Neural Network Method of Controllers’ Parametric Optimization with Variable Structure and Semi-Permanent Integration Based on the Computation of Second-Order Sensitivity Functions

Abstract

This article presents a method for researching processes in automatic control systems based on the operator approach for modelling the control object and the controller. Within the method framework, a system of equations has been developed that describes the relations between the control error, the reference and control action, the output coordinate and the controller and the control object operators. The traditional PI controller modification, including a switching function for adaptation to operating conditions, allows for the system’s effective control in real time. The controller optimization algorithm is based on a functional expression with weighting coefficients that take into account control errors and the control action. To train the neural network through implementing the proposed method, a multilayer architecture was used, including nonlinear activation functions and a dynamic training rate, which ensure high accuracy and accelerated convergence. The TV3-117 turboshaft engine was chosen as the research object, which allows the method to be demonstrated in practical applications in aviation technology. The experimental results showed a significant improvement in control characteristics, including a reduction in the gas-generator rotor speed parameter transient time to ≈1, which is two times faster than the traditional method, where the transient process reaches ≈0.5. The model achieved a maximum accuracy of 0.993 with 160 training epochs, minimizing the error function to 0.005. In comparison with similar approaches, the proposed method demonstrated better results in accuracy and training speed, which was confirmed by a reduction in the number of iterations by 1.36 times and an improvement in the mean square error by 1.86–6.02 times.

1. Introduction and Related Works

Modern automatic control systems (ACSs) [1,2] are becoming increasingly complex due to increasing requirements for their reliability, accuracy and adaptability. One of the key tasks is to optimize the controllers’ parameters to ensure the transient processes’ stability under varying external conditions. Of particular interest are controllers with a variable structure and semi-permanent integration [3,4], which allow the system parameters’ flexible adaptation depending on its current state.

Modern approaches [5,6,7] to the parametric optimization of variable-structure controllers are based on the analysis and numerical calculations of the mathematical methods used. Key areas of research are gradient optimization methods [8,9], approaches based on sensitivity theory [10,11] and the optimal control methods used [12,13]. For example, in [14,15], the research is devoted to the use of gradient algorithms, and attention is primarily paid to the computation of the quality functional gradient in ACSs using first-order sensitivity. The key drawbacks of modern approaches to parametric optimization are the first-order sensitivity limiting its applications, which reduces the tuning accuracy in systems with pronounced nonlinearity [14,15]. In addition, gradient methods often require significant computational resources, especially when working with high-dimensional systems, which limits their practical implementation [8,16]. Optimal control and sensitivity theory methods also face scalability problems and difficulties in being integrated with real-time adaptive controllers [10,11,12,13].

The indicated shortcomings [5,6,7,8,9,10,11,12,13,14,15,16] are eliminated in methods based on the computation of second-order sensitivity functions. Methods based on the computation of second-order sensitivity functions [16,17,18] allow for increasing the parametric tuning accuracy. Methods based on the computation of second-order sensitivity provide more accurate parametric tuning by taking into account the interaction between parameters and their influence on system changes [16,17]. These methods take into account changes in the system, not only depending on the current state but also on the influence of parameter interactions [16,17]. However, such approaches are often difficult to implement due to the high computational load and the need to use specialized software [16,18]. This limitation can be mitigated by using neural network technologies capable of efficiently processing data in large amounts. However, the need for specialized software tools makes integration with modern neural network architectures difficult.

Therefore, a significant direction is the development of an adaptive neural network controller, which changes the parameters that are capable of changing in real time depending on the control object dynamics [19,20,21,22,23,24]. However, most existing approaches [19,20] are limited to using static switching functions, which do not allow for the system’s nonlinear behaviour to be fully considered. This reduces adaptation efficiency in complex transient conditions [20] or causes extensive time delays [21]. Known neural network controllers are developed based on the following neural network architectures: RBF networks [20], Jordan neural networks [21], Elman neural networks [22], Hopfield neural networks [23] and Hamming neural networks [24]. Known neural network controllers’ key drawback is their limited ability to integrate sensitivity functions that are sensitive to changes in the control object’s dynamics. For example, RBF networks do not adapt effectively enough to rapidly changing conditions, and Elman and Jordan neural networks require significant computing resources to account for complex nonlinear dynamics in real time. Thus, the issue of integrating sensitivity functions with modern neural network technologies has not been sufficiently overcome.

