Pressure Loop Limit-Cycling After a Positioner Overhaul
A PI controller in a PLC regulates a 6 bar compressed-air header by throttling a 150 mm butterfly valve with a pneumatic positioner. A step test gives a process gain of 0.8 bar per percent of valve opening, a time constant of 12 s and a dead time of 4 s. The controller has Kc = 2.0 %/% and Ti = 15 s, a 200 ms scan, and the loop was stable with a gain margin near 10 dB and a phase margin near 65°. After a shutdown, three things changed: the transmitter damping was increased from 1 s to 5 s, the PLC scan was raised to 1 s, and the positioner was overhauled with its linkage re-set by a contractor. The loop now cycles continuously with a period of about 30 s and a roughly constant ±0.35 bar amplitude; the trend shows the controller output moving in regular ramps and steps, with the pressure following in a near-triangular wave. The shift engineer wants to halve Kc and double Ti to calm it down.
1. Estimate the phase and gain margin before and after the changes, and say whether the linear loop alone can explain the oscillation.
Read the worked answer
Use L(s) = Kc·Kp(1 + 1/(Ti·s))·e^(−θs)/(τs + 1), with Kc·Kp = 1.6, Ti = 15 s, τ = 12 s, θ = 4 s.
Before the shutdown (1 s damping, 0.2 s scan): gain crossover is at about 0.125 rad/s. Phase there is roughly −28° (PI) − 56° (process lag) − 29° (dead time) = −113°, giving a phase margin near 67°. The −180° crossing is at about 0.40 rad/s, where |L| ≈ 0.33, giving a gain margin near 9.6 dB. Healthy.
After the shutdown (5 s damping, 1 s scan): the damping adds a pole at 1/5 rad/s contributing about −arctan(5ω) and 1/√(1 + 25ω²) in magnitude, and the slower scan adds roughly 0.5 s of average transport lag. Gain crossover drops to about 0.11 rad/s, and the phase there becomes about −31° − 53° − 29° (dead time) − 25° (damping) − 3° (scan) ≈ −141°, so phase margin falls to about 39°. The −180° crossing moves to about 0.19 rad/s, where |L| ≈ 0.48, so gain margin falls to roughly 6.4 dB, and the ultimate period 2π/0.19 is about 32 s.
Interpretation: the loop is still linearly stable, but the margin has been cut roughly in half, and the observed 30 s period is essentially the ultimate period. Something is driving the loop around its resonance. The most likely candidate is the overhauled positioner, which commonly returns with higher effective gain after a linkage re-set and with measurable dead band or backlash.
2. How would you separate a valve-induced limit cycle from a plain tuning problem? Give the tests and the evidence you would look for.
Read the worked answer
Test 1 - valve signature. Put the controller in manual and step the output in 5% increments up and down across the operating point, recording actual stem travel. Look for hysteresis between up and down, and for a jump at each reversal. A dead band above about 1% of stroke on a high-gain butterfly valve is enough to produce a limit cycle.
Test 2 - hold the output. In manual, hold the controller output constant. If the pressure still cycles, or if the valve slowly slips and then jumps, the cause is downstream of the controller. If everything settles, the controller is at least part of the loop.
Test 3 - gain dependence. Temporarily vary Kc between about 1.0 and 3.0 while watching the amplitude and period. A tuning oscillation grows smoothly with gain and is roughly sinusoidal in both PV and output. A stiction limit cycle keeps a nearly constant amplitude and period across a wide gain range and produces a distinctly non-sinusoidal output - ramps and steps - with the PV following in a triangular wave.
Test 4 - operating point. Check where the butterfly actually sits at normal demand. A 150 mm butterfly running below about 20–30% open has a very high and rapidly changing gain; the 0.8 bar/% identified at mid-stroke may be a serious underestimate there, which alone would explain the loss of margin.
The evidence already in hand - regular ramp-and-step output, triangular PV, constant amplitude, period at the ultimate value - points strongly to the valve, not to the tuning.
3. Assess the proposal to halve Kc and double Ti and give the corrective plan you would actually implement.
Read the worked answer
The proposal works, but only as an anaesthetic. Halving Kc doubles the gain margin from about 6.4 dB to about 12 dB and doubling Ti reduces the integral contribution, so the amplitude will drop. What it does not do is remove the dead band or backlash. A sticky valve with reduced controller gain still limit-cycles, just at lower amplitude, and the slower integral action lengthens the recovery from every load change and leaves offset standing for longer. It also masks the fact that the positioner was returned in worse condition than it left.
Plan, in order:
- Restore the signal path. Return the transmitter damping from 5 s to 1–2 s and the scan from 1 s to 0.2–0.5 s. That recovers roughly 25–30° of phase at crossover at zero cost, and neither change was necessary.
- Fix or replace the positioner. Stroke-test it, measure dead band and hysteresis, and if the linkage re-set has left excessive backlash, correct it or replace the positioner. If the valve runs nearly closed at normal demand, consider a reduced trim or a smaller valve; no tuning can fix an operating point where the process gain is five times the identified value.
- Retune on the corrected loop. With Kc ≈ 1.2–1.4 and Ti ≈ 20–25 s, verify a phase margin of at least 45° and a gain margin of 6 dB or more by step testing at the normal operating point, not at mid-stroke.
- If the valve cannot be fixed immediately, accept a detuned controller temporarily, but record it as a temporary measure with a date and an owner, because detuning hides a mechanical fault that will keep getting worse.