Controller tuning: Ziegler-Nichols and Cohen-Coon

Why it matters

A PID controller is only as good as its three numbers. Ziegler–Nichols and Cohen–Coon are the classic recipes that turn a quick plant test into starting values for K_c, τ_I and τ_D. They are still used on real plants as first settings, and they are a staple of GATE numericals because they tie together dead time, ultimate gain and the process reaction curve.

Key ideas

What tuning tries to achieve. A fast response to set-point and load changes, with acceptable overshoot and good robustness to model error. The classic rules aim at a quarter decay ratio (each oscillation peak a quarter of the previous one), which is fairly aggressive; in practice the settings are often detuned afterwards (for example, K_c reduced by a factor of about 2).

Ziegler–Nichols closed-loop (ultimate-gain, continuous-cycling) method.

1. With the loop in automatic, remove integral and derivative action (P only).
2. Raise K_c in small steps until the controlled variable cycles with constant amplitude. This gain is the ultimate gain K_cu, and the period of oscillation is the ultimate period P_u.
3. Read the settings from the table below. K_cu and P_u can also be calculated from a model by the Routh test (no dead time) or by the Bode criterion (K_cu = 1/AR_p at the phase-crossover frequency, P_u = 2π/ω_co). Drawbacks: the plant must be driven to the edge of instability, which is often unsafe, and the result is aggressive.

Process reaction curve (open-loop) methods.

1. With the controller in manual, make a small step change M in the controller output.
2. Record the measured variable (the curve includes valve, process and sensor dynamics). Fit a first-order-plus-dead-time (FOPDT) model K·e^(−θs)/(τs + 1):
   • K = (final change in measurement)/M, in consistent units (often % of transmitter span per % of controller output);
   • θ and τ from the tangent at the inflection point (θ where the tangent cuts the initial value, τ from there to where it reaches the final value), or from the two-point method: τ = 1.5·(t₆₃ − t₂₈), θ = t₆₃ − τ, where t₂₈ and t₆₃ are the times to reach 28.3 % and 63.2 % of the final change.
3. Apply the Ziegler–Nichols open-loop or the Cohen–Coon formulas. The test is safe (open loop, small step) but sensitive to noise and disturbances during the test.

Cohen–Coon. Derived for FOPDT processes, also aiming at quarter decay. It includes the ratio θ/τ explicitly, so it accounts better for large dead time than Ziegler–Nichols open-loop, though it is also aggressive. Different textbooks print algebraically equivalent forms; the form below is widely used.

Guidance. All these rules degrade as θ/τ grows beyond about 1; then model-based rules (IMC, ITAE) or a Smith predictor are better. Always check tuned settings for stability margin (GM about 2, PM 30–45°).

Formulas

Ziegler–Nichols closed-loop:

• P: K_c = 0.5·K_cu
• PI: K_c = 0.45·K_cu, τ_I = P_u/1.2
• PID: K_c = 0.6·K_cu, τ_I = P_u/2, τ_D = P_u/8 K_cu dimensionless (or controller units), P_u in s or min.

FOPDT fit (two-point): K = Δy_∞/M, τ = 1.5·(t₆₃ − t₂₈), θ = t₆₃ − τ

Ziegler–Nichols open-loop (reaction curve):

• P: K_c = τ/(K·θ)
• PI: K_c = 0.9·τ/(K·θ), τ_I = 3.33·θ
• PID: K_c = 1.2·τ/(K·θ), τ_I = 2·θ, τ_D = 0.5·θ

Cohen–Coon (r = θ/τ):

• P: K_c = (1/K)·(τ/θ)·(1 + r/3)
• PI: K_c = (1/K)·(τ/θ)·(0.9 + r/12), τ_I = θ·(30 + 3r)/(9 + 20r)
• PID: K_c = (1/K)·(τ/θ)·(4/3 + r/4), τ_I = θ·(32 + 6r)/(13 + 8r), τ_D = 4θ/(11 + 2r)

K process gain (measurement units per controller-output unit); τ, θ in s or min.