Despite the success in the application of second-order sensitivity methods, the task of developing algorithms that would be researched effectively under extensive delay conditions, strong nonlinearities and high uncertainty remains relevant. In addition, most research [5,8,9,10,13,14,16,19,20] is aimed at analyzing stable control objects, which limits the application of existing methods to systems with changing characteristics.

Another unsolved problem is the selection and optimization of the controller switching function. The influence of different function options on the quality of transient processes has not been researched sufficiently [6,7,12,15,17], which limits the controllers’ use in dynamically changing conditions.

The search-free method for a controller’s parametric adaptation described in [25,26,27] has a number of advantages, including the use of controller adaptive parameters, which improves the system’s tuning accuracy and takes into account the interactions between the parameters. Regularization and gradient procedures with the inclusion of a dynamic training rate make this method especially useful for systems with variable structures. The main disadvantage of the search-free method [25,26,27,28] is its limited ability to take into account strong nonlinearities and a control object’s extensive time delay. The high computational complexity associated with the analytical calculation of second-order sensitivity functions complicates its application to systems with a variable structure [5,6] and dynamically changing characteristics [7].

Table 1. The developed algorithm for the proposed neural network training.

Approach	Advantages	Disadvantages	Studies
Gradient optimization methods	High accuracy in calculating the gradient quality functional in ACSs using first-order sensitivity.	Limited applications in systems with pronounced nonlinearity, high computational complexity in multidimensional systems.	[14,15]
Methods of sensitivity theory	Takes into account parameter interactions and their influence on changes in the system by calculating second-order sensitivity functions.	High computational load, implementation complexity, need for specialized software.	[16,17,18]
Optimal control methods	Effective for stable control objects; they take into account control errors and the control signal impact.	Scalability issues, difficulties in integration with real-time adaptive controllers.	[10,11,12,13]
Neural networks with fixed switching functions	Ability to process large amounts of data and change parameters in real time.	Limited ability to take into account nonlinearities and changes in the dynamics of the control object.	[19,20,21,22,23,24]
Controllers with variable structure and semi-permanent integration	Flexible adaptation of system parameters depending on the current state; improved tuning accuracy.	Complexity in setting up switching functions, high computational complexity in analytical calculation of second-order sensitivity functions.	[3,4]

provides a comparison of the approaches considered.

To improve the search-free method [25,26,27,28], it is necessary to integrate neural network approaches for automated modelling and the numerical computation of second-order sensitivity functions, which will reduce the computational complexity and increase accuracy. It is vital to take into account the nonlinearities and the control object’s delays by implementing adaptive switching functions in the controller operator. It is also necessary to automate the adaptation of controller parameters using error backpropagation methods modified for dynamic systems and implement a dynamic training rate mechanism to accelerate convergence. To improve the optimization stability, regularization should be included in the computation of the Hessian matrix and the optimality criterion, which will prevent overtraining and enhance system reliability in uncertain and noisy environments.

Thus, this research aims to develop a neural network method for variable-structure controllers with semi-permanent integration parametric optimization, taking into account nonlinearities, delays and high uncertainty, to improve the accuracy and stability of the ACS. The research object is an ACS with variable-structure controllers and semi-permanent integration. The research subject is parametric optimization methods for controllers based on calculating second-order sensitivity functions and using neural network approaches to take into account the control object’s nonlinear and dynamic characteristics. This research’s main contribution is the development of a neural network method for variable-structure controllers with the semi-permanent integration of parametric optimization, which takes into account nonlinearities, delays and high uncertainty. This allows for improving the ACS’s accuracy and stability, increasing its adaptability in the face of complex transient processes and objects’ dynamically changing characteristics.

