Exploring Prognostic and Diagnostic Techniques for Jet Engine Health Monitoring: A Review of Degradation Mechanisms and Advanced Prediction Strategies

: Maintenance is crucial for aircraft engines because of the demanding conditions to which they are exposed during operation. A proper maintenance plan is essential for ensuring safe ﬂights and prolonging the life of the engines. It also plays a major role in managing costs for aeronautical companies. Various forms of degradation can affect different engine components. To optimize cost management, modern maintenance plans utilize diagnostic and prognostic techniques, such as Engine Health Monitoring (EHM), which assesses the health of the engine based on monitored parameters. In recent years, various EHM systems have been developed utilizing computational techniques. These algorithms are often enhanced by utilizing data reduction and noise ﬁltering tools, which help to minimize computational time and efforts, and to improve performance by reducing noise from sensor data. This paper discusses the various mechanisms that lead to the degradation of aircraft engine components and the impact on engine performance. Additionally, it provides an overview of the most commonly used data reduction and diagnostic and prognostic techniques.


Introduction
Gas turbines (GTs) convert chemical energy from burned fuels into mechanical power. They use high temperatures and pressures in a combustion chamber to generate thrust, which is necessary for sustained flight. However, these extreme working conditions, including high temperatures, pressures, and rotational speeds, lead to gradual degradation of the turbine's components over time. The degradation of the submachines constituting an aircraft engine leads, in turn, to a reduction in engine performance [1] and flight safety and in an increase in fuel consumption and pollutants. In ref. [2] is presented an estimation model for the prediction of the effect of different forms of degradation on the creep life of gas turbines. Interesting research on the state of the art of aircraft and airport emissions is available in [3]. A maintenance plan is crucial for aeronautical companies. To enhance engine reliability, flight safety, and effectively manage costs, current research efforts are primarily focused on optimizing maintenance plans [4,5]. In the early days of aircraft engines, defects were identified and repaired through Corrective Maintenance (CM). However, this approach was later replaced by Preventive Maintenance (PM), in which inspections and tasks are scheduled according to the probability of failure for specific components and usage conditions. However, less frequent PM reduces safety and more frequent PM increases costs. In [6], concepts such as mean time to failure and mean time between failures are introduced and a study on the effect on the reliability of PM application is conducted. However, because PM is based on probability rather than the actual condition of the components, it can result in non-optimized costs. This approach can lead to the maintenance of components that have not yet reached a critical state, or to the failure of components that have already degraded beyond repair. To optimize maintenance plans, modern approaches such as Condition-based Maintenance (CBM) or predictive probability rather than the actual condition of the components, it can result in nonoptimized costs. This approach can lead to the maintenance of components that have not yet reached a critical state, or to the failure of components that have already degraded beyond repair. To optimize maintenance plans, modern approaches such as Conditionbased Maintenance (CBM) or predictive maintenance are increasingly being used in the aviation industry, which take into account the actual condition of the engine components [7,8]. As reported in [9], the implementation of Condition-based Maintenance (CBM) involves three primary stages: data acquisition, data processing, and maintenance decision making. CBM has the potential to reduce maintenance costs by 25−35%, eliminate breakdowns by 70−75%, reduce breakdown time by 35−45%, and increase production by 25−35% [10]. However, a maintenance strategy may be the best for a certain component and the worst for another one [11]. To continuously monitor the condition of engines, appropriate sensors are installed throughout the powertrain to gather useful information. Figure 1 illustrates the different maintenance strategies, and Table 1 provides their characteristics.  No data to analyze.
Maintenance is performed only after a detected failure by a visual inspection.
Faults cannot be predicted and are not avoided by scheduled maintenance; Higher maintenance costs; Lower reliability compared to other maintenance strategies. [4,12,13].

PM
Time-based; Scheduled. Maintenance interval depends on the probability of failure occurring; Preventive approach.
Historical data on components failure or test data.
A reliability theory based on bathtub curve assumptions.
Maintenance is performed after predetermined time intervals, obtained from statistical analysis of available historical data about failure of components or test data.
The maintenance plan is based on statistical assumptions. A component could be overhauled even if in good condition or could be subject to a fault before being overhauled; Higher maintenance costs and lower reliability compared to the predictive strategy. [5,6,12,13] .  Maintenance is performed when monitored parameters indicate an impending failure in one or more components.
More complex approach; Require more efforts and time for strategy development. [5,[7][8][9][10]12,13] The engine's performance is contingent on the performance of its individual components, which can be evaluated using parameters such as temperature, pressure, rotational speed, torque, and fuel flow. Additionally, the presence of contaminants in oil and exhaust gases, or high levels of vibration and noise from the components, may also indicate deteriorating performance [14]. For example, Blade Tip Timing (BTT) is a non-contact measuring technique commonly used today to monitor the strains and vibrations of blades. This technique addresses some of the limitations of strain gauges [15], such as the inability to use them on in-service engines because they need to be affixed to the blade airfoil. In BTT, sensors are typically mounted on the casing to detect the vibrations of the blades as they pass by [16]. The BTT technique serves as the foundation for a proposed method to detect crack propagation in rotor blades, as described in [17] This method involves monitoring the natural frequencies of cracked blades and comparing them to those of a healthy blade. In [18], a method that utilizes the BTT technique for dynamic strain reconstruction in rotating blades experiencing multi-mode vibration is presented. Dynamic strain measurement is essential for monitoring the health status of rotating blades [19]. However, the BTT technique can encounter certain issues. Mohamed et al. [20] propose an improvement of the technique. Computer algorithms are commonly used for diagnostic and prognostic tools, which are the key components of CBM. In recent years, the concept of biomimicry has been gaining momentum, driving the development of improved algorithms and materials that draw inspiration from nature itself [21,22]. Diagnostics involves determining the current condition of a system based on sensor-provided data on its physical parameters. Prognostics, on the other hand, focuses on predicting the remaining lifespan of a system using historical and current data on its condition, which are also collected by sensors. These topics will be further explained and illustrated with examples in the following sections. Section 2 of this paper covers the main degradation phenomena that affect gas turbines and their impact on engine performance. Section 3 focuses on the data acquisition and processing stage, including methods for reducing data dimensionality and noise. Section 4 provides an overview of the commonly used diagnostic and prognostic techniques. Finally, Section 5 concludes the review and offers suggestions for future research. Currently, many studies are being conducted on diagnostic and prognostic systems for aircraft engines, and examples can be found in the literature in [23][24][25][26][27][28].

Degradation Mechanisms of an Aircraft Engine
Aircraft engines, like any machine, experience degradation over time. Compressors and turbines are particularly vulnerable to degradation, which can manifest in various forms. The occurrence of degradation is a random event influenced by factors such as the operating environment [29] and type of mission (e.g., high-or low-power requirements), with unexpected events such as collisions with objects also playing a role.
Research on the impact of the operating environment on degradation can be found in [30,31]. In particular, the study reported in [30] describes two experiments in which sand is intentionally introduced into the inlet of an aircraft engine. In the first experiment, a substantial quantity of sand is rapidly introduced, while in the second, the amount of sand injected simulates what an engine would experience in real-world conditions. Both experiments result in a decrease in thrust and an increase in fuel consumption per unit of thrust. In [31], computational fluid dynamics is utilized to investigate the corrosive effect of volcanic ash on fan blades, using two different methodologies: one that assumes a constant rate of erosion and the other that adapts the erosion rate. The simulations in both cases demonstrate that erosion is more severe moving from the midspan towards the tip. On the suction side, the most significant loss of material is concentrated around the leading edge in the vicinity of the hub and extending towards the midchord for the outer midspan sections. On the pressure side, the material removal is less severe and primarily located near the midchord of the tip. The most prevalent forms of degradation are fouling, erosion, corrosion, and an increase in blade tip clearance. Severe but random events such as collisions with objects from the external environment or the engine itself can also occur. The impact of degradation on components results in a decline in performance, as evidenced by changes in Performance Parameters (PPs), which are all operating parameters that are influenced by the condition of the parts and impact engine thrust and fuel consumption [32]. Changes in non-measurable PPs can lead to deviations in measurable PPs that are dependent on the former, which can serve as an indicator of the engine's health. Table 2 provides a summary of the PPs commonly used for both stationary and aircraft engines. For compressors and turbines, flow capacity and efficiency are commonly used as PPs. These parameters are not directly measurable, but they influence the inlet and outlet temperatures and pressures of the same components [33]. A degraded state can also be diagnosed by monitoring other characteristics, such as component vibration and noise, or the presence of debris in the oil and gas path [32,[34][35][36][37][38]. A description of the main degradation mechanisms mentioned above follows.

