Chapter 6
Pole Place Synthesis
©
National Instruments Corporation
6-5
Xmath Interactive Control Design Module
State-Space Interpretation
In a state-space framework, it is common to classify the closed-loop poles
as
n
“control eigenvalues” and
n
“estimator eigenvalues.” But, in fact, it
makes no difference in the final controller transfer function how you
classify the closed-loop poles.
In other words, in a state-space framework, swapping a “control
eigenvalue” and an “estimator eigenvalue” will result in different feedback
and estimator gains, but the same final controller.
Opening the Pole Place Window
The Pole Place window can accept any controller with n poles, or
n
+ 1
poles provided the controller has at least one pole at
s
= 0. The
Integral
Action
toggle button will be properly set. In particular, it accepts all LQG
and H
∞
controllers. This allows the user to manually tune the closed-loop
poles in a design that was, originally, LQG or H
∞
. In this case, you cannot
read the resulting controller back into the LQG or window since the
controller no longer has this special form.
Manipulating the Closed-Loop Poles
The closed-loop poles and zeros can be dragged and edited interactively.
Refer to the
Graphically Manipulating Poles and Zeros
section of
, for a general discussion of
manipulating poles graphically.
Time and Frequency Scaling
The slider and variable-edit box show the average value of the closed-loop
pole magnitudes, and therefore can be interpreted as, roughly, the
bandwidth of the closed-loop system. The average frequency is given by:
F
avg
=
⏐λ
1
λ
2
... λ
2
n
⏐
1/2
n
where
λ
1
, ...,
λ
2
n
are the closed-loop poles. Notice that the average is
geometric.
You can change
F
avg
by dragging the slider or typing into the variable edit
box. The effect is that the closed-loop poles are all multiplied by a scale
factor in such a way that becomes the requested value. Therefore, by
changing
F
avg
, you are time- or frequency-scaling the closed-loop
dynamics.