Worked examples

Example 1 (standard): Ziegler–Nichols from the ultimate gain. For the loop G_p = 1/((s + 1)(2s + 1)(4s + 1)) (min), the Routh and Bode tests gave K_cu = 11.25 and P_u = 6.72 min. Find ZN P, PI and PID settings.

1. P: K_c = 0.5·11.25 = 5.63.
2. PI: K_c = 0.45·11.25 = 5.06; τ_I = 6.72/1.2 = 5.60 min.
3. PID: K_c = 0.6·11.25 = 6.75; τ_I = 6.72/2 = 3.36 min; τ_D = 6.72/8 = 0.84 min.

PID: K_c = 6.75, τ_I = 3.36 min, τ_D = 0.84 min

Example 2 (GATE level): reaction curve, Cohen–Coon and ZN open-loop. With the controller in manual, its output is stepped by M = 5 %. The measurement settles 10 % (of transmitter span) higher; it reaches 28.3 % of this change at 3.0 min and 63.2 % at 6.0 min. Find the FOPDT model and the Cohen–Coon PI and PID settings, and compare with ZN open-loop PID.

1. K = 10/5 = 2.0 (%/%).
2. τ = 1.5·(6.0 − 3.0) = 4.5 min; θ = 6.0 − 4.5 = 1.5 min; r = θ/τ = 0.333; τ/θ = 3.0.
3. Cohen–Coon PI: K_c = (1/2)·3.0·(0.9 + 0.333/12) = 1.5·0.9278 = 1.39; τ_I = 1.5·(30 + 1.0)/(9 + 6.667) = 1.5·31/15.667 = 2.97 min.
4. Cohen–Coon PID: K_c = 1.5·(1.3333 + 0.0833) = 1.5·1.4167 = 2.13; τ_I = 1.5·(32 + 2.0)/(13 + 2.667) = 3.26 min; τ_D = 4·1.5/(11 + 0.667) = 0.514 min.
5. ZN open-loop PID: K_c = 1.2·4.5/(2·1.5) = 1.80; τ_I = 2·1.5 = 3.0 min; τ_D = 0.5·1.5 = 0.75 min.

FOPDT: K = 2, τ = 4.5 min, θ = 1.5 min. Cohen–Coon PID: K_c ≈ 2.13, τ_I ≈ 3.26 min, τ_D ≈ 0.51 min (ZN open-loop: 1.80, 3.0, 0.75 min)

Common mistakes

• Using the PID τ_I = P_u/2 for a PI controller (PI uses P_u/1.2).
• Using the ultimate frequency in place of the ultimate period; P_u = 2π/ω_u.
• Forgetting to make K dimensionless and consistent (both input and output in % of span) before applying reaction-curve formulas.
• Measuring θ and τ from the input step time rather than from the actual start of the step response, or mixing units of seconds and minutes.
• Treating ZN or Cohen–Coon settings as final; they give quarter-decay, aggressive responses that usually need detuning.

For GATE CH

Expect direct numericals: ZN settings from K_cu and P_u (often after a Routh or Bode calculation), FOPDT parameters from a reaction curve, and Cohen–Coon or ZN open-loop settings. Conceptual questions ask what each test requires and the quarter-decay target. Memorise the ZN closed-loop table exactly; Cohen–Coon formulas are normally given or should be learnt in one textbook form.

Quick check

1. K_cu = 8 and P_u = 4 min. Give the ZN PI settings.
2. What decay ratio do ZN settings aim for?
3. A reaction curve gives t₂₈ = 2 min and t₆₃ = 4 min. Find τ and θ.
4. ZN open-loop: K = 1, τ = 6 min, θ = 2 min. What is K_c for a P controller?

Answers: 1. K_c = 3.6, τ_I = 3.33 min. 2. One quarter. 3. τ = 3 min, θ = 1 min. 4. 3.