Fouling
Fouling is the adhesion of contaminants present in the operative flow on the surfaces of components, such as dust, sand, dirt, ash, oil droplets, water mists, hydrocarbons, and industrial chemicals [39]. Due to their position in the gas path, compressors are the components most affected by fouling. In addition, compressor fouling is the most common degradation problem [40] and is usually caused by particles smaller than 2 to 10 microns [41]. The adhesion of dirt particles on the surfaces leads to a change in the aerodynamic shape and inlet angle of the airfoil, an increase in roughness, and a decrease in the opening of the airfoil [42,43], resulting in a decrease in efficiency, flow capacity, and pressure ratio [44]. The consequence on the overall GT is a decrease in output power and thermal efficiency [40]. Some of the power loss caused by compressor fouling can be recovered by washing [45,46]. The three main types of washing systems, i.e., on-line washing system, off-line crank washing system, and manual hand crank washing system, are described in [47]. Fouling has a greater effect on the first compressor stages, while it is limited in the backward stages [48,49]. Experiments conducted by Kurz et al. [50] show that the amount of dust picked up by the blades varies depending on their surface characteristics. In [49], an analytical method is proposed for investigating the fouling phenomenon in the axial compressors of industrial gas turbines. The results reveal a relationship between the axial compressor and flow characteristics, such as chord length, solidity, number of stages, particle size, flow rate and inlet flow angles, and the fouling phenomenon. Specifically, the authors of the study conclude that the number of particles adhering to the blades increases with decreasing chord length and an increase in solidity, fouling accelerates with decreasing flow rate and inlet flow angle, and fouling is more prevalent in the forward stages. Some examples of the effects of fouling on different GTs are given below. In [40], a simulation was conducted in which there was a 6% reduction in mass flow and a 5% decrease in compressor efficiency, resulting in a 55 kW drop in power and an increase of 850 Btu/kW hr in heat rate. Additional simulation results are also provided in the same study. A 5% decrease in flow capacity and 2.5% decrease in compressor efficiency results in a 10% reduction in output power [51]. A decrease of 7% in output power, accompanied by a 2.5% increase in heat rate, results from a 5% decrease in compressor inlet mass flow rate and a 1.8% decrease in efficiency, as reported in [42]. In [48], a decrease in stage pressure ratios is shown with an increase in operating hours for different compressor stages. Figure 2, taken from [52] (a and b) and from [53] (c), shows some examples of fouling. In detail, in Figure 2a,b is visible the fouling phenomena on Low-Pressure Compressor (LPC) stage blades and on Low-Pressure Turbine (LPT) vane, respectively, while in Figure 2c shows deposits of volcanic ash on the High-Pressure Turbine (HPT) nozzle guide vanes.
acteristics. In [49], an analytical method is proposed for investigating the fouling phenomenon in the axial compressors of industrial gas turbines. The results reveal a relationship between the axial compressor and flow characteristics, such as chord length, solidity, number of stages, particle size, flow rate and inlet flow angles, and the fouling phenomenon. Specifically, the authors of the study conclude that the number of particles adhering to the blades increases with decreasing chord length and an increase in solidity, fouling accelerates with decreasing flow rate and inlet flow angle, and fouling is more prevalent in the forward stages. Some examples of the effects of fouling on different GTs are given below. In [40], a simulation was conducted in which there was a 6% reduction in mass flow and a 5% decrease in compressor efficiency, resulting in a 55 kW drop in power and an increase of 850 Btu/kW hr in heat rate. Additional simulation results are also provided in the same study. A 5% decrease in flow capacity and 2.5% decrease in compressor efficiency results in a 10% reduction in output power [51]. A decrease of 7% in output power, accompanied by a 2.5% increase in heat rate, results from a 5% decrease in compressor inlet mass flow rate and a 1.8% decrease in efficiency, as reported in [42]. In [48], a decrease in stage pressure ratios is shown with an increase in operating hours for different compressor stages. Figure 2, taken from [52] (a and b) and from [53] (c), shows some examples of fouling. In detail, in Figure 2a,b is visible the fouling phenomena on Low-Pressure Compressor (LPC) stage blades and on Low-Pressure Turbine (LPT) vane, respectively, while in Figure 2c shows deposits of volcanic ash on the High-Pressure Turbine (HPT) nozzle guide vanes.

Erosion
Erosion is the gradual loss of material from the gas path due to impact with hard particles that contaminate the operational flow, such as sand, dust, dirt, ash, carbon particles, and water droplets. Unlike fouling, in erosion, the interaction between larger particles of contaminants and blades causes the latter to lose a small portion of material. This leads to a change in the aerodynamic shape and roughness, and reduces performance [54]. More in detail, erosion causes a drop in efficiency and flow capacity in an affected compressor, while manifests itself in turbines causing a decrease in efficiency and an increase in flow capacity [55]. On the overall GT, erosion leads to a decrease in output power and an increase in heat rate [42]. Otherwise, the breaks on the blades create stress concentration zones, which reduces fatigue strength. Despite the fact that erosion can occur anywhere in the gas path, turbines are more susceptible to this phenomenon than compressors [39]. Erosion is a problem most commonly encountered in aircraft engines than in industrial ones, due to more effective filtration systems used in this latter environment [56]. In [57], erosion in turbomachinery is predicted by means of an analytical model developed in the same paper. Furthermore, at the end of the study, the authors conclude that, in general, in an axial flow compressor, erosion manifests itself by cutting back the leading edge and thinning the trailing edge of the blade. They also state that, especially after impacting a rotor, particles will centrifuge radially, concentrating on the tips of the blades and causing severe damage in this location. Erosion damages can be reduced by applying a coating on compressor and turbine blades, in order to protect them from the impact of hard and large contaminants particles which cause erosion. In [58], the protective capabilities against erosion and corrosion on compressor blades of various coatings are investigated. The results conclude that coatings obtained by physical vapor deposition show the best results in both erosion and corrosion protection, particularly the (Ti, Al)N coatings. Electron beam physical vapor deposition thermal barrier coatings have been used for protection purposes for several years, and more studies on their erosion and corrosion mechanisms and resistance have been conducted [59][60][61][62]. More information on coatings and erosion are reported in [63][64][65][66][67][68][69]. Figure 3, taken from [52], portrays the erosion effect on the leading edge of fan blades.
compressors [39]. Erosion is a problem most commonly encountered in aircraft engines than in industrial ones, due to more effective filtration systems used in this latter environment [56]. In [57], erosion in turbomachinery is predicted by means of an analytical model developed in the same paper. Furthermore, at the end of the study, the authors conclude that, in general, in an axial flow compressor, erosion manifests itself by cutting back the leading edge and thinning the trailing edge of the blade. They also state that, especially after impacting a rotor, particles will centrifuge radially, concentrating on the tips of the blades and causing severe damage in this location. Erosion damages can be reduced by applying a coating on compressor and turbine blades, in order to protect them from the impact of hard and large contaminants particles which cause erosion. In [58], the protective capabilities against erosion and corrosion on compressor blades of various coatings are investigated. The results conclude that coatings obtained by physical vapor deposition show the best results in both erosion and corrosion protection, particularly the (Ti, Al)N coatings. Electron beam physical vapor deposition thermal barrier coatings have been used for protection purposes for several years, and more studies on their erosion and corrosion mechanisms and resistance have been conducted [59][60][61][62]. More information on coatings and erosion are reported in [63][64][65][66][67][68][69]. Figure 3, taken from [52], portrays the erosion effect on the leading edge of fan blades.

Corrosion
Corrosion is a chemical reaction between some reactive particles contained in the operational flow and the material that constitutes the components encountered in the gas path. "Cold corrosion" refers to the corrosion reaction that occurs in compressor sections, due to wet deposits of salts, acid, and aggressive gases such as chlorine and sulfides. On the other hand, the corrosion occurring in the turbine is known as "hot corrosion," which is a form of accelerated oxidation caused by molten salts that chemically react with the surfaces of the components on which they rest. Hot corrosion is further dividId into two classes: high-temperature hot corrosion (or type I) that occurs at a tem-

Corrosion
Corrosion is a chemical reaction between some reactive particles contained in the operational flow and the material that constitutes the components encountered in the gas path. "Cold corrosion" refers to the corrosion reaction that occurs in compressor sections, due to wet deposits of salts, acid, and aggressive gases such as chlorine and sulfides. On the other hand, the corrosion occurring in the turbine is known as "hot corrosion," which is a form of accelerated oxidation caused by molten salts that chemically react with the surfaces of the components on which they rest. Hot corrosion is further dividId into two classes: high-temperature hot corrosion (or type I) that occurs at a temperature range of 730 to 950 • C, and low-temperature hot corrosion (or type II) that occurs at a temperature range of 550 to 730 • C [70]. More information is available in [71]. Corrosion is one of the main reasons for blade failure [72]. A series of examples of aircraft failures due to corrosion are available in [73], as well as further information on the phenomenon of corrosion. RD-33 turbofan engine is presented in [74], and the effect of corrosion on it is shown. In [75], eddy current testing is presented as a means of detecting defects resulting from corrosion around the rivets of supersonic aircraft intakes. As reported in the paper, this type of corrosion is caused by the presence of moisture and salinity in the atmosphere, and eddy current testing is a more suitable technique for detecting deeper defects. Corrosion is a phenomenon that must be carefully monitored, as it can result in significant cost increases. Today, various non-destructive techniques are used to detect ongoing corrosion processes [76]. Also for corrosion, compressor and turbine blades can be protected by means of adequate coatings [77]. The effect of corrosion on the compressor is a decrease in efficiency and flow capacity, while on the turbine, corrosion leads to a decrease in efficiency and an increase in flow capacity [78], resulting in a decrease in engine performance. In Figure 4, taken from [52], are visible the effects of corrosion on an LPT vane. it can result in significant cost increases. Today, various non-destructive techniques are used to detect ongoing corrosion processes [76]. Also for corrosion, compressor and turbine blades can be protected by means of adequate coatings [77]. The effect of corrosion on the compressor is a decrease in efficiency and flow capacity, while on the turbine, corrosion leads to a decrease in efficiency and an increase in flow capacity [78], resulting in a decrease in engine performance. In Figure 4, taken from [52], are visible the effects of corrosion on an LPT vane.

Abrasion
Abrasion is a loss of material due to the rubbing of one surface against another. The removed material typically leads to an increase in seal or tip gaps. Minimizing clearances between blade tips and casing is crucial for reducing the aerodynamic losses that occur in rotating machines, as the overall engine efficiency is negatively impacted by these losses [79]. In many cases, abradable surfaces are present. In these engines, a certain amount of rubbing is allowed during the run-in of the engine to establish proper clearances [80]. In [81], a study is conducted to investigate the high-speed scraping behavior between a vulcanized silicone rubber and a Ti-6Al-4V fan blade. Blade tip rubbing can occur due to an unbalanced or misaligned rotor due to the shaft bearing damage, engine rotor flexing at heavy operating loads, or the thermal growth of blades [82]. The result is an increase in blade tip clearance. Another phenomenon that leads to tip rubbing is the engine casing distortion due to flight loads. The principal effect of this is a reductIon in the efficiency of the damaged component [78], which results in a drop in engine performance. An example of abrasion on a High-Pressure Compressor (HPC) blade is depicted in Figure 5, taken from [52].

Abrasion
Abrasion is a loss of material due to the rubbing of one surface against another. The removed material typically leads to an increase in seal or tip gaps. Minimizing clearances between blade tips and casing is crucial for reducing the aerodynamic losses that occur in rotating machines, as the overall engine efficiency is negatively impacted by these losses [79]. In many cases, abradable surfaces are present. In these engines, a certain amount of rubbing is allowed during the run-in of the engine to establish proper clearances [80]. In [81], a study is conducted to investigate the high-speed scraping behavior between a vulcanized silicone rubber and a Ti-6Al-4V fan blade. Blade tip rubbing can occur due to an unbalanced or misaligned rotor due to the shaft bearing damage, engine rotor flexing at heavy operating loads, or the thermal growth of blades [82]. The result is an increase in blade tip clearance. Another phenomenon that leads to tip rubbing is the engine casing distortion due to flight loads. The principal effect of this is a reductIon in the efficiency of the damaged component [78], which results in a drop in engine performance. An example of abrasion on a High-Pressure Compressor (HPC) blade is depicted in Figure 5, taken from [52].

