The actuator that arrives late

The mass that helps by being late found what a tuned mass damper is: a small mass on a spring, tuned so that it moves a quarter of a cycle behind the structure, which puts its force on the structure in phase with the structure’s velocity and so takes energy out of it. Three per cent of a building’s mass cut its peak response sevenfold. The essay ended on two directions, one of them upward in sophistication: replace the spring and the dashpot with an actuator that pushes wherever a controller says, unlimited by tuning and by the size of the mass, and with a failure mode no passive device has — a control system that adds energy at the wrong moment.

This essay is that failure mode, measured. The simplest active damper there is — an actuator that pushes against the structure’s velocity in proportion to it — is perfect on paper and unconditionally stable. Put a delay between the measurement and the push, as every real sensor, computer and hydraulic valve does, and the arithmetic changes character. The quarter-cycle that makes a passive mass work is exactly the delay that makes an active one fail, and the delay a real loop has is a fixed time while the quarter-cycle it must stay under is a fraction of each structure’s own period.

The same push, arriving later #

The structure throughout is a single mode with 1 per cent of critical damping, the value the only thing that stops it found typical of a steel or concrete frame in service. Under a harmonic force at its natural frequency it responds 50 times as much as under the same force applied slowly. The actuator applies a force $-g\,v(t - \tau)$: proportional to the velocity the sensor measured a time $\tau$ ago. The gain is expressed as the damping ratio the force would add with no delay, $\zeta_a = g/2m\omega$, so that a gain of 0.10 is an actuator that, on time, would take the mode from 1 to 11 per cent of critical.

At the natural frequency a delay $\tau$ is a phase lag of $2\pi\tau/T$. The delayed force can be split into a part in phase with the velocity, which is damping, and a part in phase with the displacement, which is stiffness.

The damping share is the cosine of the phase lag. A delay of a tenth of the period costs a fifth of the damping, a fifth costs two thirds of it, and a quarter costs all of it: an actuator told to oppose the velocity and arriving a quarter-cycle late is opposing the displacement instead, which is a spring, and a spring removes no energy. Past a quarter-cycle the cosine is negative and the force is in phase with the velocity it was meant to oppose. The actuator is then doing positive work on the structure every cycle — the definition of a motion that feeds itself.

The comparison with the tuned mass is exact and inverted. The passive mass’s force is in phase with the structure’s velocity because the mass is a quarter-cycle behind the structure’s displacement: displacement lags velocity by a quarter-cycle, so a mass lagging the displacement by another quarter pushes against the velocity. The actuator measures the velocity directly and is supposed to push against it at once; every quarter-cycle of lag it acquires moves its force a quarter-cycle away from where it should be. The passive device is designed around a quarter-cycle of lag; the active device is destroyed by one.

A blow, and the energy the controller adds #

The plainest demonstration is a free vibration. Strike the mode and let the feedback act. On time, a gain of 0.10 takes the motion down to 7 per cent of its starting amplitude within a dozen cycles, as an 11 per cent damped structure should. A quarter-period late, the same gain leaves the motion growing — 1.3 times larger at the end than at the start — because the feedback’s damping is nil, its stiffness has moved the mode’s frequency up to where the lag is more than a quarter-cycle, and the small negative damping there has overwhelmed the structure’s own 1 per cent. The controller is supplying the energy it was installed to remove.

Every delay sets a largest gain #

The growth in the second trace is an instability, and it has a threshold. For each delay there is a gain above which the controlled structure vibrates by itself, and below which it is stable however badly it performs. The threshold follows from asking when the loop can sustain a vibration with no force at all: at some frequency the feedback’s negative damping must exactly cancel the structure’s own, and its added stiffness must exactly account for the frequency. Two conditions, one unknown frequency and one gain, which together give the critical gain in closed form for each delay.

With no delay there is no limit. With a delay of a twentieth of the period the limit is 2.41 — a gain no actuator would be given, so the delay is harmless. At a tenth of the period the limit is 1.06, still comfortably above any practical gain. Then it falls quickly: 0.25 at a fifth, 0.079 at a quarter, 0.028 at three tenths, where it has nearly reached the mode’s own 1 per cent. A delay of a quarter-period does not destabilise every gain, as the phase argument alone would suggest, because the feedback’s stiffness pushes the frequency up and the instability happens at a frequency where the lag is larger than a quarter-cycle; but it caps the gain at under a tenth of critical, which is where a practical design would want to be.

