Discrete-time systems, Discrete-time systems -21 – National Instruments NI MATRIXx Xmath User Manual
Page 44
Chapter 2
Additive Error Reduction
© National Instruments Corporation
2-21
2.
Find a stable order ns – 2 approximation G
ns – 2
of G
ns – 1
(s), with
3.
(Step ns–nr):
Find a stable order nsr approximation of G
nsr + 1
,
with
Then, because
for
,
for
, ..., this being a property of the algorithm, there follows:
The only difference that arises when singular values have multiplicity in
excess of 1 is that the degree reduction at a given step is greater. Thus, if
σ
ns
(G) has multiplicity k, then G(s) is approximated by G
ns – k
(s) of degree
ns – k.
No separate optimization of the D matrix of G
nsr
is required. The
approximation error bound is the same as for the first algorithm. The actual
approximation error may be different. Notice that this second algorithm
does not calculate an unstable G
u
(s) such that
Discrete-Time Systems
No special algorithm is presented for discrete-time systems. Any stable
discrete-time transfer-function matrix G(z) can be used to define a stable
continuous-time transfer-function matrix by a bilinear transformation, thus
when
α is a positive constant.
G
ns 1
–
j
ω
( ) G
ns 2
–
j
ω
( )
–
∞
σ
ns 1
–
G
ns 1
–
(
)
=
.
.
.
G
nsr 1
+
j
ω
( ) G
nsr
j
ω
( )
–
∞
σ
nsr 1
+
G
nsr 1
+
(
)
=
σ
i
G
ns 1
–
(
) σ
i
G
( )
≤
i ns
<
σ
i
G
ns 2
–
(
) σ
i
G
ns 1
–
(
)
≤
i s i
–
≤
G j
ω
( ) G
nsr
j
ω
( )
–
σ
nsr 1
+
G
nsr 1
+
(
) ... σ
ns
+
+
G
( )
≤
σ
i
G
( )
i
nsr 1
+
=
ns
∑
≤
G j
ω
( ) G
nsr
j
ω
( )
–
G
u
j
ω
( )
–
∞
σ
nsr 1
+
=
H s
( )
G α
s
+
α s
–
------------
⎝
⎠
⎛
⎞
=