Thermal Distortion
Thermal distortions typically occur at combustor exit and are due to some problems located in the previous stations, such as the cocking of fuel nozzles, change in fuel spray angles, change in compressor performance, and warping of the combustor liners, which in turn result in a change in radial and circumferential temperature patterns at the combustor exit. The results could be temporary or permanent damage in the downstream components, such as crack, bending, warp, burn in turbine nozzle guide vanes, and change in nozzle guide vanes areas. The principal effect is a drop in turbine efficiency [26,78], which is responsible for a reduction in engine performance. Figure 6, taken from [52], reports a burned HPT blade.

Thermal Distortion
Thermal distortions typically occur at combustor exit and are due to some problems located in the previous stations, such as the cocking of fuel nozzles, change in fuel spray angles, change in compressor performance, and warping of the combustor liners, which in turn result in a change in radial and circumferential temperature patterns at the combustor exit. The results could be temporary or permanent damage in the downstream components, such as crack, bending, warp, burn in turbine nozzle guide vanes, and change in nozzle guide vanes areas. The principal effect is a drop in turbine efficiency [26,78], which is responsible for a reduction in engine performance. Figure 6, taken from [52], reports a burned HPT blade.

Thermal Distortion
Thermal distortions typically occur at combustor exit and are due to some problems located in the previous stations, such as the cocking of fuel nozzles, change in fuel spray angles, change in compressor performance, and warping of the combustor liners, which in turn result in a change in radial and circumferential temperature patterns at the combustor exit. The results could be temporary or permanent damage in the downstream components, such as crack, bending, warp, burn in turbine nozzle guide vanes, and change in nozzle guide vanes areas. The principal effect is a drop in turbine efficiency [26,78], which is responsible for a reduction in engine performance. Figure 6, taken from [52], reports a burned HPT blade.

Foreign Object Damage and Domestic Object Damage
Foreign Object Damage (FOD) and Domestic Object Damage (DOD) refer to damage caused by impact with objects from the external environment and from the GT itself, respectively. For example, FOD can be caused by impact with ingested birds, ice, stones, debris, and gravel on runways, while DOD can be caused by impact with fragments detached from blades or large carbon particles from the fuel nozzles [39]. FOD and DOD damages are comprised between nonrecoverable (with washing) fault and a catastrophic engine fault [42]. FOD is one of the biggest problems and can lead to catastrophic failures [83]. The effects on performance primarily focus on the efficiency variation of the affected component. The changes in flow capacity depend on how the objects causing the FOD/DOD interact with the component. If the impact results in a lost blade, flow capacity increases, while if the object gets stuck and obstructs the flow, flow capacity decreases [78]. The cases shown in Figure 7, both taken from [52], depict damage that is typically due to the impact with a sharp object (a) and the breaking of a blade of a HPC (b).

Foreign Object Damage and Domestic Object Damage
Foreign Object Damage (FOD) and Domestic Object Damage (DOD) refer to damage caused by impact with objects from the external environment and from the GT itself, respectively. For example, FOD can be caused by impact with ingested birds, ice, stones, debris, and gravel on runways, while DOD can be caused by impact with fragments detached from blades or large carbon particles from the fuel nozzles [39]. FOD and DOD damages are comprised between nonrecoverable (with washing) fault and a catastrophic engine fault [42]. FOD is one of the biggest problems and can lead to catastrophic failures [83]. The effects on performance primarily focus on the efficiency variation of the affected component. The changes in flow capacity depend on how the objects causing the FOD/DOD interact with the component. If the impact results in a lost blade, flow capacity increases, while if the object gets stuck and obstructs the flow, flow capacity decreases [78]. The cases shown in Figure 7, both taken from [52], depict damage that is typically due to the impact with a sharp object (a) and the breaking of a blade of a HPC (b).

Increased Blade Tip Clearance
Due to phenomena that can remove material, such as erosion, impact with contaminants, or thermal and centrifugal expansions, the clearance between moving blades and stationary blades with casing and a rotating hub, respectively, increases [33,84]. The power elaborated by the compressor and turbine depends on the amount of air flowing through the area occupied by blades, so the increase in blade tip clearance results in more flow passing through the compressor and turbine sections without being processed by the stage, resulting in a decrease in the efficiency of the rotating machine [85]. Stall margin is also affected by an increase in blade tip clearance in axial compressors [86]. An increase in blade tip clearance of about 0.125 mm in HPC or HPT can reduce the related efficiencies by 0.5%. Furthermore, an increase in engine specific fuel consumption by 0.2% and 0.25% is present when HPC is affected and when HPT is affected,

Increased Blade Tip Clearance
Due to phenomena that can remove material, such as erosion, impact with contaminants, or thermal and centrifugal expansions, the clearance between moving blades and stationary blades with casing and a rotating hub, respectively, increases [33,84]. The power elaborated by the compressor and turbine depends on the amount of air flowing through the area occupied by blades, so the increase in blade tip clearance results in more flow passing through the compressor and turbine sections without being processed by the stage, resulting in a decrease in the efficiency of the rotating machine [85]. Stall margin is also affected by an increase in blade tip clearance in axial compressors [86]. An increase in blade tip clearance of about 0.125 mm in HPC or HPT can reduce the related efficiencies by 0.5%. Furthermore, an increase in engine specific fuel consumption by 0.2% and 0.25% is present when HPC is affected and when HPT is affected, respectively [87]. Other examples show a reduction of 3% and 2% in flow capacity and isentropic efficiency, respectively, due to an increase in tip clearance of 0.8% [88]. In [89], the results of experiments on the effect of compressor rotor tip clearance on the performance of a turboshaft engine are reported. The experiments were conducted on a combined five-stage axial and one-stage centrifugal compressor. Specifically, the experiments were conducted by simulating an increase in tip clearance, first only at the first axial stage, and subsequently, only at the fifth axial stage for different severity levels in both cases; then, finally simulating equal damage severity at all five axial stages simultaneously. All the experiments show a reduction in overall engine performance. In more detail, the results show that a small increase in tip clearance throughout all stages is responsible for a greater loss in performance than a larger increase in tip clearance at only one stage. A further study on the effect of increased blade tip clearance has been conducted by Gourdain et al. where three different configurations were considered for a multi-stage high-pressure compressor, one of which considered a casing treatment based on a honeycomb design. More information is available in [90]. The increase in blade tip clearance has a significant impact on the aero-elastic stability of the blades, which is an important aspect that cannot be ignored. The reason for this is the profound impact that aero-elastic phenomena have on blade health status and, consequently, engine performance. Aero-elastic phenomena such as flutter and forced vibration, which are caused by the interaction between unsteady aerodynamic forces and blade motion, or by upstream and downstream vane wakes or inlet distortion, respectively, can lead to high cycle fatigue and fractures [91]. In [92], the correlation between the flow of tip clearance and the behavior of another aero-elastic phenomenon called non-synchronous vibration is investigated. Non-synchronous vibration manifests itself similarly to flutter, but it occurs in a region that is far removed from it. In extreme cases, non-synchronous vibration can also result in blade loss [93]. An interesting work is present in [94], in which experiments were conducted to excite aero-elastic phenomena, such as blade forced response, flutter, and acoustic resonance, on an aero-engine's fan. Acoustic measurements were used as a means of recognizing the presence of a particular aero-elastic phenomenon through the appropriate acoustic features. Interested readers can find additional information on blade tip clearance and the technologies used to measure it in [85,95]. A damaged tip (together with the effect of erosion) is visible in Figure 8, taken from [96]. used as a means of recognizing the presence of a particular aero-elastic phenomenon through the appropriate acoustic features. Interested readers can find additional information on blade tip clearance and the technologies used to measure it in [85,95]. A damaged tip (together with the effect of erosion) is visible in Figure 8, taken from [96].  Table 3 summarizes the degradation phenomena described earlier, along with the effects on the affected components and on the overall GT, based on data available in the literature.   Table 3 summarizes the degradation phenomena described earlier, along with the effects on the affected components and on the overall GT, based on data available in the literature. Change in the aerodynamic shape of blades. Increase in roughness.

Summary of Degrading Phenomena
Corroded compressor: decrease in η c and f c ; Corroded turbine: decrease in η t and increase in f t .
Drop in output power and increase in heat rate.
Compressors and turbines.
Increase in seal and tip gaps.
Drop in output power. [78][79][80][81][82] Thermal distortion Damages in parts located at burner exit due to a change in radial and circumferential temperature patterns.
Generation of cracks, bends, warpings, burns in turbine nozzle guide vanes and change in nozzle guide vanes areas.
Decrease in η t . Drop in output power. [26,78] FOD/DOD Impact with an ingested object or with detached ones from engine components, respectively.
Compressors and turbines.
Bends and breaks in blades.
Affected compressor: decrease in η c and increase or decrease in f c ; Affected turbine: decrease in η t and increase or decrease in f t .
Drop in output power, increase in heat rate. [39,42,78,83] Increase in blade tip clearance Increase in clearance between moving blades and stationary blades with casing and rotating hub, respectively.
Compressors and turbines.
Tip breaks and consequent increase in clearance.
Affected compressor: decrease in η c and f c ; Affected turbine: decrease in η t and f t .

Data Acquisition and Processing
The methodologies for prognosis and diagnosis are based on a set of information that assesses the health of an engine and/or its components. In an aircraft engine, various data can be collected in real time to inform diagnostic or prognostic algorithms, such as pressures and temperatures at the inlet or outlet of compressors, turbines, and burners, fuel flow, shaft speed, or torque. To accomplish this, appropriate sensors must be installed throughout the engine. To enhance the performance of the diagnostic or prognostic system, the collected data can be analyzed to eliminate noise measurements, disregard measurements from faulty sensors, and discard redundant information that unnecessarily increases the dataset. This section introduces the concept of data cleaning and provides examples of data reduction techniques used.

