=Paper=
{{Paper
|id=Vol-3628/paper5
|storemode=property
|title=Neuro-Fuzzy System for Detection Fuel Consumption of Helicopters Turboshaft Engines
|pdfUrl=https://ceur-ws.org/Vol-3628/paper5.pdf
|volume=Vol-3628
|authors=Serhii Vladov,Ruslan Yakovliev,Oleksandr Hubachov,Juliia Rud
|dblpUrl=https://dblp.org/rec/conf/ittap/VladovYHR23
}}
==Neuro-Fuzzy System for Detection Fuel Consumption of Helicopters Turboshaft Engines==
https://ceur-ws.org/Vol-3628/paper5.pdf
Neuro-Fuzzy System for Detection Fuel Consumption of
Helicopters Turboshaft Engines
Serhii Vladov 1, Ruslan Yakovliev 1, Oleksandr Hubachov 1 and Juliia Rud 1
1
Kremenchuk Flight College of Kharkiv National University of Internal Affairs, Peremohy street 17/6,
Kremenchuk, 39605, Ukraine
Abstract
The work is dedicated to the development of a neuro-fuzzy system for detection and control
the fuel consumption of helicopters turboshaft engines at flight modes. The realization of the
developed system was performed on the basis of ANFIS – an adaptive neuro-fuzzy system that
implements the Sugeno fuzzy inference system in the form of a five-layer neural network of
direct signal propagation, the first layer of which contains terms of input variables (helicopters
turboshaft engines thermogas-dynamic parameters current values and their delayed values). In
the process of forming the model, the sample of initial data was divided into two parts: training
and testing. To estimate the effectiveness of using the ANFIS network for intelligent control
of the specific fuel consumption of helicopter gas turbine engines, a training sample was used.
The sample contains data on air consumption in the combustion chamber, specific engine
power, the ratio of fuel and air consumption in the combustion chamber, calculated (in absolute
units) using a helicopters turboshaft engines neural network model. It has been experimentally
proven that for adapting the neuro-fuzzy network ANFIS and the Sugeno zero-order fuzzy
inference system to solving the problem of control the fuel consumption of helicopters
turboshaft engines, it is effective to use a hybrid training method, 2...3 delayed inputs and two
two-way Gaussian membership functions. It was found that the two-sided Gaussian
membership function provides the smallest network training error, equal to 3.28·10–3,
compared to others, which give the largest neural network training error – 0.138. Increasing
the number of outputs of the ANFIS neuro-fuzzy network and expanding the base of fuzzy
rules makes it possible to improve the decision-making logic and, as a result, expand the range
of calculation of activity levels of the rules. Prospects for further research is the introduction
of the developed neuro-fuzzy system for detection and control the fuel consumption into a
closed on-board neural network control system for helicopters turboshaft engines.
Keywords 1
neuro-fuzzy network, helicopters turboshaft engines, fuel consumption, ANFIS neuro-fuzzy
network, Gaussian membership function, thermogas-dynamic parameters, Sugeno fuzzy
inference system
1. Introduction
To effectively manage the quality of intricate technical entities [1, 2], such as helicopter turboshaft
engines, a thorough investigation into the phenomena occurring at each stage is essential. This is crucial
for establishing the correlation between operational factors and the inherent characteristics of the
engines.
Proceedings ITTAP’2023: 3rd International Workshop on Information Technologies: Theoretical and Applied Problems, November 22–24,
2023, Ternopil, Ukraine, Opole, Poland
EMAIL: ser2610196@gmail.com (S. Vladov); director.klk.hnuvs@gmail.com (R. Yakovliev); oleksandrgubachov@gmail.com
(O. Hubachov); juliarud25@gmail.com (J. Rud)
ORCID: 0000-0001-8009-5254 (S. Vladov); 0000-0002-3788-2583 (R. Yakovliev); 0000-0002-1826-259X (O. Hubachov);
0000-0002-0328-5895 (J. Rud)
©️ 2023 Copyright for this paper by its authors.
Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
CEUR Workshop Proceedings (CEUR-WS.org)
CEUR
ceur-ws.org
Workshop ISSN 1613-0073
Proceedings
An essential concern in helicopter flight operation is the selection of optimal flight modes to enhance
fuel efficiency [3, 4]. The most rational approach in this context involves the utilization of theoretical
methods, particularly numerical modeling and intelligent information technologies [5, 6]. To implement
such an approach, having a mathematical model of helicopter dynamics, which incorporates a module
for determining fuel consumption, is necessary.
A promising avenue for advancing tools for controlling helicopter operation involves integrating
components of artificial intelligence, such as production rules [7], fuzzy logic [8], artificial neural
networks [9], hybrid neuro-fuzzy architectures [10], and genetic algorithms [11].
Analysis of patent materials and recent publications indicates that considerable attention is given
internationally to creating intelligent control methods for complex technical entities [12, 13]. The
methodological approach to synthesizing intelligent systems relies on the technology of distributed
expert and neural network systems. A fundamental aspect of developing intelligent methods,
particularly those incorporating elements based on neural network structures, is the formulation of
principles for constructing an identification block and a knowledge base.
Hence, enhancing economic efficiency and maintaining a high level of operational reliability for
helicopters in special operational situations, through the development of theoretical foundations,
methods, and intelligent management means, constitutes a pressing scientific and applied challenge.
2. Related works
It is recognized that the control of helicopter operation heavily relies on the continuous monitoring
of its engine’s operational status. The engine is a complex nonlinear dynamic system influenced by the
interplay of gas-dynamic and thermophysical processes in its various units (air inlet section,
compressor, combustion chamber, compressor turbine, free turbine, exhaust unit) [14].
To regulate these intricate processes, the prevalent approach involves employing a mathematical
framework, specifically in the form of artificial neural networks. A literature review indicates that
neural networks are widely applied to tackle diverse tasks, demonstrating notable accuracy, particularly
in modeling and detecting complex technical systems [15].
In [16, 17], the development of a neural network model for a small-sized gas turbine engine is
discussed, utilizing a recurrent neural network, along with the simulation results.
In [18, 19], the specifics of constructing a real-time neural network model for operating gas turbine
engines are outlined, accompanied by an evaluation of the model’s effectiveness.
However, a survey of the literature regarding neural network methods for controlling gas turbine
engines, including helicopter turboshaft engines, reveals the absence of current methods and models for
controlling specific fuel consumption. Hence, addressing this gap is the pertinent objective addressed
in this research.
3. Proposed technique
The mathematical model, outlined in [20], is designed for calculating the specific fuel consumption
of turboshaft engines in helicopters, taking the TV3-117 engine as an example, which forms part of the
power plant for the Mi-8MTV helicopter. As per the findings in [20], the specific fuel consumption of
turboshaft engines in helicopters is contingent upon factors such as air consumption in the combustion
chamber, specific engine power, and the ratio of fuel to air consumption in the combustion chamber.
This particular parameter is directly influenced by the type of aviation fuel in use.
The computation of thermogas-dynamic parameters for turboshaft engines in helicopters, including
air consumption in the combustion chamber, specific engine power, and the ratio of fuel to air
consumption in the combustion chamber, is performed utilizing a neural network model for helicopters
turboshaft engines developed by our team of authors, as detailed in [21, 22] (fig. 1).
Figure 1: An overview of a segment of the mathematical model for turboshaft engines in helicopters
within the Matlab/Simulink program, wherein 11 thermogasdynamic parameters of the engine
operating process are computed [21, 22]
Addressing the challenge of intelligent control over the specific fuel consumption of helicopter
turboshaft engines through neural networks holds promise and is timely. However, selecting a neural
network architecture that aligns seamlessly with the quality requirements for controlling the specific
fuel consumption of these engines constitutes an independent challenge, contingent upon the unique
aspects of the problem at hand. Thus, the resolution of this challenge involves the following sequence
of actions: computing thermogas-dynamic parameters of helicopter turboshaft engines (including air
consumption in the combustion chamber, specific engine power, and the ratio of fuel to air consumption
in the combustion chamber); determining fuel consumption values; and classifying the extent of
deviation of fuel consumption from the norm. These tasks are intricately interconnected and can be
functionally segmented, necessitating the development (or refinement) of appropriate methods and tools
implemented through various neural network models. Schematically, this progression appears as
follows: task – functions – methods (tools) – implementation.
