To expand on Peter's remarks:
The open loop gain of a system is the product (GH) of the forward
(G) and feedback (H) transfer functions.
You can alter static scaling terms without changing the
P,I,D values as long as the loop gain (GH) _product_ is the same.
The encoder_scale (units/encoder_counts) is a reciprocal term in
the feedack transfer function, e.g., H is proportional to
(1/encoder_scale).
Similarly, there is usually an amplifier scale factor that is
a reciprocal term in the forward gain, e.g., G is proportional to
1/amplifier_scale.
For a position loop with an interior velocity loop (or something
as common as a voltage-controlled permanent magnet dc motor where
no-load speed is approximately proportional to the input voltage),
it is convenient to make the amplifier_scale equal to the maximum
system output velocity in units/second. This scaling makes the
derivative feed forward term (FF1) nominally equal to one.
For example, if you have the following position servo system
with an 800 count encoder mounted on the motor, a 6000rpm=36000
degs/sec maximum motor speed, and choose to scale in units of
motor degrees:
units: degree
encoder_scale: 800/360 = 2.222 (counts/degree)
amplifier_scale: 36000 (degrees/sec)
product: (1/2.222) * (1/36000) = 1/80000
The same P,I,D gain terms would apply for the system re-scaled
with engineering units==revolutions with new encoder and
amplifier scalings:
units: revolutions
encoder_scale: 800/1 = 800 (counts/revolution)
amplifier_scale: 100 (revolutions/sec)
product: (1/800) * (1/100) = 1/80000
Similarly, if there were a belt drive of 1:4 and a screw with
0.2 lead and you wanted the output scaled in inches traveled
by a nut on the screw:
units: inches
encoder_scale: 800 counts/motorturn * 4 motorturns/screwturn * 1
screwturn/0.2 in
= 800 * 4 / 0.2 = 16000 counts/inch
amplifier_scale: 5 (in/sec)
product: (1/16000) * (1/5) = 1/80000
When rescaling, deadband (in the engineering units) would
typically also be re-adjusted as a few counts * (1/encoder_scale).
Note also that the equivalence of P,I,D terms for different scalings
assumes that the system dynamic characteristics (inertia and damping)
are _not_ modified.
--
Dewey Garrett
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