Sensor Measurements
Sensors are the tools used to collect crucial information about the engine's health and the status of its components and subsystems [97]. The use of sensors is very important to increase flight safety, and good knowledge in this field is fundamental to obtain a modern and safe engine [98]. Sensors are distributed not only in the engines but throughout the entire aircraft. Modern aircrafts have hundreds or even thousands of sensors [99]. Choosing which sensors to install is not a trivial task, and performing a thorough analysis to understand the advantages of various sensors can greatly aid in the decision-making process [100]. Like all other components, sensors can experience faults. A faulty sensor will supply incorrect data to the diagnostic or prognostic algorithm, which may subsequently lead to an inaccurate assessment of the engine's health. To enhance the performance of the EHM system, processing the data obtained from the sensors can be beneficial. For example, in [101], a sensor Fault Detection, Isolation, and Identification (FDII) technique was developed to detect the presence of single or multiple sensor faults. Another noteworthy study was conducted by Sadough Vanini et al. [102], in which they developed an FDII model that handles various degradation scenarios, accounts for concurrent and sensor faults, and is able to rectify erroneous readings from a faulty sensor using the readings of the remaining functional sensors. A similar but older work is presented in [103]. Sensor measurements are also affected by noise and bias, which can increase over time due to sensor aging [104]. Signal filtering or smoothing are helpful approaches to improve the quality of information provided by sensors. Figure 9 illustrates the difference between two different approaches. sensors [99]. Choosing which sensors to install is not a trivial task, and performing a thorough analysis to understand the advantages of various sensors can greatly aid in the decision-making process [100]. Like all other components, sensors can experience faults.
A faulty sensor will supply incorrect data to the diagnostic or prognostic algorithm, which may subsequently lead to an inaccurate assessment of the engine's health. To enhance the performance of the EHM system, processing the data obtained from the sensors can be beneficial. For example, in [101], a sensor Fault Detection, Isolation, and Identification (FDII) technique was developed to detect the presence of single or multiple sensor faults. Another noteworthy study was conducted by Sadough Vanini et al. [102], in which they developed an FDII model that handles various degradation scenarios, accounts for concurrent and sensor faults, and is able to rectify erroneous readings from a faulty sensor using the readings of the remaining functional sensors. A similar but older work is presented in [103]. Sensor measurements are also affected by noise and bias, which can increase over time due to sensor aging [104]. Signal filtering or smoothing are helpful approaches to improve the quality of information provided by sensors. Figure 9 illustrates the difference between two different approaches. Signal denoising is a very active area of research [105]. As reported in [106], the presence of noise in a sensor measurement can be modeled as follows: where ( ) denotes the measured noisy signal, ( ) denotes the true signal, and ( ) represents the sum of all the noise present in the measurement. For more detailed information on sensor noise, the interested reader is referred to the same paper. A technique employed for noise reduction is the wavelet transform [107,108]. The wavelet transform technique is capable of expressing a given signal as a linear combination of basis functions called wavelets. Wavelets are obtained starting from a base wavelet ( ) called the "mother wavelet", by scaling and translation: where a is the scale factor and b is the translation factor. Further details on wavelet theory are available in [109].

Data Reduction Techniques
Diagnostic and prognostic algorithms process a large amount of data from the engine, which contains information about certain monitored parameters used to assess the condition of the components. To improve performance in terms of required computer Signal denoising is a very active area of research [105]. As reported in [106], the presence of noise in a sensor measurement can be modeled as follows: where y M (t) denotes the measured noisy signal, y T (t) denotes the true signal, and x(t) represents the sum of all the noise present in the measurement. For more detailed information on sensor noise, the interested reader is referred to the same paper. A technique employed for noise reduction is the wavelet transform [107,108]. The wavelet transform technique is capable of expressing a given signal as a linear combination of basis functions called wavelets. Wavelets are obtained starting from a base wavelet Ψ(t) called the "mother wavelet", by scaling and translation: where a is the scale factor and b is the translation factor. Further details on wavelet theory are available in [109].

Data Reduction Techniques
Diagnostic and prognostic algorithms process a large amount of data from the engine, which contains information about certain monitored parameters used to assess the condition of the components. To improve performance in terms of required computer memory, time, and effort, it is beneficial to analyze the data and use appropriate data reduction techniques to reduce the dataset size [110][111][112]. In this section, some techniques used to perform a data reduction process, learned from [112], are presented. The notation used in the description of the data reduction methods is the same used in [112], from which information was mainly taken and is reported in Table 4 for simplicity. Table 4. Notation adopted in the description of the following data reduction methods [112]. Principal Component Analysis (PCA) is an unsupervised, linear transformation algorithm that produces the so-called Principal Components (PCs) (which represents the new features) based on the variance of the data. The PCA algorithm works by projecting the highly dimensional dataset into a new subspace in which the orthogonal axes, the PCs, represent the direction of the maximum data variance. The first of the PCs is the one with the highest variance, which decrease for the next PCs. The algorithm followed by the PCA is:
Build the covariance matrix XX T ; 2.
Obtain Eigen values and Eigen vectors by applying Eigen decomposition to XX T ; 3.
Sort Eigen values from the higher to the lower; 4.
Build the dxk transformation matrix W with k top Eigen vectors; 5.
Uses W to transform X to obtaining the new subspace Y = XW.
The PCA algorithm uses a transformation matrix W with which it maps the original d-dimensional space X to a new k-dimensional space Y, with k ≤ d code. The PCs are the eigen vectors of the covariance matrix XX T , obtained together with the eigen vectors by applying the linear eigen decomposition method to the covariance matrix. Eigen vectors and eigen values show the data direction and the data magnitude, respectively. Each column of the matrix W represents a different eigen vector, and they are ordered from the one with the highest associated eigen value to that with the lowest. The PCA technique is suitable for linear problems, but it does not operate well when used for non-linear data. In this case, the Kernel Principal Component Analysis, which represents a version of PCA suitable for non-linear scenarios, is a better candidate. PCA is used in [113] to conduct a data reduction process to lower the dimensionality of a dataset used for aircraft engine diagnostics. The dimensionality reduction capabilities of PCA are also employed in [114] with the goal of predicting the Remaining Useful Life (RUL) of airplane engines.

KPCA
The KPCA is a non-linear adaptation of the common PCA. KPCA operates by mapping the X matrix to a higher feature space φ(X) using polar coordinates. This projection is made by using the kernel method: The kernel function avoids the mapping process with the function φ, which makes the process cheaper. The most common kernels exploited in KPCA are reported in Table 5. Table 5. Commonly used kernels in KPCA.

Kernel Description
Linear: k(x a , x b ) = x a T x b + constant Generally used when data are linearly separable. Polynomial: Shows the similarity among data in a feature space over the polynomials of actual variables.
Used when data are non-linearly separable.
Mostly used in neural networks.
KPCA first applies the kernel method and subsequently uses the common PCA algorithm on the new linearly separable data. The algorithm followed is: Produce linear data X with kernel mapping function K to X; 2.
Uses the common PCA to X , obtaining reduced space Y.
KPCA is employed in [115] as a dimensionality reduction technique for datasets used in aircraft engine diagnostics. In [116], the data reduction capabilities of the KPCA are used to develop a method used for face recognition purposes. Another example of the application of the KPCA as an element to perform a data reduction process in a diagnostic tool used for aircraft engines is available in [117].

LDA
Linear Discriminant Analysis (LDA) is another technique used for FE, which works by identifying a new feature space by projecting data with the goal of maximizing the classes' separability. Starting from the d independent features of a dataset, the LDA extracts k new independent features that separate the dependent features the most, producing a number of components smaller than the number of classes-1. The algorithm followed is: Obtain two scatter matrices of X: in-between-class and within-class; 2.
Calculate the eigen values and eigen vectors of the scatter matrices; 3.
Rank eigen vectors in descending order, based on eigen values; 4.
Build the transformation dxk matrix W with k top eigen vectors; 5.
Uses W to transform X to obtaining the new subspace Y = XW.
The two scatter matrices are built with: where SM b and SM w are the in-between-class and the within-class matrices, respectively, which compute the distance between the mean of each class and the distance between the mean of each class together with the data within the class. Furthermore, in Equations (4) and (5), m represents the number of classes, µ denotes the overall mean, µ k and N k are the mean and sizes of the respective classes, respectively, and finally, µ k is the mean vector of a class. The final part of the algorithm represents the final part of the PCA algorithm. The LDA algorithm is used in [118] for FE purposes to develop a diagnostic tool for rolling bearings.

MDS
The Multi-Dimensional Scaling (MDS) method performs an FE process by focusing on the relationships between data in multi-dimensional space. Two main versions of the MDS algorithm exist: the metric MDS (or classic) and the non-metric MDS. The type here reported is the classic one. The algorithm followed by the classic MDS is: 1.
Obtain the dissimilarity matrix d X X; 2.
Compute K by utilizing the centering matrix H; 3.
Uses eigen decomposition with K to have top k Eigen values and corresponding vectors; 4.
Provide Y using diagonal matrix K and top k Eigen vectors.
MDS computes the dissimilarity matrix such that similar data are near and less similar ones are far apart. Starting from dissimilarity matrix d X , MDS search the output Y for which the similarity between d X and d Y is maximized, represent the distance between any two points a and b. The kernel matrix K is obtained with the following equation: in which the centering matrix is H = I − 1 n ee T , where I is the identity matrix and e is a column vector of 1 s. Applying eigen decomposition to K, the matrices D and B containing its eigen values and eigen vectors are obtained. After the selection of the top k eigen values, removing the rest of the eigen values and corresponding eigen vectors, matrices D and B becomeD ∈ R kxk and B 1 ∈ R nxk , representing the diagonal matrix of top k eigen values and the matrix containing the top k eigen vectors, respectively. Finally, the following equation: performs the mapping of the data to a low dimension. MDS is used in [119], located in a health monitoring system developed to monitor the health status of the landing gear retraction/extension system.

SVD
Singular Value Decomposition (SVD) is able to provide a representation of a dataset as a matrix with any number of dimensions. The precision of the representation of the SVD depends on the number of dimensions (components) chosen; the greater they are, the better the representation. In SVD, the following law is used to choose the top k largest singular values: where X is the original matrix; N is an nxk matrix of orthogonal unit vector columns, ordered by importance; S is a kxk diagonal matrix; and Z is a kxd orthogonal matrix. The SVD uses the following equation: to obtain a reduced matrix Y in which only the top k singular values are retained. In Equation (9), N k , S k , and Z T k are a truncation of N, S, and Z T . The diagonal entries of S are called the singular values. They are positive and arranged in descending order. The algorithm followed by SVD is:
Obtain Y by selecting k top singular values from S.
SVD is used in [120] together with PCA as a data reduction tool, which represents part of a more complex system developed to perform RUL estimation in aircraft engines.