Gain helps until the delay takes it back #

Plotted against the gain, the peak response shows what the limit costs before it is reached. With no delay more gain is always better: at a gain of 0.30 the peak amplification is 1.7, against 50 uncontrolled. With a delay of a tenth of the period the curve is almost the same. At 0.15 it turns up before the end of the range, and at 0.20 it has a clear minimum — 15.5, at a gain around 0.12 — beyond which more gain makes the peak worse and then, at 0.25, makes the structure unstable. For a delayed loop the gain is a design choice with an optimum, not a knob to turn up, and the optimum is set by the delay.

The worse peak at high gain is not at the natural frequency.

A delayed loop grows a resonance of its own. As the gain rises the natural peak is flattened, as intended, but a second peak appears above it — at 1.3 to 1.7 times the natural frequency for this delay — where the lag is past a quarter-cycle and the feedback is adding energy. At a gain of 0.50 that second peak is at 11, higher than the natural one it was installed to suppress; at 0.55 it becomes an unbounded oscillation. The instability is not a failure of the structure’s mode but the birth of a new one, belonging to the structure and the controller together, and it happens at a frequency the structure alone would not have.

What a bounded actuator can buy #

A passive tuned mass is limited by its mass: the essay on it found an optimally tuned 3 per cent mass takes the peak amplification to 8.2 and a 1 per cent mass to 14.2. An actuator is limited by its force, which caps the useful gain. With the gain capped at 0.30 of critical — a strong actuator — the comparison depends entirely on the delay.

With no delay the bounded actuator reaches a peak of 1.7, five times better than a 3 per cent passive mass. At a tenth of the period it is almost as good, 1.8. Then the curve rises steeply: past 0.18 of the period the actuator is worse than a 3 per cent passive mass, past a fifth worse than a 1 per cent one, and at a quarter it achieves 44.7 against an uncontrolled 50 — nothing, at the cost of an actuator, a power supply and a controller. The whole advantage of active control lies in the first tenth of a period of delay.

One loop, a tower and a floor #

A delay is a fixed time — the sum of a sensor’s filter, a controller’s cycle and an actuator’s response, a few tens of milliseconds for a fast hydraulic system — and a period is a property of the structure. The same loop is therefore early or late depending on what it is attached to.

A tall building swaying at 0.2 Hz has a period of five seconds, and even a slow loop is a hundredth of a period late: the actuator works as designed. A footbridge at 2 Hz has a half-second period, and a 25 ms loop is a twentieth of it — still fine. It is on tall buildings and towers, with periods of seconds, that active and hybrid mass dampers have mostly been installed. A floor at 8 Hz has a period of 125 ms; a 25 ms loop is a fifth of that, and the peak amplification rises from 4.6 to 19; a 50 ms loop cannot be used at all, since the mode is unstable at that gain from 4.9 Hz upward. A floor that is strong and unusable is precisely the structure an engineer would like to fix without adding steel, and precisely the one a slow active system cannot help.

The frequency at which a given loop stops helping is a fixed fraction of one over its delay: about a fifth of $1/\tau$ for the damping to have mostly gone. That is the rule the figure compresses, and it is why the same controller that is the right answer for a skyscraper’s sway is the wrong one for a laboratory floor.

Where the delay comes from #

No single component of a control loop is slow. The delay is a sum of small ones, and each is small only against a long period. A digital controller samples the sensor and holds its output for a sample interval, which behaves like half an interval of pure delay. The sensor’s signal is filtered before it is sampled, to keep high-frequency noise from folding down into the band being controlled, and a filter is a phase lag that grows with frequency. If the sensor is an accelerometer, as it usually is, the velocity is obtained by integrating its signal, and the integration has to be filtered at low frequencies to stop it drifting, which adds lag of its own. And the actuator does not produce its force the instant it is told: a hydraulic ram responds through a servo-valve and the compressibility of its oil, an electric one through its inductance and its gearing, and either is a lag of milliseconds to tens of milliseconds.

