2.6. MODEL REDUCTION 65
2.6.1 Truncation and Residualization
The simplest form of model reduction is state truncation. Considera system, P(s), with
a partitioned state matrix,
P(s)=
A
11 A12 B1
A21 A22 B2
C1C2D
.
Truncating the states associated with A22 results in,
Ptrun(s)=A
11 B1
C1D.
In any practical application we would order the states so that those truncated do not
significantly affect the system response. For example, to truncate high frequency modes,
Ais transformed to be diagonal (or with 2×2 blockson the diagonal) and the
eigenvalues are ordered in increasing magnitude. This results in A21=0andA
22
corresponds to the high frequency modes.
Truncation also affects the zero frequency response of the system. Residualization
involves truncating the system and adding a matrix to the Dmatrix so that the zero
frequency gain is unchanged. This typically gives a closer approximation to the original
system at low frequency. If the original system rollsoff withfrequency, the low order
residualized approximation will usually not share this characteristic. Using theabove
P(s), the result is,
Presid(s)=A
11 −A12A−1
22 A21 B1−A12A−1
22 B2
C1−C2A−1
22 A21 D−C2A−1
22 B2.
2.6.2 Balanced Truncation
Consider a stable state-space system,
P(s)=AB
C D.