CHEN90032 · Process Simulation and Control
PID Controller Tuning
PID Controller Tuning is the Module-8 payoff of University of Melbourne CHEN90032 Process Simulation and Control: it turns the stability analysis of the earlier weeks into the three numbers — controller gain Kc, integral time TI and derivative time TD — you actually dial into a loop. You learn the classical continuous-cycling test that reads the ultimate gain Kcu and ultimate period Pu at the edge of stability, then the Ziegler–Nichols and Tyreus–Luyben rules (both provided on the exam formula sheet, both written in Kcu and Pu) that convert them into settings. You also meet the safer experimental alternatives — relay auto-tuning and step-reaction-curve fits (Cohen–Coon) — and model-based tuning (direct synthesis), which computes the controller straight from a process model.
What this chapter covers
- 01Why tuning matters: choosing Kc, TI, TD to balance a fast response against stability and robustness
- 02The continuous-cycling test (Ziegler–Nichols, 1941): P-only, raise Kc until a constant-amplitude oscillation, read Kcu and Pu
- 03The ultimate quantities from a model: Kcu = 1/AR_p(ωc) and Pu = 2π/ωc, where the open-loop phase φ = −180° at ωc
- 04Why only a process with dead time (or three-plus lags) has a finite Kcu — a bare first- or second-order lag has no stability limit
- 05The Ziegler–Nichols rules: P (0.5 Kcu), PI (0.45 Kcu, Pu/1.2) and PID (0.6 Kcu, Pu/2, Pu/8), aimed at a quarter-decay response
- 06The Tyreus–Luyben rules: PI (0.31 Kcu, 2.2 Pu) and PID (0.45 Kcu, 2.2 Pu, Pu/6.3) — more conservative and robust
- 07Relay (on-off) auto-tuning: a bounded limit cycle gives Pu and Kcu on-line without driving the plant to true instability
- 08Step-reaction-curve fits (Cohen–Coon): fit a first-order-plus-dead-time model, then tune from the θ/τ ratio
- 09Model-based tuning by direct synthesis: pick a desired closed-loop time constant τmc and back out an ideal PID from the process model
- 10The Ziegler–Nichols P gain margin of 2 as a built-in sanity check on your Kcu
Apply the Ziegler–Nichols and Tyreus–Luyben rules from Kcu and Pu
- +1Ziegler–Nichols PID gain: Kc = 0.6 × Kcu = 0.6 × 8.0 = 4.8 (dimensionless). Read the PID row of the Z–N table, not the PI row.
- +1Ziegler–Nichols PID times: TI = Pu/2 = 6.0/2 = 3.0 min; TD = Pu/8 = 6.0/8 = 0.75 min. Keep the units of Pu (minutes) on TI and TD.
- +1Tyreus–Luyben PI: Kc = 0.31 × Kcu = 0.31 × 8.0 = 2.48; TI = 2.2 × Pu = 2.2 × 6.0 = 13.2 min. (T–L PI has no derivative term.)
- +1Compare: the Tyreus–Luyben gain (2.48) is about half the Ziegler–Nichols gain (4.8) and its integral time (13.2 min) is over four times longer than the Z–N value (3.0 min). Tyreus–Luyben is therefore the calmer, more robust design with better set-point tracking; Ziegler–Nichols is faster but oscillatory (quarter-decay).
Key terms
- Controller tuning
- Choosing the controller settings — gain Kc, integral (reset) time TI and derivative time TD in the PID law Gc(s) = Kc(1 + 1/(TI s) + TD s) — so the closed loop is fast enough but still stable and robust to model error and disturbances.
- Ultimate gain Kcu / ultimate period Pu
- The proportional gain that makes the closed loop critically stable (sustained, constant-amplitude oscillation) and the period of that oscillation. From a model, Kcu = 1/AR_p(ωc) and Pu = 2π/ωc, where ωc is the frequency at which the open-loop phase reaches −180°.
- Continuous cycling
- The Ziegler–Nichols (1941) on-line test: run P-only control and slowly raise Kc until the loop oscillates with constant amplitude. That gain is Kcu and the oscillation period is Pu. It only works for self-regulating processes that have a real stability limit (dead time or three-plus lags).
- Ziegler–Nichols rules
- Tuning relations that convert Kcu and Pu into settings: P uses Kc = 0.5 Kcu; PI uses Kc = 0.45 Kcu, TI = Pu/1.2; PID uses Kc = 0.6 Kcu, TI = Pu/2, TD = Pu/8. They target a quarter-decay response — fast but oscillatory, with weak set-point tracking.
