# Non-Invasive Testing of Physical Systems Using Topological Sensitivity
## 5 On Inverse Problems Associated with Diagnosis/Prognosis of Engineering Devices  
_Diagnosis_ and _prognosis_ are concepts that arised in the medical field, in which diagnosis stands for identifying a present disease or illness in a patient [92] using symptoms, while prognosis [93] consists of predicting the likely future evolution of the diagnosed disease or illness, which intends determining a good present medical treatment to allow for a good evolution of the illness. These concepts have been extended to the technological field to, e.g., first performing diagnosis (which involves solving an inverse problem) in an engineering device, such as an aeroengine [94], and then using this to apply prognosis, which permits anticipating future failures in the device. The latter is known in the field as _predictive maintenance_[95], which anticipates failures, saving cost and time in the needed maintenance of the device. Concentrating in _diagnosis of an aeroengine_, a detailed model for the system behavior is not possible. This is because such detailed model should account for the various, extremely complex, fluid dynamic, chemical, thermal, and elastic physical processes occurring inside the engine. Thus, a black-box model for the engine operation must be used to obtain the outcome of the direct problem, as formulated in (1). There are several black-box models in the field, which are based on phenomenological descriptions of the main engine subsystems (e.g., the fan and the low and high pressure compressors and turbines) that give very good results. One such model is built using commercial software [96]. Thus, the simplest formulation of the inverse problem described in Section 2 must be applied, in which the outcome of the direct problem is written according to Equation (1), which as explained in Section 2, requires that the sizes of the sensor and degradation vectors be both moderate. Fortunately enough, the number of components in the degradation vector is typically of the order of 10 components in the case of an aeroengine. Thus, the black-box solver can be used to solve problem (1), and the counterpart of the objective function (5) minimized, which has been done in Reference [97]. Following the idea developed in Reference [79] to increase the amount of available data, sensor measurements at two flight conditions are used, which multiplies by two the amount of available data. In addition, it must be taken into account that, in an aeroengine, the sensors outcomes account for various temperatures (measured in K), pressures (measured in Pa), rotational velocities (measured in rpm), and flow rates (measured in kg/s), which exhibit very disparate values, namely up to five orders of magnitude apart from one another. Thus, a convenient scaling of the sensors outcomes is a must. Let us note here that scaling is often quite important in optimization processes [76; 77]. Still, flight conditions can only be'measured' (in fact, estimated) in practice with low accuracy, which means that their precise values must be calculated, along with the degradations, when solving the inverse problem by minimizing the objective function. Using an obvious (somewhat simple) scaling, the minimization of the objective function (to compute the degradations and the associated flight conditions) was performed in Reference [97], obtaining reasonably good results. However, the aforementioned optimization process was quite computationally expensive (requiring several CPU hours in a standard PC) because the Hessian of the objective function is quite ill-conditioned, which was to be expected, as anticipated in Section 2. More recently, such large CPU time has been highly decreased (to a few CPU minutes) in Reference [98], where a suitable, somewhat subtle _scaling of the sensor outcomes_ is performed. Moreover, the counterpart of the topological sensitivity (as defined in (7)) approach, gives very good results when degradations are very small, which is the usual case except if some important problem is occurring in the engine.