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# PID Integral Time/Derivative Time

## PID Integral Time/Derivative Time

(OP)
This is a historical question.  In the standard ISA PID Equation (and usually in older texts/papers), Ti is in the equation as 1/Ti and Td is, well Td.  What was the reasoning for Integral Time to be a 1/x function?

### RE: PID Integral Time/Derivative Time

Integral units can be either minutes or repeats per minute, either of which is the reciprocal of the other:
2 rpm = 0.5 min.

So the 1/Ti is probably in units of repeats per minute, whereas Rate is usually in units of minutes.

I'm too young to know which units were initially used for integral, minutes or repeats per minute, but I'm sure someone here knows.

Dan

### RE: PID Integral Time/Derivative Time

Simple
The integrator integrates error over time.  The units are error_units x time.  When one divides by the integrator time constant the result is now error_units.  The derivative term multiplies the derivative of the error by the derivative time constant.  error_units/time x time = error_units.  The error units for each term are then summed and multiplied by the controller gain which has units of control_output_units/error_unit.

Peter Nachtwey

### RE: PID Integral Time/Derivative Time

What has been said above is true. But is it the real *reason* for this convention?

I have, myself, always thought that it reflects the use of Laplace transforms where integral is 1/s and derivation is *s.

It can also simply be the fact that "integral gain" is proportional to 1/Ti while derivative gain is proportional to Td, which is in line with Peter's thinking.

Gunnar Englund
www.gke.org
--------------------------------------
100 % recycled posting: Electrons, ideas, finger-tips have been used over and over again...

### RE: PID Integral Time/Derivative Time

I was trying to keep it simple. For those of you in the s domain...

PID(s)=K(1+1/(ti*s)+td*s)

Think of s as a frequency or 1/time then the frequency times time = 1 and the gain still has units of control_output_units/error_unit).

### RE: PID Integral Time/Derivative Time

(OP)
Thanks PNatchwey.  I had always assumed the Laplace reasoning but the dimentional analysis had eluded me.  The servo industry almost always uses gain values Kp, Ki, Kd) so that's what I'm use to.

### RE: PID Integral Time/Derivative Time

#### Quote:

Thanks PNatchwey.  I had always assumed the Laplace reasoning but the dimentional analysis had eluded me.  The servo industry almost always uses gain values Kp, Ki, Kd) so that's what I'm use to.
Then

Kp=K
Ki=K/Ti
Kd=K*Td

I too am a motion person.  I don't understand why the process industry uses terms like reset or repeats.  I never have understood what is reseting or repeating.

I still think in terms of Kp/Ki=Ti since this is a useful number for figuring out how long it will take the integrator to wind up to a steady state value.   ( 5 x Ti )

### RE: PID Integral Time/Derivative Time

History - or Pseudo History
I once heard that PID control originated with battle-ship gun controls in time for use during WWII. The process industry uses the terms that were applied by manufacturers of pneumatic PID controllers.

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