2. Materials and Methods

According to [25,26,27,28], delays are often encountered in industrial processes such as transportation and substance mixing. Provided that the ratio ${\displaystyle \frac{{\tau }_{ob}}{T_{ob}^{\left(m\right)}}}>1$ (where τ_ob is the delay time and $T_{ob}^{\left(m\right)}$ is the object’s maximum time constant), standard controllers such as P, PI and PID do not provide the required quality in transient processes. One of the control methods involves PI controllers with a delay, which can be used in the structure. Semi-constant integration in the PI controller is also used, which allows for avoiding an increase in the number of adjustable parameters. These controllers with a variable structure are difficult to adjust, which requires approximate methods, reducing their efficiency. In this regard, algorithmic tuning methods are necessary. Positive results from the gradient algorithm applied based on sensitivity theory [10,11] to calculate the proportional controller’s adjustable parameters confirm its applicability for solving the parametric optimization issue, which consists of minimizing a given optimization criterion. The parametric optimization issue is to find values for the controller’s adjustable parameters, vector θ = (θ₁, θ₂), so that the minimum optimization criterion is met. Thus, according to [10,11,25,26,27,28], the processes occurring in the system are described by the following equation:

$$\epsilon (t, \overline{\theta })=r(t)-x(t); u(t)=G_c(s, \overline{\theta }) · \epsilon (t, \overline{\theta }); x\left(t\right)=G_p\left(p\right) · u\left(t\right),$$

(1)

where ε(t, $\overline{\theta }$) is the control system error; r(t) is the reference action; u(t) is the control action; x(t) is the ACS output coordinate; G_c(s, $\overline{\theta }$) is the controller operator; G_p(s) is the control object operator; and $s={\displaystyle \frac{d}{dt}}$ is the differentiation operator (according to [25,28]).

The control object operator, G_p(s), in this research is chosen in such a form that it is possible to describe the most complex dynamic object processes:

$$G_p\left(s\right)={\displaystyle \frac{k_{ob}}{\left(T_{ob}^{\left(1\right)}·s+1\right)·\left(T_{ob}^{\left(2\right)}·s+1\right)}}·e^{-{\tau }_{ob}·s}+f_{nonline}\left(x,u\right),$$

(2)

where k_ob is the static gain; $T_{ob}^{\left(1\right)}$ and $T_{ob}^{\left(2\right)}$ are the time constants; τ_ob is the delay time, and f_nonline(x, u) is the nonlinear part depending on the system’s state.

The proposed controller operator G_c(s, $\overline{\theta }$), which is a traditional PI controller extension including a switching function, has the form:

$$G_c\left(s\right)=k_p·\left(1+{\displaystyle \frac{1}{T_i·s}}\right)·g\left(t,\overline{\theta }\right),$$

(3)

where k_p is the proportionality coefficient that determines the controller’s response intensity to a deviation from the set value, T_i is the integration time that characterizes the influence of previous errors on the controller’s action, and g(t, $\overline{\theta }$) is the switching function that depends on time t and the parameter $\overline{\theta }$, which is responsible for the controller’s dynamic adaptation.

The function g(t, $\overline{\theta }$) regulates changes in the controller’s behaviour depending on the system’s current operating conditions, based on [25], and it is represented by one of the following possible expressions:

$$g\left(t,\overline{\theta }\right)=\left\{\begin{array}{c}{k}_1,\epsilon \left(t,\overline{\theta }\right)·\dot{\epsilon }\left(t,\overline{\theta }\right)\ge 0,\\ k_2·e^{-\alpha ·t}+k_3,\epsilon \left(t,\overline{\theta }\right)·\dot{\epsilon }\left(t,\overline{\theta }\right)<0,\end{array}\right.$$

(4)

where k₁, k₂, k₃, and α are adjustable parameters depending on θ.

