National Instruments Xmath Interactive Control Design Module ICDM - Manual (Page 59)

Chapter 6 Pole Place Synthesis
Xmath Interactive Control Design Module 6-4 ni.com
We can write this polynomial equation as follows:
These 2n linear equations are solved to find the 2n controller parameters
x
1
, ..., x
n
and y
1
, ..., y
n
.
Integral Action Mode
The degree (number of poles) of the controller is fixed and equal to n + 1,
so there are a total of 2n + 1 closed-loop poles. In this case, the 2n + 1
degrees of freedom in the closed-loop poles, along with the constraint that
the controller must have at least one pole at s = 0, exactly determine the
controller transfer function. In fact, the closed-loop poles give a complete
parameterization of all controllers with at least one pole at s = 0, and n or
fewer other poles.
Equations similar to those shown in the Normal Mode section are used to
determine the controller parameters given the closed-loop pole locations.
b
0
0 … 0
b
1
b
0
… 0
b
2
b
1
… 0
… …
b
n 1 –
b
n 2 –
b
0
b
n
b
n 1 –
b
1
0 b
n
b
2
0 0 b
3
… …
0 0 … b
n
x
1
·
·
·
x
n
1 0 … 0
a
1
1 … 0
a
2
a
1
… 0
… …
a
n 1 –
a
n 2 –
1
a
n
a
n 1 –
a
1
0 a
n
a
2
0 0 a
3
… …
0 0 … a
n
+
y
1
·
·
·
y
n
+
a
1
…
a
n
0
…
0
α
1
…
α
2n
= +