Currently, the integration of neural networks with fuzzy logic has the potential to significantly
enhance the effectiveness of automatic control systems employing neuro-fuzzy networks. This is
attributed to the fact that the drawbacks inherent in one technology are offset by the advantages of the
other [23]. Specifically, neural networks demonstrate a strong training capability, albeit with a complex
training process. On the other hand, systems utilizing fuzzy logic provide clear explanations for
conclusions but have limitations on the number of input variables. Consequently, the development of
neuro-fuzzy networks becomes feasible, wherein conclusions are drawn based on fuzzy logic, and
membership functions are adjusted using a neural network. The advantage of such systems is evident:
the constructed structure not only utilizes a priori information but also has the capacity to acquire new
knowledge while maintaining logical transparency [24].
Presently, a variety of hybrid neuro-fuzzy networks exist, exhibiting diverse architectures,
capabilities, and methodologies [23]. A comprehensive analysis reveals several key properties,
including the ability to automatically generate decision rules, the flexibility to employ different training
algorithms, the option for online training during data reception, the capability to modify the structure,
and the preservation of knowledge within the system through parametric optimization or training new
rules. In [25], essential characteristics of various hybrid neuro-fuzzy systems are outlined, accompanied
by recommendations for their selection based on the nature of the problem at hand. According to the
data presented in [25], it is recommended to employ an adaptive neuro-fuzzy inference system (ANFIS),
such as the Adaptive Network-based Fuzzy Inference System, for developing a process control system.
Compared to alternative methods, ANFIS stands out for its rapid training, algorithmic simplicity, and
optimal integration with the MatLab mathematical modeling system. In the ANFIS system, conclusions
are drawn using fuzzy logic apparatus, and membership function parameters are adjusted through the
error backpropagation algorithm or a hybrid method during neural network training. This approach
facilitates the identification of patterns and the discovery of new dependencies. Following the
recommendations mentioned in this study, simulations were conducted in the MatLab environment
using the Fuzzy Logic Toolbox extension package. Within the Fuzzy Logic Toolbox package of the
MatLab system, the ANFIS adaptive neuro-fuzzy inference system is characterized as a hybrid network
– a multilayer neural network with a unique structure and no feedback. This network utilizes standard
(non-fuzzy) signals, weights, and activation functions, employing fixed T-norm, T-conorm, or another
continuous operation for summation. In this context, the values of inputs, outputs, and weights in the
hybrid neural network fall within the real number range [0, 1].
Commonly recognized inference algorithms, such as Mamdani, Sugeno, Tsukamoto, and Larsen,
can serve as solutions for the inference algorithm [26]. These algorithms implement fuzzy logical
inference in varying manners, yet they do not exhibit significant differences. However, the precision of
the resulting control signal can be enhanced by selecting an appropriate output algorithm. A
comparative analysis of these algorithms, as indicated in [25], reveals that, all else being equal, the error
in function approximation using the Sugeno algorithm is somewhat lower than when using the Mamdani
algorithm. Furthermore, the Sugeno algorithm is computationally simpler than the Mamdani algorithm
and requires 50 to 100 times less computation time. Therefore, for constructing a fuzzy controller for
the control system of helicopter turboshaft engines, the Sugeno algorithm is employed [26]. The ANFIS
network training technique for determining the parameters of the membership functions in Sugeno-type
fuzzy inference systems can utilize either the backpropagation algorithm or a hybrid training algorithm.
Let's conduct a comparative analysis of training a four-layer neuro-fuzzy network with different
membership functions using both the backpropagation method and the hybrid method. The training of
an ANFIS network can follow typical neural network training procedures since it employs only
differentiable functions. This usually involves a combination of gradient descent in the form of
backpropagation and least squares.
