User guide

Appendix C: Calculating Your Own Gains 123
The PIV tuning transfer function is a third-order system with
a single zero. We want to fit the classical second-order
system equation used for PV tuning to this transfer
function. The typical second-order system is of the form:
H(S) =
ωn
2
S
2
+ 2ξωnS + ωn
2
Where, ξ = damping ratio
ωn = natural frequency
We can rewrite our transfer function in this form:
θ
a
θ
c
=
(2ξωnσ + ωn
2
)S + ωn2σ
(S + σ) (S
2
+ 2ξωnS + ωn
2
)
NOTE: We have added a pole at s and a zero which
result from adding the integral term. This will
affect our second-order system response.
We want the characteristic equation in the form
S
3
+ b
1
S
2
+ b
2
S + b
3
. Therefore, we must multiply through
to get a characteristic equation of :
S
3
+ (σ + 2ξωn)S
2
+ (2ξωnσ + ωn
2
)S + ωn
2
σ.
We can now equate the constants of our position loop
transfer function characteristic equation to the desired
characteristic equation:
b
1
= σ + 2ξωn =
K
D
K
T
J
K
V
K
P
b
2
= 2ξωnσ + ωn
2
=
K
D
K
T
J
K
P
b
3
= ωn
2
σ =
K
D
K
T
J
K
P
K
I
We will choose the same response criteria as for the PV
case:
ξ = 0.9 TS = 30 ms ωn = 148.15 rad/sec
We must also choose a value for
σ. This is the pole that
was introduced by the integral gain term. This obviously will
affect the response of our system such that it will not be
identical to a second-order system. Depending on the
choice of
σ, the response will be affected differently. By
setting
σ to zero, we will introduce the transfer function to
the second-order system seen earlier. This equivalent to
setting
K
I
equal to zero by having a σ which is much larger
than
ωn (σ >> ωn). We will cause all of the PIV gains to be
very large; in fact, they will be so large as to saturate the
DAC output, making the system non-linear. The root Locus
plot of the PV versus the PIV system is shown below.
((Bob—the next page from your original handywork appears to
be missing. Please check your desk.))
...... term has been provided to address this tradeoff. The
integral limit allows you to have a larger integral gain and will
reduce the overshoot caused by this gain. The larger gain
makes the system more responsive, settling to a zero
position error at a faster rate. To summarize the effects of
the selection of
σ, the following table is provided.
Overshoot Settling Time
σ << ωn small, eliminated greatly increased
σ >> ωn large overshoot fast