smart technologies for safety engineering

96 Smart Technologies for Safety Engineering
In the DC case (real distortions), h corresponds to the gradient of objective function ∇g.Inthe
AC case (complex distortions), h is equal to the directional derivative and formally cannot be
associated with the gradient of g, since the objective function takes only real values and has
no derivatives with respect to complex distortions. In every iteration of the steepest-descent
method, distortions are updated according to the following formula:
ε 0(p+1) = ε 0(p) − λ (p) h (120)
where p denotes the number of iteration steps and λ (p) is a nonnegative factor that normalizes
and scales the value of distortion updating. An approximation procedure for λ (p) is
λ (p) = [∇d]H d 2
2 ∇d [∇d] H d 2
(121)
Constraints are usually defined with respect to the modification parameter μ. Constraints
include physical conditions (μk >0) but can also be defined in relation to the specific type of
defects. In the case of breaks μk ∈[ 0; 1] and in the case of short-circuits μk 1. To impose
constraints, a corresponding vector of modification parameters needs to be calculated after
every distortion updating.
The gradient-based optimization method also usually demands that the number of distortion
locations should not be larger than the number of independent reference responses. This ensures
that a generation of certain false configurations of distortions during the optimization process
is restrained. Generally, when the mentioned condition is not fulfilled, the vector of distortions
obtained from the optimization procedure can consist of three components:
ε 0 = ε 0 (μ) + ε 0(i=0) + ε 0(u=0)
(122)
where ε 0 (μ) represents the appropriate solution associated with the modeled modifications.
The remaining components are called impotent states of distortions because they generate only
a voltage (ε 0(i=0) ) or a current (ε 0(u=0) ) response. Impotent states occur only in the specific
configurations: ε 0(u=0) in all elements of a loop, while ε 0(i=0) in all elements connected with
a node.
3.5.4.2 Defect Identification in Dynamics
The procedure of defect identification in the transient state is also based on the steepestdescent
method, but the gradient of the objective function will be calculated with respect to
the modification parameters μ. The vector of the distance functions is defined for all discrete
time instants:
d(t) = f(μ, t) − f ref (t) (123)
The objective function g is defined as a sum of squares of all distance functions:
g =
di(t)di(t) (124)
t