Cognitive Method for Synthesising a Fuzzy Controller Mathematical Model Using a Genetic Algorithm for Tuning

Abstract

In this article, a fuzzy controller mathematical model synthesising method that uses cognitive computing and a genetic algorithm for automated tuning and adaptation to changing environmental conditions has been developed. The technique consists of 12 stages, including creating the control objects’ mathematical model and tuning the controller coefficients using classical methods. The research pays special attention to the error parameters and their derivative fuzzification, which simplifies the development of logical rules and helps increase the stability of the systems. The fuzzy controller parameters were tuned using a genetic algorithm in a computational experiment based on helicopter flight data. The results show an increase in the integral quality criterion from 85.36 to 98.19%, which confirms an increase in control efficiency by 12.83%. The fuzzy controller use made it possible to significantly improve the helicopter turboshaft engines’ gas-generator rotor speed control performance, reducing the first and second types of errors by 2.06…12.58 times compared to traditional methods.

1. Introduction

1.1. Relevance of the Research

In modern dynamic control systems, intelligent controller development that is capable of adapting to changing conditions and minimising control errors is of great importance [1,2]. In conditions of high uncertainty and nonlinear processes, traditional approaches to control are often insufficiently effective, which emphasises the need to use a fuzzy approach capable of ensuring control flexibility and stability. One promising solution for this class of problems is the use of fuzzy [3,4] and neuro-fuzzy [5,6] controllers, which combine the advantages of neural networks and fuzzy logic. This combination ensures high accuracy (98% and above) and controls stability under uncertainty [7,8,9,10]. However, setting up complex controllers often requires significant computing resources and time costs, reducing their effectiveness when implemented in natural systems. The cognitive approach in the controller mathematical model synthesis allows one to consider the system features and automate the optimisation process, minimising operator intervention and accelerating the setup.

The relevance of this research is due to the growing need for autonomous control systems that adapt to unstable operating conditions, such as aviation [11,12,13] and robotics [14,15,16,17]. For example, the research [18] presents a fuzzy control strategy based on the Takagi–Sugeno fuzzy systems, including a switching model and a control strategy with a damping factor, to ensure stable and cost-effective operation of a dynamic supply network under uncertain demand and time delays. Meanwhile, the research [19] proposes a new fuzzy robust H∞ control strategy for a nonlinear supply chain system with time delays based on the Takagi–Sugeno model. It provides asymptotic stability, soft switching between subsystems, and reduced oscillations in system variables, which is confirmed by numerical simulation. In [20], a fuzzy robust control strategy for supply chain restoring after disruptions based on the Takagi–Sugeno model is investigated. It provides soft switching between base models, reduces costs, and ensures the system’s stable operation under emergencies, which is confirmed by simulation.

The above indicates the relevance of using fuzzy control with cognitive components in the controller’s mathematical model synthesis. It allows for increasing the stability, adaptability, and efficiency of control systems in uncertainty conditions and complex dynamic processes.

1.2. State-of-the-Art

The genetic algorithm used [21,22] for tuning a fuzzy controller is an innovative solution that optimises the controller parameters and increases adaptive capabilities. Since genetic algorithms imitate natural selection processes, their use allows for exploring possible solutions in large spaces and finding optimal settings considering multi-criteria constraints [23,24,25,26]. The cognitive method development for synthesising a fuzzy controller with genetic optimisation aims to create more stable, efficient, intelligent control systems for complex technical objects.

The development of fuzzy controllers is an actively developing research area in the intelligent adaptive control systems field, which requires high adaptability to changing conditions and the complexity of control objects [27,28,29,30,31]. Neuro-fuzzy controllers [5,6,32,33] combine the training and prediction capabilities of neural networks with fuzzy logic systems, which allows the efficient processing of imprecise and incomplete data. Traditional approaches [34,35,36] to setting up neuro-fuzzy controllers are usually based on the use of expert systems and the manual optimisation of parameters, which is often insufficient for complex dynamic systems and requires significant computational costs.

According to the research review [37,38,39,40,41,42,43], various optimisation algorithms, including genetic algorithms, differential evolution, and particle swarm methods, are increasingly used to tune fuzzy and neuro-fuzzy systems. Genetic algorithms, based on natural selection and evolution principles, have proven to be a powerful tool for finding optimal parameters in complex multidimensional spaces. Research [37,38] has shown that using genetic algorithms to tune the neuro-fuzzy controllers’ parameters is advantageous compared to traditional optimisation methods in reducing tuning time and achieving results with a given accuracy context. However, the studies reviewed do not consider the influence of changing environmental conditions on the controller settings, which, in natural systems, can lead to instability and reduced control efficiency.

A key problem of many modern researchers is the insufficient attention to the fuzzy controller’s adaptability and cognitive aspects. Most approaches [39,41,42,43] assume static parameter adjustment, effective only under constant or slightly changing external conditions. Cognitive computing [44,45], based on perception imitation, adaptation, and training processes, offers a new perspective on the creation of intelligent controllers that can independently adjust their parameters depending on current conditions. The adaptive adjustment of smart controller parameters is necessary for complex systems such as aircraft control [46,47] and aircraft engines [48,49], where constant environmental changes require high flexibility and self-learning ability.

Machine learning [50] is another promising method for improving the quality of neuro-fuzzy controllers, allowing the processing of extensive data sets to process automation to find optimal solutions. Machine learning uses deep learning methods [51], recurrent neural networks [49], and other approaches [50,51,52,53,54], which are capable of adaptability at high levels and controller accuracy. However, machine learning integration with neuro-fuzzy systems remains limited due to the high complexity and training such models resource-intensiveness and the unified approaches joint optimisation lack. Current research shortcomings indicate the need for more intelligent methods, such as cognitive computing and genetic algorithms, to overcome these limitations.

Among the existing approaches, analytical methods for ensuring the stability of fuzzy and neuro-fuzzy controllers against random noise and interference in the complex systems’ actual operating conditions are also insufficiently developed. Most models—for example, those described in [55,56]—have uncertainties and time delays, which require the use of fuzzy robust control to ensure stability and minimise oscillations. This is due to the lack of filtering and error compensation mechanisms, which reduces their accuracy and reliability by 15…20% on average. The cognitive approach, in particular, can allow the model development to consider not only current data—for example, the data recorded by sensors at the current moment [57,58]—but also their historical patterns, allowing them to predict possible failures and adapt to them. In this regard, genetic algorithms can filter parameters optimally, increasing the control stability and reliability.

The genetic algorithm used in this context also lies in its ability to find optimal solutions in multi-criteria optimisation conditions, which is difficult to achieve using traditional methods. Cognitive computing provides real-time model adaptation, while machine learning methods allow the analysis of large datasets to predict and optimise system behaviour. These combined approaches open up new prospects for creating intelligent control systems operating in incomplete conditions or with inaccurate information and a changing environment. Thus, the critical gaps in existing research include insufficient adaptability, limited self-learning capabilities, and resistance to external interference and noise. The cognitive computing and genetic algorithms used in the neuro-fuzzy controller synthesising process can overcome these limitations, ensuring the high accuracy and reliability of controls in complex dynamic systems.

1.3. Main Attributes of the Research

Based on the above, the research aim is to develop a method for synthesising a fuzzy controller mathematical model using cognitive computing and a genetic algorithm for automated tuning and adaptation to changing environmental conditions. The research object is fuzzy controllers used to control complex dynamic systems, and the research subject is methods and algorithms for the fuzzy controller’s cognitive synthesis and automatic tuning using a genetic algorithm and machine learning elements. The scientific novelty of the developed method lies in the integration of mental components and machine learning methods for adjusting a fuzzy controller, which ensures its adaptation based on the analysis of previous errors and automatic updating of rules and parameters in real-time, in contrast to the traditional approach using a genetic algorithm [59]. The research contribution is to develop a method for synthesising a fuzzy controller using cognitive computing and a genetic algorithm for automated tuning and adaptation to changing environmental conditions, which significantly improves the control of complex objects and systems. It increases the control system efficiency by 12.83% compared to traditional methods [59].

1.4. Organisation of the Research

This article consists of an Introduction, “Materials and Methods”, “Results”, “Discussion”, Conclusions, and References. The Introduction Section substantiates the research relevance, provides an overview of existing research, highlights unresolved issues, and formulates the research aim, object and subject. The “Materials and Methods” Section directly develops a method for synthesising a fuzzy controller mathematical model using cognitive computing and a genetic algorithm. The “Results” Section shows a computational experiment assessing the helicopter turboshaft engines (TE) gas-generator rotor speed fuzzy controller quality, synthesised using the developed method implemented in the neuro-fuzzy network form. The qualitative comparison with the closest analogue [59] results is also presented. The “Discussion” Section presents the main advantages of the results obtained, highlights the limitations, and describes the prospects for further research. The conclusions briefly outline the study’s primary results, summarise the achieved aim, and identify directions for further improvement of the developed fuzzy controller cognitive synthesis method.