LLE
Locally Linear Embedding (LLE) assumes that initial data lie on a smooth nonlinear manifold embedded in a high feature space to produce low-dimensional global coordinates. During feature reduction, LLE preserves the local properties of a data point x a by means of the linear combination of the reconstruction matrix W ab with the c-nearest neighbors x b . LLE uses the Euclidean distance to find the c-nearest neighbors of each data. Subsequently, by reducing the reconstruction error (Equation (10)), the local weights for each data point (X a ), which are an optimal representation of the data (X a ) as a linear combination of the neighbors, are calculated. Next, the Eigen vector-based optimization is used to obtain a new vector space Y. The weights with which the vectors (data points) are rebuilt from their nearest neighbors are computed. The reconstruction error is: which is minimized under the condition that all data (X a ) are rebuilt only from the nearest neighbors (W ab = 0 when X a and X b are not neighbors) and that each row of the weight matrix gives a sum of 1, i.e., ∑ b W ab = 1. The mapping of the data X a from d-dimension to Y a in k-dimension is obtained by minimizing the cost: The algorithm performed by the LLE is the following:
Find the c-nearest neighbors for each data point of X; 2.
Calculate local weights W with which data are best reconstructed (X ) from their neighbors; 3.
Use the weights from the previous step to map X to Y on k-dimensions by minimizing the cost.
An example of the application of the LLE for health monitoring purposes is available in [121], where it is present as a part of a diagnostic approach applied to a case of a gearbox.

ISOMAP
Isometric Mapping (ISOMAP) is a method that overcomes the problem of classical scaling that they are not able to operate with the nonlinear structure in a manifold. The first step determines the geodesic distances to obtain a neighborhood graph of the data, in which data x a and their nearest neighbors are connected. Next, the algorithm finds the shortest path for each pair of data in the graph to approximate the geodesic distance between them, obtaining a pairwise geodesic distance matrix d G . Subsequently, the MDS method with kernel matrix is applied to d G . Equation (6) is applied to the matrix d G , and finally, ISOMAP finds the low-dimension embedding of a data point by applying Equation (7).
Obtain the neighborhood graph of X; 2.
Compute the matrix d G of the geodesic distances; 3.
Use MDS method with d G to obtain the new space Y.
A practical example of the application of ISOMAP is available in [122], where it is used as a nonlinear data reduction tool for a diagnostic system developed to detect faults in rotating machines.

LE
Laplacian Eigenmap (LE) operates similarly to the LLE, searching low-dimension data by maintaining the local properties of a manifold. Initially, LE creates a neighborhood graph G in which the data x a and the related nearest neighbors are linked to each other. A weighted edge connects all the x a and x b data in G , and the Gaussian kernel function is used where the weights follow this equation: where WE ab is the weight of the connection between x a and x b , and σ is the variance of the Gaussian leading to an adjacency matrix (W). The dimensionality reduction is performed by minimizing a cost function: The cost function above can be reformulated as the Eigen decomposition problem: where L = W − D is the Laplacian matrix of G , D is a diagonal matrix whose elements are the row sums of W, and the W is a matrix with entries equal to 1 or 0 and with the diagonal entries equal to 0. The W entries suggests if a pair of vertices are adjacent or not. The FE technique is based on the following algorithm: • Input: X ∈ R nxd ; Output: Y ∈ R nxk ; 1.
Construct the neighborhood graph G of X using adjacency matrix W; 2.
Compute the weights of the edges of G ; 3.
Optimize the cost function to obtain the new space Y.
LE is applied in [123], where a tool is developed to perform fault diagnosis and prognosis on turbofan engines.

ICA
Independent Component Analysis (ICA) is a linear and supervised technique. ICA can reduce the higher and second order dependencies of a dataset to create new features that are statistically independent. The ICA technique looks for directions that are independent from each other. ICA performs a decomposition of the matrix X: where A is the mixing matrix, while S is the independent components. Subsequently, create the Y matrix by selecting the top k independent components: The algorithm of the ICA follows these steps: • Input: X ∈ R nxd ; Output: Y ∈ R nxk ; 1. Performs a decomposition of X to A and S; 2.
Select top k independent components; 3.
Obtain Y exploiting the k components.
The ICA is used in [124], where a technique able to detect the presence of debris in the exhaust gases of aircraft engines by means of electrostatic induction is presented. Another example of the application of the ICA is available in [125], where an intelligent system to detect and diagnose induction motor faults is developed.

t-SNE
t-Distributed Stochastic Neighbor Embedding (t-SNE) is a nonlinear, unsupervised, and manifold-based reduction method in which data mapping from a higher to a low dimension takes place by preserving the significant structure of the original data. T-SNE performs a conversation of the high-dimensional Euclidean distances into conditional probabilities representing similarities for every pair of data. The equation: where P a|b is the conditional probability which represents the similarity of data x a to data x b and provides information about the proximity of the two data points x a and x b , considering a Gaussian distribution around x b with a variance σ 2 . Subsequently, the set of probability Q a|b in the low-dimensional space is computed by means of a Student's t-distribution with one degree of freedom. Finally, the method minimizes the difference between the two cited probabilities from the low-dimensional to the high-dimensional spaces. The difference is given by the optimization of the cost function φ: The steps of the t-SNE algorithm are: • Input: X ∈ R nxd ; Output: Y ∈ R nxk ; 1.
Compute the conditional probabilities P a|b and Q a|b ; 2.
Minimize the difference between P a|b and Q a|b to perform a mapping process of X to Y.
The properties of the t-SNE algorithm are exploited in [126,127] in the development of an engine fault diagnosis tool. Table 6, which is built with information available in the tables present in [111,112], summarizes the data reduction techniques reported together with our addition of some Refs, in which their application for health monitoring purposes is shown. Table 6. Collection of the data reduction methods listed in this paper and taken from [112].  Requires more memory and less efficient with noised data.

Method
[121] ISOMAP Unsupervised Not specified classifier. Relationship between data points is preserved.
Suffers from topological instability. [122] LE Unsupervised SVM * for diagnostics and PHM * for prognostics. Gives a unique solution.
Is able to filter noise from the signal. Long training time. [124,125] t-SNE Unsupervised No diagnostic or prognostic algorithms. Based on spatial structural characteristics of QAR * data [126]; ANN [127].
Works efficiently with nonlinear data.
Provides only 2 or 3 features. Computationally complex.

Diagnostic and Prognostic Techniques
As previously discussed, CBM is a maintenance-optimization strategy utilized in the aviation industry to achieve cost savings, improve flight safety, and streamline maintenance management [129]. The principal disciplines constituting CBM are diagnostics and prognostics [26]. Diagnostics is the field that deals with assessing the condition of a system based on information provided by its intrinsic physical parameters. In the context of aircraft engines, diagnostics refers to the process of determining the condition of the engine and/or its components using data from sensors or information gathered from visual inspections [26]. Prognostics, on the other hand, is the field that deals with forecasting the remaining lifespan of a system, based on historical and current data related to the system's condition. In the aviation industry, prognostics is widely used to estimate the RUL of engines by analyzing data from previous flights and in-flight data obtained from sensors [130]. Figure 10 presents an illustration of diagnostic and prognostic approaches. These methods can be divided into two main categories: model-based methods and data-driven methods. Model-based methods are based on a mathematical model of the engine, which links the non-measurable and measurable parameters of the engine with mathematical and thermodynamic equations that describe the engine's behavior and from which the degradation process can be predicted. On the other hand, data-driven methods do not rely on mathematical models of the engine, which can be difficult to obtain due to a lack of information about the gas turbine parameters and component maps. Instead, they are based on a set of historical data on gas turbine measurements under various operating conditions, from which they can identify trends. In recent years, research has also focused on developing hybrid methods [26,32,131,132]. In the present section, a list of the most used diagnostic and prognostics technique is presented and explained [25,39,[133][134][135].

Gas Path Analysis (GPA)
The objective of the GPA is to detect the occurrence of a fault in a component by monitoring certain parameters related to its condition [136,137]. As reported in [32], GPA was introduced by Urban [138] and later made more popular in the 1980s thanks to Saravanamuttoo et al. [139,140]. methods do not rely on mathematical models of the engine, which can be difficult to obtain due to a lack of information about the gas turbine parameters and component maps. Instead, they are based on a set of historical data on gas turbine measurements under various operating conditions, from which they can identify trends. In recent years, research has also focused on developing hybrid methods [26,32,131,132]. In the present section, a list of the most used diagnostic and prognostics technique is presented and explained [25,39,[133][134][135].

