Appendix C: Calculating Your Own Gains
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Appendix C: Calculating Your Own Gain Values
This appendix explains how to calculate the 6250's servo system transfer functions in generic
polynomial terms (as an alternative to letting Motion Architect® calculate them for you).
Step-by-step procedures are provided to calculate the gains for two basic types of control systems:
❏
PV (proportional and velocity feedback) control system
❏
PIV (proportional, integral and velocity feedback) control system
In both cases, gains are calculated for torque drive and motor drive applications.
The 6250 has a unique servo algorithm that allows you to
calculate the gains for a specific desired response using
classical control techniques. Furthermore, these
calculations can be performed in the continuous time
domain. The 6250 performs the necessary conversions to
the discrete time domain so that the desired resulting
response will be what was predicted in the continuous time
domain calculations. What follows is an exercise in
calculating these gains. The 6250's block diagram is shown
below.
Control
Algorithm
-
Motor/
Driver
θ
c
θ
a
θ
c
= Commanded position or trajectory
= Actual position
θ
a
We will now show this basic control system in terms of the
polynomials of the system (see illustration below).
P
L
θ
c
θ
a
F
θ
a
N
A
B
polynomial represents the Feedforward gains
F
P
L
polynomials comprise the Proportional & Itegral gains
A
B
polynomial represents the Velocity Feedback gain
N
polynomials define the motor/drive system
The next step is to derive the position loop transfer function
in terms of the polynomials:
θ
a
θ
c
=
P A - F L A
B L + P A + N L A
Notice that the feedforward terms are in the numerator only;
consequently, they do not affect the system dynamic
response, which is determined by the poles of the
characteristic equation. We can therefore simplify the
placement of our poles by setting the feedforward gains to
zero for the PIV calculations:
θ
a
θ
c
=
PA
B L + P A + N L A
PV System Gain Calculations
For the majority of applications, the only gains required will
be the proportional (P) gain and the velocity feedback (V)
gain.
The following calculations show how to determine gains for
the PV system only. After that, the integral (I) gain will be
added and its affects will be explained. For a PV controller,
the polynomials are as follows:
P
L
=
K
P
I
N = K
P
K
V
S
Therefore:
θ
a
θ
c
=
K
P
∗
A
B + K
P
A + K
P
K
V
S A
PV System — Torque Drive Gain Calculations
At this point we must take into account whether we are
controlling a velocity drive or a torque drive. We then
provide the appropriate drive transfer function to the
equation above.
For a torque drive:
A
B
=
K
D
K
T
JS
2
Where,
K
D
= drive gain in amps/volt
K
T
= torque constant
J
= load in rotor inertia of the motor
For a velocity drive:
A
B
=
2
π
K
A
a
S ( S + a )
Where,
K
A
= drive gain in revs/sec/volt
a
= pole in radians of the drive/motor system
(We have assumed a first-order response to a step input
for the drive/motor system.)
The torque drive control system has this overall transfer
function:
θ
a
θ
c
=
K
P
∗
K
D
K
T
JS
2
+ K
P
K
D
K
T
+ K
P
K
V
SAK
D
K
T
θ
a
θ
c
=
K
P
K
D
K
T
J
S
2
+ K
P
K
V
K
D
K
T
J
S + K
P
K
D
K
T
J
This is in the form of a well-known second-order system for
which classical control techniques have been applied in
