smart technologies for safety engineering

Dynamic Load Monitoring 141
small singular values, i.e. the values below the threshold level defined by the expected relative
measurement accuracy δ ≥ 0 (see References [21], [31] and [32]). In this way the modified
diagonal and system matrices Σ δ , D fδ are obtained and Equation (47) takes the following
regularized form
D fδ = U Σ δ V T
Note that with δ = 0 all singular values are preserved: Df0 = Df .
denote the two matrices
δ , where the number of columns of V1 equals
the number of positive singular values of Dfδ , i.e. the number of positive diagonal values of
are mutually orthonormal vectors,
Let F be the linear space of all possible loadings f. Let Vδ 1 and Vδ 2
composing together the matrix V = Vδ 1
Vδ 2
Σ δ . The matrix V is unitary; thus the columns of Vδ 1 and Vδ 2
constitute an orthonormal basis in F and hence span two orthogonal and complementary linear
subspaces Fδ 1 and Fδ 2 :
Due to Equation (48), F δ 2 = ker Dfδ , i.e.
(48)
F = F δ 1 × Fδ 2 , Fδ 1 = span Vδ 1 , Fδ 2 = span Vδ 2 (49)
D fδ V δ 2
= 0 (50)
and hence the regularized system transfer matrix Dfδ is a linear measurement operator, which
effectively: (1) transforms F orthonormally, (2) projects it on to Fδ 1 , losing a part of the load
information, (3) rescales along the basis directions by Σ δ and finally (4) transforms again
orthonormally via U. Therefore, with respect to Dfδ , Fδ 1 is the reconstructible subspace and Fδ 2
is the unreconstructible subspace of F. In other words, given the relative measurement accuracy
δ ≥ 0, each load f can be uniquely decomposed into a sum of two orthogonal components:
f = VVTf = Vδ 1Vδ T δ
1 f + V2Vδ T
2 f
= Vδ 1 m1 + Vδ 2 m2 = fδ R + Vδ 2 m2
where the first component fδ R = Vδ 1Vδ T δ
1 f = V1 m1 is a linear combination of the columns of
Vδ 1 , and hence fully reconstructible from the noisy measurements εM = Dff, while the second
component Vδ 2 m2 is a linear combination of the columns of Vδ 2 , and hence unreconstructible,
since all respective information is lost in the noisy measurement process represented by the
linear operator D fδ (due to Equation (50) D fδ V fδ
2 m2 = 0) and thus not retained in the noisy
measurements above the required relative degree of accuracy δ ≥ 0.
Reconstructible Load Component
Given the noisy measurements εM and the regularized system matrix Dfδ , the unique corresponding
reconstructible load component fδ R = Vδ 1mδ 1 can be found either
directly by the standard pseudoinverse matrix D fδ + of the regularized system matrix D fδ ,
f δ R = Dfδ+ ε M = V Σ δ+ U T ε M
(51)
(52)