.02; at 0.01, 2.002. As the damping falls the ratio settles on two: the mode’s two nearly equal σ cost, together, what one of them would. At ζ = 0.01 the last mode’s σ are 4.28 and 4.19, the bound is 16.9, and the error is 8.46 — twice their average.

The reason is the same as the one-state result’s. In balanced coordinates the error from removing a set of states is the gain of the discarded part, and the one-state result says a single balanced state’s contribution peaks at exactly twice its σ. A lightly damped mode’s two states are not independent contributions that peak separately: they are one rotation, whose gain at the mode’s frequency is the gain of a single resonance. Their two σ describe how much the rotating state is worth along two orthogonal axes; the resonance they make together is worth about twice either. The sum treats them as two things, and they are one.

On the heat model every mode is one state with one σ, and there is nothing to double-count. That is the whole of the difference between the earlier essays’ equality and this essay’s factor of two.

Where the error sits #

The frequency at which the error peaks says the same thing from the other side. The six modes sit at 1.00, 1.97, 3.90, 7.70, 15.2 and 30.0. At ζ = 0.03 the error of the order-1 model peaks at 1.00; of orders 2 and 3, at 1.97; of orders 4, 5 and 6, at 3.88; of order 7 at 4.28, just beside the third mode; of orders 8 and 9 at 15.17; and of orders 10 and 11 at 29.98. Every peak is on a mode, or beside one, and within each run of orders the peak stays on the same mode while the order changes — the error is the gain of the largest resonance the reduced model has lost or damaged, and cutting a further state elsewhere does not move it.

The error of a reduced model of a lightly damped system is a resonance, and a resonance is the gain of a mode, not of a state. That is why the balanced coordinates’ own bookkeeping, one σ per state, over-counts: the quantity the error is made of comes one per mode.

The order of the modes in the Hankel singular values is not their order in frequency, which is worth noticing. The largest σ belong to the lowest frequencies here, because the input’s weights are comparable and a low-frequency mode’s resonant gain, $1/(2\zeta\omega^2)$ for a unit mass, is larger; the two σ pairs near 23 at ζ = 0.01 are the third and fourth modes, whose gains happen to be close. Small compared to what made the general point that a σ needs the system’s scale beside it before it decides anything; on oscillators it also needs the mode it belongs to.

Why the over-statement reaches five #

A factor of two per mode explains the last mode and orders that cut whole modes. The larger factors, up to five, come from orders that cut several modes, or that cut a mode in half. At ζ = 0.01, order 4 keeps the two largest modes and removes the four others, and the bound over the error is 5.00: eight σ are summed and the error is the largest single resonance among the discarded modes, roughly twice the largest discarded σ. Removing several modes whose resonances sit at different frequencies costs the largest of them, not their sum, because a gain over frequency is a maximum and the discarded modes peak at different places. The earlier essays met this as the interaction between removals; on oscillators the interaction is not a correction but the main effect, since each discarded mode has its own frequency and the maxima do not add.

Odd orders cut a mode in half, keeping one of its two nearly equal σ. At ζ = 0.01, order 3 keeps the first mode and half of the second, and the ratio is 2.98: the half-mode left behind is a state the reduced model cannot use well, and the error is dominated by the second mode’s resonance, which neither side of the cut now reproduces.

One resonance, not a sum: the reading checked at every even order #

Those two observations make a rule that can be checked against the sweep without running anything new, because the sweep already records, at every order, the bound and its ratio to the error, and so the error itself. At an even order balanced truncation cuts whole pairs. Take the discarded pairs, and charge only the largest: twice the larger σ of the biggest pair cut. Call it the resonance reading. Set it beside the measured error at orders 2, 4, 6, 8 and 10.

At ζ = 0.01 it is 127.8 against a measured 125.7, 48.8 against 49.1, 46.4 against 46.2, 20.2 against 19.9, and 8.56 against 8.46 — within two per cent at every order, where the bound was out by factors of 2.0 to 5.0. At ζ = 0.03 it is within five per cent at every even order, from 1.008 to 1.053 times the error. Charging every discarded pair once, rather than only the largest, does much worse — between 1.45 and 2.52 times the error at ζ = 0.01 — which is the second observation in numbers: the maxima over frequency do not add.