Gas Path Analysis (GPA)
The objective of the GPA is to detect the occurrence of a fault in a component by monitoring certain parameters related to its condition [136,137]. As reported in [32], GPA was introduced by Urban [138] and later made more popular in the 1980s thanks to Saravanamuttoo et al. [139,140].
The goal of detecting, isolating, and quantifying certain faults can be achieved if the fault in question results in a change in the measurable (i.e., dependent) variables [141]. Independent variables, such as flow capacity and efficiencies, which can serve as a health indicator for the components, are not directly measurable, but are physically linked to the measurable (or dependent) variables, such as temperatures or pressures. Therefore, a change in independent variables (not measurable) results in a change in dependent (measurable) variables. In an arbitrary GT configuration, dependent variables are expressed as a function, such as: where denotes the input vector, i.e., the operating conditions; the functions of ambient temperature and pressure, and of power-setting parameters. Meanwhile, is the independent variables vector containing the PPs of the components such as flow The goal of detecting, isolating, and quantifying certain faults can be achieved if the fault in question results in a change in the measurable (i.e., dependent) variables [141]. Independent variables, such as flow capacity and efficiencies, which can serve as a health indicator for the components, are not directly measurable, but are physically linked to the measurable (or dependent) variables, such as temperatures or pressures. Therefore, a change in independent variables (not measurable) results in a change in dependent (measurable) variables. In an arbitrary GT configuration, dependent variables Z are expressed as a function, such as: where I denotes the input vector, i.e., the operating conditions; the functions of ambient temperature and pressure, and of power-setting parameters. Meanwhile, H is the independent variables vector containing the PPs of the components such as flow capacities and efficiencies [141]. Considering a very small change in health parameters and a fixed operating point, a linear approximation can be performed: This is the equation for Linear GPA, ignoring measurement noise and bias. ICM stands for Influence Coefficient Matrix, which can be obtained by solving a set of differential equations expressing the thermodynamic relationship between dependent and independent variables, or with an approximate approach that involves perturbing the independent variables one at a time and observing their effect on the dependent variables [141]. Inverting the ICM, the Fault Coefficient Matrix (FCM) is obtained, and changes in independent variables are obtainable by the change in dependent ones as: However, a linear approximation is suitable for a fixed operating condition and assuming little change in dependent and independent variables. When large variations in operating conditions and severe degradation are taken into account, Linear Gas Path Analysis (LGPA) becomes less reliable. These limitations can be overcome by introducing Non-Linear Gas Path Analysis (NLGPA), which can be obtained through successive iterations using LGPA and the Newton-Raphson method until an exact solution of the non-linear equation 19 is obtained [137]. As mentioned in [137], each independent parameter can be expressed individually as a unique function of the dependent parameters. Denoting with: the relation between dependent and independent variables and considering a small change in H: F(H + δH) = Z + δZ (23) which becomes: Using Taylor series expansion, the term f (H + δH) can be rewritten as: Jacobian notation is used to refer to the first derivative of the matrix function F(H + δH) in the Taylor Series, that is: and: where h i with i = 1, 2, . . . , m are the independent parameters and f j with j = 1, 2, . . . , n are the relations between dependent and independent parameters. For small variation in H, HOTs can be neglected and Equation (25) becomes: Combining Equations (24) and (28): By inverting J, as described by Donaghy [142]: the correction δH to the independent variable is added to the solution vector: The iterative process is repeated until a state of convergence is achieved. More exhaustive information is available in [137]. In [143] are presented some examples on GPA application for different purposes, such as component condition monitoring, the analysis and validation of essential performance parameters used for EHM, and the prediction of the effect of component performance restoration. The results demonstrate a good diagnostic performance and the utility of performing a performance restoration effect analysis. Zhou et al. [144] proposed a fault diagnosis and prognosis method based on a long short-term memory neural network and GPA algorithms. More specifically, in the work, the health parameters diagnosed by means of GPA are used to predict the future degradation of components. The results show that the Root Mean Squared Error (RMSE) of the diagnosed health parameters remains below 0.193%, and equals 0.232%, 0.029%, 0.069%, and 0.043% for the predicted health parameters of each component, considering the best prediction model tested. Figure 11, adapted from [39], shows an illustration useful to compare LGPA and NLGPA approaches. on a long short-term memory neural network and GPA algorithms. More specifically, in the work, the health parameters diagnosed by means of GPA are used to predict the future degradation of components. The results show that the Root Mean Squared Error (RMSE) of the diagnosed health parameters remains below 0.193%, and equals 0.232%, 0.029%, 0.069%, and 0.043% for the predicted health parameters of each component, considering the best prediction model tested. Figure 11, adapted from [39], shows an illustration useful to compare LGPA and NLGPA approaches.

Kalman Filters (KFs)
KFs were presented by Kalman in 1960 [145]. KFs are a specific type of Bayesian recursive filter and are particularly well-suited for state estimation in cases of linear systems and Gaussian noise [146]. Kalman Filters (KFs) are widely used in applications such as satellites, spacecrafts, and aircrafts for the purpose of automatic control of the systems [147]. KFs are a widely used technique in aircraft engine diagnostics. In [148], a comparison of the capabilities of KFs and neural networks to isolate a single GT fault is presented. The results demonstrate a comparable level of accuracy between the

Kalman Filters (KFs)
KFs were presented by Kalman in 1960 [145]. KFs are a specific type of Bayesian recursive filter and are particularly well-suited for state estimation in cases of linear systems and Gaussian noise [146]. Kalman Filters (KFs) are widely used in applications such as satellites, spacecrafts, and aircrafts for the purpose of automatic control of the systems [147]. KFs are a widely used technique in aircraft engine diagnostics. In [148], a comparison of the capabilities of KFs and neural networks to isolate a single GT fault is presented. The results demonstrate a comparable level of accuracy between the methods, with slightly better performance from the KF approach. In [149][150][151], Kobayashi and Simon investigate the capabilities of KFs for sensor and actuator fault detection and isolation, as well as the detection of component faults. The accuracy of the developed system is evaluated through a series of simulations of various engine faults. A similar research is proposed in [152]. For interested readers, other examples are available in [153][154][155]. KFs utilize a recursive solution, meaning each updated estimate of the state is calculated from the previous estimate and the new input data [156]. According to Luppold et al., Kalman Filters (KFs) are a data processing algorithm utilized to estimate both system states and unmeasurable parameters in real time [157]. KFs are suitable for linear systems. Adaptations for nonlinear systems include Extended KFs (EKFs) and Unscented KFs (UKFs). A more detailed explanation of KFs, EKFs, and UKFs can be found in [158]. In [159], KFs are used to develop a tool for prognosis purposes with multiple sources of uncertainty. The method has been tested on an aircraft fuel feeding system case and has shown good results. Figure 12 illustrates the engine health monitoring approach using KFs [160].

Genetic Algorithms (GAs)
Developed by John Holland at the University of Michigan, GAs are algorithms that take inspiration from the mechanism of natural selection, where the "fittest" individuals pass on their characteristics to the next generation [161]. utilized to estimate both system states and unmeasurable parameters in real time [157]. KFs are suitable for linear systems. Adaptations for non-linear systems include Extended KFs (EKFs) and Unscented KFs (UKFs). A more detailed explanation of KFs, EKFs, and UKFs can be found in [158]. In [159], KFs are used to develop a tool for prognosis purposes with multiple sources of uncertainty. The method has been tested on an aircraft fuel feeding system case and has shown good results. Figure 12 illustrates the engine health monitoring approach using KFs [160].

Genetic Algorithms (GAs)
Developed by John Holland at the University of Michigan, GAs are algorithms that take inspiration from the mechanism of natural selection, where the "fittest" individuals pass on their characteristics to the next generation [161].
In a GA, a set of solutions (referred to as "individuals") for a specific problem is initially generated randomly. This set of solutions is referred to as the "population", and each of them represents the current iteration's generation. The fitness of each candidate solution is determined by evaluating an objective function of the optimization process that needs to be solved. The more fit solutions are selected and used to form the next generation. The solutions in this new generation are subject to the same process as their "parents". The iterative process continues until a stopping criterion is met, such as reaching a maximum number of generated generations or achieving a target level of fitness [26,[162][163][164]. Each iteration of a GA comprises three main steps: selection, crossover, and mutation. In the selection phase, individuals are chosen for the next generation based on their level of fitness. In the crossover phase, information is exchanged between different individuals to create fitter individuals. Finally, in the mutation phase, certain existing individuals are randomly altered within predetermined upper and lower thresholds to create new individuals [165]. More exhaustive treatises on GAs are available in [166,167]. GAs are exploited in [168] to detect and estimate sensor bias, in order to enhance the estimation of health parameters using a neural network. In [169], a diagnostic tool based on GAs is used for a turbofan engine to detect In a GA, a set of solutions (referred to as "individuals") for a specific problem is initially generated randomly. This set of solutions is referred to as the "population", and each of them represents the current iteration's generation. The fitness of each candidate solution is determined by evaluating an objective function of the optimization process that needs to be solved. The more fit solutions are selected and used to form the next generation. The solutions in this new generation are subject to the same process as their "parents". The iterative process continues until a stopping criterion is met, such as reaching a maximum number of generated generations or achieving a target level of fitness [26,[162][163][164]. Each iteration of a GA comprises three main steps: selection, crossover, and mutation. In the selection phase, individuals are chosen for the next generation based on their level of fitness.
In the crossover phase, information is exchanged between different individuals to create fitter individuals. Finally, in the mutation phase, certain existing individuals are randomly altered within predetermined upper and lower thresholds to create new individuals [165]. More exhaustive treatises on GAs are available in [166,167]. GAs are exploited in [168] to detect and estimate sensor bias, in order to enhance the estimation of health parameters using a neural network. In [169], a diagnostic tool based on GAs is used for a turbofan engine to detect power-setting sensor faults, also in combination with engine component degradation and sensor faults. An interesting application of GAs is reported in [170], where they were used for the optimization of neural network topology. Similarly, GAs are used in [171] to optimize the parameters of decision tree-based methods for RUL prediction. Figure 13, adapted from [39], illustrates an example of a GA reproduction cycle.

Artificial Neural Networks (ANNs)
ANNs are algorithms designed to mimic the behavior of the human nervous system, enabling them to learn from experience [172]. These algorithms can find a correlation between two sets of variables, the input and the output, and use this learned correlation to predict a new set of output variables based on a new set of input variables. Artificial neural networks (ANNs) consist of layers and neurons. The first and last layers are known as the input and output layers, respectively. Each layer is composed of a certain number of neurons. The input and output layers contain several neurons equal to the number of input and output variables, respectively. The layers located between the input and output layers are known as hidden layers, and they contain hidden neurons. The performance of the network depends on the number of hidden layers and neurons, which can be challenging to determine [173].
power-setting sensor faults, also in combination with engine component degr and sensor faults. An interesting application of GAs is reported in [170], whe were used for the optimization of neural network topology. Similarly, GAs are [171] to optimize the parameters of decision tree-based methods for RUL pre Figure 13, adapted from [39], illustrates an example of a GA reproduction cycle.