The backpropagation algorithm adjusts the parameters of the antecedents of the rules, specifically
the membership functions. Meanwhile, the coefficients for rule conclusions are determined through the
least square’s method, given their linear relationship to the network output. The tuning procedure
comprises two steps during each iteration. In the initial phase, a training sample is inputted to the
network, and optimal parameters for the nodes of the fourth layer are determined using the iterative
least squares method, based on the residual between the desired and actual network behavior.
Subsequently, in the second stage, the residual discrepancy is propagated from the network output to
the inputs, and the parameters of the nodes in the first layer are modified using the error backpropagation
method. It's crucial to note that the rule conclusion coefficients identified in the first stage remain
unchanged. This iterative tuning process persists until the residual surpasses a predetermined threshold.
Based on the aforementioned information, the system developed was implemented using ANFIS, an
adaptive neuro-fuzzy system that employs the Sugeno fuzzy inference system in the structure of a five-
layer neural network with direct signal propagation (see Figure 2). The initial layer of this network
encompasses terms related to the input variables, representing both the current values of thermogas-
dynamic parameters of helicopter turboshaft engines and their delayed values. During the model
formation, the initial dataset was partitioned into two segments: one for training and another for testing.
Input Layer 1
A1
A2
X1
A3
A4
Layer 2 Layer 3 Layer 4 Layer 5 Output
B5 π N
B6 π N
X2 Σ Q
B7 π N
B8 π N
C9 X1X2X3
Q 1- fuel consumption is normal
C10
Q - allowable increase in fuel
X3 2
consumption
C11 Q3- critical increase in fuel
consumption
C12
Figure 2: A five-layer neural network utilizing the Sugeno fuzzy inference system with direct signal
propagation
To assess the efficacy of employing the ANFIS network for intelligent control of the specific fuel
consumption of helicopter turboshaft engines, a training sample was utilized. This sample comprises data
on air consumption in the combustion chamber, specific engine power, and the ratio of fuel to air
consumption in the combustion chamber. These values were calculated in absolute units using a neural
network model for helicopter turboshaft engines, as per [21, 22].
In the proposed neuro-fuzzy system, the transformation performed by a typical neuron with two
inputs has the form y = (1 x1 + 2 x2 ) , where f ( ) – sigmoid function. In order to generalize it, you
need to imagine that the weight of the neuron does not necessarily have to be multiplied by the value
of the corresponding input, but some other operation can be used here. Further, the summation of effects
can also be replaced by some other action. Finally, instead of the sigmoid function, the potential of the
neuron can be transformed in some new way. In fuzzy logic, the multiplication operation is replaced
for Boolean variables by the AND operation, and for numerical ones by the operation of taking the
minimum (min). The summation operation is replaced, respectively, by OR operations and taking the
maximum (max).
If we perform the corresponding changes in the transformation carried out by the neuron we know,
and put f ( z ) = z (linear output) in it, then we will get the so-called fuzzy OR-neuron:
y = max min (1 , x1 ) min (2 , x2 ) min (3 , x3 ). (1)
For fuzzy neurons, it is assumed that the values of the inputs and weights are in the interval [0, 1],
so the output of the OR neuron will also belong to the same interval. Using the opposite substitution
(multiplication max), (addition min) we get a transformation characteristic of a fuzzy AND-neuron:
y = min max (1 , x1 ) max (2 , x2 ) max (3 , x3 ). (2)
The architecture of the proposed neuro-fuzzy system is an isomorphic fuzzy knowledge base. It uses
differentiable implementations of triangular norms (multiplication and probabilistic OR), as well as
smooth membership functions. This makes it possible to use fast neural network training algorithms
based on the method of inverse error propagation to adjust neural fuzzy networks. The proposed neuro-
fuzzy system implements Sugeno fuzzy inference system in the form of a five-layer neural network of
direct signal propagation. The purpose of the layers is as follows: first layer – terms of the input
variables; second layer – antecedents (premises) of fuzzy rules; the third layer – rules fulfillment
degrees normalization; fourth layer – rules conclusions; fifth layer – aggregation of the result obtained
according to different rules, wherein 2 terms are used for linguistic evaluation of input variables x1, x2,
x3 [27]. The general fuzzy rule with serial number k has the form:
Rk : If x1 = a1,k and x2 = a2,k and x3 = a3, k then y = b0, k + b1, k x1 + b2, k x2 + b3, k x3 ; (3)
where m – rules number, k = 1, m ; ai,k – fuzzy term with a membership function μk(xi), used for linguistic
evaluation of a variable xi in the k-th rule ( k = 1, m , i = 1, n ); bq,k – real numbers in the conclusion of the
k-th rule ( k = 1, m , q = 0, n ).