The switching function g(t) in (4) is chosen to ensure the adaptive behaviour of the controller in a dynamically changing control environment [8], which is confirmed by both theoretical and empirical considerations. On the one hand, the presence of adjustable parameters (k₁, k₂, k₃, α) allows for fine-tuning the controller’s gain, where the exponential factor $e^{-\alpha ·t}$ reduces the influence of sharp disturbances, preventing overgain and oscillations, which ensures an optimal balance between the system’s speed and stability. On the other hand, empirical data on the gas-generator part of the TV3-117 engine operation’s modelling [13] demonstrate that the use of g(t) allows for reducing the transient process time by half compared to traditional methods and improves the optimization algorithm’s convergence, reducing the number iterations and minimizing the loss function to MSE ≈ 0.005. Experimental research [8,13,25,26,27,28] confirms that the function’s selected form ensures the controller’s reliable adaptation in real time, the system parameters’ correct restoration, and a smoother and more predictable transient process, which is confirmed by the Hessian matrix’s positive definiteness and the stability of the system’s dynamics.

The object operator parameters’ fundamental values (2) are taken, taking into account the significant delay condition, i.e., ${\displaystyle \frac{{\tau }_{ob}}{T_{ob}^{\left(m\right)}}}>1$.

Taking into account the expression that determines the control action, u(t), values, similarly to [25], the analytical expression describing u(t) is presented as follows:

$$u\left(t\right)=\left\{\begin{array}{c}{\theta }_1·\epsilon \left(t,\overline{\theta }\right)+{\displaystyle \frac{{\theta }_2}{s}}·\epsilon \left(t,\overline{\theta }\right),g\left(t,\epsilon \left(t,\overline{\theta }\right),\dot{\epsilon }\left(t,\overline{\theta }\right)\right)=1,\\ {\theta }_1·\epsilon \left(t,\overline{\theta }\right)+{\displaystyle \frac{{\theta }_2}{s}}·0\left(t\right),g\left(t,\epsilon \left(t,\overline{\theta }\right),\dot{\epsilon }\left(t,\overline{\theta }\right)\right)=0.\end{array}\right.$$

(5)

Similarly to [25], this research sets the task of investigating the influence of the different variants of the switching function (4) in the controller (5) on the transient processes’ quality. Figure 1 shows the controller’s structural diagram under research [25].

As a criterion for optimizing the automatic parametric optimization-generated algorithm, based on [29], the functionality widely used in the practice of automatic regulation was selected:

$$J=\underset{0}{\overset{\infty }{\int }}\left(a_1·{\epsilon }^2\left(t,\overline{\theta }\right)+a_2·u^2\left(t\right)\right)dt,$$

(6)

where a₁ and a₂ are weighting coefficients.

In the formation of the automatic parametric optimization algorithm in this research, sensitivity theory methods [10,11] for a system’s continuous class were used [18,19] since the control action, u(t), discontinuity point’s coordinate t_m = (1, 2, …, m) does not depend on $\overline{\theta }$ [21,25,26].

Similarly to [25,26,27,28], in this research, a model was obtained for the computation of the controller’s adjustable parameters’ (5) sensitivity functions:

$${\xi }_i\left(t\right)=G_p\left(s\right)·{\displaystyle \frac{\partial u\left(t\right)}{\partial {\theta }_i}}, i=\mathrm{1,2},$$

(7)

$$\begin{gathered} {\displaystyle \frac{\partial u\left(t\right)}{\partial {\theta }_i}}={\displaystyle \frac{\partial G_c\left(s,\overline{q}\right)}{\partial {\theta }_i}}·\epsilon \left(t,\overline{\theta }\right)-{\xi }_i\left(t\right)·G_c\left(s,\overline{\theta }\right); \\ {\displaystyle \frac{\partial u\left(t\right)}{\partial {\theta }_1}}=\epsilon \left(t,\overline{\theta }\right)-{\xi }_i\left(t\right)·G_c\left(s,\overline{\theta }\right); \end{gathered}$$

(8)

$${\displaystyle \frac{\partial u\left(t\right)}{\partial {\theta }_2}}=\left\{\begin{array}{c}{\displaystyle \frac{1}{p}}·\epsilon \left(t,\overline{\theta }\right)-{\xi }_2\left(t\right)·G_c\left(s,\overline{\theta }\right),g\left(t,\epsilon \left(t,\overline{\theta }\right),\dot{\epsilon }\left(t,\overline{\theta }\right)\right)=1,\\ {\displaystyle \frac{1}{p}}·0\left(t\right)-{\xi }_2\left(t\right)·G_c\left(s,\overline{\theta }\right),g\left(t,\epsilon \left(t,\overline{\theta }\right),\dot{\epsilon }\left(t,\overline{\theta }\right)\right)=0.\end{array}\right.$$