The reading is not a bound, and the sweep says so on both sides. At ζ = 0.01 and order 4 it is 0.993 of the error, below it by less than a per cent but below it. And at ζ = 0.1 it falls apart: 1.24, 1.13, 0.54, 1.17 and 1.14 at the five even orders. At that damping the σ are no longer paired — 13.2, 10.4, 6.15, 5.35, 2.78, 1.95, 1.94, 1.79 — so “the larger of the pair” names two values that belong to different modes, and at order 6 the pairs the rule sees are not the modes the cut removes. It needs the pairs to be pairs, which is to say it needs the damping to be light, and it needs the resonances to sit apart; at ζ = 0.1 neither is quite true of this system.

That is worth stating plainly, because the rule is tempting. A formula accurate to two per cent at light damping would let the order be read off a tolerance as on the heat model; but its failure at ζ = 0.1 has no warning in it except the σ themselves, and a reader has to look at them to see whether they arrive in pairs before trusting it. The bound needs no such look, which is the whole of what it is for.

What the other methods would have done #

Balanced truncation is not the only way to reduce these systems, and the pairing bears on the alternatives too. Exact at the points that were named described the other kind of reduction, interpolation at chosen points, which bounds nothing and costs a linear solve per point; interpolating at the model’s own poles found its best-known variant within half a per cent of balanced truncation on the heat model. On a lightly damped system, interpolation’s natural points are the resonances themselves, one per mode, which is the counting the error follows; an interpolant that matches the model at each kept mode’s frequency keeps each kept resonance exactly, and its error is then the discarded resonances, as balanced truncation’s is. Whether the two methods are as close on these systems as on the heat model is not measured here.

What is measured is that the bound, the one thing balanced truncation offers that interpolation does not, is weaker here than on the heat model by the factors above. Why a Gramian can be truncated at all found the Gramians’ eigenvalues falling off a cliff on the heat model; on lightly damped oscillators they fall in steps of two, and the bound sums each step twice.

What the bound is for, on a system that rings #

None of this makes the bound wrong; it is a theorem and it holds at every order here. What changes is the use the earlier essay found for it. On the heat model, a bound equal to the error lets the order be read off from a tolerance: pick the smallest r whose tail sum is below the tolerance, and the error will be exactly that. On six oscillators the same rule picks an order up to five times too conservative in the error and never too lax, and the over-statement is predictable: about two for every lightly damped mode cut, and at most the number of modes cut when their resonances sit apart.

The resonance reading above is the sharper rule the measurement suggests, and it shows both sides of the trade: within two per cent at ζ = 0.01, half the error at one order at ζ = 0.1. It is a reading of what these measurements show, not a bound, and the bound’s over-statement is the price of never needing the reading’s precondition. Bracketing an error nobody can measure found the Gramian’s own error sitting a steady factor below its rational bound; this is the same lesson for the reduction’s error — a bound worth calibrating against the kind of system it is used on.

Six modes, one damping, one input #

One system of six modes, one input and one output, frequencies spread over a factor of thirty, and every mode with the same damping. Modes with different dampings would pair their σ unevenly; modes at nearly the same frequency, like the third and fourth here, would make the “one resonance per mode” reading blur into one resonance for two modes. The weights coupling input and output to the modes have mixed signs, which the state that is removed is not a mode found to matter on real poles; whether all-positive coupling changes the factors here is not measured.

The H∞ error is found on a grid of six thousand frequencies with a golden-section refinement at the best, which puts several points inside the narrowest resonance; a peak narrower than the grid’s spacing near a frequency the grid does not favour would be under-measured, and the one-state ratios of exactly 1.000 at every damping are the check that it is not.

Still open: where the pairing ends, and a system with modes in a cluster #

When the pairs stop being pairs. Twice the largest discarded pair’s larger σ matched the error to two per cent at ζ = 0.01 and five at 0.03, and missed it by half at 0.1, where the σ no longer pair. Somewhere between those dampings the reading stops working, and where depends on how far apart the modes’ resonances sit against their widths, $2\zeta\omega$. Whether the crossing can be predicted from the σ alone — from how closely consecutive values agree — is the measurement that would say when a lightly damped system has a formula for its reduction error, as the heat model did.

Modes in a cluster. The third and fourth modes here have σ within a few per cent of each other across the pair — four nearly equal values. A system built with several modes at nearly the same frequency, as a periodic structure has, would make the pairing ambiguous and the resonances overlap. Whether the bound’s over-statement there grows with the size of the cluster, or collapses back towards the pair’s factor of two, is the case a structural engineer would meet first.