PID tuning

Q: How do I tune a PID from a reaction curve?

With the controller in manual, step the output 5% to 10% and record PV until it settles. Fit the model (K, τ, θ) from the 28.3% and 63.2% points and apply a rule. In the example, K ≈ 1.6 %/%, τ ≈ 42.5 s and θ ≈ 20.6 s give a lambda PI (λ = τ) with Kc ≈ 0.42 and Ti ≈ 42.5 s.

Q: Ziegler-Nichols or lambda?

Ziegler-Nichols aims for a fast quarter-decay response, accepting 20% to 60% overshoot. Lambda with λ = τ gives a robust response with no overshoot; it is preferred for flow, level and temperature in continuous processes.

Q: What is the difference between ideal and parallel forms?

In the ideal (ISA) form, Kc multiplies all three terms and Ti, Td are times. In the parallel form the gains are independent: Ki = Kc ÷ Ti and Kd = Kc · Td. With Kc = 0.42 and Ti = 42.5 s, Ki ≈ 0.0099 1/s.

Q: When should I use derivative action?

On slow, low-noise processes such as temperature, with significant dead time. On liquid flow and pressure, measurement noise makes D harmful: use PI.

Paste the PV response to a step in controller output, or enter K, τ and θ, to get controller settings from three rules and see how each responds to a setpoint change.

Pre-sizing and estimating tool: results do not replace design, calculations or reports signed by a licensed professional, nor verification against the applicable codes and standards.

Preliminary estimate

Process

Model source

Pasted curve: time (s) and PV, one row per sample

0	79.93
2	79.86
4	80.06
6	79.83
8	80.01
10	79.95
12	79.82
14	80
16	79.81
18	79.97
20	79.83
22	79.84
24	80.12
26	80.67
28	80.98
30	81.75
32	82.75
34	83.78
36	84.59
38	85.51
40	86.75
42	87.38
44	88.71
46	89.47
48	90.38
50	91.31
52	92.3
54	93.39
56	93.99
58	94.98
60	95.79
62	96.45
64	97.25
66	97.75
68	98.42
70	99.12
72	99.92
74	100.4
76	100.91
78	101.55
80	102.01
82	102.43
84	103.08
86	103.48
88	103.72
90	104.25
92	104.61
94	105.11
96	105.39
98	105.54
100	106.13
102	106.08
104	106.48
106	106.88
108	106.89
110	107.27
112	107.32
114	107.79
116	108.04
118	108.16
120	108.47
122	108.42
124	108.75
126	108.87
128	109.02
130	109.12
132	109.41
134	109.58
136	109.52
138	109.72
140	109.59
142	109.96
144	110.04
146	110.28
148	110.3
150	110.17
152	110.3
154	110.49
156	110.31
158	110.56
160	110.51
162	110.56
164	110.6
166	110.94
168	110.74
170	110.84
172	110.95
174	111.19
176	110.92
178	111.12
180	111.2
182	111.37
184	111.38
186	111.44
188	111.24
190	111.33
192	111.33
194	111.57
196	111.63
198	111.34
200	111.37
202	111.42
204	111.44
206	111.57
208	111.63
210	111.52
212	111.43
214	111.62
216	111.61
218	111.71
220	111.88
222	111.79
224	111.73
226	111.79
228	111.82
230	111.59
232	111.94
234	111.9
236	111.95
238	111.93
240	111.77
242	111.79
244	111.68
246	111.9
248	111.67
250	111.68
252	111.75
254	111.74
256	111.81
258	111.7
260	111.69
262	111.75
264	111.74
266	111.85
268	111.72
270	112.06
272	111.96
274	111.78
276	111.83
278	111.87
280	111.88
282	111.78
284	112.08
286	112.14
288	111.93
290	111.94
292	111.78
294	111.79
296	111.89
298	111.86
300	112.09
302	111.83
304	111.77
306	112.15
308	111.98
310	111.83
312	111.99
314	111.78
316	111.98
318	112.17
320	112.12
322	112.05
324	111.88
326	111.93
328	111.85
330	112.09
332	111.99
334	112.09
336	111.92
338	111.87
340	112.11
342	112.18
344	112.13
346	112.11
348	112.12
350	112.08
352	111.88
354	112
356	111.93
358	111.8
360	111.8

Load .xlsx or .csv

Number format

See what was read

Step applied to output (MV)%

Time of the steps

Transmitter calibrated span (URV − LRV)PV units

Model fit

Example: oven at 80 °C, 0 to 200 °C range, output stepped from 40% to 50% at t = 10 s. With PV already in % of span, use 100.

Tuning

Controller

Rule for the main result

Desired λ ÷ τ

Setpoint step in the simulation% of span

——

—

Reaction curve and fitted model

Points: measured PV. Lines: first-order-plus-dead-time models from both fitting methods.

Tap the chart to see the values.