The proposed neuro-fuzzy system functions as follows.
Layer 1. Each unit in the first layer represents one term with a Gaussian membership function. The
network inputs are connected only to their terms. The number of units in the first layer is equal to the
sum of the cardinalities of the term sets of the input variables. The output of the unit is the degree to
which the value of the input variable belongs to the corresponding fuzzy term:
1
k ( xi ) = ; (4)
xi − C
2B
1+
A
where A, B and C – configurable parameters of the membership function.
Layer 2. The number of units in the second layer is m. Each node in this layer corresponds to one
fuzzy rule. The node of the second layer is connected to those nodes of the first layer that form the
antecedents of the corresponding rule. Therefore, each node in the second layer can receive from 1 to n
input signals. The layers output is the degree of rule realization, which is calculated as the product of
the input signals. Let us denote the outputs of the nodes of this layer by τk ( k = 1, m ).
Layer 3. The number of units in the third layer is also m. Each unit of this layer calculates the relative
degree of fulfillment of the fuzzy rule:
k
k* = . (5)
j
j =1, m
Layer 4. The number of units in the fourth layer is also m. Each unit is connected to one unit in the
third layer, as well as to all network inputs. The fourth layer node calculates the contribution of one
fuzzy rule to the network output:
yk = k* (b0,k + b1,k x1 + b2,k x2 + b3,k x3 ). (6)
Layer 5. A single unit in this layer summarizes the contributions of all rules:
y = y1 + ... + yk + ... + ym . (7)
4. Experiment
The authors group conducted the analysis and initial processing of the input data, as detailed in [28,
29]. The input parameters for the mathematical model of helicopter turboshaft engines include
atmospheric values (h – flight altitude, TN – temperature, PN – pressure, ρ – air density). Parameters
measured on board the helicopter (nTC – gas generator rotor speed, nFT – free turbine rotor speed, TG –
gas temperature in front of the compressor turbine) were converted to absolute values using the theory
of gas-dynamic similarity developed by Professor Valery Avgustinovich (refer to table 1). In this work,
we assume that atmospheric parameters remain constant (h – flight altitude, TN – temperature, PN –
pressure, ρ – air density) [28, 29].
Table 1
Part of training set (author's development, described in [28, 29])
Number TG nTC nFT
1 0.932 0.929 0.943
2 0.964 0.933 0.982
3 0.917 0.952 0.962
4 0.908 0.988 0.987
5 0.899 0.991 0.972
6 0.915 0.997 0.963
7 0.922 0.968 0.962
8 0.989 0.962 0.969
9 0.954 0.954 0.947
10 0.977 0.961 0.953
… … … …
256 0.953 0.973 0.981
Ensuring the homogeneity of the training and test samples is a crucial consideration in the evaluation
process. To address this, we employ the Fisher-Pearson criterion χ2 with degrees of freedom equal to
r – k – 1 [28, 29]:
r
m − npi ( )
2 = min i ; (8)
i =1 npi ( )
where θ – represents the maximum likelihood estimate determined from the frequencies m1, …, mr; n –
denotes the number of elements in the sample, and pi(θ) – refers to the probabilities of elementary
outcomes up to a certain indeterminate k-dimensional parameter θ.
The concluding step in the statistical data processing involves normalization, which can be
accomplished using the following expression:
y − yi min
yi = i ; (9)
yi max − yi min
where y i – dimensionless quantity in the range [0; 1]; yimin and yimax – minimum and maximum values
of the yi variable.
