# Fuzzy Logic in Aircraft Onboard Systems Reliability Evaluation—A New Approach

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## Abstract

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## 1. Introduction

- Failure risk analysis and assessment.Fuzzy logic is mainly used to estimate risk and determine the probability of damage. An example research article describing this trend is a study showing the effective use of fuzzy numbers as inputs and outputs in the isobutane cylinder rupture risk analysis based on Fault Tree and Event Tree methods [10]. Other studies based on real source data of two Italian industrial plants (a tyre manufacturing company and a chemical plant) indicate that fuzzy logic can be successfully applied to quantify the risk of accidents at work [11]. Fuzzy logic was also used in aviation [12,13]. It was used to assess the risk of a helicopter crash depending on two factors: intensity of operations and the probability of a crash [14].

- Human factors leading to damage.Fuzzy logic is used to model the uncertainty associated with human error factors. Research [15] has indicated the possibility of using fuzzy logic to define quality standards for operations, maintenance, and production activities, which can significantly reduce errors made by oil refinery personnel. Fuzzy logic and expert judgement have also been successfully used to determine the probability of human error among nuclear plant operators. The research results presented in [16] demonstrate the effectiveness of using fuzzy logic to determine the significance of risk of human error. Other studies which use fuzzy logic, taking into account human error and uncertainty in failure data, aim at assessing the imprecise failure probability of level crossing systems in Morocco [17]. A further example of the application of fuzzy logic in aviation is the study of the influence of human factors on damage. It has been successfully used to assess basic event failure rates for safety-critical avionics systems [18].

- Adequate planning of maintenance, and consequently, prevention of damage.Fuzzy logic has also been successfully used for proper planning of maintenance, and thus, to prevent damage. In article [19], fuzzy logic was successfully applied to model imprecise answers in a reliability-centered maintenance (RCM) diagram to answer questions on the causes, symptoms, and types of failure. Additionally, a maintenance-oriented milling machine reliability study using fuzzy logic and comparing it with the conventional method allowed a more accurate determination of the causes and consequences of failure [20]. Fuzzy logic was also used to determine specific maintenance tasks used to make reactions in chemical plants, based on equipment operating data. Along with neural networks, fuzzy logic complemented the RCM strategy [21].

## 2. Research Methodology

- Analysing the available literature with a particular emphasis on the applications of fuzzy logic and the reliability issues of technical systems [9];
- Analysing available methods for estimating the systems reliability in a mathematical approach;
- Computing the reliability indicators of a selected airborne armament system, by means of a mathematical approach (based on statistical data from an IT-based aircraft reliability analysis system);
- Developing a reliability model of a selected airborne armament system by means of fuzzy logic;
- Assessing the reliability indicators of the aircraft armament system on the basis of the developed model, by means of fuzzy logic (using input signals from an IT system for the aircraft reliability analysis);
- Comparing the results obtained in the mathematical approach and the fuzzy set theory approach;
- Formulating and presenting the conclusions.

## 3. Object of Research

## 4. Scientific Approach

#### 4.1. Analysis of Statistical Data

#### 4.2. Probabilistic Model of Reliability of the Gun/Cannon Subsystem

#### 4.2.1. Alignment of Measurement Results

#### 4.2.2. Fitting the Distribution of a Random Variable

- shape parameter α—2.2664
- scale parameter β—318.78

- function of the intensity of the damage λ(t)$$\lambda \left(t\right)=\frac{\alpha}{{\beta}^{\alpha}}{t}^{\alpha -1};\alpha ,\beta 0;t0$$
- density distribution function f(t)$$f\left(t\right)=\frac{\alpha}{\beta}{\left(\frac{t}{\beta}\right)}^{\alpha -1}{e}^{-{\left(\frac{t}{\beta}\right)}^{\alpha}}$$
- random variable distribution F(t)$$Q\left(t\right)=F\left(t\right)=1-{e}^{-{\left(\frac{t}{\beta}\right)}^{\alpha}}$$
- the reliability function R(t).$$R\left(t\right)={e}^{-{\left(\frac{t}{\beta}\right)}^{\alpha}}$$

