Tuning a Furnace Temperature Loop with 30 s Dead Time
A reheat furnace temperature loop runs at a 1 s controller sample. The identified process from controller output (%) to temperature (°C) is first-order plus dead time: gain 2.5 °C/%, time constant 180 s, effective dead time 30 s (24 s transport plus 6 s sensor lag). The present PI settings are Kp = 1.8 (controller gain, %/°C) and Ti = 60 s. On a setpoint step from 800 °C to 850 °C the loop overshoots by 12% and settles in about 25 minutes, and the controller output sits at its 100% limit for the first 4 minutes of the step. Product quality allows no more than 2% overshoot and requires the temperature to be within ±3 °C of setpoint within 10 minutes. The shift engineer wants to raise Kp to speed the loop up, while the plant manager is asking for a quote on model-predictive control. A valve actuator stroke time of 1.5 s is not included in the identified model.
1. Estimate the ultimate gain and period of the process and use them to judge the stability margin of the existing Kp = 1.8. Is the loop on the edge of instability?
Read the worked answer
For proportional-only control the phase crossover satisfies:
30ω + atan(180ω) = π
Solving numerically gives ωu ≈ 0.0557 rad/s, so Tu = 2π/ωu ≈ 113 s, and
Ku = √(1 + (180ωu)²)/2.5 = √(1 + 10.03²)/2.5 = 10.08/2.5 ≈ 4.0
At Kp = 1.8 the proportional gain margin is 20 log(4.0/1.8) = 7 dB. The PI term adds |1 + 1/(jωuTi)| = √(1 + 1/(0.0557 × 60)²) ≈ 1.04 (0.4 dB) and about 17° of extra lag at ωu, so the realistic gain margin is roughly 6.5 dB.
The loop is therefore stable with a reasonable margin - the 12% overshoot is not incipient instability. It comes from two things: Ti = 60 s is only one third of the 180 s process time constant (far too much integral action for a process with 30 s dead time), and the output saturates for 4 minutes at the start of the step, so the integrator winds up and has to be unwound after the temperature crosses setpoint. Raising Kp would reduce the gain margin towards 4 dB and make the overshoot worse, not better; Kp is already close to the 6 dB practical ceiling of about 2.0.
2. Compute SIMC/IMC PI settings for a closed-loop time constant of 30 s and check them against the overshoot and settling requirements.
Read the worked answer
SIMC rules for a first-order-plus-dead-time model:
Kc = τ / (K(τc + θ)) = 180 / (2.5 × (30 + 30)) = 1.2 Ti = min(τ, 4(τc + θ)) = min(180, 240) = 180 s
So the gain drops from 1.8 to 1.2 and the integral time triples from 60 s to 180 s. The dominant change is the integral: at ωu the old Ti = 60 s contributed about 17° of extra phase lag, which the new value cuts to about 6°.
Predicted response: with τc = θ = 30 s the closed loop behaves approximately as a 30 s first-order lag plus the 30 s dead time, so 2% settling ≈ 4 × 30 + 30 = 150 s (2.5 min), well inside the 10-minute requirement. SIMC with τc = θ on a FOPDT process typically gives only a few percent overshoot, so the 2% target is reachable - but only if wind-up is fixed, because the output is saturated at 100% for the first four minutes. Add anti-windup (integrator hold or back-calculation) and pre-load the integrator with the pre-step output value. Derivative action is not needed and would amplify sensor noise on a thermocouple loop with 30 s dead time.
3. The unmodelled 1.5 s valve stroke and the 1 s sample are raised as arguments for buying model-predictive control. Give your engineering judgement and your recommendation.
Read the worked answer
The extra dynamics do not justify MPC. The 1.5 s stroke is 5% of the 30 s dead time - inside the identification uncertainty and far below what the loop can resolve. The 1 s sample gives 30 samples per dead time, which is more than adequate for a PI loop (a common rule of thumb is 10 or more). MPC, gain scheduling and adaptive control are all aimed at multivariable interaction, constraints that must be handled predictively, or processes whose dynamics drift over time. None of those apply to a single SISO temperature loop with a well-behaved FOPDT model.
Recommended action, in order of payoff:
- Retune to Kc = 1.2, Ti = 180 s.
- Fix anti-windup - the four minutes at 100% output is the dominant cause of the 12% overshoot, and no tuning change will fix it.
- Add feedforward from line speed or load so the loop does not have to correct every load change through the dead time.
- Re-verify the sensor lag. A fouled thermowell can silently increase the 6 s sensor lag, changing θ and invalidating the tuning; that is a far more likely source of future drift than the valve stroke time.
MPC could be revisited if the furnace later needs coordinated control of zone temperature, excess O₂ and fuel flow with hard constraints.