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Control engineering

PID gains from integral and derivative times

Converts the integral and derivative time form of a PID controller into the parallel form with separate proportional, integral and derivative gains.

Ref: CTL-09
Rev: v1
Field: Control engineering
Result: K_i [1/s]

K_{i} = \frac{K_{p}}{T_{i}} K_{d} = K_{p} T_{d}

K_p(1 + 1/(T_is) + T_ds) is the integral-and-derivative-time form of a PID controller. Expanding it gives the parallel form K_p + K_i/s + K_ds with K_i = K_p/T_i and K_d = K_pT_d. T_i and T_d are in seconds, so K_i has the units of K_p per second and K_d the units of K_p times seconds.

Converts the integral and derivative time form of a PID controller into the parallel form with separate proportional, integral and derivative gains. The time form is K_p(1 + 1/(T_is) + T_ds) and the parallel form is K_p + K_i/s + K_ds.

Calculator

Solve for

K_iIntegral gainComputed

1/s

K_pProportional gain

0.510

T_iIntegral time

0.520

T_dDerivative time

0.012

Derivative gain, K_d = K_pT_d0.2 s
Integral corner frequency, 1/T_i0.25 rad/s
Derivative corner frequency, 1/T_d10 rad/s
Ratio T_i / T_d40

What drives the result

Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.

Integral gain (1/s) current value 0.5 1/s

T_iIntegral timeKey driver

0.14

K_pProportional gain

0.1252.5

T_dDerivative time

At the defaults K_i = 0.5 1/s and K_d = 0.2 s. With a constant error of 0.1 and a 20 ms sample interval the integral term grows by K_i T_s e = 0.001 per sample.
Doubling K_p to 4 while keeping the times changes K_i to 1 1/s and K_d to 0.4 s. In the parallel form the three gains are independent, so the same change would leave the others alone.

Chart data as tables

Ranked variables

Variable	Typical range	Lowest result	Highest result	Swing (1/s)	Local step	Change in result
T_i Integral time	0.5 to 20 s	0.1	4	3.9	+10%	−9.09 %
K_p Proportional gain	0.5 to 10	0.125	2.5	2.375	+10%	+10 %
T_d Derivative time	0.01 to 2 s	0.5	0.5	0	+10%	0 %

Sweep of Integral time

Integral time (s)	Integral gain (1/s)
0.5	4
2.938	0.6809
5.375	0.3721
7.813	0.256
10.25	0.1951
12.69	0.1576
15.13	0.1322
17.56	0.1139
20	0.1

Sweep one variable

Vary

Integral gain (1/s)Hover, touch, or use the arrow keys on the chart

Variables and how each one contributes

K_i

Integral gain

Result · 1/s

What the equation gives you. Everything else feeds it.

K_p

Proportional gain

· typical 0.5 to 10

The proportional gain. In the time form it multiplies all three terms. Directly proportional: K_i and K_d both scale with K_p in this convention, so retuning K_p alone changes all three gains. Raising it by 10 % (to 2.2) moves the result by +10 %, from 0.5 to 0.55 1/s.

T_i

Integral time

s · typical 0.5 to 20

The integral time in s: roughly the time the integral term takes to repeat the proportional action for a constant error. A smaller T_i means stronger integral action. Inversely proportional: halving T_i doubles the integral gain. Raising it by 10 % (to 4.4) moves the result by −9.09 %, from 0.5 to 0.4545 1/s.

T_d

Derivative time

s · typical 0.01 to 2

The derivative time in s: how far ahead the derivative term looks along the error trend. No effect on K_i. It sets the derivative gain instead: K_d = K_pT_d = 0.2 s at the defaults.

Worked example with the values above

Integral gain: K_i = K_p / T_i = 2 / 4 = 0.5 1/s.
Derivative gain: K_d = K_p T_d = 2 × 0.1 = 0.2 s.
The parallel controller is C(s) = 2 + 0.5/s + 0.2s, with K_i = 0.5 1/s.

Assumptions and limits

The ideal (non-interacting) time form, with the same K_p multiplying the integral and derivative terms.
Consistent units: T_i and T_d in seconds (not minutes or repeats per minute).

Common mistakes

Entering T_i where K_i is expected. The two act in opposite directions: a larger T_i means weaker integral action.
Mixing the ideal form with a series (interacting) form, where the same numbers give a different controller.

Statistics and uncertainty

K_i is a ratio, so independent relative errors in K_p and T_i add in quadrature: 5 % in each gives about 7 % in K_i. Integral times quoted in minutes, rather than seconds, are a common source of a 60-fold error, and no statistics will catch that.

Where it is used

Moving settings between software tools and tuning rules that use different PID conventions.
Comparing a hardware controller that quotes integral time with a simulation that uses an integral gain.

Maths you need

PID controller: proportional, integral and derivative terms
Units of the gains: K_i is K_p per second, K_d is K_p times seconds

Related formulas

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