Hankel singular values, Hankel singular values -8 – National Instruments NI MATRIXx Xmath User Manual
Page 15
Chapter 1
Introduction
1-8
ni.com
•
The controllability grammian is also E[x(t)x
′(t)] when the system
has been excited from time –
∞ by zero mean white
noise with
.
•
The observability grammian can be thought of as measuring the
information contained in the output concerning an initial state.
If
with
x(0) = x
0
then:
Systems that are easy to observe correspond to Q with large
eigenvalues, and thus large output energy (when unforced).
•
lyapunov(A,B*B')
produces P and
lyapunov(A',C'*C)
produces Q.
For a discrete-time G(z) = D + C(zI-A)
–1
B with |
λ
i
(A)|<1, P and Q are:
P – APA
′ = BB′
Q – A
′QA = C′C
The first dot point above remains valid. Also,
•
and
with the sums being finite in case A is nilpotent (which is the case if
the transfer-function matrix has a finite impulse response).
•
[I–A
⊗ A] vec P = vec (BB′)
lyapunov( )
can be used to evaluate P and Q.
Hankel Singular Values
If P, Q are the controllability and observability grammians of a
transfer-function matrix (in continuous or discrete time), the Hankel
Singular Values are the quantities
λ
i
1/2
(PQ). Notice the following:
•
All eigenvalues of PQ are nonnegative, and so are the Hankel singular
values.
•
The Hankel singular values are independent of the realization used to
calculate them: when A,B,C,D are replaced by TAT
–1
, TB, CT
–1
and D,
then P and Q are replaced by TPT
′ and (T
–1
)
′QT
–1
; then PQ is replaced
by TPQT
–1
and the eigenvalues are unaltered.
•
The number of nonzero Hankel singular values is the order or
McMillan degree of the transfer-function matrix, or the state
dimension in a minimal realization.
x·
Ax Bw
+
=
E w t
( )w′ s
( )
[
]
I
δ t s
–
(
)
=
x·
Ax
=
y
,
Cx
=
y
′ t
( )y t
( )dt
0
∞
∫
x
′
0
Qx
0
=
P
A
k
BB
′A′
k
k
0
=
∞
∑
=
Q
A
k
C
′CA′
k
k
0
=
∞
∑
=