Specifications

Scaling
11-15
Scaling
Statespaceisthepreferred modelforLTIsystems, especially withhigher order
models. Even with state-space models, however, accurate results are not
guaranteed, because of the finite-word-length arithmetic of the computer. A
well-conditioned problem is usually a prerequisite for obtaining accurate
results.
You should generally normalize or scale the matrices of a system
to improve their conditioning. An example of a poorly scaled problem might be
a dynamic system where two states in the state vector have units of light years
and millimeters. You would expect the matrix to contain bothvery largeand
very small numbers. Matrices containing numbers widely spread in value are
oftenpoorlyconditionedbothwithrespecttoinversionandwithrespecttotheir
eigenproblems, and inaccurate results can ensue.
Normalization also allows meaningful statements to be made about the degree
of controllability and observability of the various inputs and outputs.
A set of matrices can be normalized using diagonal scaling
matrices , , and to scale u, x,andy.
so the normalized system is
where
Choose the diagonal scaling matrices according to some appropriate
normalization procedure.Onecriterionisto choose themaximumrangeofeach
of the input, state, and output variables. This method originated in the days of
analog simulation computers when , , and were forced to be between
Volts. A second method is to form scaling matrices where the diagonal
ABCD
,,,()
A
ABCD
,,,()
N
u
N
x
N
y
uN
u
u
n
= xN
x
x
n
= yN
y
y
n
=
x
·
n
A
n
x
n
B
n
u
n
+=
y
n
C
n
x
n
D
n
u
n
+=
A
n
N
x
1–
AN
x
= B
n
N
x
1–
BN
u
=
C
n
N
y
1–
CN
x
= D
n
N
y
1–
DN
u
=
u
n
x
n
y
n
10±