The figure’s three loops — 10, 25 and 50 ms — span that range from a fast, carefully built system to an ordinary one. None of it matters to a tower. All of it matters to a floor, and the component that dominates is usually the actuator, which is also the component chosen last, by force and stroke, with its speed an afterthought.

The crowd that is a controller with the wrong sign #

The quarter-cycle has already been met in this collection, from the other side. The bridge that was pushed by its own sway found that people walking on a deck that sways sideways adjust their stride to it, and that the sideways force their feet then apply is partly in phase with the deck’s velocity. In the language of this essay, the crowd is a velocity-feedback controller whose gain has the wrong sign: it adds negative damping in proportion to the number of people, and above a critical number it overwhelms the deck’s own damping and the sway grows by itself.

The two cases are the same equation. An actuator with a positive gain and a delay beyond a quarter-cycle puts its force in phase with the velocity; a crowd whose response to the sway happens to put its force in phase with the velocity does the same without any delay at all. Either way the structure’s energy balance acquires a term that pumps rather than drains, and the threshold — a critical gain, a critical crowd — is where the pumping first equals the structure’s own losses. The cure that worked for the bridge was dampers, passive ones, which added enough positive damping to move the threshold beyond any crowd the deck could hold. An active cure would have had to be fast enough that its own lag stayed well under a quarter of the sway’s period, which at the bridge’s sway frequency of about 1 Hz was the easy case; for a crowd jumping on a grandstand at several hertz it would not be.

The quarter-period, by hand #

For the floor at 8 Hz and a 25 ms loop: the period is 125 ms, so the delay is $0.025/0.125 = 0.20$ of it and the phase lag at the natural frequency is $2\pi \times 0.20 = 1.26$ radians, 72°. A gain of 0.10 then adds damping of $0.10 \cos 72° = 0.031$ — less than a third of what it would add on time — and stiffness in proportion to $\sin 72° = 0.95$, which raises the mode’s frequency slightly and makes the lag slightly worse. The mode’s total damping is about $0.01 + 0.031 = 0.041$, so its resonant amplification is roughly $1/(2 \times 0.041) = 12$; the full calculation, which follows the peak to the frequency the stiffness has moved it to, gives the figure’s 19.

For the tower at 0.2 Hz the same 25 ms is $0.025/5 = 0.005$ of the period, a lag of 1.8°, and the added damping is $0.10 \cos 1.8° = 0.0999$: all of it.

One mode, a pure delay and a linear actuator #

The structure is one mode. A real building or floor has others, and an actuator that damps the first can destabilise a higher one, whose period is shorter and whose quarter-cycle is reached by a smaller delay — the next mode up is the first to go. That makes every number here optimistic for a structure with more than one period.

The delay is pure and the feedback is velocity. Real controllers filter their signals, which adds lag that grows with frequency, and use feedback laws designed to compensate for a known delay by predicting ahead. Compensation works to the extent that the delay and the structure’s frequency are known; a floor whose frequency changes with its occupancy, or a delay that varies with the controller’s load, defeats it.

The actuator is linear and unlimited in stroke. A real one saturates, and a saturating actuator in a loop near its stability limit behaves in ways no linear calculation describes.

What the pictures cannot show #

What happens when the controller fails. A passive tuned mass that loses its damper becomes a worse tuned mass; an active damper that loses its power leaves the structure with its own damping, which is the design case most codes insist on. But an active damper that keeps running with a sensor fault, a mis-set gain or a slower loop than it was designed for can be worse than none, and the figures above are the measure of how much worse: a factor of three to four in peak response, or an oscillation that did not exist before the controller was installed.

Still open: the hybrid that keeps a passive mass underneath #

The practical answer to all of this has been to put the actuator on a tuned mass, not in place of one: the passive mass does most of the work and is stable on its own, and the actuator only nudges it. Whether a delay that would destabilise pure velocity feedback is harmless in a hybrid — because the mass’s own quarter-cycle of lag and the actuator’s add up to something the mass can absorb — or whether the hybrid inherits the same limit a little later, is the question that decides what an active system can safely be allowed to do to a floor.