(9)

During the optimization process, the adjustable parameter vector $\overline{\theta }$ is changed in accordance with the gradient procedure [25] (Figure 2):

$$\overline{\theta }\left[l\right]=\overline{\theta }\left[l-1\right]-{\eta }_l·I·{\nabla }_{\overline{\theta }}I\left(\epsilon \left(t,\overline{\theta }\left[l-1\right]\right)\right), l=\mathrm{1,2},\dots ,$$

(10)

where Γ = diag(γ) = {δ_ij, γ_ij} is the weight coefficient diagonal matrix; δ_ij is the Kronecker delta; ${\nabla }_{\overline{\theta }}I\left(\epsilon \left(t,\overline{\theta }\left[l-1\right]\right)\right)$ is the gradient vector (6); and η_l is the dynamic training rate depending on the change in J. The definition of the gradient vector ${\nabla }_{\overline{\theta }}I\left(\epsilon \left(t,\overline{\theta }\left[l-1\right]\right)\right)$ in this research is associated with the computation of the sensitivity function vector Ξ for the variable-structure controller under consideration:

$$\begin{gathered} \Xi =\left({\xi }_1\right(t),{\xi }_2(t\left)\right), \\ {\displaystyle \frac{\partial \mathsf{\Xi }}{\partial t}}=\left({\displaystyle \frac{\partial {\xi }_1\left(t\right)}{\partial {\overline{\theta }}_1}},{\displaystyle \frac{\partial {\xi }_2\left(t\right)}{\partial {\overline{\theta }}_2}}\right), \\ {\displaystyle \frac{{\partial }^2\mathsf{\Xi }}{\partial t^2}}=\left({\displaystyle \frac{\partial {\xi }_1^2\left(t\right)}{\partial {\overline{\theta }}_1^2}},{\displaystyle \frac{\partial {\xi }_2^2\left(t\right)}{\partial {\overline{\theta }}_2^2}}\right). \end{gathered}$$

(11)

The implementation of the automatic parametric optimization-generated algorithm on a computer requires stopping the formation of the optimization process; in [25,28,30,31], the following condition was applied and tested for the automatic parametric optimization algorithms based on sensitivity theory:

$$S_j\left[l\right]=S_J\left[l+1\right]+\left\{\begin{array}{c}1 \mathrm{at} ∆J\left[l\right]\times ∆J\left[l-1\right]<0,\\ 0 \mathrm{at} ∆J\left[l\right]\times ∆J\left[l-1\right]>0,\end{array}\right.$$

(12)

$$\begin{gathered} S_{\partial J_j}\left[l\right]=S_{\partial J_j}\left[l+1\right]+\left\{\begin{array}{c}1 \mathrm{at} {\displaystyle \frac{\partial J\left({\overline{\theta }}^l\right)}{\partial {\overline{\theta }}_j}}\times {\displaystyle \frac{\partial J\left({\overline{\theta }}^{l-1}\right)}{\partial {\overline{\theta }}_j}}<0,\\ 0 \mathrm{at} {\displaystyle \frac{\partial J\left({\overline{\theta }}^l\right)}{\partial {\overline{\theta }}_j}}\times {\displaystyle \frac{\partial J\left({\overline{\theta }}^{l-1}\right)}{\partial {\overline{\theta }}_j}}>0,\end{array}\right. \\ j=\overline{\mathrm{1,2}}. \end{gathered}$$

(13)

where $∆J\left[l\right]=J\left({\overline{\theta }}^l\right)-J\left({\overline{\theta }}^{l-1}\right)$ is the difference between the optimality criteria in the next two iterations.