#### 4.3. Fuzzy Logic Reliability Model of a Firing Subsystem

#### 4.4. Model Tests

#### 4.5. Comparison of Research Findings

## 5. Conclusions

- Fatigue processes are an important causal group of damage to the shooting armament of the TS-11 “Iskra” aircraft.
- The differences between the parameters of empirical Weibull distribution from the operational data and the graphic method of determining the parameters, and theoretical Weibull distribution are within the 95% confidence interval, which means that with a probability of 95%, reliability indicators from both the empirical and theoretical distribution can be used in further studies on the armament reliability.
- The interpretation and comparison of the results obtained from the reliability analysis in the classical approach and the reliability analysis in the fuzzy set theory approach allowed formulating the following conclusions:
- In order to develop a reliability model using fuzzy logic, access to reliable expert knowledge is necessary.
- Fuzzy logic offers a possibility to determine reliability based on various parameters and also allows an analysis and interpretation of the relationship between input parameter values and reliability.
- The developed reliability model using fuzzy logic can be used to assess the reliability of various systems without the need for knowledge of an extensive mathematical apparatus.
- The controller pattern, designed with a fuzzy logic reliability model, can be easily upgraded by changing the membership functions (shapes and limits) and the deduction principles.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Conflicts of Interest

## References

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**Figure 1.**Percentage of gun/cannon damage in the total number of air armament damage from 2014 to 2018.

**Figure 7.**Comparison of empirical and theoretical reliability function with marked 95% confidence interval.

**Figure 13.**Reliability dependence on the number of flying hours and shooting hours with the centre of sums method.

**Figure 14.**Reliability dependence on the number of flying hours and shooting hours with the middle of maximum method.

**Figure 15.**Reliability dependence on the number of flying hours and shooting hours with the first maximum method.

**Figure 16.**Reliability functions from a fuzzy logic model using the centre of gravity and the centre of sum defuzzification methods.

**Figure 18.**Discrepancy of the reliability function values obtained from the classical and the “fuzzy” model.

**Figure 19.**Reliability functions obtained from the classical and fuzzy model with marked 95% confidence interval.

**Table 1.**Basic tactical and technical data of the TS-11 “Iskra” aircraft [24].

Type | Data |
---|---|

Length | 11.15 m |

Wingspan | 10.06 m |

Height | 3.50 m |

Lifting surface | 17.50 m² |

Unloaded | 2560 kg |

Gross weight | 3724 kg |

Maximum take-off weight | 3840 kg |

Engine | WSK SO-3W with a thrust of 10.80 kN (1100 kG) |

Maximum speed | 720 km/h in 5000 m |

Range | 1260 km |

Operational ceiling | 11,000 m |

Climb rate | 14.8 m/s |

Wing loading | 213 kg/m² |

Thrust-to-weight ratio | 1:3.4 |

Type | Data |
---|---|

Calibre | 23 mm |

Rate of fire | 500–590 rounds/min |

Gun weight | 37.5–38.2 kg |

Length | 1985 mm |

Width | 164 mm |

Height | 256 mm |

Barrel length | 1450 mm |

Minimum air pressure required for reloading | 30 kG/cm^{2} |

No | Observation | Flying Hours from 2014 to 2018 | T (Flying Hours Until Malfunction [h]) | Censored Observations |
---|---|---|---|---|