2. Materials and Methods

Neuro-fuzzy and adaptive fuzzy controllers differ in their approaches to training and adaptation: neuro-fuzzy systems integrate neural networks and fuzzy logic, where the neural network automatically adjusts membership functions, rules, and weights using a dependency processing layer with an activation function f(g), which provides a high degree of self-adjustment and generalisation; in contrast, adaptive fuzzy controllers rely solely on fuzzy logic, adapting rules and parameters to changes in the environment using methods such as gradient descent or heuristic algorithms. Neuro-fuzzy systems are suitable for complex, nonlinear problems with the need for automatic rule generation. While adaptive controllers are more straightforward, their functionality is limited to predefined rules and dynamic adjustment. Adaptive fuzzy controllers are appropriate for use in control problems whose implementation require ease, solutions interpretability, and the ability to adjust rules in the presence of limited data or when the system expresses weak nonlinearity.

This article proposes a method for developing a fuzzy controller with parameter tuning using a genetic algorithm and cognitive computing. For this aim, a traditional closed-loop control system for a dynamic object (Figure 1) is investigated—a stabilisation system [59,60], in which u(t) is a given control; e(t) is a control error (mismatch); u_p(t) is a control signal from the controller output; and y(t) is the stabilisation system output value.

To synthesise a fuzzy controller mathematical model with its parameter’s adjustment by a genetic algorithm and cognitive computing with the structure shown in Figure 1, a method consisting of 12 stages is proposed.

Stage 1. The development of the control object mathematical model. The controlled object mathematical model is developed at this stage, and the controller synthesising problem using the classical method is solved. The controller structure (P-, PI-, PID-controllers, etc.) is selected, after which its coefficients are calculated or selected. Let, as described in [61,62], a differential equations system describe the control object:

$$\begin{gathered} \dot{x}\left(t\right)=A·x\left(t\right)+B·u\left(t\right), \\ y\left(t\right)=C·x\left(t\right)+D·u\left(t\right). \end{gathered}$$

(1)

The P-controllers’ output u(t) is proportional to the error e(t):

u(t) = K_p · e(t).

(2)

The PI controller includes proportional and integral components, that is:

$$u\left(t\right)=K_p·e\left(t\right)+K_i·\int e\left(t\right)dt.$$

(3)

The PID controller includes proportional, integral and differential components, that is:

$$u\left(t\right)=K_p·e\left(t\right)+K_i·\int e\left(t\right)dt+K_d·{\displaystyle \frac{de\left(t\right)}{dt}}.$$

(4)

To select the controller coefficients in this study, the Ziegler–Nichols method is used, which allows you to adjust the coefficients K_p, K_i, and K_d based on the analysis of the oscillatory response of the system, using the critical gain coefficient K_critical and the oscillation period T_critical. The method consists of the controller integral and differential components disabling, increasing the proportional gain coefficient until the critical oscillation amplitude is reached, after which the oscillation period is measured to the controller optimal parameters determined [63,64]. In this case, the proportional gain coefficient K_p is gradually increased to K_critical, at which the system oscillates with constant amplitude (on the stability verge). After reaching the critical gain, the oscillation period T_critical is measured. Based on the K_p and T_critical obtained values, the coefficients for the controllers’ different types are determined (

Table 1. Analytical expressions for calculating coefficients for the controllers’ different types.

Controller Type	Analytical Equations
P-controller	Kp = 0.5 · Kkp
PI-controller	Kp = 0.45 · Kkp, $K_i={\displaystyle \frac{K_p}{T_i}},$Ti = 0.83 · Tkp
PID-controller	Kp = 0.6 · Kkp, $K_i={\displaystyle \frac{K_p}{T_i}},$Ti = 0.5 · Tkp, Kd = Kp ⋅ Td, Ti = 0.125 · Tkp

).

The calculated coefficients K_p, K_i, and K_d allow the controller to provide the systems’ required dynamics, being selected so that the system becomes stable and quickly reaches the set value without overshooting or minimal overshooting.

Stage 2. Testing the stabilisation system. The constructed stabilisation system is tested (based on the stage 1 results) in various operating modes to confirm stability. The aim is to check the regulation stability and quality. Let the equation describe the closed systems’ dynamics:

e(t) = r(t) − y(t).

(5)

It is known that for a stable system, the Lyapunov stability condition [65] is satisfied if there is a positive definite function V(x) (Lyapunov function) such that:

$$\dot{V}\left(x\right)<0 \mathrm{when}x\ne 0.$$

(6)

Based on the stabilisation system testing data (parameters e(t), u(t), and y(t)), a training data set (

Table 2. The training dataset general view.

The Stabilisation System Circuits’ Registered Signals	Signal Values Recorded at Time t
e(t)	e1(t)	e2(t)	…	ei(t)	…
$\dot{e}\left(t\right)$	${\dot{e}}_1\left(t\right)$	${\dot{e}}_2\left(t\right)$	…	${\dot{e}}_i\left(t\right)$	…
u(t)	u1(t)	u2(t)	…	ui(t)	…

) is formed for the fuzzy controller synthesising subsequent procedure. Pairs (e(t), $\dot{e}\left(t\right)$, u(t)) are recorded for different operating modes:

$$Training dataset={\left\{\left(e_i,{\dot{e}}_i,u_i\right)\right\}}_{i=1}^N.$$

(7)

Stage 3. The fuzzy controller structure defining. Based on the obtained training data set size, a decision is made on the fuzzy controllers’ structure (for example, one input – one output (see Figure 2), where μ(x) is the membership function of each term, T₁, T₂, …, T_N, [x_min, x_max] are the term set definition domain, x₁, x₂, …, x_N are the term boundaries, determined based on training data set [64]) and a linguistic description is selected for the input and output variables (fuzzification). The fuzzy controllers’ structure is determined based on the requirements for the system and the nature of the controlled object. In general, the fuzzy controllers’ input parameters are the error e(t) and the error change rate $\dot{e}\left(t\right)$, and the output is the control action u(t). At the next stage of the fuzzy controllers’ structure determination, the input parameters e(t) and $\dot{e}\left(t\right)$ fuzzy description and the output parameter u(t) are introduced in the linguistic variables term set form. In the most common case, the input and output parameters’ fuzzy descriptions are given in the form:

$$\begin{gathered} e\left(t\right)=\left\{\mathrm{“}\mathrm{n}\mathrm{e}\mathrm{g}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{v}\mathrm{e}\mathrm{”},\mathrm{“}\mathrm{z}\mathrm{e}\mathrm{r}\mathrm{o}\mathrm{”},\mathrm{“}\mathrm{p}\mathrm{o}\mathrm{s}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{v}\mathrm{e}\mathrm{”}\right\}, \\ \dot{e}\left(t\right)=\left\{\mathrm{“}\mathrm{d}\mathrm{e}\mathrm{c}\mathrm{r}\mathrm{e}\mathrm{a}\mathrm{s}\mathrm{e}\mathrm{s}\mathrm{”},\mathrm{“}\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{t}\mathrm{”},\mathrm{“}\mathrm{i}\mathrm{n}\mathrm{c}\mathrm{r}\mathrm{e}\mathrm{a}\mathrm{s}\mathrm{e}\mathrm{s}\mathrm{”}\right\}, \\ u\left(t\right)=\{\mathrm{“}\mathrm{d}\mathrm{e}\mathrm{c}\mathrm{r}\mathrm{e}\mathrm{a}\mathrm{s}\mathrm{e}\mathrm{”},\mathrm{“}\mathrm{m}\mathrm{a}\mathrm{i}\mathrm{n}\mathrm{t}\mathrm{a}\mathrm{i}\mathrm{n}\mathrm{”},\mathrm{“}\mathrm{i}\mathrm{n}\mathrm{c}\mathrm{r}\mathrm{e}\mathrm{a}\mathrm{s}\mathrm{e}\mathrm{”}\}. \end{gathered}$$

(8)

The input parameters’ e(t) and $\dot{e}\left(t\right)$ each value is transformed into the corresponding membership degrees (fuzzification) to the given terms using the membership function μ. For the error e(t):

μ_zero(e) = membership function for “zero”,
μ_negative(e) = membership function for “negative”,
μ_positive(e) = membership function for “positive”.