- Tyreus–Luyben rules
- More conservative tuning relations: PI uses Kc = 0.31 Kcu, TI = 2.2 Pu; PID uses Kc = 0.45 Kcu, TI = 2.2 Pu, TD = Pu/6.3. The much larger integral time gives a more robust loop that tracks set points better than Ziegler–Nichols.
- Relay (on-off) auto-tuning
- An automatic method that replaces the controller with an on-off element switching the manipulated variable between two levels. The loop settles into a small, bounded limit cycle near the critical frequency, so Pu (its period) and Kcu are found on-line without driving the plant to true instability.
- Cohen–Coon (step-reaction-curve tuning)
- An empirical method that applies one open-loop step, fits the reaction curve to a first-order-plus-dead-time model (gain K, time constant τ, dead time θ), and maps the θ/τ ratio onto Kc, TI, TD. Gentler on the plant than continuous cycling.
- Model-based tuning (direct synthesis)
- Tuning from a trusted process model: specify a desired closed-loop response Gobj and invert the servo relation, Gmc = (1/Gp)·Gobj/(1 − Gobj). For a second-order-plus-delay process this gives an ideal PID with Kc = (τ1+τ2)/[Kp(τmc+θ)], TI = τ1+τ2, TD = τ1τ2/(τ1+τ2); the desired time constant τmc sets the speed.
PID Controller Tuning FAQ
How do I get the ultimate gain Kcu and ultimate period Pu without running a plant to instability?
Two safe routes. From a model, compute them by frequency response: find the critical frequency ωc where the open-loop phase equals −180°, then Kcu = 1/AR_p(ωc) (the reciprocal of the process amplitude ratio there) and Pu = 2π/ωc. Experimentally, use relay auto-tuning: an on-off element forces a small, bounded limit cycle whose period is Pu, so you never push the plant to true instability. Note that a pure first- or second-order lag with no dead time has no finite Kcu at all — it is the time delay (or three-plus lags) that creates a stability limit.
When should I use Ziegler–Nichols versus Tyreus–Luyben, and why not just pick the highest gain?
Both convert the same Kcu and Pu into settings, but they aim for different behaviour. Ziegler–Nichols targets a quarter-decay response: fast, but oscillatory and poor at set-point tracking, with only a small stability margin. Tyreus–Luyben uses a smaller gain and a much larger integral time (TI = 2.2 Pu), giving a more robust, better-tracking loop that tolerates model error — the usual modern default. A higher gain is not automatically better: past the ultimate gain the loop goes unstable, and even below it a too-high gain leaves the response ringing. Match the rule to whether speed or robustness matters more for the loop.
Can AI help me with PID controller tuning in CHEN90032?
Yes, as a study aid. Sia can explain the method step by step — how to find the critical frequency and read off Kcu and Pu, which Ziegler–Nichols or Tyreus–Luyben row to use, how relay auto-tuning bounds the cycle, and how direct synthesis backs a PID out of a process model — and it can check your working and units on practice problems. Treat it as a tutor that walks you through the reasoning, not a source of ready-made assessment answers: it does not sit your exam or guarantee a mark, and doing the derivations and calculations yourself is what earns credit and clears the exam hurdle.
Studying with AI? Sia — free AI chemical engineering tutor works through CHEN90032 step by step.
Exam move
Drill the pipeline until it is automatic: open-loop phase = −180° gives ωc, then Kcu = 1/AR_p(ωc) and Pu = 2π/ωc, then look up the named rule on the formula sheet and write Kc, TI, TD with units. Practise both Ziegler–Nichols and Tyreus–Luyben on the same Kcu and Pu so you can see how much T–L detunes the loop, and keep the PI and PID rows straight (their integral times differ). Rehearse the safe alternatives conceptually — relay auto-tuning bounds the cycle, Cohen–Coon fits a step reaction curve, direct synthesis needs a model — and remember the Cohen–Coon and relay coefficients are not on the provided formula sheet, so use whatever a question supplies rather than reciting numbers. The written final is 4 questions and 100 marks over 3 hours, about 1.8 minutes per mark, so an 8-mark 'find Kcu, Pu and tune' part is roughly 14 minutes; show the phase equation and the ωc you solved, since the marks are for working. It sits in the Semester 1 exam period, around June (confirm the date on Canvas), and the final is a hurdle you must pass, so this formulaic topic is easy marks worth banking. On every practice question, watch the units (this subject mixes minutes and seconds) and check the direction: more dead time lowers ωc and Kcu, so a good rule gives a smaller gain.
