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Chapter 6 State-Space Design
Xmath Control Design Module 6-38 ni.com
accurate. Given a variable Sys built from the matrices {A,B,C,D}, the
modal decomposition
SysMod is built from T
–1
AT, T
–1
B, CT, and D, where
T is the transformation matrix to modal form. If you have complex poles,
then T
–1
AT is in block diagonal form. Initial conditions X
0
also are
transformed to T
–1
X
0
.
modal( ) does not accept input systems in transfer-function form, as the
concept of modes applies only to a state-variable system representation and
modes and poles are not interchangeable terms. The poles of a transfer
function always correspond to the system modes (eigenvalues of the system
A matrix).
mreduce( )
SysRed = mreduce(Sys, keep)
The mreduce( ) function computes a reduced-order form of a given
system by retaining the states indicated within the vector
keep. States not
specified within this vector are eliminated to obtain a lower-order model
SysRed.
mreduce( ) is implemented by partitioning the state vector x into two
subvectors, x
1
(states to be retained in the reduction) and x
2
(states to be
eliminated in the reduction), so that:
Similarly, the A, B, and C matrices are partitioned according to this state
partition:
The model reductions differ for the continuous and discrete-time cases
because the updates for the states being eliminated are handled differently
in the respective differential and difference equations. In both cases, the
eliminated states are taken to be constant over time. In the continuous case,
x
x
1
x
2
=
A
A
11
A
12
A
21
A
22
= B
B
1
B
2
= C
C
1
C
2
=