(9)

For the error derivative $\dot{e}\left(t\right)$:

$$\begin{gathered} {\mu }_{\mathrm{d}\mathrm{e}\mathrm{c}\mathrm{r}\mathrm{e}\mathrm{a}\mathrm{s}\mathrm{e}\mathrm{s}}\left(\dot{e}\right)=\mathrm{m}\mathrm{e}\mathrm{m}\mathrm{b}\mathrm{e}\mathrm{r}\mathrm{s}\mathrm{h}\mathrm{i}\mathrm{p} \mathrm{f}\mathrm{u}\mathrm{n}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n} \mathrm{f}\mathrm{o}\mathrm{r} \mathrm{“}\mathrm{d}\mathrm{e}\mathrm{c}\mathrm{r}\mathrm{e}\mathrm{a}\mathrm{s}\mathrm{e}\mathrm{s}\mathrm{”}, \\ {\mu }_{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{t}}\left(\dot{e}\right)=\mathrm{m}\mathrm{e}\mathrm{m}\mathrm{b}\mathrm{e}\mathrm{r}\mathrm{s}\mathrm{h}\mathrm{i}\mathrm{p} \mathrm{f}\mathrm{u}\mathrm{n}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n} \mathrm{f}\mathrm{o}\mathrm{r} \mathrm{“}\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{t}\mathrm{”}, \\ {\mu }_{\mathrm{i}\mathrm{n}\mathrm{c}\mathrm{r}\mathrm{e}\mathrm{a}\mathrm{s}\mathrm{e}\mathrm{s}}\left(\dot{e}\right)=\mathrm{m}\mathrm{e}\mathrm{m}\mathrm{b}\mathrm{e}\mathrm{r}\mathrm{s}\mathrm{h}\mathrm{i}\mathrm{p} \mathrm{f}\mathrm{u}\mathrm{n}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n} \mathrm{f}\mathrm{o}\mathrm{r} \mathrm{“}\mathrm{i}\mathrm{n}\mathrm{c}\mathrm{r}\mathrm{e}\mathrm{a}\mathrm{s}\mathrm{e}\mathrm{s}\mathrm{”}. \end{gathered}$$

(10)

Stage 4. The fuzzy base rules formation. Based on the training data set, logical inference schemes are created the fuzzy rule base use corresponding to the linguistic description. The rule base formation includes dividing the error values and their derivatives into several fuzzy sets (e.g., “small”, “medium”, and “large”), based on which a ruleset for control is then created. Thus,

$$\begin{gathered} \mathrm{I}\mathrm{f} e =\mathrm{“}\mathrm{s}\mathrm{m}\mathrm{a}\mathrm{l}\mathrm{l}\mathrm{”} \mathrm{a}\mathrm{n}\mathrm{d} \dot{e}=\mathrm{“}\mathrm{s}\mathrm{m}\mathrm{a}\mathrm{l}\mathrm{l}\mathrm{”},\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{n} u =u_1, \\ \mathrm{I}\mathrm{f} e =\mathrm{“}\mathrm{l}\mathrm{a}\mathrm{r}\mathrm{g}\mathrm{e}\mathrm{”} \mathrm{a}\mathrm{n}\mathrm{d} \dot{e}=\mathrm{“}\mathrm{l}\mathrm{a}\mathrm{r}\mathrm{g}\mathrm{e}\mathrm{”},\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{n} u =u_2, \\ \dots \end{gathered}$$

(11)

Stage 5. Selecting a defuzzification method. At this stage of developing a fuzzy controller, after applying fuzzy rules to the input parameters, defuzzification is performed to obtain the specific value of output parameter u(t):

u(t) = defuzzify({μ_decrease, μ_maintain, μ_increase}).

(12)

Many methods are used for defuzzification, such as the centre of gravity (COG) method, which calculates the output value by finding the resulting membership function mass centre, thus providing the systems’ output with the most accurate and smooth representation [66]. In [67], a COG method modification is shown, which consists of taking into account x_j is the category A_j centre, the category centres and their membership degrees weighted products sum $\sum _j{w}_j·x_j·{\mu }_j\left(u\right)$, as well as the weighted membership degrees sum $\sum _j{w}_j·{\mu }_j\left(u\right)$. In the COG method, classical analytical expression $y={\displaystyle \frac{\sum _j{x}_j·{\mu }_j\left(u\right)}{\sum _j{\mu }_j\left(u\right)}}$, all fuzzy inference points are taken into account with the same weight. It leads to underestimation or specific categories’ importance overestimation. For example, if the one category significance is higher than another, this leads to the defuzzification result distortion. The proposed modified analytical expression for the COG method

$$y={\displaystyle \frac{\sum _j{w}_j·x_j·{\mu }_j\left(u\right)}{\sum _j{w}_j·{\mu }_j\left(u\right)}}$$

(13)

allows you to consider each category’s significance degree during defuzzification, which will improve the outputs’ accuracy and assesses the weighting factors’ influence on the final result.

Stage 6. The fuzzy controller testing. The stabilisation system is tested with a fuzzy controller, with a subsequent conclusion about the systems’ stability. In instability, the terms’ x_i boundaries are adjusted until the system’s stable operation is achieved. For the fuzzy controller test, a controlled object’ system consisting and a fuzzy controller is used, described as follows:

y(t) = G(u(t)).

(14)

Stability criteria are used to analyse the systems’ stability with a fuzzy controller. If linear differential equations describe the system, the Rubezh criterion [68] or the Nichols criterion [69] can be used. For example, method one is the closed system characteristic equation analysis, which is in the following expression form:

1 + G(s) · H(s) = 0.

(15)

The system is considered stable if all roots of the characteristic equation obtained from (15) have negative real parts.

Testing the system involves measuring the response y(t) to a step setting, as well as calculating the settling time t_s, overshoot M_p, rise rate R_t, and stability as:

t_s = time during which ∣y(t) − r(t)∣ < ϵ,

(16)

$$M_p={\displaystyle \frac{y_{\mathrm{m}\mathrm{a}\mathrm{x}}-r\left(t\right)}{r\left(t\right)}}·100\%,$$

(17)

R_t = t₉₀ – t₁₀.

(18)

If testing shows that the system is not stable (e.g., if M_p > 20% or the settling time t_s is too long), then the terms ξ boundary in the fuzzification for the input and output variables are adjusted. For example,

ξ_new = ξ ± Δξ,

(19)

The system is retested after adjusting the term boundaries to check for stability. The process is repeated until the systems’ required stability characteristics are achieved. If stability is achieved after adjustments, the system can finally be accepted for operation, i.e.,:

Stability is achieved at (t_s, M_p) within the permissible values.

(20)

Stage 7. The genetic algorithm development. A genetic algorithm is created to the fuzzy controllers’ parameters, which includes the initial population (the input and output variables’ terms’ boundaries) formation, the current population generation, the genetic operators’ application, the formation of new chromosomes, checking the stopping condition, the completion of evolution, and obtaining an optimal solution [70,71,72].

The initial population is generated randomly within the input and output variables’ terms’ given boundaries, that is:

P₀ = {ξ⁽¹⁾, ξ⁽²⁾, …, ξ^(N)}.

(21)

In this case, the terms’ boundaries are specified for each input variable e(t) and the error derivative $\dot{e}\left(t\right)$ and the output parameter u(t).

To generate the current population, at each iteration step, the current population is updated by applying genetic operators, that is:

$$P_k=\left\{{\xi }_k^{\left(1\right)},{\xi }_k^{\left(2\right)},\dots {\xi }_k^{\left(N\right)}\right\}.$$

(22)

For the current population, the selection, crossover, and mutation genetic operators are applied to form a new population according to the expressions:

S_k = Selection(P_k),

(23)

C_k = Crossover(S_k),

(24)

M_k = Mutation(C_k),

(25)

after which a new population is formed:

P_{k+ 1} = M_k.

(26)

After applying the operators, each chromosome quality is assessed based on the integral quality criterion Q as:

$$Q\left({\xi }^{\left(i\right)}\right)=f\left({\xi }^{\left(i\right)}\right).$$

(27)

For example, the integral quality criteria optimal form is an analytical expression of the type:

$$Q\left({\xi }^{\left(i\right)}\right)=\underset{0}{\overset{T}{\int }}\left(w_1·{\left(e\left(t\right)\right)}^2+w_2·{\left(\dot{e}\left(t\right)\right)}^2+w_3·{\left(∆u\left(t\right)\right)}^2\right)dt.$$

(28)

The function $f\left({\xi }^{\left(i\right)}\right)$ in (28) evaluates the fuzzy controllers’ quality through the integral quality criterion, which considers both the control error, the control error change rate depreciation, and the control signal oscillations minimisation. The stopping condition of the genetic algorithm is set based on the integral quality criterion Q_max limit value or the iterations number k_max in the form:

$$\mathrm{Stop}, \mathrm{if} \underset{i}{\max }Q\left({\xi }^{\left(i\right)}\right)-Q_{optimal}<ϵ \mathrm{or} k \ge k_{\max }$$

(29)

After completing the tuning process, the fuzzy controllers’ optimal parameters are selected as follows:

$${\xi }^{*}=\arg \underset{i}{\max }Q\left({\xi }^{\left(i\right)}\right) \mathrm{for} i = 1, 2,\dots , N$$

(30)

The conditions for stopping the genetic algorithm were chosen, taking into account the need for the fuzzy controller parameters to achieve optimal values within a given accuracy or iteration maximum number. The algorithm can be stopped if the difference between the integral quality Q(ξ⁽ⁱ⁾) current value and the optimal value Q_optimal becomes less than the specified threshold ϵ. It indicates that an optimisation acceptable level has been achieved, or if the iteration number reaches the limit value k_max, which prevents an infinite calculation process.