The χ2 statistics mentioned above, given the stated assumptions, enable the testing of the hypothesis
regarding the representability of sample variances and the covariance of factors within the statistical
model. The domain for accepting the hypothesis is defined as 2 n − m , , where α – denotes the
significance level of the criterion. The computed results based on equation (8) are presented in table 2
Table 2
Part of the training sample during the operation of helicopters TE (on the example of TV3-117 TE)
(author's development, described in [28, 29])
Number P(TG) P(nTC) P(nFT)
1 0.561 0.109 0.652
2 0.588 0.155 0.574
3 0.542 0.128 0.515
4 0.612 0.147 0.655
5 0.644 0.121 0.612
… … … …
256 0.537 0.098 0.651
To assess the representativeness of the training and test samples, an initial data cluster analysis was
conducted (table 2), revealing the identification of eight classes (fig. 3, a). Subsequently, following a
randomization procedure, the actual training (control) and test samples were chosen in a 2:1 ratio,
corresponding to 67 % and 33 %, respectively. The clustering process applied to both the training
(fig. 3, b) and test samples indicated that, like the original sample, each contains eight classes. The
distances between the clusters closely match in each of the examined samples, affirming the
representativeness of both the training and test samples [28, 29].
a b
Figure 3: Clustering results: a – initial experimental sample (I…VIII – classes); b – training sample
(author's development, described in [28, 29])
A fragment of the expert knowledge matrix for intelligent control of helicopters TE specific fuel
consumption is given in table 3.
Table 3
Expert knowledge matrix for intelligent control of helicopters TE specific fuel consumption (on the
example of the TV3-117 engine)
Rule IF weight
specific engine
combustion consumption in the
power
chamber combustion chamber
1 0.985 0.995 0.990 1 1
2 0.975 0.995 0.985 1 1
3 0.965 0.995 0.980 1 1
4 0.940 0.980 0.975 2 1
5 0.930 0.980 0.970 2 1
6 0.920 0.980 0.965 2 1
7 0.895 0.960 0.960 3 1
8 0.875 0.960 0.955 3 1
9 0.855 0.960 0.950 3 1
First, the neuro-fuzzy network was trained using the error backpropagation algorithm (fig. 4). After
2200 training epochs, the training error was δ = 5.86 · 10–4. According to the results of training by the
hybrid method (fig. 5), the training error after 20 epochs was δ = 1.29 · 10–4, that is, 4.53 times less than
when training with the backpropagation algorithm.
Figure 4: Diagram of the results of training a neuro-fuzzy network by the backpropagation algorithm
Figure 5: Diagram of training results for a neuro-fuzzy network by the hybrid method
Fig. 4 and 5 indicate that the hybrid method requires 105 times fewer epochs to train the network
compared to the error backpropagation algorithm. Consequently, the hybrid method was employed for
training the neural network in this work.
In the development of a neuro-fuzzy network, the selection of the membership function holds
significant importance. Table 4 displays the network training error achieved for various membership
functions during the network training using the hybrid method.
Table 4
Neural network training error obtained with different membership functions
Membership function name Training Number of Training time,
error epochs seconds
triangular (trimf) 0.007418 25 5.4715
trapezoidal (trampf) 0.018642 45 6.3604
generalized bell-shaped (gbellmf) 0.000216 28 5.2499
symmetrical gaussian (gaussmf) 0.000367 20 5.1689
two-sided gaussian (gauss2mf) 0.000129 20 5.1837
membership function as difference between two
0.002436 80 7.6352
sigmoid functions (dsigmf)
product of two sigmoid membership functions (psigmf) 0.002277 80 7.9781
The result of tuning the membership functions in the case of two and three terms of input variables
[30] is shown in fig. 6.
