I'm trying to understand subspace algorithms. I have read a number of references, but still have a question.
For a free vibration problem, with no noise,
x(k+1)=Ax(k)
y(k) = Cx(k)
k = 1 to S
for
yhat(k,N) = [y(k), y(k+1), y(k+2), ... y(k+N-1)]
Y = [yhat(1); yhat(2); yhat(3); ...;yhat(M)]
S = M + N - 1
Its easy to show that
Y = Gamma xhat(1,N)
Gamma = [C; CA; CA^2; ...; CA^(M-1)]
and Gamma can be calculated from singular values of Y
Y = P Sigma transpose(V)
then
Gamma = P
where the dimension of P can be reduced based on a plot of the singular values.
I understand the math up to this point
but why does
x(1) = sum ( Sigma(i,i) * V
,i)) ?
For a free vibration problem, with no noise,
x(k+1)=Ax(k)
y(k) = Cx(k)
k = 1 to S
for
yhat(k,N) = [y(k), y(k+1), y(k+2), ... y(k+N-1)]
Y = [yhat(1); yhat(2); yhat(3); ...;yhat(M)]
S = M + N - 1
Its easy to show that
Y = Gamma xhat(1,N)
Gamma = [C; CA; CA^2; ...; CA^(M-1)]
and Gamma can be calculated from singular values of Y
Y = P Sigma transpose(V)
then
Gamma = P
where the dimension of P can be reduced based on a plot of the singular values.
I understand the math up to this point
but why does
x(1) = sum ( Sigma(i,i) * V
,i)) ?