Artificial Neural Networks (ANNs)
ANNs are algorithms designed to mimic the behavior of the human n system, enabling them to learn from experience [172]. These algorithms can correlation between two sets of variables, the input and the output, and use this correlation to predict a new set of output variables based on a new set of input va Artificial neural networks (ANNs) consist of layers and neurons. The first and las are known as the input and output layers, respectively. Each layer is compos certain number of neurons. The input and output layers contain several neurons the number of input and output variables, respectively. The layers located betw input and output layers are known as hidden layers, and they contain hidden n The performance of the network depends on the number of hidden layers and n which can be challenging to determine [173].
Each neuron in the hidden and output layers is connected to the neuron preceding layer and each connection has a weight. The purpose of these neuro process information received from the preceding neurons, and in turn, information that is dependent on the input, as described by the following equatio = + where represents the number of previous neurons from which information is re represents the neuron output, represents the connection weight with previous neuron, represents the information deriving from the i-th p neuron, and represents an added bias. The output of a neuron is limited activation function. There are many types of neural networks today, which broadly divided into two main categories: Feed-Forward Neural Networks ( [174] and Recurrent Neural Networks (RNN) [175][176][177]. FFNNs are simple n composed of layers and neurons, as previously described. If each neuron in eac Figure 13. Reproduction cycle of a GA [39].
Each neuron in the hidden and output layers is connected to the neurons in the preceding layer and each connection has a weight. The purpose of these neurons is to process information received from the preceding neurons, and in turn, provide information that is dependent on the input, as described by the following equation: where l represents the number of previous neurons from which information is received, O represents the neuron output, W i represents the connection weight with the i-th previous neuron, I NF i represents the information deriving from the i-th previous neuron, and b represents an added bias. The output of a neuron is limited by an activation function.
There are many types of neural networks today, which can be broadly divided into two main categories: Feed-Forward Neural Networks (FFNNs) [174] and Recurrent Neural Networks (RNN) [175][176][177]. FFNNs are simple networks composed of layers and neurons, as previously described. If each neuron in each layer has a connection with every neuron in the next layer, the FFNN is referred to as fully connected. On the other hand, if some connections are missing, the FFNN is referred to as partially connected. Recurrent neural networks (RNNs) are designed to learn sequential or time-varying patterns [176] and differ from FFNNs due to the presence of at least one feedback loop. This feature provides a great deal of flexibility and allows RNNs to approximate arbitrary dynamic systems with high precision. A single-loop feedback network is a type of RNN that has a single feedback loop [172,175]. Figure 14 illustrates the difference between the typical structures of a FFNN (7.a) and a RNN (7.b). ANNs were used in [178] for the purpose of predicting performance and engine health status. The results obtained from applying the ANN-based diagnostic tool on a single spool turbojet experiencing compressor and turbine degradation were compared to the results obtained from another diagnostic technique that used a support vector machine (SVM) developed in the same study. Both the ANN and SVM techniques yielded very good and comparable results in predicting the EGT and fuel flow during a specific flight, with an RMSE ranging between 0.0089 and 0.0191. However, when used for engine health prediction, the SVM technique performed extremely well, while the ANN technique showed a less efficient performance. In [179], ANNs form the basis of a prognostic method. results obtained from another diagnostic technique that used a support vector machine (SVM) developed in the same study. Both the ANN and SVM techniques yielded very good and comparable results in predicting the EGT and fuel flow during a specific flight, with an RMSE ranging between 0.0089 and 0.0191. However, when used for engine health prediction, the SVM technique performed extremely well, while the ANN technique showed a less efficient performance. In [179], ANNs form the basis of a prognostic method. The study focused on predicting the remaining useful life (RUL) of a fan module. ANNs were used to estimate current health parameters, such as efficiency and flow capacity, and the future trajectory of these parameters was estimated using an autoregressive model. RUL estimation was performed based on the estimated trajectory of the health parameters using moving window and progressive window approaches, which differ in how data are fed to the autoregressive model. It was found that the progressive window approach yielded the best results.

Bayesian Belief Networks (BBNs)
A belief network is a graphical representation of a probability distribution that depicts the cause-and-effect relationships among predisposing factors, faults, and symptoms [180]. The graph represents a set of variables indicated by nodes and directed The study focused on predicting the remaining useful life (RUL) of a fan module. ANNs were used to estimate current health parameters, such as efficiency and flow capacity, and the future trajectory of these parameters was estimated using an autoregressive model. RUL estimation was performed based on the estimated trajectory of the health parameters using moving window and progressive window approaches, which differ in how data are fed to the autoregressive model. It was found that the progressive window approach yielded the best results.

Bayesian Belief Networks (BBNs)
A belief network is a graphical representation of a probability distribution that depicts the cause-and-effect relationships among predisposing factors, faults, and symptoms [180]. The graph represents a set of variables indicated by nodes and directed edges indicating their dependencies. The level of relationship between variables is represented by a conditional probability [39,181]. In a BBN graph, three types of nodes can be distinguished: child nodes, which are nodes with edges directed towards them; parent nodes, which are nodes from which edges originate; and root nodes, which are nodes without any edges directed towards them. In an aircraft engine diagnostic approach, by setting independent variables such as flow capacities and efficiencies as parent nodes, and dependent variables such as temperatures, pressures, etc., as child nodes, the BBN approach expresses the diagnostic problem as: where h i and z j are the i-th independent and j-th dependent variables, respectively; P h i /z j is the probability of h i given z j ; P z j /h i is the probability of z j given h i ; P(h i ) is the probability of the independent parameter h i ; and P z j is the probability of the dependent parameter z j [39]. Figure 15, adapted from [39], shows an example of a BBN structure used for GT diagnostic purposes. where ℎ and are the i-th independent and j-th dependent variables, respectively; ℎ / is the probability of ℎ given ; /ℎ is the probability of given ℎ ; (ℎ ) is the probability of the independent parameter ℎ ; and ( ) is the probability of the dependent parameter [39]. Figure 15, adapted from [39], shows an example of a BBN structure used for GT diagnostic purposes. Figure 15. Typical BBN structure for GT diagnostic purposes. Adapted from [39].
BBNs were first applied for gas turbine diagnostics in the 1990s by Breese et al. [182]. In [183], a BBN that includes 11 health parameters and seven measured parameters is used to develop a diagnostic tool for detecting faults in a turbofan engine. The BBN considers deviation in flow parameters and efficiencies as potential indicators of faults. Mast et al. [184] applied a series of Bayesian Belief Networks (BBNs) for diagnostic purposes for a turbofan engine and conducted tests under different flight conditions. They found that fault detection is easier during cruise phases compared to take-off and climb phases. Ferreiro et al. [185] employed Bayesian Belief Networks (BBN) as a means to develop a prognostic technique for predicting brake wear on an aircraft. The paper describes the use of two different BBN-based models.

Expert Systems (ESs)
ES are computer programs designed to mimic the decision-making processes of an expert human in solving problems or providing advice [164]. The typical components of an ES are: a user, a developer, an inference engine, and a knowledge base. The knowledge base stores the information necessary to solve a specific problem, and it serves as the substitute for the human expert. The inference engine acts as a bridge of Figure 15. Typical BBN structure for GT diagnostic purposes. Adapted from [39].
BBNs were first applied for gas turbine diagnostics in the 1990s by Breese et al. [182]. In [183], a BBN that includes 11 health parameters and seven measured parameters is used to develop a diagnostic tool for detecting faults in a turbofan engine. The BBN considers deviation in flow parameters and efficiencies as potential indicators of faults. Mast et al. [184] applied a series of Bayesian Belief Networks (BBNs) for diagnostic purposes for a turbofan engine and conducted tests under different flight conditions. They found that fault detection is easier during cruise phases compared to take-off and climb phases. Ferreiro et al. [185] employed Bayesian Belief Networks (BBN) as a means to develop a prognostic technique for predicting brake wear on an aircraft. The paper describes the use of two different BBN-based models.

Expert Systems (ESs)
ES are computer programs designed to mimic the decision-making processes of an expert human in solving problems or providing advice [164]. The typical components of an ES are: a user, a developer, an inference engine, and a knowledge base. The knowledge base stores the information necessary to solve a specific problem, and it serves as the substitute for the human expert. The inference engine acts as a bridge of communication between the user and the "virtual expert" (the knowledge base) by linking the user's queries with the relevant answers stored in the knowledge base. This task is performed by utilizing appropriate programmed rules to establish a relationship between multiple items. Finally, the developer designs the expert system and supplies the necessary knowledge [186]. Basic ES utilize Boolean logic or decision trees, which can present limitations in accurately representing real-life problems due to their simplicity. More advanced ES employ inference rules and chaining, which are programmed as IF-THEN statements [39]. Figure 16, adapted from [39], shows a classical structure of an ES. ES have been utilized to create a variety of diagnostic systems for different types of engines [39,135], such as TEXMAS (Turbine Engine Expert Maintenance Advisor System) for T53 engines [187]; HELIX (HELicopter Integrated eXpert) [188] for helicopter engines; XMAN, a tool for automated jet engine diagnostics [189]; TIGER (Testability Insertion Guidance Expert System) [190]; IFDIS (Interactive Fault Diagnosis and Isolation System); and SHERLOCK, for helicopter engines.
programmed as IF-THEN statements [39]. Figure 16, adapted from [39], shows a classical structure of an ES. ES have been utilized to create a variety of diagnostic systems for different types of engines [39,135], such as TEXMAS (Turbine Engine Expert Maintenance Advisor System) for T53 engines [187]; HELIX (HELicopter Integrated eXpert) [188] for helicopter engines; XMAN, a tool for automated jet engine diagnostics [189]; TIGER (Testability Insertion Guidance Expert System) [190]; IFDIS (Interactive Fault Diagnosis and Isolation System); and SHERLOCK, for helicopter engines.

Fuzzy Logic (FL)
Introduced by L. Zedah in the 1960s [191,192], fuzzy sets, which are sets formed by elements that have degrees of membership, were thought to mathematically model the uncertainty of natural language [193]. Fuzzy logic (FL), which is based on fuzzy sets, is used to simplify complex phenomena. The quality of the simplification is dependent on the availability and quality of the rules used [194]. A general FL system comprises several essential components, including fuzzy rules, a fuzzifier, inference engine, and a defuzzifier. The fuzzifier is responsible for the process of fuzzification, where input signals are converted into fuzzy sets. The inference engine establishes the relationships between the various fuzzy sets. The defuzzifier performs the process of defuzzification, resulting in the output signals. Finally, the fuzzy rules represent a collection of IF-THEN statements utilized to implement FL [39,135]. Figure 17, adapted from [194], shows a typical FL system structure. FL has been extensively employed for fault diagnosis in aircraft engines. In [195], a FL system is implemented for diagnostic purposes on a military turbofan engine. In this

Fuzzy Logic (FL)
Introduced by L. Zedah in the 1960s [191,192], fuzzy sets, which are sets formed by elements that have degrees of membership, were thought to mathematically model the uncertainty of natural language [193]. Fuzzy logic (FL), which is based on fuzzy sets, is used to simplify complex phenomena. The quality of the simplification is dependent on the availability and quality of the rules used [194]. A general FL system comprises several essential components, including fuzzy rules, a fuzzifier, inference engine, and a defuzzifier. The fuzzifier is responsible for the process of fuzzification, where input signals are converted into fuzzy sets. The inference engine establishes the relationships between the various fuzzy sets. The defuzzifier performs the process of defuzzification, resulting in the output signals. Finally, the fuzzy rules represent a collection of IF-THEN statements utilized to implement FL [39,135]. Figure 17, adapted from [194], shows a typical FL system structure.
classical structure of an ES. ES have been utilized to create a variety of diagnostic systems for different types of engines [39,135], such as TEXMAS (Turbine Engine Expert Maintenance Advisor System) for T53 engines [187]; HELIX (HELicopter Integrated eXpert) [188] for helicopter engines; XMAN, a tool for automated jet engine diagnostics [189]; TIGER (Testability Insertion Guidance Expert System) [190]; IFDIS (Interactive Fault Diagnosis and Isolation System); and SHERLOCK, for helicopter engines.