1 | O10 | 313 | 65 | complete |

2 | O9 | 80 | 80 | censored |

3 | O14 | 95 | 95 | censored |

4 | O5 | 249 | 106 | complete |

5 | O8 | 391 | 113 | complete |

6 | O3 | 148 | 133 | complete |

7 | O11 | 305 | 138 | complete |

8 | O1 | 296 | 148 | complete |

9 | O28 | 310 | 156 | complete |

10 | O24 | 247 | 175 | complete |

11 | O7 | 204 | 204 | censored |

12 | O21 | 410 | 205 | complete |

13 | O4 | 246 | 214 | complete |

14 | O12 | 232 | 232 | censored |

15 | O22 | 243 | 243 | censored |

16 | O26 | 251 | 251 | censored |

17 | O29 | 423 | 252 | complete |

18 | O23 | 314 | 257 | complete |

19 | O2 | 278 | 268 | complete |

20 | O25 | 278 | 278 | censored |

21 | O30 | 294 | 294 | censored |

22 | O27 | 297 | 297 | censored |

23 | O16 | 453 | 302 | complete |

24 | O6 | 307 | 307 | censored |

25 | O13 | 310 | 310 | censored |

26 | O15 | 442 | 328 | complete |

27 | O20 | 490 | 376 | complete |

28 | O18 | 448 | 386 | complete |

29 | O19 | 477 | 395 | complete |

30 | O17 | 519 | 465 | complete |

No | Observation | Flying Hours from 2014 to 2018 | Flying Hours Until Malfunction (h) | η |
---|---|---|---|---|

1 | O10 | 313 | 65 | 1.495158054 |

2 | O5 | 249 | 106 | 1.136449054 |

3 | O8 | 391 | 113 | 1.075206054 |

4 | O3 | 148 | 133 | 0.900226054 |

5 | O11 | 305 | 138 | 0.856481054 |

6 | O1 | 296 | 148 | 0.768991054 |

7 | O28 | 310 | 156 | 0.698999054 |

8 | O24 | 247 | 175 | 0.532768054 |

9 | O21 | 410 | 205 | 0.270298054 |

10 | O4 | 246 | 214 | 0.191557054 |

11 | O29 | 423 | 252 | 0.140904946 |

12 | O23 | 314 | 257 | 0.184649946 |

13 | O2 | 278 | 268 | 0.280888946 |

14 | O16 | 453 | 302 | 0.578354946 |

15 | O15 | 442 | 328 | 0.805828946 |

16 | O20 | 490 | 376 | 1.225780946 |

17 | O18 | 448 | 386 | 1.313270946 |

18 | O19 | 477 | 395 | 1.392011946 |

19 | O17 | 519 | 465 | 2.004441946 |

d K-S | K-S p | |
---|---|---|

Weibull | 0.080785 | 0.980681 |

Generalised extreme value | 0.080995 | 0.980185 |

Normal | 0.083536 | 0.973478 |

Gaussian mixture | 0.083917 | 0.972358 |

Johnson SB | 0.086214 | 0.964943 |

Rayleigh | 0.136471 | 0.584155 |

Log-normal | 0.141784 | 0.536058 |

Triangular | 0.161554 | 0.373656 |

Generalised Pareto | 0.166667 | 0.337101 |

Semi-normal | 0.240961 | 0.051135 |

**Table 6.**Differences in scale and shape parameters for empirical and theoretical Weibull distributions.

Scale (β) | Shape (α) | |
---|---|---|

Empirical Weibull distribution | 318.78 | 2.2664 |

Theoretical Weibull distribution | 318.06 | 2.5030 |

**Table 7.**Values of the distribution density f(t) and reliability function R(t) for empirical and theoretical Weibull distribution.

Flying Hours Until Malfunction (h) | Empirical Density of Distribution f(t) | Theoretical Density of Distribution f(t) | Empirical Reliability R(t) | Theoretical Reliability R(t) |
---|---|---|---|---|

