PID tuning rules: Ziegler–Nichols, Chien/Hrones/Reswick, Samal and T-sum rule

What is calculated?

Empirical tuning rules do without an exact plant model (Zacher/Reuter sec. 8.2). In the inflection-tangent method the plant step response is approximated by K_S·e^(−s·Tu)/(1 + s·Tg): the tangent at the inflection point cuts the delay time Tu from the time axis and the rise time Tg from the final-value line (Bild 8.5). The ratio Tg/Tu is the controllability – the larger, the stronger the controller may be set. Ziegler–Nichols aims at good disturbance rejection with roughly quarter-amplitude decay (D ≈ 0.2–0.3), Chien/Hrones/Reswick distinguish setpoint and disturbance behaviour as well as aperiodic response and 20 % overshoot, Samal halves the gain for sluggish plants, and Kuhn's T-sum rule uses the sum time constant T_Σ. In the oscillation test a P controller is raised until sustained oscillation; K_PR,crit and the period T_crit give the Ziegler–Nichols values. The critical values can also be estimated from Tu and Tg with eq. 8.10 or computed with the Bode/root-locus calculator.

Equations

G_S(s) ≈ K_S·e^(−s·Tu)/(1 + s·Tg) (Bild 8.5)

Ziegler-Nichols (Sprung): K_PR·K_S·Tu/Tg = 1 | 0,9 | 1,2; Tn = 3,3·Tu | 2·Tu; Tv = 0,5·Tu

Ziegler-Nichols (Schwingversuch): K_PR = 0,5 | 0,45 | 0,6·K_krit; Tn = 0,83·T_krit | 0,5·T_krit; Tv = 0,125·T_krit

CHR: K_PR·K_S·Tu/Tg nach Tabelle (0,3 … 1,2), Tn = 1,2·Tg … 2,0·Tu, Tv = 0,42 … 0,5·Tu

Samal: K_PR = Tg/(2·K_S·Tu); Gl. 8.10: K_krit ≈ (Tg/(2·Tu) + 1)/K_S

Assumptions and typical mistakes

Valid for self-regulating plants that can be approximated by PT1 + dead time (Zacher/Reuter eq. 8.4); not for integrating, oscillatory or unstable plants. The table values come from simulations with simple models; real plants with actuator limits or nonlinearities need re-tuning. Heinrich Tab. 7.9 quotes slightly different factors for the oscillation test (0.455/0.588 instead of 0.45/0.6); the calculator follows Zacher/Reuter.

Swapping Tu and Tg or drawing the tangent away from the inflection point. Using disturbance settings although setpoint steps matter (overshoot). Transferring Tn/Tv of the additive form into a controller with a multiplicative (series) structure without conversion.

Which method when?

Ziegler–Nichols: fast disturbance rejection, strong, often too oscillatory for sluggish processes. CHR aperiodic: no overshoot, slower – for setpoints that must not be exceeded. CHR 20 %: compromise between speed and damping. Samal: slow plants in plant engineering (Heinrich sec. 7.4.2). T-sum rule: when the tangent is hard to draw but the area above the step response can be evaluated.

What you enter – and where the values come from

Method

'Inflection tangent' uses K_S, Tu, Tg read from the step response; 'oscillation test' uses K_PR,crit and T_crit measured on the closed loop. Both give the Ziegler–Nichols table; only the tangent method also gives CHR, Samal and Kuhn.

Plant gain K_S

Final change of the controlled variable divided by the step size of the manipulated variable, from the step test. Must be > 0; invert the controller for negative signs.

Delay time Tu

Time from the step to the intersection of the inflection tangent with the initial line (Bild 8.5). Contains dead time and small time constants. Typically seconds to minutes; > 0.

Rise (balancing) time Tg

Time between the intersections of the inflection tangent with the initial and final lines. Corresponds to the dominant time constant. Must be > 0; Tg/Tu = 3…10 counts as well controllable.

Sum time constant T_Σ

Only for the T-sum rule: instant at which the areas above and below the step response are equal (Bild 8.6); approximately the sum of all time constants plus dead time. 0 = skip the rule.

Critical gain K_PR,crit

Controller gain at which the loop with a pure P controller just oscillates permanently. From a plant test, from the gain margin (K_crit = K·A_R) or from the root locus.

Critical period T_crit

Period of the sustained oscillation at K_PR,crit, in seconds. T_crit = 2π/ω_π with the phase crossover from the Bode plot.

What each value means

Tuning table

K_PR, Tn and Tv per rule and controller type in the time unit of the inputs. Tn and Tv refer to the additive form K_PR·(1 + 1/(s·Tn) + s·Tv). The values are starting points for commissioning, not an optimum.

Controllability Tg/Tu

Index per Zacher/Reuter sec. 8.2.1: > 10 very good, 3…10 good, < 3 poor controllability. Poor controllability calls for small gains and possibly dead-time compensation.

Estimated K_crit (eq. 8.10)

Critical P gain from the rule of thumb K_crit ≈ (Tg/(2·Tu) + 1)/K_S, useful when no oscillation test on the plant is possible.

Worked example from the literature

Zacher/Reuter Beispiel 8.3: for the PT2 dead-time plant (K_S = 0.8, T₁ = 5 s, T₂ = 6 s, Tt = 2 s) the simulation gives K_PR,crit = 7.9 and T_crit ≈ 15 s. The Ziegler–Nichols table yields the PI controller K_PR = 0.45·7.9 = 3.55 and Tn = 0.83·15 s = 12.45 s. The loop tuned this way overshoots by 50 %; the book improves it by simulation to K_PR = 1, Tn = 8 s.

What this calculator is for

Initial commissioning of temperature, pressure, level and flow controls in process engineering, drive and heating controllers, lab exercises. The rules deliver reliable starting values when only a step response is available.

How to proceed

1. Record the plant step response and draw the inflection tangent; read K_S, Tu and Tg (or run the oscillation test).
2. Check controllability and choose the rule by goal: disturbance rejection (Ziegler–Nichols, CHR disturbance) or setpoint tracking (CHR setpoint).
3. Transfer the settings to the step-response calculator and check overshoot and settling time.
4. Fine-tune carefully on the plant; Ziegler–Nichols values are deliberately aggressive.

Technical basis

Zacher/Reuter, Regelungstechnik für Ingenieure, 17th ed. 2024: sec. 8.2.1 inflection-tangent method, Ziegler–Nichols, Samal, controllability, Chien/Hrones/Reswick, T-sum rule (eq. 8.4–8.11, tables), Beispiel 8.3. Heinrich, Grundlagen Regelungstechnik, 6th ed., sec. 7.4, Tab. 7.9 and 7.10. Dubbel, ch. X, sec. 5.3.

The sources support the equations and worked examples; the reference examples are recomputed in the calculator's automated tests. The calculator does not replace a simulation with the complete nonlinear plant model.

Last updated: 2026-09-20

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