Fuzzy Logic (FL)
Introduced by L. Zedah in the 1960s [191,192], fuzzy sets, which are sets formed by elements that have degrees of membership, were thought to mathematically model the uncertainty of natural language [193]. Fuzzy logic (FL), which is based on fuzzy sets, is used to simplify complex phenomena. The quality of the simplification is dependent on the availability and quality of the rules used [194]. A general FL system comprises several essential components, including fuzzy rules, a fuzzifier, inference engine, and a defuzzifier. The fuzzifier is responsible for the process of fuzzification, where input signals are converted into fuzzy sets. The inference engine establishes the relationships between the various fuzzy sets. The defuzzifier performs the process of defuzzification, resulting in the output signals. Finally, the fuzzy rules represent a collection of IF-THEN statements utilized to implement FL [39,135]. Figure 17, adapted from [194], shows a typical FL system structure. FL has been extensively employed for fault diagnosis in aircraft engines. In [195], a FL system is implemented for diagnostic purposes on a military turbofan engine. In this Figure 17. The typical structure of an FL system. FL has been extensively employed for fault diagnosis in aircraft engines. In [195], a FL system is implemented for diagnostic purposes on a military turbofan engine. In this case study, the FL system developed predicts the condition of the intermediate pressure compressor with an average error of approximately 7.5%. Another application of FLs for fault identification is shown by Ganguli [196,197], which utilizes FLs to develop a fault isolation system for a turbofan engine. The system is able to recognize faults in the fan, lowpressure compressor, high-pressure compressor, high-pressure turbine, and low-pressure turbine based on deviation from the baseline (when the engine is in a healthy condition) of certain parameters, such as exhaust gas temperature, fuel flow, and low and high rotor speed. Similarly, in [198], a case study of a three-spool engine is analyzed. Fuzzy Logic (FL) is used to create single fault isolation and partial multiple fault isolation systems (with only two components considered faulty). Sensor noise and biases are added, and the results demonstrate excellent performance in each case studied. In [199], FL is used for the same purposes. The developed system is tested using flight data, and the results show the detection of a fault in the high-pressure spool two months before it occurred.

Support Vector Machine (SVM)
SVMs are a type of learning system based on statistical learning theory, and they are commonly used for classification or regression tasks [32,200]. Like ANNs, SVM algorithms require a set of inputs (or features) and a corresponding set of outputs for the training phase. Each input can be thought of as a dimension of a hyperplane. The goal of SVM is to create a hyperplane that separates the input space into two or more classes, with the goal of achieving the maximum separation between the classes. By doing this, the expected generalization error is minimized, meaning the probability of incorrectly classifying new, unclassified features during the classification process is reduced. The process of maximizing separation is achieved by creating two parallel hyperplanes, known as the "bounding planes," on either side of the separating hyperplane. The distance between these bounding planes is called the "margin." The learning process of an SVM involves finding a hyperplane with the greatest margin and the least misclassification errors [201]. Figure 18 illustrates an example of the structure of SVMs used for fault classification [202]. In the aforementioned paper, SVMs were used in conjunction with Artificial Neural Networks (ANNs) to develop a diagnostic system that can effectively classify single, double, and triple component faults, and subsequently predict the magnitude of the faults with high accuracy. During the training process of an SVM, a Kernel function and a set of parameters known as "Kernel parameters" must be selected. The performance of the SVM is highly dependent on the choice of these parameters [203]. In [200], the authors investigate the impact of various kernel functions (linear, polynomial, radial basis, and sigmoidal) and training sample size on the performance of an SVM for gas turbine fault diagnosis. They compare these results to those obtained using a neural network. Two samples of small size have been used. The results indicate that the sample size has a minimal effect on performance when using the radial basis kernel function, which shows the best results. However, the effect of sample size on performance is more significant when using other kernel functions. Additionally, the study also examines the impact of including the compressor outlet temperature in the training dataset, considering different kernel functions and sample sizes. Finally, the performance of the SVMs trained with the four different kernel functions was compared to the performance of a neural network. The results showed that the SVM trained with the radial basis function performed better than the neural network for various sample sizes. Furthermore, increasing the sample size led to improved performance for all four of the tested kernel functions when compared to the neural network. SVMs are used in [204] to perform multi-class classification for fault diagnosis in a JT9D aero engine. Various approaches are compared to one another, and noise at different magnitudes is added to the data. As expected, the results show a decrease in accuracy with an increase in noise level. An example of the application of SVMs for prognostic purposes can be found in [205]. In this paper, the authors propose a method for classifying the health of an aircraft component, using a training dataset that includes flight data and maintenance logs. The method was applied to classify the health state of a bleed valve in the air management system as either healthy or unhealthy, in terms of a low or high probability of failure over a specific period of time.

Final Summary
As it is evident, numerous studies are focused on developing diagnostic and prognostic techniques to monitor the health status of aeronautical engines and other systems. Each study employes its own diagnostic or prognostic tool, developed using different techniques and approaches, and utilizing different types of data. In conclusion, Tables 7 and 8 present a comprehensive collection of some of the works available in the literature. For a more in-depth understanding, it is recommended to refer to the corresponding references. Table 7. List of some works present in the literature with relative characteristics. For a more detailed description, read the corresponding references. Continues in Table 8.  [195] FL DD * Diagnostics Aircraft turbofan engine Model data Yes - [196] * MB: Model-based; DD: Data-driven. FL -Average percentage error between predicted and target 7.5% IPC * fault [195] FL -% of correct fault isolation Between 89% and 100% FAN, LPC *, HPC *, HPT * and LPT * fault [196] * LPC: low-pressure compressor; HPC: high-pressure compressor; HPT: high-pressure turbine; LPT: low-pressure turbine; IPC: intermediate-pressure compressor.

Conclusions
Modern GTs are highly complex machines composed of a large number of components, each of which may be susceptible to various types of damage. As the number and complexity of parts increases, it leads to a higher probability of failure and a decrease in reliability. For GTs used in aircraft engines, reliability is of paramount importance. The reliability of a system is closely linked to the maintenance plan it is subject to. To enhance reliability and reduce maintenance costs, an effective maintenance plan is crucial, which can accurately predict the optimal time for performing maintenance. As a result, current research is focused on implementing CBM strategies, which schedule maintenance interventions based on the actual health status of the monitored system, rather than relying on traditional corrective and preventive maintenance strategies that are based on visual inspections and the probability of potential failures. This paper aimed to examine the primary degradation mechanisms to which gas turbines are typically subjected, highlighting the effect each of them has on the machine's performance. Subsequently, the paper presented the main techniques currently used for the implementation of the two key disciplines of CBM: diagnostics and prognostics. These techniques are based on the real-time health status of the monitored component, which is obtained through the use of appropriate sensors that provide useful information. Sensor measurements are then utilized to implement the algorithms for the developed diagnostic or prognostic technique. However, it should be noted that sensor measurements are not always free of noise and bias, which can negatively impact the performance of the techniques used to monitor the health status. A significant portion of research efforts are directed towards developing techniques for improving the quality of sensor measurements, in order to reduce the amount of noise present in the sensed information, and to detect and isolate faulty sensors. In some cases, these techniques may also reconstruct the values lost due to incorrect detection by a damaged sensor. The performance of health monitoring techniques is dependent on the amount and accuracy of information collected from the monitored system. As the amount of data increases, so does the computational time and effort required. To improve performance, researchers are also developing techniques for data reduction. There are many types of computer algorithms currently used for diagnostic and prognostic purposes, each with their own advantages and limitations. Hybrid techniques, which combine multiple approaches, are an area of ongoing research, and future efforts are focused in this direction. In recent years, there has been growing scientific interest in the use of hybrid engines to reduce their environmental impact, which is an extremely important topic. However, most of the literature on degrading phenomena, data reduction techniques, and diagnostic and prognostic algorithms related to the aeronautical world has focused on thermal engines. Hybrid engines are composed of both thermal and electric components, and an increase in the number of components can lead to a decrease in reliability. Therefore, there is a need to study the application of CBM to hybrid engines, which represent the future of aviation and beyond. It is crucial to fully understand the degradation mechanisms of the electrical portion of a hybrid engine (i.e., related to components such as batteries, electric motors, dynamos, etc.) and their effect on engine performance to properly develop modern CBM systems suitable for hybrid engines.
Finally, regarding degradation health monitoring techniques, there are several gaps in the existing literature that require further exploration and research. These gaps include: i.
Lack of standardization: The field currently lacks standardized techniques and metrics for monitoring degradation, which can lead to inconsistencies across studies. Developing standardized techniques could help improve the reliability and consistency of results. ii. Limited real-world testing: Most studies on degradation health monitoring techniques have been conducted in laboratory or simulated environments, which may not accurately represent real-world conditions. Further testing in real-world environments could provide more relevant and realistic data. iii. Lack of comparative studies: While many studies have evaluated the performance of individual monitoring techniques, there is a lack of comparative studies that assess the strengths and weaknesses of different techniques for monitoring the same type of degradation. iv. Insufficient consideration of uncertainty: Many monitoring techniques provide point estimates of degradation levels without accounting for the uncertainty associated with these estimates. Considering uncertainty could improve the reliability of degradation predictions. v.
Limited consideration of multiple degradations: Many studies focus on monitoring a single type of degradation, but real-world systems often experience multiple degradations simultaneously. Exploring how to monitor multiple degradations could provide a more comprehensive understanding of system health.

Data Availability Statement:
The data presented in this study are available on request from the corresponding author.