The automatic parametric optimization-generated algorithm for computation-adjustable parameters is considered complete when the following condition is met:

$$\left(S_J\left[l\right]\ge n_{SJ}\right)\vee \left(\left(S_{\partial J_1}\ge n_{\partial J_1}\left[l\right]\right)\wedge \left(S_{\partial J_2}\ge n_{\partial J_2}\left[l\right]\right)\right),$$

(14)

where n_SJ, $n_{\partial J_1}$, and $n_{\partial J_2}$ are given positive values characterizing the corresponding number of changes in the sign of the optimality criterion increments and the gradient components of each sign. The specific values of the parameters n_SJ, $n_{\partial J_1}$, and $n_{\partial J_2}$ were obtained as a preliminary research result.

For the automatic parametric optimization-generated algorithm, in the next stage, the analytical dependencies were determined to achieve the research aim of checking the reliability of the adjustable parameters ${\overline{\theta }}^{*}$ computed based on criterion (6) for the minimum value.

In this research, the Hessian matrix $H\left(J\right)={\displaystyle \frac{{\partial }^{2}J}{\partial {\overline{\theta }}^2}}+\lambda ·J$ was calculated for this aim, where λ is the regularization coefficient, and the computational technique from [25,26,27,28] is additionally applied. As is known, a positive definite matrix, H(J), at the point ${\overline{\theta }}^{*}$ indicates that the minimum for criterion (6) has been found:

$$H\left(J\right)=\left[\begin{array}{cc}{\displaystyle \frac{{\partial }^{2}J}{\partial {\theta }_1^2}}+\lambda ·J& {\displaystyle \frac{{\partial }^{2}J}{\partial {\theta }_1\partial {\theta }_2}}+\lambda ·J\\ {\displaystyle \frac{{\partial }^{2}J}{\partial {\theta }_2\partial {\theta }_1}}+\lambda ·J& {\displaystyle \frac{{\partial }^{2}J}{\partial {\theta }_2^2}}+\lambda ·J\end{array}\right]=\left[{\displaystyle \frac{{\partial }^{2}J}{\partial {\theta }_i\partial {\theta }_j}}+\lambda ·J\right], \left(i,j=\overline{\mathrm{1,2}}\right).$$

(15)

Sylvester’s criterion for a positive definite matrix (15) is presented as follows:

$${\displaystyle \frac{{\partial }^{2}J}{\partial {\theta }_1^2}}+\lambda ·J>0, {\displaystyle \frac{{\partial }^{2}J}{\partial {\theta }_2^2}}+\lambda ·J>0, \mathrm{d}\mathrm{e}\mathrm{t}\left(H\left(J\right)\right)>0.$$

(16)

The definition of the elements in matrix (15) is based on the second-order sensitivity functions; that is,

$$\left[\begin{array}{cc}{\displaystyle \frac{\partial {\xi }_1\left(t\right)}{\partial {\theta }_1}}& {\displaystyle \frac{\partial {\xi }_1\left(t\right)}{\partial {\theta }_2}}\\ {\displaystyle \frac{\partial {\xi }_2\left(t\right)}{\partial {\theta }_1}}& {\displaystyle \frac{\partial {\xi }_2\left(t\right)}{\partial {\theta }_2}}\end{array}\right]=\left[{\xi }_{ij}\left(t\right)\right], \left(i,j=\overline{\mathrm{1,2}}\right).$$

(17)

The second-order sensitivity function characterizes a change’s effect on the change rate of another sensitivity function, allowing one to take into account the system parameters’ interactions under nonlinear dynamics [16,17]. This approach provides a more in-depth analysis of the parameters’ influence on the system’s behaviour than first-order methods and allows one to more accurately adjust the controller’s settings. This is especially important for systems with significant delays and complex nonlinearities, where traditional optimization methods may not be effective enough [16]. Empirical research has confirmed that the second-order sensitivity functions used lead to accelerated convergence of the optimization algorithm and increased system stability [18].

