smart technologies for safety engineering

128 Smart Technologies for Safety Engineering
problems exists in which such a model is not known or does not exist. In this case, in order to
estimate on-line the load acting on a mechanical system, the unknown input technique of Hou
and Müller [17] can be used. In their approach, the only required information is the model
of the system (mass and stiffness matrices) and measurements of a part of the state vector
(displacements or velocities).
Consider a dynamical system described by Equation (11). Additionally assume that m < p,
which means that the number of sensors exceeds the number of forces to be identified.
The first step of the present method is to decompose the system of Equation (11) into two
subsystems, so that the unknown input u(t) acts on one of them only. By the singular value
decomposition (SVD), the matrix B can be restated in the following form:
B = UΣV T
where U ∈ R n×n and V ∈ R m×m are unitary matrices, which satisfy the following condition:
Matrix Σ is a rectangular matrix (n × m) of the form
Σ =
(14)
UU T = I, VV T = I (15)
Σ0
=
0
⎡
σ1
⎢
0
⎢ .
⎢ .
⎢ .
⎢ 0
⎢ 0
⎢ .
⎣ .
0
σ2
.
0
0
.
···
···
. ..
···
···
. ..
⎤
0
0 ⎥
.
⎥
. ⎥
σm ⎥
0 ⎥
. ⎥
. ⎦
0 0 ··· 0
where matrix Σ0 has singular values of B on the main diagonal. By substituting Equation (14)
into (11),
˙x(t) = Ax(t) + UΣV T u(t)
Next, multiplying on the left by U T and using the orthogonality (Equation (15)),
Linear transformations of the variables,
yields the two following coupled subsystems:
U T ˙x(t) = U T Ax(t) + ΣV T u(t) (16)
x := U¯x, u := Vū (17)
˙¯x1(t) = Ā11 ¯x1(t) + Ā12 ¯x2(t) + Σ0ū(t)
˙¯x2(t) = Ā21 ¯x1(t) + Ā22 ¯x2(t)
(18)