Stage 8. The genetic algorithm testing. At this stage, the fuzzy controller parameter settings (term boundaries) testing is carried out using the test control u(t) within the specified ranges of fuzzy controller parameter (term boundary) variations. That is,

$$u\left(t\right)=\mathrm{G}\left(e\right(t), \dot{e}\left(t\right)$$

(31)

After this, the parameters are adjusted relative to the values set in the previous stage. For each adjustment, a check is made on how changing the term boundaries affects the output control. Thus,

ξ_k = [ξ_1,k, ξ_2,k, …, ξ_n,k],

(32)

Data on the systems’ output response is collected to evaluate the system’s operation with new settings. The systems’ stability is evaluated using the settling time t_s, overshoot M_p, and rise rate R_t (16)–(18) criteria. In this case, the system response is defined as:

y(t) = Response(u(t)).

(33)

The systems’ obtained performance characteristics with the new settings are compared with the benchmark values established in step 6 to determine how much the genetic algorithm has improved the system. Thus,

ΔQ = Q_new − Q_ref.

(34)

To check the genetic algorithm and its ability to achieve an optimal solution correctly, a convergence and improvement (correctness) criterion is used, defined as:

Correctness ⟺ ΔQ > 0.

(35)

The value ΔQ > 0 indicates that the genetic algorithm has successfully improved the parameters, and the system has become more efficient. Otherwise, repeated adjustments of the system are made until ΔQ > 0.

The testing iterations continue until a specific stopping condition is reached, such as an iterations’ fixed number, or the integral quality criterion limit value is reached, similar to (29), that is:

$$\mathrm{Stop}, \mathrm{if} \underset{i}{\max }\left|Q_{new}-Q_{optimal}\right|<ϵ \mathrm{or} k \ge k_{\max }$$

(36)

Stage 9. The fuzzy controllers’ parameters setting up. At this stage, the fuzzy controllers’ parameters are set up using a genetic algorithm for the input action u(t) given type. The fuzzy controllers’ structure is set, including the input and output variables number, as well as their linguistic terms:

$$G\text{:} E \times \dot{E}\to U$$

(37)

Next, the target function F is formed, which describes the fuzzy controller’s operation quality depending on the configured parameters, in the form:

F(ξ) = a₁ ⋅ Q_performance(ξ) – a₂ ⋅ R_complexity(ξ).

(38)

To optimise the objective function, a genetic algorithm is used, which includes steps similar to stage 7: population initialisation according to (21), the selection, crossover and mutation genetic operators according to (23)–(25), and population evaluation according to (27).

When testing and adjusting the fuzzy controller, the specified input action u(t) is taken into account by using test signals to check the system’s response:

$$u\left(t\right)=\mathrm{G}\left(e\right(t), \dot{e}\left(t\right), \xi ).$$

(39)

The parameter tuning continues until the stopping condition (29) is reached based on the objective function or the iterations’ maximum number improvement. After completing the tuning process, the fuzzy controllers’ optimal parameters are selected according to (30).

After this stage, the fuzzy controllers’ parameters will be finally tuned and ready for use in the control system, improving its performance and stability.

Stage 10. The cognitive component introduction. The cognitive computations integrate into the fuzzy controller with one formal neuron addition, which will allow the weight of each rule to be set when constructing the resulting output. The formal neuron introduction into the fuzzy controllers’ structure involves the weights used for each rule [73,74]. The general structure is represented as:

$$Y=\sum _{i=1}^n{w}_i·R_i.$$

(40)

In this case, the weights are determined through training based on the available data:

$$w_i={\displaystyle \frac{Q\left(R_i\right)}{{\sum }_{j=1}^n\left(R_j\right)}}.$$

(41)

The weights are adapted based on new data, which allows the controller to “train”. For example, the gradient descent algorithm with an adaptive training rate is used to update the weights:

$$w_i^{\left(k+1\right)}=w_i^{\left(k\right)}+{\eta }^{\left(k\right)}·\nabla Q\left(w_i^{\left(k\right)}\right).$$

(42)

Cognitive computing integration involves the memory used to store previous states and results and is implemented as follows:

M = {Y⁽¹⁾, Y⁽²⁾, …, Y^(k)}.

(43)

In case of receiving new data, the cognitive component updates the rules and their weights according to the expression:

$$R_i^{\left(k+1\right)}=R_i^{\left(k\right)}+\lambda ·∆R_i.$$

(44)

As a result, the resulting output will be calculated taking into account the new weights and the adaptive approach according to the expression:

$$Y^{\left(k+1\right)}=\sum _{i=1}^n{w}_i^{\left(k+1\right)}·R_i^{\left(k+1\right)}.$$

(45)

Thus, the system trains and adapts by integrating a cognitive component that uses historical data stored in memory (43). This memory stores past output values, which allows for the analysis of trends and relationships. The rules (R_i) and weights (w_i) are updated dynamically as new data arrives. The rules are updated according to (44), where λ is the learning coefficient that determines the adjustment degree. At the same time, the weights are recalculated to reflect each rule’s significance based on the training data. Convergence is ensured as follows:

• By choosing the correct learning rate. This is justified by the fact that the adaptive learning rate η^(k) decreases as k increases, for example, according to the law ${\eta }^{\left(k\right)}={\displaystyle \frac{{\eta }_0}{1+\beta ·k}}$, where η₀ is the initial rate and β is the attenuation coefficient.
• By regularising the gradient. This is explained by the fact that the error function Q is usually smooth and convex near the weights’ optimal values, which guarantees convergence to a local minimum.
• Stopping criterion. The iterations are terminated if $\left|\nabla Q\left(w_i^{\left(k\right)}\right)\right|$ becomes less than a given threshold, or the change in Q between iterations becomes insignificant.

Thus, introducing the cognitive component will provide more flexible control and the ability to train based on new data. After completing this stage, the fuzzy controller will be integrated with cognitive computing. It will allow it to adapt to changing conditions, improve the controls’ quality, and, as a result, significantly increase the fuzzy controllers’ efficiency. The resulting output constructing procedure for the m rules base is schematically presented in Figure 3, where x₀ is the neurons’ initial state, x₁, …, x_m are the rules’ base output variables, a_0, …, a_m are the weights of the rules, FN is the formal neuron, f(g) is the neuron activation function, and g is the activation function argument [59,75].

Stage 11. Adaptive control. At this stage, machine learning methods are implemented to adapt the fuzzy controller based on the previous control error analysis. It will allow the model to automatically update its rules and parameters depending on changes in the behaviour of the control object. To analyse previous control errors e(t), an analytical expression of the form is used:

e(t) = r(t) − y(t).

(46)

Based on the collected data on control errors, an update rule is formed:

$$R_i^{\left(k+1\right)}=R_i^{\left(k\right)}+\lambda ·e\left(t\right).$$

(47)

The rule weights are also adapted depending on the previous errors according to the expression:

$$w_i^{\left(k+1\right)}=w_i^{\left(k\right)}-{\eta }^{\left(k\right)}·e\left(t\right)·{\displaystyle \frac{\partial R_i^{\left(k\right)}}{\partial e}}.$$

(48)

Machine learning algorithms such as regression or neural networks are then used to model the relations between errors and the fuzzy controllers’ rules. For example, the model is represented as:

$$\widehat{R}=f\left(E,\dot{E}\right).$$

(49)

When the control object changes behaviour, the fuzzy controllers’ parameters are also updated adaptively according to the expression:

ξ^{(k + 1)} = ξ^(k) + γ ⋅ ∇L.

(50)

Adaptive control involves real-time optimisation using the data received. Thus:

$$Y^{\left(k+1\right)}=\sum _{i=1}^n{w}_i^{\left(k+1\right)}·R_i^{\left(k+1\right)}.$$

(51)

Thus, as the implementation of machine learning and adaptive control methods results, the fuzzy controller can adapt dynamically to changes in the control objects’ behaviour.

Stage 12. The neuro-fuzzy controller testing. At this stage, the fuzzy controller is tested with unit weights of rules a_j in the range from 0 to 1. If the results coincide with the results of stage 9, the weights a_j are adjusted in the specified range to improve the system’s performance quality integral criterion. If further improvement is impossible, a conclusion about the system performance quality achieved level is made. The controllers’ initial testing is performed with weights equal to one, that is:

a_j(0) = 1, ∀j.