The analytical search for sensitivity functions (15) is a labour-intensive task [8,12,14]. Therefore, in this research, a more straightforward method based on the sensitivity model [16,17,18,25,26] was used. Within this research framework, similarly to [25], a second-order sensitivity function model was obtained for system (1) with controller (5):

$$\begin{gathered} {\xi }_{ij}\left(t\right)=G_p\left(s\right)·{\displaystyle \frac{{\partial }^{2}u\left(t\right)}{\partial {\theta }_i\partial {\theta }_j}}, \left(i,j=\overline{\mathrm{1,2}}\right), \\ {\displaystyle \frac{{\partial }^{2}u\left(t\right)}{\partial {\theta }_i\partial {\theta }_j}}={\displaystyle \frac{{\partial }^2{G}_c\left(s,\overline{\theta }\right)}{\partial {\theta }_i\partial {\theta }_j}}·\epsilon \left(t,\overline{\theta }\right)-{\displaystyle \frac{\partial G_c\left(s,\overline{\theta }\right)}{\partial {\theta }_i}}·{\xi }_j\left(t\right)-{\displaystyle \frac{\partial G_c\left(s,\overline{\theta }\right)}{\partial {\theta }_j}}·{\xi }_i\left(t\right)-G_c\left(s,\overline{\theta }\right)·{\xi }_{ij}\left(t\right), \\ {\displaystyle \frac{{\partial }^{2}u\left(t\right)}{\partial {\theta }_i\partial {\theta }_j}}=\left\{\begin{array}{c}-2·{\xi }_1\left(t\right)-{\displaystyle \frac{1}{s}}·{\xi }_{11}\left(t\right),g\left(t,\epsilon \left(t,\overline{\theta }\right),\dot{\epsilon }\left(t,\overline{\theta }\right)\right)=1,\\ -2·{\xi }_1\left(t\right)-{\displaystyle \frac{1}{s}}·0\left(t\right),g\left(t,\epsilon \left(t,\overline{\theta }\right),\dot{\epsilon }\left(t,\overline{\theta }\right)\right)=0,\end{array}\right. \\ {\displaystyle \frac{{\partial }^{2}u\left(t\right)}{\partial {\theta }_1\partial {\theta }_2}}=\left\{\begin{array}{c}-{\xi }_2\left(t\right)-{\displaystyle \frac{1}{s}}·{\xi }_1\left(t\right)-G_c\left(s,\overline{\theta }\right)·{\xi }_{12}\left(t\right),g\left(t,\epsilon \left(t,\overline{\theta }\right),\dot{\epsilon }\left(t,\overline{\theta }\right)\right)=1,\\ -{\xi }_2\left(t\right)-{\displaystyle \frac{1}{s}}·0\left(t\right)-G_c\left(s,\overline{\theta }\right)·{\xi }_{12}\left(t\right),g\left(t,\epsilon \left(t,\overline{\theta }\right),\dot{\epsilon }\left(t,\overline{\theta }\right)\right)=0,\end{array}\right. \\ {\displaystyle \frac{{\partial }^{2}u\left(t\right)}{\partial {\theta }_2\partial {\theta }_1}}=\left\{\begin{array}{c}-{\xi }_2\left(t\right)-{\displaystyle \frac{1}{s}}·{\xi }_1\left(t\right)-G_c\left(s,\overline{\theta }\right)·{\xi }_{21}\left(t\right),g\left(t,\epsilon \left(t,\overline{\theta }\right),\dot{\epsilon }\left(t,\overline{\theta }\right)\right)=1,\\ -{\xi }_2\left(t\right)-{\displaystyle \frac{1}{s}}·0\left(t\right)-G_c\left(s,\overline{\theta }\right)·{\xi }_{21}\left(t\right),g\left(t,\epsilon \left(t,\overline{\theta }\right),\dot{\epsilon }\left(t,\overline{\theta }\right)\right)=0,\end{array}\right. \\ {\displaystyle \frac{{\partial }^{2}u\left(t\right)}{\partial {\theta }_2\partial {\theta }_2}}=\left\{\begin{array}{c}-{\displaystyle \frac{2}{s}}·{\xi }_2\left(t\right)-G_c\left(s,\overline{\theta }\right)·{\xi }_{22}\left(t\right),g\left(t,\epsilon \left(t,\overline{\theta }\right),\dot{\epsilon }\left(t,\overline{\theta }\right)\right)=1,\\ -{\displaystyle \frac{2}{s}}·0\left(t\right)-G_c\left(s,\overline{\theta }\right)·{\xi }_{22}\left(t\right),g\left(t,\epsilon \left(t,\overline{\theta }\right),\dot{\epsilon }\left(t,\overline{\theta }\right)\right)=0.\end{array}\right. \end{gathered}$$

