User guide

122 6250 Servo Controller User Guide
most controls textbooks. The second-order system is of
the form:
H(S) =
ωn
2
S
2
+ 2ξωnS + ωn
2
Where, ξ = damping ratio
ωn = natural frequency
The time constant of this system is
1
ωn
.
The damped frequency is
ωd = ωn√1 - ξ
2
.
For the output to settle to within 2% of its stead state value
when a step input is applied, it will take four time constants,
or TS =
4
ξωη
, to settle to within 2%.
Equating our transfer function to the second-order
equation, we find:
K
P
K
D
K
T
J
= ωn
2
K
P
K
V
K
D
K
T
J
= 2ξωn
If we select a settling time of 30 ms and a damping ratio of
0.9, we can then determine K
P
and K
V
.
TS = 0.03 sec =
4
0.9 (ωn)
= > ωn = 148.15
rad
sec
The values of K
D
, K
T
, and T can be found from the
motor/drive's user documentation.
It then follows that:
K
P
=
ωn
2
∗ J
K
D
K
T
K
V
=
2ξωn ∗ J
K
P
K
D
K
T
After you have calculated K
P
and K
V
, then you must use
the following scale factor to put it in the units of SGP
(proportional feedback gain) and SGV (velocity feedback
gain) for the 6250:
SGP = K
P
*
1000 ∗ 2π
ERES
SGV = K
V
*
10
6
∗ 2π
ERES
(ERES is the encoder resolution)
PV System — Velocity Drive Gain Calculations
For a velocity drive system, the transfer function is:
θ
a
θ
c
=
K
P
∗ 2πK
A
a
S(S+a) + K
P
2πK
A
a + K
P
K
V
S2πK
A
a
θ
a
θ
c
=
K
P
∗ 2πK
A
a
S
2
+ (a + K
P
K
V
2πK
A
a)S + K
P
2πK
A
a
Equating this with our system,
K
P
2πK
A
a = ωn
2
(a + K
P
K
V
2πK
A
a) = 2ξωn
K
P
=
ωn
2
2πK
A
a
K
V
=
[2ξωn - a]
K
P
2πK
A
a
SGP = K
P
*
1000 ∗ 2π
ERES
SGV = K
V
*
10
6
∗ 2π
ERES
(ERES is the encoder resolution)
"K
A
" and "a" can be measured using Motion Architect's
drive tuning module. In this module, you will issue a step
command to the drive system and then obtain a value for
"K
A
" and "a" for the calculations above.
PIV System Gain Calculations
If we now add an integral term to our control system, you will
find that the order is increased to 3.
The polynomials will now be added for the control algorithm:
P
L
=
K
P
S +K
P
K
I
S
N= K
P
K
V
S
F= K
AFFS
S
2
+ K
VFF
S
Note that we have set K
AFFS
(acceleration feedforward
gain) and K
VFF
(velocity feedforward gain) to zero.
The block diagram for the control algorithm is as follows:
θ
c
θ
a
K
AFF
S
2
+ K
VFF
S
θ
a
K
P
K
V
S
A
B
K
I
S
IK
P
Substituting this into our transfer function yields the
following:
θ
a
θ
c
=
A [K
P
S + K
P
K
I
]
BS + [K
P
S + K
P
K
I
]A + K
V
K
P
S ∗ S ∗ A
PIV System — Torque Drive Gain Calculations
For the torque drive system under PIV control, the position
loop transfer function is as follows:
θ
a
θ
c
=
K
D
K
T
[K
P
S + K
P
K
I
]
S
2
K
D
K
T
K
V
K
P
+ JS
2
∗ S + K
D
K
T
K
P
S + K
D
K
T
K
P
K
I
θ
a
θ
c
=
K
D
K
T
K
P
S + K
D
K
T
K
P
K
I
S
3
+
K
D
K
T
J
K
V
K
P
S
2
+
K
D
K
T
J
K
P
S +
K
D
K
T
J
K
P
K
I