(52)

The systems’ resulting output is calculated as:

$$Y^{\left(0\right)}=\sum _{j=1}^n{a}_j^{\left(0\right)}·R_j=\sum _{j=1}^n{R}_j.$$

(53)

After this, the test results are compared with the results obtained in step 9:

If Y⁽⁰⁾ ≈ Y_stage_₉, then the scale adjustment continues.

(54)

If the results match, the weights a_j are adjusted in the range from 0 to 1 using the gradient descent algorithm with an adaptive training rate:

$$a_j^{\left(k+1\right)}=a_j^{\left(k\right)}-{\eta }^{\left(k\right)}·{\displaystyle \frac{\partial L}{\partial a_j}}.$$

(55)

The system’s operation quality assessment uses the integral criterion (27). If the updated weights lead to an improvement in the integral quality criterion, then the weights are further adjusted, i.e.,:

Q^(k+1) < Q^(k) ⇒ Continue adjusting the scales.

(56)

If improvement is not possible, then the system’s operation achieved quality is recorded. That is:

Q^(k+1) ≥ Q^(k) ⇒ Record the system’s operation quality level.

(57)

Thus, if, after testing and adjusting the scales, the level of system performance achieved meets the requirements, the fuzzy controllers’ results are recorded, and the testing is completed. If necessary, additional iterations are performed to adjust the scales for the controller characteristics and further improvement.

The developed method’s scientific novelty lies in the difference between the proposed method of tuning a fuzzy controller and the traditional method with a genetic algorithm [59], which implies the integration of the cognitive components and adaptive control. The conventional method of tuning a fuzzy controller with a genetic algorithm [59] focuses on choosing the controller’s structure and adjusting its parameters using evolutionary operators. In contrast, the proposed method includes implementing machine learning methods, which allow the controller to adapt based on previous errors and automatically update its rules and parameters in real-time analysis. It provides more flexible control and improves the control quality due to continuous training on new data, an essential advantage in the systems’ dynamic conditions.

3. Results

In this article, a computational experiment was conducted on the helicopter TE gas-generator rotor r.p.m. fuzzy controller quality, which was evaluated and synthesised using the developed method implemented in the neuro-fuzzy network form [67]. The helicopter TE gas-generator rotor r.p.m. is an important parameter affecting the energy resource use efficiency and the energy system optimisation processes [76,77]. The neuro-fuzzy controller transfer function for the gas-generator rotor r.p.m., developed in [67], is presented in the analytical expression form:

$$W_{ACS}\left(z\right)={\displaystyle \frac{z}{z^2+94.781·z+6.608·{10}^3}},$$

(58)

from which it follows that K_p + s_i + s_d = 5.86, K_p + 2 · s_d = 94.781, s_i – s_d = 6.614 · 10³, K_p = 6.703 · 10³, $s_i=K_i·∆t$ = −42, $s_d={\displaystyle \frac{K_d}{∆t}}$ = −6.656 · 10³. At the same time,

$$\begin{gathered} u\left(t\right)=u\left(t+∆t\right)+K_p·\left(e\left(t\right)-e\left(t-∆t\right)\right)+s_i·\left(e\left(t\right)+e\left(t-∆t\right)+e\left(t-2·∆t\right)\right)+s_d \\ ·\left(e\left(t\right)-2·e\left(t-∆t\right)+e\left(t-2·∆t\right)\right), \end{gathered}$$

(59)

The research object for the computational experiment is the TV3-117 TE, part of the Mi-8MTV helicopter power plant and its modifications, which is widely used in civil and military aviation [78,79]. During the flight tests, the data on the gas-generator rotor r.p.m. n_TC were obtained and recorded on board the helicopter by the D-2M sensor [80] (the data were recorded in 256 s of the actual flight with a sampling period of 1 s). The TV3-117 TE gas-generator rotor r.p.m. values data were provided following the article’s author’s official request to the Ministry of Internal Affairs of Ukraine and are intended for the implementation of the project “Theoretical and Applied Aspects of the Development of the Aviation Sphere”, officially registered in Ukraine No. 0123U104884, headed by the author of this article. According to [64], 256 gas-generator rotor r.p.m. n_TC values were selected, as shown in Figure 4. Based on the gas-generator rotor r.p.m. selected values n_TC, the control error and the control error rate values (

Table 3. The training dataset fragment (author’s research]).

Number	1	…	35	…	102	…	169	…	256
nTC	0.984	…	0.987	…	0.972	…	0.985	…	0.983
$e_{n_{TC}}$	0.012	…	0.011	…	0.016	…	0.019	…	0.016
${\displaystyle \frac{d{e}_{n_{TC}}}{dt}}$	0.018	…	0.021	…	0.017	…	0.018	…	0.019

) were obtained, constituting the training dataset (Table 3). The homogeneity of the training dataset was assessed according to the Fisher–Pearson criterion [81,82] and Fisher–Snedecor [83,84] (

Table 4. The training dataset homogeneity evaluating results for parameters e(t) and ${\displaystyle \frac{de\left(t\right)}{dt}}$ using the Fisher–Pearson and Fisher–Snedecor criteria.

Input Parameter	The Criterion Value		Description
	Calculated	Critical
The Fisher–Pearson criterion
e(t)	6.483	6.6	The Fisher–Pearson test produced values for each parameter e(t) and ${\displaystyle \frac{de\left(t\right)}{dt}}$ that remained below the critical threshold. It indicating uniformity within the training dataset.
${\displaystyle \frac{de\left(t\right)}{dt}}$	6.495
The Fisher–Snedekor criterion
e(t)	2.411	2.58	The Fisher–Snedecor criterion generated values for each parameter e(t) and ${\displaystyle \frac{de\left(t\right)}{dt}}$ that fell below the critical threshold. It indicating consistency within the training dataset.
${\displaystyle \frac{de\left(t\right)}{dt}}$	2.428

).

The representativeness of the training and test datasets was evaluated using cluster analysis, where the input dataset x = (e(t), ${\displaystyle \frac{de\left(t\right)}{dt}}$) (Table 3) was divided into k predetermined clusters [75]. Each cluster contains objects that exhibit more significant similarity to each other than to those in different clusters. This process continues until minimal shifts occur in the centroids or the specified number of iterations is reached [85,86]. The training dataset (Table 4) cluster analysis identified eight clusters (I…VIII). Random sampling generated the training and test sets in a 2:1 ratio (67 and 33%, respectively). Both datasets exhibited all eight clusters, reflecting a similar structure. The distances between clusters were almost identical in both sets, confirming their comparability (Figure 5). As a result, the optimal dataset sizes were defined as follows: the training dataset contains 256 elements (100%), the validation dataset comprises 172 elements (67% of the training dataset), and the test dataset includes 84 elements (33% of the training dataset).

Based on [67], a triangular membership function of the type shown in Figure 6 is selected, which is characterised as LN (Low Negative), MN (Medium Negative), Z (Zero), MP (Medium Positive), and LP (Low Positive). Its mathematical description is presented as follows:

$${\mu }_j\left(n_{TC}\right)=\left\{\begin{array}{c}0 \mathrm{for} u_1\le a\\ {\displaystyle \frac{n_{TC}-a}{b-a}} \mathrm{for} a\le n_{TC}\le b\\ 1 \mathrm{for} b\le n_{TC}\le c\\ {\displaystyle \frac{d-n_{TC}}{d-c}} \mathrm{for} c\le n_{TC}\le d\\ 0 \mathrm{for} n_{TC}\ge d\end{array} \right.,$$

(60)

where a is the function segments’ initial point where it begins to increase and which corresponds to the LP category; b is the function vertex where its maximum value is reached and which corresponds to the MP category; c is the function segments’ central point where the function has the most excellent value and corresponds to the Z category; and d is the function segments’ final point where the function begins to decrease and which corresponds to the MN category [67].

For each output coefficient, K_p, K_i, and K_d, a fuzzy rule base developed by the author in [67] is applied, presented in

Table 5. Proposed fuzzy rules base for each output coefficient K_p, K_i, and K_d [64].

Rule	Report
Rule 1	IF	e ∈ LN	AND	de ∈ LN	THEN	$K_d\in LP,$ ${\displaystyle \frac{K_i}{K_p}}\in LP$
Rule 2		e ∈ MN		de ∈ MN		$K_d\in MP,$ ${\displaystyle \frac{K_i}{K_p}}\in MP$
Rule 3		e ∈ Z		de ∈ Z		$K_d\in Z,$ ${\displaystyle \frac{K_i}{K_p}}\in Z$
Rule 4		e ∈ MP		de ∈ MP		$K_d\in MN,$ ${\displaystyle \frac{K_i}{K_p}}\in MN$
Rule 5		e ∈ LP		de ∈ LP		$K_d\in LN,$ ${\displaystyle \frac{K_i}{K_p}}\in LN$

.