65 | 0.00092 | 0.00071 | 0.97315 | 0.98139 |

106 | 0.00162 | 0.00142 | 0.92085 | 0.93809 |

113 | 0.00174 | 0.00154 | 0.90908 | 0.92774 |

133 | 0.00205 | 0.00190 | 0.87118 | 0.89335 |

138 | 0.00212 | 0.00198 | 0.86076 | 0.88365 |

148 | 0.00226 | 0.00215 | 0.83887 | 0.86298 |

156 | 0.00236 | 0.00228 | 0.82040 | 0.84526 |

175 | 0.00257 | 0.00256 | 0.77347 | 0.79919 |

205 | 0.00281 | 0.00292 | 0.69235 | 0.71672 |

214 | 0.00286 | 0.00299 | 0.66680 | 0.69012 |

252 | 0.00294 | 0.00317 | 0.55601 | 0.57214 |

257 | 0.00293 | 0.00318 | 0.54134 | 0.55626 |

268 | 0.00291 | 0.00317 | 0.50923 | 0.52132 |

302 | 0.00274 | 0.00303 | 0.41286 | 0.41546 |

328 | 0.00254 | 0.00280 | 0.34412 | 0.33957 |

376 | 0.00205 | 0.00221 | 0.23369 | 0.21866 |

386 | 0.00194 | 0.00208 | 0.21377 | 0.19721 |

395 | 0.00184 | 0.00195 | 0.19679 | 0.17909 |

465 | 0.00109 | 0.00105 | 0.09509 | 0.07522 |

Input Signal | Membership Functions | |
---|---|---|

Flying hours [h] | VERY SMALL [0 0 105] SMALL [0 105 205] MEDIUM [105 205 305] LARGE [205 305 410] VERY LARGE [305 410 500 580] |

**Table 9.**Membership functions and boundaries of fuzzy sets of the input signal “Number of firing hours”.

Input Signal | Membership Functions | |
---|---|---|

Firing hours [number of shots] | VERY SMALL [0 0 400] SMALL [0 400 800] MEDIUM [400 800 1300] LARGE [800 1300 1800] VERY LARGE [1300 1800 2000 2000] |

Input Signal | Membership Functions | |
---|---|---|

Corrosion [mm] | VERY SMALL [0 0 0.02] SMALL [0 0.02 0.04] MEDIUM [0.02 0.04 0.06] LARGE [0.04 0.06 0.075] VERY LARGE [0.06 0.075 0.1 0.1] |

Output Signal | Membership Functions | |
---|---|---|

Reliability [P] | VERY SMALL [0 0 0.2] SMALL [0 0.2 0.4] MEDIUM [0.2 0.4 0.6] LARGE [0.4 0.6 0.8] VERY LARGE [0.6 0.8 1] OPTIMUM [0.8 1 1] |

No | Flying Hours (h) | Shots | Corrosion (mm) | R(t) |
---|---|---|---|---|

1 | 65 | 0 | 0 | 0.9300 |

2 | 106 | 0 | 0 | 0.9100 |

3 | 113 | 0 | 0 | 0.8860 |

4 | 133 | 0 | 0 | 0.8430 |

5 | 138 | 0 | 0 | 0.8350 |

6 | 148 | 0 | 0 | 0.8220 |

7 | 156 | 0 | 0 | 0.8130 |

8 | 175 | 0 | 0 | 0.7950 |

9 | 205 | 0 | 0 | 0.6990 |

10 | 214 | 0 | 0 | 0.6650 |

11 | 252 | 0 | 0 | 0.5680 |

12 | 257 | 0 | 0 | 0.5560 |

13 | 268 | 0 | 0 | 0.5310 |

14 | 302 | 0 | 0 | 0.4160 |

15 | 328 | 0 | 0 | 0.3460 |

16 | 376 | 0 | 0 | 0.2470 |

17 | 386 | 0 | 0 | 0.2170 |

18 | 395 | 0 | 0 | 0.1790 |

19 | 465 | 0 | 0 | 0.0633 |

**Table 13.**Values of the reliability function from the fuzzy model for different defuzzification methods.

No | Flying Hours [h] | Shots | Corrosion [mm] | R(t) Centre of Gravity Defuzzification Method | R(t) Method of Defuzzification of the Centre of Sums |
---|---|---|---|---|---|