Simulated response to a setpoint step

Closed loop on the process model, for the three rules with the chosen controller. Derivative on PV with filter (N = 10), no output limits.

Tap the chart to see the values.

Parameters by rule and controller

Ideal (ISA) form: Kc, Ti, Td. Parallel form: Kp, Ki, Kd. Proportional band = 100 ÷ Kc.

Next step

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Method, assumptions and references

Equations used

Model: G(s) = K · e^(−θs) ÷ (τs + 1); K = (ΔPV ÷ span × 100) ÷ ΔMV

Smith: τ = 1.5 · (t63.2 − t28.3); θ = t63.2 − τ. Tangent: θ = where the steepest tangent crosses the initial PV; τ = ΔPV ÷ maximum slope

Ziegler-Nichols: P Kc = τ/(Kθ); PI Kc = 0.9τ/(Kθ), Ti = 3.33θ; PID Kc = 1.2τ/(Kθ), Ti = 2θ, Td = 0.5θ

Cohen-Coon (r = θ/τ): P Kc = τ/(Kθ)·(1 + r/3); PI Kc = τ/(Kθ)·(0.9 + r/12), Ti = θ(30 + 3r)/(9 + 20r); PID Kc = τ/(Kθ)·(4/3 + r/4), Ti = θ(32 + 6r)/(13 + 8r), Td = 4θ/(11 + 2r)

Lambda/IMC: PI Kc = τ/[K(λ + θ)], Ti = τ; PID Kc = (τ + θ/2)/[K(λ + θ/2)], Ti = τ + θ/2, Td = τθ/(2τ + θ); P uses the PI Kc

Parallel: Kp = Kc; Ki = Kc ÷ Ti; Kd = Kc · Td; proportional band = 100 ÷ Kc (%)

Assumptions and limits

Self-regulating process, approximated by first order plus dead time; open-loop test with a single step, PV steady before the step and at the end of the curve.
Discrete simulation (6,000 zero-order-hold steps) of the identified model, ideal controller with derivative on PV and Td/10 filter, without output saturation or anti-windup. In the field, fine-tune while watching the real loop.
Ziegler-Nichols and Cohen-Coon target a 1/4 decay ratio and work best for 0.1 < θ/τ < 1; Kc is dimensionless (% output per % PV).

Standards and technical documents

Ziegler, J. G.; Nichols, N. B. Optimum settings for automatic controllers. Transactions of the ASME, 64, 1942
Cohen, G. H.; Coon, G. A. Theoretical consideration of retarded control. Transactions of the ASME, 75, 1953
Smith, C. L. Digital Computer Process Control. Intext, 1972 (método de dois pontos)
Rivera, D. E.; Morari, M.; Skogestad, S. Internal model control: PID controller design. Ind. Eng. Chem. Process Des. Dev., 25, 1986
Seborg, D. E.; Edgar, T. F.; Mellichamp, D. A.; Doyle, F. J. Process Dynamics and Control, 4th ed. Wiley, 2016
ISA-5.1-2022: Instrumentation Symbols and Identification; IEC 61131-3: linguagens de programação de CLP (bloco PID)

U.S. regulations

OSHA 29 CFR 1910.119 (PSM): process safety information and management of change for control loop settings
ANSI/ISA-61511-1-2018: Functional Safety — Safety Instrumented Systems for the Process Industry Sector
NFPA 85: Boiler and Combustion Systems Hazards Code (combustion control loops)

Frequently asked questions

How do I tune a PID from a reaction curve?

With the controller in manual, step the output 5% to 10% and record PV until it settles. Fit the model (K, τ, θ) from the 28.3% and 63.2% points and apply a rule. In the example, K ≈ 1.6 %/%, τ ≈ 42.5 s and θ ≈ 20.6 s give a lambda PI (λ = τ) with Kc ≈ 0.42 and Ti ≈ 42.5 s.

Ziegler-Nichols or lambda?

Ziegler-Nichols aims for a fast quarter-decay response, accepting 20% to 60% overshoot. Lambda with λ = τ gives a robust response with no overshoot; it is preferred for flow, level and temperature in continuous processes.

What is the difference between ideal and parallel forms?

In the ideal (ISA) form, Kc multiplies all three terms and Ti, Td are times. In the parallel form the gains are independent: Ki = Kc ÷ Ti and Kd = Kc · Td. With Kc = 0.42 and Ti = 42.5 s, Ki ≈ 0.0099 1/s.

When should I use derivative action?

On slow, low-noise processes such as temperature, with significant dead time. On liquid flow and pressure, measurement noise makes D harmful: use PI.

Reaction curve and fitted model?

Simulated response to a setpoint step?

Parameters by rule and controller?

Next step

Method, assumptions and references

Equations used

Assumptions and limits

Standards and technical documents

U.S. regulations

Frequently asked questions

Reaction curve and fitted model

Simulated response to a setpoint step

Parameters by rule and controller