To perform testing of the stabilisation system with a fuzzy controller functioning (stage 6), a fuzzy controllers’ software module was created and converted into the stabilisation systems’ functional block format, to which the inputs of the arrays of term parameters are fed.

A₀[4] = [A₁, A₂, A₃, A₄],
B₀[4] = [B₁, B₂, B₃, B₄].

(61)

 

The stabilisation system with fuzzy controller operation testing showed that the system with a fuzzy controller without tuning is stable according to the Nyquist criterion (Figure 7).

Figure 7 shows a graphical representation of the guaranteed stability margin in amplitude and phase for the engine control system based on the gas-generator rotor speed: M = 1.097, the “dangerous” regions’ circle radius is r = 41.915, and the distance of its centre from the origin is R = 42.418. In Figure 7, the “black curve” is the hodograph, the “blue circle” is the unit circle, the “red line” is the “dangerous” region boundary with the radius r = 41.915 and the centre at the point (−42.418, j0), and the “green dotted line” is the ray from the origin through the hodographs’ intersection point with the unit circle. The amplitude and phase stability reserves are 0.95 and 75°, respectively, indicating the possibility of changing the output signal amplitude by ±0.95 from the input signal amplitude and changing the phase stability by ±75° from the critical point.

A genetic algorithm was developed using the method described in the previous section to optimise the fuzzy controllers’ parameters. The initial population was formed based on the initial data (61), considering the symmetry in the linguistic terms’ description (see Figure 6). For the fuzzy controllers’ input and output variables, the chromosomes’ initial populations are presented as follows:

(a₁₀, a₂₀, a₃₀), (b₁₀, b₂₀, b₃₀).

(62)

According to chromosomes (62), the fitness function value is calculated according to (28). To determine the ranges of values for genes, it is assumed that the value of each gene in chromosomes (60) is a number in the given intervals [59]:

$$a_1^{\mathrm{m}\mathrm{i}\mathrm{n}}\le a_1\le a_1^{\mathrm{m}\mathrm{a}\mathrm{x}},a_2^{\mathrm{m}\mathrm{i}\mathrm{n}}\le a_2\le a_2^{\mathrm{m}\mathrm{a}\mathrm{x}},a_3^{\mathrm{m}\mathrm{i}\mathrm{n}}\le a_3\le a_3^{\mathrm{m}\mathrm{a}\mathrm{x}},$$

(63)

$$b_1^{\mathrm{m}\mathrm{i}\mathrm{n}}\le b_1\le b_1^{\mathrm{m}\mathrm{a}\mathrm{x}},b_2^{\mathrm{m}\mathrm{i}\mathrm{n}}\le b_2\le b_2^{\mathrm{m}\mathrm{a}\mathrm{x}},b_3^{\mathrm{m}\mathrm{i}\mathrm{n}}\le b_3\le b_3^{\mathrm{m}\mathrm{a}\mathrm{x}}.$$

(64)

A random selection of values from intervals (63) and (64) forms a new chromosome. To generate a population, three chromosomes of type (62) are created randomly within the values’ ranges (63) and (64). An example of chromosome construction is presented in

Table 6. The initial data for creating the current population [59].

Chromosomes’ Number	Chromosomes’ Designation
	(a1, a2, a3)	(b1, b2, b3)
1	a1 = (a11, a12, a13)	b1 = (b11, b12, b13)
2	a2 = (a21, a22, a23)	b2 = (b21, b22, b23)
3	a3 = (a31, a32, a33)	b3 = (b31, b32, b33)

.

The algorithm’s steps continue until the maximum value of the integral criteria (28) is reached according to (30), that is, for all chromosomes’ combinations, and until the fitness indicator k becomes equal to 0 for all offspring.

Since chromosome a forms the fuzzy controllers’ input variables and chromosome b the output variables, nine combinations (each with each) are created based on the chromosomes shown in Table 6. The fitness level of the current population is assessed by calculating chromosomes’ fitness values and survival rates as:

$$\begin{gathered} \mathrm{I}\mathrm{f} I_0<I_{ij},\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{n} k_{ij}=1\left(i=\overline{\mathrm{1,3}},\mathrm{j}=\overline{\mathrm{1,3}}\right), \\ {k}_{ij}=0 \mathrm{o}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{w}\mathrm{i}\mathrm{s}\mathrm{e}. \end{gathered}$$

(65)

$$\begin{gathered} \mathrm{I}\mathrm{f} k_{ij}=1,\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{n}{\mathsf{\Delta }}_n=I_0–I_{ij}\left(n=\overline{\mathrm{1,9}}\right), \\ {\mathsf{\Delta }}_n=0 \mathrm{o}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{w}\mathrm{i}\mathrm{s}\mathrm{e}. \end{gathered}$$

(66)

The survival rate is determined by the following expression, based on the selecting chromosomes task in the population with the highest survival rate based on the fitness criterion:

$$B_n={\displaystyle \frac{{∆}_n}{{\sum }_{n=1}^9{∆}_n}}·100\%.$$

(67)

To develop a population pair, the chromosome combinations’ pairs are selected.

Table 7. An example of the new populations’ formation.

Chromosome Combination	Combination Number	Combination
For father	8	a3b2
	4	a2b1
	8	a3b2
For mother	4	a2b1
	3	a1b3
	8	a3b2

shows an example of forming a new population.

To optimise the fuzzy controllers’ parameters using the genetic algorithm, the following parameters are used: the initial population is formed randomly based on symmetry in the linguistic terms description, including three chromosomes of the type (a₁, a₂, a₃) and (b₁, b₂, b₃), the gene values are limited by the intervals [$a_i^{\mathrm{m}\mathrm{i}\mathrm{n}}$, $a_i^{\mathrm{m}\mathrm{a}\mathrm{x}}$] and [$b_i^{\mathrm{m}\mathrm{i}\mathrm{n}}$, $b_i^{\mathrm{m}\mathrm{a}\mathrm{x}}$] (where i = 1, 2, 3). The population size includes all possible combinations of input and output chromosomes (nine combinations). The fitness level is assessed based on the integral quality criterion, and the individual’s survival rate is determined through normalised indicators of the difference between the current population and the base value integral criterion. Pairs for forming a new population are selected based on a roulette mechanism, where the selection probability is proportional to the survival rate. Setting up the fitness function involves calculating the quality criterion using a (65), where the minimum difference between the I₀ value and the current I_ij criterion indicates a high fitness level. The population size is maintained constant at each stage. The mutation rate is regulated by randomly changing genes within given intervals with low probability to ensure a balance between searching for a global optimum and local adaptation. The optimisation process continues until the maximum value of the quality criterion is reached or the results stabilise. The developed genetic algorithms efficiency was tested on the gas-generator rotor speed n_TC stabilisation system. An example of the fuzzy controller parameter optimisation process using the genetic algorithm is given in

Table 8. Data from the output file of the software module implementing the genetic algorithm for the fuzzy controller tuning.

Algorithms’ Steps	Integral Quality Criterion, %	The Fuzzy Controllers’ Input Variable Terms Boundaries				The Fuzzy Controllers’ Output Variable Terms Boundaries
		A1	A2	A3	A4	B1	B2	B3	B4
0	85.36	0.980	0.961	0.943	0	0.923	0	0	0
1	90.97	0.993	0.983	0.969	0	0.951	0	0	0
2	98.19	0.993	0.995	0.985	0	0.978	0	0	0
3	98.19	0.993	0.996	0.997	0	0.995	0	0	0
4	98.19	0.993	0.996	0.997	0	0.995	0	0	0

.

At step i = 0 (before the genetic algorithm starts working), the term boundaries correspond to the initial values (61). At i = 1 (the genetic algorithm’s first step), the term boundaries changed within the ranges (63) and (64), according to Figure 3, towards improving the stabilisation system with a fuzzy controller performance. The criterion’s (28) value increased from 85.36 to 90.97. The genetic algorithm finished working at the fourth step when the stabilisation system performance increased to Q = 98.19; improving the result was impossible. At i = 4, the result was duplicated, and the genetic algorithm stopped working.

The fuzzy controller, tuned by a genetic algorithm synthesis algorithm, is performed by the scheme shown in Figure 3. A formal neuron is built into the fuzzy inference algorithm, with the help of which the fuzzy controller is fine-tuned in each weight’s terms of the inference rules in the base (see Table 5).

The fuzzy controller with tuning using a genetic algorithm operation testing in a gas-generator rotor speed stabilisation system was carried out, considering the random impact on the control object in the “white noise” and random pulse form (Figure 8).

