Chapter 1 Introduction
Xmath Model Reduction Module 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) = x0 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)–1B with |λi(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 λi1/2(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+=
Ewt()w′s()[]Iδts–()=
x
·Ax=y,Cx=
y′t()yt()dt
0
∞
∫x′0Qx0
=
PA
kBB′A′k
k0=
∞
∑
=QA
kC′CA′k
k0=
∞
∑
=