smart technologies for safety engineering

VDM-Based Health Monitoring 95
locations, which supposedly contains all unknown locations of defects. In the most general
case, the set of distortion locations includes all resistive elements of the circuit.
3.5.4.1 Defect Identification in the Steady State
In the steady-state cases, the problem can be solved by transforming Equation (108), in which
modeled responses are substituted by the responses from reference points:
D f ε 0 = f ref − f L
(115)
Obviously, to obtain a unique solution, the influence matrix D f , assembled for the selected
reference points and distortion locations, needs to be square and nonsingular. This means that
the number of independent reference responses cannot be lower than the assumed number
of distortion locations. The mutual independence of responses is determined by distortion
locations and Kirchhoff’s laws. In a circuit consisting of n elements and K nodes, there are at
most (K-1) independent current Kirchhoff’s laws and (n − K + 1) voltage Kirchhoff’s laws.
When all resistive elements are considered as possible defect locations, in order to obtain a
nonsingular influence matrix, the set of reference points must include a current reference in
every independent loop of the circuit and voltage reference in all but one node. With every
element excluded from the set of distortion locations, the number of unknowns is reduced and
one reference response becomes redundant.
An alternative approach uses iterative methods based on gradient optimization. Although
more computationally demanding, they enable constraints to be imposed. The evolution of
optimized parameters can be actively controlled, ensuring more reliable results (when reference
responses are perturbed by sloppy measurement or noise). In some cases a reduction in the
number of necessary reference responses is possible. The proposed procedure, valid for both
the DC and AC cases, is based on the steepest-descent method, with the objective function
defined as the least square problem and distortions chosen as optimization variables.
The vector of distance functions d describes differences between the modeled responses
(defined by the actual state of distortions) and responses obtained from the reference points:
d = f(ε 0 ) − f ref
(116)
Components of the gradient of distance functions with respect to distortions (complex derivatives
in the AC case) are equal to entries of the influence matrix:
The objective function g is defined as
∇d = ∂di
∂ε 0 j
= D f
ij
(117)
g = didi = (d) H d (118)
where (·) H denotes the conjugate transpose. Function g is a real-valued function of real or
complex arguments. With respect to the objective function, the direction of steepest descent h
is defined as
h = 2[∇d] H d (119)