Figure 9 shows diagrams of the change in the output parameter time of the stabilisation system in the random pulse disturbance presence (see Figure 6) on the control object when modelling an inaccurate mathematical description of the control object is the helicopter TE, where the “black curve” shows the result obtained using a fuzzy controller with a genetic algorithm setting, the blue curve is the result obtained using a fuzzy controller without a genetic algorithm setting [67], and the red curve is the result obtained using a traditional PID controller.

The helicopter TE gas-generator rotor speed developed fuzzy controller efficiency was assessed in comparison with the fuzzy controller developed in [67] and the traditional PID controller according to the following quality metrics: overshoot (O), steady-state value (ϵ_st), and transient time (t_trans) [67,80] (

Table 9. The transient process controls quality parameters and calculates results.

Parameter	The Proposed Fuzzy Controller Without a Genetic Algorithm Tuning	The Fuzzy Controller Without a Genetic Algorithm Tuning [67]
Overshoot (O)	1.104%	2.026%
Steady-state value (ϵst)	0.052	0.107
Transient time (ttrans)	0.319	0.483

), calculated using the following expressions:

$$O={\displaystyle \frac{n_{TCmax}-n_{TCst}}{n_{TCst}}}·100\%, {ϵ}_{st}=n_{TCst}-n_{TC}\left(t\right), t_{trans}=t_2-t_1.$$

(68)

where (according to [62,67]) n_TCmax denotes the gas-generator rotor speed’s highest recorded value in the transient process, n_TCst is the gas-generator rotor speed steady-state (target) value, and n_TC(t) is the gas-generator rotor speed value at a specific time t after the transient process end. In this case, t₁ indicates when the monitored parameter first reaches or enters a specified proximity (%) of the steady-state value n_TCst. At the same time, t₂ denotes the moment when the gas-generator rotor speed value again approaches this proximity and stabilises after the transition.

The comparative analysis showed that the proposed fuzzy controller without genetic tuning outperforms a similar controller from [67] and a traditional PID controller in the transient process control quality indicators terms. In particular, the developed controller demonstrates a lower overshoot value (1.104% versus 2.026%), a minor steady-state error (0.052 versus 0.107), and a reduced transient process time (0.319 versus 0.483). The proposed approach improves the gas-generator rotor speed n_TC control efficiency in transient modes: overshoot is reduced by approximately 1.84 times, the steady-state error is reduced by 2.06 times, and the transient process time is reduced by 1.51 times compared to the controller from [67].

The first and second types of errors are calculated (

Table 10. The first and second types of errors calculating results.

Error Type	The Proposed Fuzzy Controller Without a Genetic Algorithm Tuning	The Fuzzy Controller Without a Genetic Algorithm Tuning [64]	Traditional PID Controller
The first type of error	0.353	0.728	3.992
The second type of error	0.197	0.415	2.479

) in the helicopter TE gas-generator rotor speed-controlling issue using a fuzzy controller with genetic algorithm tuning, a fuzzy controller without genetic algorithm tuning [67], and a traditional PID controller. In the helicopter TE gas-generator rotor speed controlling problem, the first and second errors are associated with the system states’ incorrect classification. The case number determines the first type of error (false positive) when the system erroneously determines the speed deviation presence, although there is no deviation. The second type of error (missed deviation) is the situation number when the system does not detect an actual frequency deviation, leaving it unnoticed. These errors are estimated based on the event’s total number in which a frequency deviation occurs, regardless of whether it was recognised. The first type of error reduction allows for avoiding false alarms, increasing the control efficiency, and the second type of error reduction improves the reliability and safety of engine operation. Thus [87,88],

$$\begin{gathered} \alpha ={\displaystyle \frac{FP}{P}},\beta ={\displaystyle \frac{FN}{P}}, \\ \mathsf{\alpha }=P\left(\mathrm{d}\mathrm{e}\mathrm{c}\mathrm{i}\mathrm{s}\mathrm{i}\mathrm{o}\mathrm{n} \mathrm{o}\mathrm{n} \mathrm{t}\mathrm{h}\mathrm{e} \mathrm{d}\mathrm{e}\mathrm{v}\mathrm{i}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{’} \mathrm{p}\mathrm{r}\mathrm{e}\mathrm{s}\mathrm{e}\mathrm{n}\mathrm{c}\mathrm{e} |\mathrm{n}\mathrm{o} \mathrm{d}\mathrm{e}\mathrm{v}\mathrm{i}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{s}\right), \\ \beta =P(\mathrm{d}\mathrm{e}\mathrm{c}\mathrm{i}\mathrm{s}\mathrm{i}\mathrm{o}\mathrm{n} \mathrm{o}\mathrm{n} \mathrm{t}\mathrm{h}\mathrm{e} \mathrm{d}\mathrm{e}\mathrm{v}\mathrm{i}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{’} \mathrm{a}\mathrm{b}\mathrm{s}\mathrm{e}\mathrm{n}\mathrm{c}\mathrm{e} | \mathrm{d}\mathrm{e}\mathrm{v}\mathrm{i}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{s} \mathrm{e}\mathrm{x}\mathrm{i}\mathrm{s}\mathrm{t}\mathrm{s}). \end{gathered}$$

(69)

In the helicopter TE gas-generator rotor speed controlling problem context, the false positive (FP) rate is the number of times the system erroneously signals the need for corrective action due to an assumed rotor speed deviation when there is no deviation, which corresponds to a false alarm. The missed deviation (FN) rate is the number of times the system does not detect an actual rotor speed deviation, leaving it unnoticed and not signalling for correction. It reduces control accuracy and may adversely affect engine reliability or performance, representing a false negative.

As can be seen from Table 10, the developed fuzzy controller with genetic algorithm tuning used for the helicopter TE gas-generator rotor speed made it possible for the first and second types errors to reduce by 2.06…2.11 times compared to the fuzzy controller developed in [67] and by 11.31…12.58 times compared to the traditional PID controller.

In the experimental research course, a comparative analysis of the developed controller operation was carried out in comparison with the fuzzy controller developed in [67] and the traditional PID controller under white noise conditions (with zero mathematical expectation M = 0 and values σ = 0.01; 0.03; 0.05). The results of these methods are presented in

Table 11. The first and second types of errors calculating results under white noise conditions.

Error Type	The Proposed Fuzzy Controller Without a Genetic Algorithm Tuning				The Fuzzy Controller Without a Genetic Algorithm Tuning [67]				Traditional PID Controller
	σ = 0	σ = 0.01	σ = 0.03	σ = 0.05	σ = 0	σ = 0.01	σ = 0.03	σ = 0.05	σ = 0	σ = 0.01	σ = 0.03	σ = 0.05
The first type of error	0.353	0.367	0.388	0.403	0.728	0.787	0.856	0.893	3.992	4.112	4.632	5.228
The second type of error	0.197	0.201	0.219	0.227	0.415	0.479	0.511	0.585	2.479	2.883	3.099	3.947

.

The analysis results showed that the developed fuzzy controller demonstrates the greatest resistance to noise, providing the first and second types errors’ minimal values at all levels of σ. Thus, at σ = 0.05, the first type error was 0.403, which is significantly lower than the values for the controller from [67] (0.893) and the PID controller (5.228). Similarly, the second type error for the developed controller was 0.227, which is also significantly less than that of the controller from [67] (0.585) and the PID controller (3.947). These results indicate high stability and lower sensitivity of the developed controller to disturbances, which allows more accurate control under noisy conditions.

4. Discussion

4.1. Evaluation of Results

This article is further research [67,76] in the fuzzy and neuro-fuzzy controllers development field for controlling the helicopter TE thermogas-dynamic parameters, including the gas-generator rotor speed.

Based on previous research [67], the article develops a fuzzy controller mathematical model synthesising method using cognitive computing and a genetic algorithm for automated tuning and adaptation to changing environmental conditions, consisting of 12 stages (see the “Materials and Methods” Section).

The method proposed for the fuzzy controller synthesising combines genetic algorithms and cognitive computing to adjust the controller parameters, allowing it to be adapted to the control systems’ specific requirements. In the first stage, a control objects’ mathematical model is created, and using classical methods, such as the Ziegler–Nichols method, the controllers’ proportional, integral and differential components coefficients are adjusted. This approach provides an initial setting that makes the system stable and minimises oscillations (overshoot does not exceed 1.3%, and the transient process time does not exceed 0.032 s), which is critical for dynamic systems with high requirements for accuracy and response speed (see Figure 5).

The developed method includes creating a database used for the fuzzy controller synthesis. Based on the training sample, the structure of the controller is determined, where the errors and their derivative parameters are fuzzified to the deviations, and their rate levels are reflected. Fuzzification represents the input and output variables as terms, such as “negative”, “zero”, “positive”, etc. (8)–(10), which simplifies the development of logical rules (see Figure 4). A modified centre of gravity method is used at the defuzzification stage, where each category’s significance is considered (13). For the fuzzy controllers’ final tuning and testing, an analysis is carried out based on stability criteria, such as the Nichols criterion. During testing, key response indicators such as time settling, overshoot, and slew rate are measured, which helps adjust the term boundaries in fuzzification. The developed method differs from the fuzzy controllers’ tuning traditional approach with a genetic algorithm [59] due to the integration of cognitive components and machine learning methods for adaptive control, which allows the controller to automatically update rules and parameters based on error analysis in real-time.

