## 2 Theoretical Background  
An operating gas turbine engine is a system that relates quantities that define its operating point (inputs) to measured quantities that describe its operation (outputs). This interrelation depends on the condition of its components and can be expressed as a mathematical function. A diagnostic method using steady-state data for the generation of the health indices of the components of a gas turbine uses, in one way or another, the inversion of the nonlinear relation, \(F_{SS}\), between measured quantities \((Y)\), the engine operating condition \((u)\), and component health parameters \((f)\):  
\[Y=F_{SS}(u,f)\to f=F_{SS}^{-1}(u,Y) \tag{1}\]  
When the engine operates in a transient condition, the equations that formulate the vector function (F) change. The following relationships change:  
* The energy balance equation between components on the same shaft, due to the existence of the shaft acceleration or deceleration power;
* The energy equation of the components, due to the presence of heat transfer between the fluid and the component solid walls;
* The continuity and momentum equation of the components, due to the inertia and compressibility of the fluid within the component.  
The relative significance in terms of the magnitude of these different factors depends on the particular engine (for example, the component size and type) and the form of transient operation. For example, the inertia of the fluid within blade passages contributes measurably to the response when the transient is \"fast\".  
In any case, the interrelation between measurements and component health factors is represented by a different vector function, \(F_{TR}\):  
\[Y^{\prime}=F_{TR}(u,f) \tag{2}\]This implies that if the steady-state function \((F_{SS})\) is inverted instead of the transient function \((F_{TR})\) for calculating the health coefficients, a set that is different from the actual one will be derived:  
\[f^{\prime}=F_{SS}^{-1}(u,\gamma^{\prime}) \tag{3}\]  
Deviations in the health indices of the components could thus mistakenly be interpreted as changes in the component condition.  
We note here that all the terms that are modifying the steady-state functional interrelation, to make it represent the transient behavior, are time derivatives. In the power balance equation, for example, the derivative of angular velocity is contained in the acceleration or deceleration power term:  
\[P_{tur}=P_{comp}+P_{load}+P_{acc}=P_{comp}+P_{load}+\frac{1}{2}I\frac{d(\omega ^{2})}{dt} \tag{4}\]  
Similarly, when considering the heat transfer between metal and gas, the metal temperature is described by the following dynamic equation [19]:  
\[\frac{d(T_{m})}{dt}=\frac{h\cdot A}{m\cdot Cp}\cdot(Tt_{g}-Tm) \tag{5}\]  
This means that when the values of these derivatives become comparatively small, then the transient behavior will be well approximated by the steady-state relations.  
This observation will form the basis of the criteria that will be presented later in the paper, which will then be used for deciding whether an operating condition can be approximated by the steady-state model with sufficient accuracy.  
Before continuing, it should be noted that gas turbine and jet engine transient performance modelling capabilities have reached a high level of maturity [19; 20]. A particular aspect that is very important when control or monitoring applications are to be supported is the ability to adapt the models to specific engine units. An example of such an approach, taking advantage of modern data-driven technologies, has been presented in [21].