Chapter 6 State-Space Design
© National Instruments Corporation 6-39 Xmath Control Design Module
the derivative of x
2
is set to zero, resulting in reduced-order state equations
of the form:
In the discrete case, x
2k + 1
is taken to be equal to x
2k
so that the state
equations become:
When using
mreduce( ), remember to remove states corresponding to
complex conjugate poles. Not doing so—that is, eliminating only one pole
in a pair—will produce a meaningless system.
More complex model reduction algorithms, which are intended to model
complete system dynamics in the absence of one of more states, are
available with the Xmath Model Reduction Module, as shown in Figure 6-8
and in Example 6-13.
Example 6-13 Model Reduction Module
A= [0.37,0.26,0.22,0.67;
0,0.52,0.63,0.20;
0,0,0.76,0.39;
0,0,0.04,0.83]
B = [0,1.7e-5,0,0.0004]'
C = [1,0,1,0]
D = 0
Sys = system(A,B,C,D,{dt = 0.2});
[SysM, T] = modal(Sys)
SysM (a state space system) =
A
0.37 0 0 0
0 0.52 0 0
0 0 0.665289 0
x
·
1
A
11
A
12
A
22
1–
A
21
–()x
1
B
1
A
12
A
22
1–
B
2
–()u+=
yC
1
C
2
– A
22
1–
A
21
()x
1
DC
2
A
22
1–
B
2
–()u+=
x
1k 1+
A
11
A
12
A
22
I–()
1–
– A
21
[]x
1k
+=
yC
1
C
2
– A
22
I–()
1–
A
21
[]x
1k
+=
B
1
A
12
A
22
I–()
1–
B
2
–[]u
k
DC
2
A
22
I–()
1–
B
2
–[]u
k