A computational experiment was conducted to determine the helicopter TE gas-generator rotor speed transient process based on the data obtained in actual conditions during a helicopter flight (see Table 1). An example of adjusting the fuzzy controller parameters using a genetic algorithm is given (see Table 7 and Table 8). It is shown that the genetic algorithm used in changing the fuzzy controllers’ parameters made it possible to increase the integral quality criterion from 85.36% (0.8536) to 98.19% (0.9819) (see Table 9), which improves the control efficiency under dynamic conditions by 12.83%.

The computational experiments’ results showed that the fuzzy controller with genetic algorithm tuning use led to an improvement in the control quality indicators for the helicopter TE gas-generator rotor speed parameter compared to the fuzzy controller developed in [67] and the traditional linear PID controller from 1.51 to 2.06 times. The fuzzy controller with genetic algorithm tuning used for the helicopter TE gas-generator rotor speed made it possible to reduce the first and second types of errors by 2.06…2.11 times compared to the fuzzy controller [67] and by 11.31…12.58 times compared to the traditional PID controller.

4.2. Limitations and Future Research

It is noted that the research has limitations related to the specific conditions in which the experiment was conducted. In particular, the results obtained based on actual helicopter flight data may not fully reflect the system behaviour in various operating conditions and operating ranges. In addition, the proposed fuzzy controller tuning method requires high-quality training datasets, which may limit its applicability in situations with insufficient data or information’ incomplete sets on the system characteristics. It should be noted that the controllers’ successful adaptation and training also depend on the correctness of the genetic algorithms’ selected parameters. It may require additional research and calibration in the future.

Also, this research’s limitations are related to the limited training datasets required for fine-tuning the fuzzy controller, which may limit its applicability in systems with insufficient data or incomplete information about the control object characteristics. In addition, the proposed method requires correct tuning of the genetic algorithm, which may require additional research and calibration in the future. It is essential to take into account that the experimental results based on actual helicopter flight data may not fully reflect the system’s behaviour in various operating conditions and operating ranges, as noted in [89,90].

In addition, there is a limitation associated with the method’s scalability for multivariate systems since an increase in the input and output variables can significantly complicate the tuning and improve the controller efficiency process. It is also worth noting the computational complexity that may arise when applying the proposed method to real-time applications that require fast decision-making and minimal response time, which may require additional optimisation of algorithms and computing resources.

Future research will be aimed at expanding the developed method application to control other dynamic systems [91,92,93,94,95], not limited to monitoring the helicopter TE thermogas-dynamic parameters in flight mode. One of the promising areas is the database structure optimisation to improve the training efficiency and more complex machine learning methods introduction that can improve the controllers’ adaptability. An important step will be the various environmental parameters that influence the study on the control systems performance, as well as the new criteria development for assessing the control quality, which will improve the fuzzy and neuro-fuzzy controller’s stability in various operating conditions.

For a more detailed analysis, a comparison with modern adaptive control methods, such as particle swarm optimisation (PSO), model predictive control (MPC), reinforcement training, and hybrid neuro-fuzzy controllers will be conducted. This research will allow us to more accurately identify the strengths and weaknesses of the proposed method in contexts other than popular approaches [96,97]. For example, PSO and MPC methods can offer high accuracy in real-time predictions and controls but may require significant computational resources and difficulty in tuning. In turn, reinforcement training can significantly improve the controllers’ adaptability, but it requires a long time of training [98]. Hybrid neuro-fuzzy controllers, combining the neural networks and fuzzy logic capabilities, are able to quickly adapt to changing conditions but may suffer from difficulty in training and the need for fine-tuning [99,100]. The comparison of these methods with the developed approach will allow us to identify optimal directions for further improvement and controller adaptation in various applications.

To eliminate these limitations and expand the application of the developed method, the following roadmap for future research can be proposed (

Table 12. The proposed roadmap for future research.

Limitation	Aim	Steps
The method’s scalability for multidimensional systems	Develop methods that will allow the approach to be effectively scaled to systems with input and output variables in large numbers.	Explore methods for system decomposition and the multi-group genetic algorithms used for multiple submodels’ parallel optimisation.
		Develop algorithms to reduce computational costs as the variables increase in number, such as dimensionality reduction methods or the subspaces used.
		Conduct experiments with large multidimensional systems, such as robotic systems or systems with multiple sensors and actuators.
Optimising computational complexity for real-time applications	Reduce the computational load when using the method in real-time, where minimal latency is essential.	Apply accelerated data processing methods, such as using optimisation algorithms with fast computation (for example, the stochastic gradient descent method).
		Use parallel computing and optimisation of work with computing resources at the processors or graphics accelerators (GPU) level.
		Conduct tests on real systems with a high refresh rate to evaluate response time and tune the method to real-time requirements.
Database structure optimisation and training methods improvement	Improve the controller training efficiency by optimising the data structure and implementing new machine learning methods.	Apply active training techniques to reduce the need for large amounts of training data.
		Explore the deep neural networks used for more complex models and adaptive training methods such as reinforcement training.
		Develop approaches to optimising the database structure to minimise data redundancy and improve training speed.
Research into the various external parameters that influence the control system performance	To evaluate how various external and system parameters affect the controllers’ stability and efficiency under different operating conditions.	Conduct experiments simulating various real-world conditions series, including noise, data uncertainties, and environmental changes.
		Develop and implement adaptive methods that can adjust the controllers’ behaviour based on changing environmental conditions.
Comparison with modern adaptive control methods (PSO, MPC, reinforcement training, hybrid neuro-fuzzy controllers)	To evaluate the advantages and disadvantages of the proposed method in comparison with other modern control methods.	Conduct thorough comparative research with PSO, MPC, reinforcement training, and hybrid neuro-fuzzy controllers on key criteria: accuracy, adaptation speed, computational load, and robustness.
		Determine in applications which types and operating conditions each approach is most effective, and identify opportunities to combine methods to improve results.
		Develop hybrid methods that leverage each approach’s strengths, such as combining MPC with neuro-fuzzy methods for prediction and adaptation.
Long-term adaptation and calibration of genetic algorithms	Improve the accuracy and adaptability of genetic algorithms used to tune controllers.	Study the genetic algorithms’ various parameters impact, such as population size, mutation and crossover probability, on the system performance.
		Develop methods for automatic calibration of genetic algorithms using self-learning approaches or meta-learning to optimise these parameters.
		Conduct iterative studies to improve the adaptation of genetic algorithms in real-world applications.

).

5. Conclusions

A fuzzy controller mathematical model synthesising method has been developed, which uses cognitive computing and a genetic algorithm for automated adjustment and adaptation to changing environmental conditions. The method includes 12 stages, from the creation of the control object mathematical model to the final testing of the controllers, which ensures high system stability (overshoot does not exceed 1.104%, and the transient process time does not exceed 0.319 s) and minimises fluctuations in dynamic conditions.

The developed method combines genetic algorithms and cognitive computing to adjust the controller parameters, allowing the system to effectively adapt to specific control requirements (the integral quality criterion is 98.19%). The classical methods used, including the Ziegler–Nichols method, at the adjustment stage ensure the systems’ stability (the first type errors do not exceed 0.728%, and the second type errors do not exceed 0.415%), which is critical for the helicopter TE thermogas-dynamic parameters controlling in helicopter flight mode.

The computational experiments’ results showed improved control quality indicators: the integral quality criterion increased from 85.36 to 98.19%, corresponding to an increase in control efficiency under dynamic conditions by 12.83%. The fuzzy controller with genetic algorithm tuning use made it possible to reduce the first and second types of errors by 2.06…2.11 times compared to the previously proposed fuzzy controller and by 11.31…12.58 times compared to the traditional PID controller, which confirms the developed methods’ effectiveness.

Future research will be aimed at expanding the application of the developed method to control other dynamic systems in addition to monitoring the helicopter TE thermogas-dynamic parameters. Key areas will be database structure optimisation to improve training efficiency and the introduction of more complex machine learning methods that would improve controllers’ adaptability. The various environmental parameters that influence the control systems’ performance will also be studied, and new criteria for assessing the control quality will be developed. A critical stage will involve a comparison with modern adaptive control methods, such as particle swarm optimisation (PSO), model predictive control (MPC), reinforcement training, and hybrid neuro-fuzzy controllers, which will reveal the relative strengths and weaknesses of each approach, as well as determine the most effective ways to improve the controllers’ stability and adaptability under various conditions.