Figure 6: Membership functions of terms of input variables [30]
In the course of the study, it was observed that the two-sided Gaussian membership function yields
the smallest training error (fig. 7 [30]), with δ = 1.29 · 10–4 and N = 20 epochs. This is in comparison
to the symmetric Gaussian membership function, which results in an error of δ = 3,67 · 10–4 with N = 20
epochs, and the trapezoidal membership function, which produces the highest training error of
δ = 1.86 · 10–2 with N = 45 epochs. Therefore, when adapting a neuro-fuzzy network to address a
control task, it is recommended to use a two-sided Gaussian membership function.
Figure 7: Type of fuzzy inference surfaces for different types of membership functions [30]
Table 5 presents the outcomes of evaluating the impact of the number of membership functions on
the performance indicators of the task being addressed [30].
Table 5
Neural network training error obtained with different membership functions
Number of membership functions Training error Execution time, seconds
1 0.000129 5.46
2 0.000367 48.92
3 0.000492 1566.08
The investigation into the impact of the number of delayed inputs on the efficiency indicators of
identification [29] revealed, as shown in table 6, that the optimal results are attained with 2 to 3 delayed
inputs.
Table 6
Neural network training error obtained with different membership functions
Number of inputs Training error Training time, seconds
1 0.000326 3.18
2 0.000129 8.44
3 0.000156 17.56
4 0.000286 49.35
5 0.000422 77.93
6 0.000575 185.72
5. Results
In order to implement the above proposed method, a universal program (fig. 8) was created for
generation according to the given values of the knowledge base matrix for the fuzzy controller. This
program works as follows:
– saved matrix of elements is called;
– largest modulo values of each of the inputs and outputs are determined;
– range of values (from –max to +max) of each input and output is divided into terms with a step
specified by the user;
– according to the affiliation of the variable values (from the resulting matrix), at each step, “input–
output” rules are formed to one or another term;
– rules are combined if the output does not change its values when the inputs change;
– same rules are also combined;
– after executing this method, a rule base is formed.
To load the knowledge base that is necessary for the operation of the neuro-fuzzy system, we used
the command fuzzy2=readfis('fuzzy2777').
Figure 8: A universal program structure
We accept X1 – air flow in the combustion chamber; X2 – specific engine power; X3 – ratio of fuel
and air consumption in the combustion chamber. Fig. 9 shows the response curves of the FIS block for
generating variables X1, X2, X3. The combined diagram of the initial data and the results of controlling
the specific fuel consumption, as well as the control error, are shown in fig. 10 and 11. Fig. 12 – 14
shows scatter diagrams of the output parameters: fuel consumption on the training sample in the interval
from 0 to 5 seconds, fuel consumption on the test sample in the interval from 0 to 5 seconds; resulting
training error of the neuro-fuzzy system in the interval from 0 to 5 seconds, respectively.
a b c
Figure 9: Response curves of the variable generation block: a – X1; b – X2; c – X3
Figure 10: The result of checking the model on train (1) and test (2) data
Figure 11: Model error result
Figure 12: The scatter diagrams on training data
Figure 13: The scatter diagrams on test data
Figure 14: The scatter diagrams model error
Also, using the neuro-fuzzy system developed above, data on specific fuel consumption in various
operating modes of the TV3-117 engine at H = 0, V = 0 were obtained. The initial data for the calculation
were taken from [32], where the engine’s nameplate data is also presented, with which the calculation
results were compared. The calculation results and passport data are summarized in table 7 and are
presented in fig. 15 (curve 1 – calculated data, curve 2 – passport data, curve 3 – calculated data obtained
in [31]).
Figure 15: Comparison of the obtained values of specific fuel consumption of the TV3-117 engine with
the passport characteristics and results obtained in [31]
Table 7
Comparison of the obtained values of specific fuel consumption of the TV3-117 engine with the passport
characteristics and results obtained in [31]
Engine operating mode Output shaft power, hp Specific fuel consumption, g/hp.h
Calculated Passport [31]
Emergency 2200 232 230 263
Takeoff 2000 235 236 272
Nominal 1700 249 248 295
I cruising 1500 260 258 307
II cruising 1200 280 278 330
The adequacy of a mathematical model is determined by its accuracy and consistency with
experimental data [31]. In terms of consistency, it can be argued that the considered mathematical model
of specific fuel consumption does not contradict the passport data, since the nature of the calculated
dependence and the dependence constructed from the passport data coincide. In terms of accuracy, we
can note the presence of a systematic error, the reasons for which will be investigated further.