1 | 65 | 0 | 0 | 0.9300 | 0.9400 |

2 | 106 | 0 | 0 | 0.9100 | 0.9300 |

3 | 113 | 0 | 0 | 0.8860 | 0.9200 |

4 | 133 | 0 | 0 | 0.8430 | 0.8600 |

5 | 138 | 0 | 0 | 0.8350 | 0.8400 |

6 | 148 | 0 | 0 | 0.8220 | 0.8200 |

7 | 156 | 0 | 0 | 0.8130 | 0.8100 |

8 | 175 | 0 | 0 | 0.7950 | 0.7900 |

9 | 205 | 0 | 0 | 0.6990 | 0.7600 |

10 | 214 | 0 | 0 | 0.6650 | 0.7400 |

11 | 252 | 0 | 0 | 0.5680 | 0.5200 |

12 | 257 | 0 | 0 | 0.5560 | 0.5000 |

13 | 268 | 0 | 0 | 0.5310 | 0.4700 |

14 | 302 | 0 | 0 | 0.4160 | 0.4000 |

15 | 328 | 0 | 0 | 0.3460 | 0.3700 |

16 | 376 | 0 | 0 | 0.2470 | 0.2600 |

17 | 386 | 0 | 0 | 0.2170 | 0.1400 |

18 | 395 | 0 | 0 | 0.1790 | 0.1000 |

19 | 465 | 0 | 0 | 0.0633 | 0.0500 |

**Table 14.**Values of the reliability functions from the classical model (empirical distribution and theoretical distribution) and the “fuzzy” model.

No | Flying Hours (h) | R(t)—Classical Model (Empirical) | R(t)—Classical Model (Theoretical) | R(t)—Fuzzy Model |
---|---|---|---|---|

1 | 65 | 0.97315 | 0.98139 | 0.9300 |

2 | 106 | 0.92085 | 0.93809 | 0.9100 |

3 | 113 | 0.90908 | 0.92774 | 0.8860 |

4 | 133 | 0.87118 | 0.89335 | 0.8430 |

5 | 138 | 0.86076 | 0.88365 | 0.8350 |

6 | 148 | 0.83887 | 0.86298 | 0.8220 |

7 | 156 | 0.82040 | 0.84526 | 0.8130 |

8 | 175 | 0.77347 | 0.79919 | 0.7950 |

9 | 205 | 0.69235 | 0.71672 | 0.6990 |

10 | 214 | 0.66680 | 0.69012 | 0.6650 |

11 | 252 | 0.55601 | 0.57214 | 0.5680 |

12 | 257 | 0.54134 | 0.55626 | 0.5560 |

13 | 268 | 0.50923 | 0.52132 | 0.5310 |

14 | 302 | 0.41286 | 0.41546 | 0.4160 |

15 | 328 | 0.34412 | 0.33957 | 0.3460 |

16 | 376 | 0.23369 | 0.21866 | 0.2470 |

17 | 386 | 0.21377 | 0.19721 | 0.2170 |

18 | 395 | 0.19679 | 0.17909 | 0.1790 |

19 | 465 | 0.09509 | 0.07522 | 0.0633 |

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**MDPI and ACS Style**

Żyluk, A.; Kuźma, K.; Grzesik, N.; Zieja, M.; Tomaszewska, J.
Fuzzy Logic in Aircraft Onboard Systems Reliability Evaluation—A New Approach. *Sensors* **2021**, *21*, 7913.
https://doi.org/10.3390/s21237913

**AMA Style**

Żyluk A, Kuźma K, Grzesik N, Zieja M, Tomaszewska J.
Fuzzy Logic in Aircraft Onboard Systems Reliability Evaluation—A New Approach. *Sensors*. 2021; 21(23):7913.
https://doi.org/10.3390/s21237913

**Chicago/Turabian Style**

Żyluk, Andrzej, Konrad Kuźma, Norbert Grzesik, Mariusz Zieja, and Justyna Tomaszewska.
2021. "Fuzzy Logic in Aircraft Onboard Systems Reliability Evaluation—A New Approach" *Sensors* 21, no. 23: 7913.
https://doi.org/10.3390/s21237913