6. Discussions
During the experiment, the accuracy of the ANFIS neuro-fuzzy control algorithm was assessed in
comparison to other machine learning techniques, including Random Forest, ExtraTree, and the
Multilayer Perceptron Classifier (MLP) [32] (table 8).
Table 8
Results of a comparative analysis of the effectiveness of control methods
Methods Accuracy F1-measure Completeness
ANFIS 0.992 0.905 0.976
Random Forest 0.862 0.754 0.811
ExtraTree 0.975 0.907 0.977
MLP 0.723 0.698 0.634
The overall performance of the ANFIS neuro-fuzzy control system demonstrated commendable
generalization capabilities compared to MLP. However, the ExtraTree algorithm exhibited an advantage
in terms of the F1-measure, given its ensemble nature. The outcomes detailing errors of the 1st and 2nd
kind concerning the fuel consumption of helicopter turboshaft engines are provided in table 9.
Table 9
The results of determining errors of the 1st and 2nd kind
Neural networks architectures Probability of error in determining the
optimal value of fuel consumption
Type 1st error Type 2nd error
ANFIS (proposed) 0.76 0.47
Multilayer perceptron 1.04 0.63
Radial basis function network 1.05 0.66
Hopfield neural network 1.38 0.82
Elman neural network 1.77 1.01
Jordan neural network 1.96 1.12
Hamming neural network 2.13 1.25
LSTM-network 2.69 1.43
Kohonen network 3.17 1.84
Adaptive resonance network 3.93 2.36
The comparative analysis of the results obtained (table 9) affirms that the developed neuro-fuzzy
method for controlling fuel consumption in helicopter turboshaft engines achieves the minimum error in
addressing the task during helicopter flight operations.
The results of comparison of the obtained results of the specific fuel consumption of the TV3-117
engine with the passport data, as well as with studies conducted in [31] (table 7) are presented in table 10.
Table 10
The comparative analysis results
Engine operating mode Specific fuel consumption absolute error, g/hp.h
Results obtained with passport Results obtained with results
data obtained in [31]
Emergency 2 31
Takeoff 1 37
Nominal 1 46
I cruising 2 47
II cruising 2 50
As can be seen from table 10, the developed neuro-fuzzy system most accurately determines the
specific fuel consumption of the TV3-117 engine in various modes of its operation.
7. Conclusions
1. For the first time, an on-board method for neuro-fuzzy control of the fuel consumption of
helicopters turboshaft engines has been developed, which, through the use of an adaptive neuro-fuzzy
system that implements the Sugeno fuzzy inference system in the form of a five-layer neural network
of direct signal propagation, allows control the fuel consumption of helicopters turboshaft engines in
real time with an accuracy of 99.2 %.
2. It has been experimentally proved that in order to adapt the ANFIS neuro-fuzzy network and the
zero-order Sugeno fuzzy inference system to solving the problem of controlling the fuel consumption
of helicopters turboshaft engines, it is effective to use a hybrid training method, 2...3 delayed inputs and
two two-sided Gaussian membership functions.
3. The implementation of the neuro-fuzzy method for control the fuel consumption of helicopters
turboshaft engines provides a decrease in errors of the first and second kind compared to other neural
networks architectures from 1.34 to 5.17 times and also that the hybrid method training the neural
network for the number of epochs 105 times less than the backpropagation algorithm.
4. It has been established that the two-sided Gaussian membership function provides the smallest
network training error, equal to 1.29 · 10–4, compared with others that give the largest neural network
training error 5.86 · 10–4.
5. Increasing the outputs number of the ANFIS neuro-fuzzy network and expanding the base of
fuzzy rules makes it possible to improve the decision-making logic and, as a result, expand the range
of rule activity levels calculating.
6. The prospect of further research is the introduction of the developed methods and algorithms into
a modified closed onboard helicopters turboshaft engines automatic control system [28, 29].
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