(18)

The matrix H(J) for model (17) according to criteria (6) is presented similarly to in [25]:

$$H\left(J\right)=\left[\begin{array}{cc}\lambda ·J+2·\underset{0}{\overset{\infty }{\int }}\left({\xi }_1^2\left(t\right)-{\xi }_{11}\left(t\right)·\epsilon \left(t,\overline{\theta }\right)\right)dt& \lambda ·J+2·\underset{0}{\overset{\infty }{\int }}\left({\xi }_1\left(t\right)·{\xi }_2\left(t\right)-{\xi }_{12}\left(t\right)·\epsilon \left(t,\overline{\theta }\right)\right)dt\\ \lambda ·J+2·\underset{0}{\overset{\infty }{\int }}\left({\xi }_1\left(t\right)·{\xi }_2\left(t\right)-{\xi }_{21}\left(t\right)·\epsilon \left(t,\overline{\theta }\right)\right)dt& \lambda ·J+2·\underset{0}{\overset{\infty }{\int }}\left({\xi }_2^2\left(t\right)-{\xi }_{22}\left(t\right)·\epsilon \left(t,\overline{\theta }\right)\right)dt\end{array}\right]$$

(19)

The optimization process is complete if the following conditions are met:

$$\left|I^{\left(l\right)}-I^{\left(l-1\right)}\right|<ϵ \mathrm{a}\mathrm{n}\mathrm{d} n_{SJ}\le N,$$

(20)

where ϵ is the permissible error and n_SJ is the gradient sign change number.

As noted above, in addition to testing the automatic parametric optimization-generated algorithm’s performance and, accordingly, testing the computation (19) correctness, a technique oriented towards a computational experiment was used. The idea behind this technique is that the automatic parametric optimization algorithm is launched with at least three different initial values for the adjustable parameters ${\overline{\theta }}_k^0=\left({\theta }_{1k}^0,{\theta }_{2k}^0\right)$ and $k=\overline{\mathrm{1,2},3}$; then, at the adjustable parameters’ space points ${\overline{\theta }}_k^{*}=\left({\theta }_{1k}^{*},{\theta }_{2k}^{*}\right)$, the necessary condition for the optimality criterion extremum for ${\overline{\theta }}^{*}$ is checked:

$${\displaystyle \frac{\partial J\left({\overline{\theta }}^{*}\right)}{\partial \overline{\theta }}}=\left(0\pm ∆\right),$$

(21)

where Δ is the calculation error.

Additionally, to check the automatic parametric optimization-generated algorithm’s results reliability, the following condition is used:

$$\left|J\left({\overline{\theta }}^{*}\right)-J\left(\overline{\theta }\right)\right|<\delta , \forall i,j$$

(22)

for ${\overline{\theta }}^{*}$ within the range of adjustable parameters used in this research, where δ is the permissible error.

Based on the above, we summarize that, in this research, the automatic parametric optimization-generated algorithm’s performance indicator comprises criteria (16), (21), and (22) for the fulfilment for ${\overline{\theta }}^{*}$.

In the proposed algorithm, the controller’s parameters are updated according to (10): θ₍ₗ₊₁₎ = θ₍ₗ₎ − ηₗ · (W · ∇J), where the dynamic training rate, ηₗ, is adjusted based on the change in the optimization criterion, which allows for adapting the optimization step as it approaches the minimum of the function J (see (10)–(12)). The stopping conditions specified in (12)–(14) ensure that the iterative process is completed when the difference between the J values at successive steps becomes lower than a given threshold, which prevents overtraining. Regularization is used to stabilize the algorithm: adding the coefficient λ to the Hessian matrix, H(J) in (15), and checking for positive definiteness using Sylvester’s criterion (16) minimizes the influence of noise and nonlinearities. In addition, the choice of switching function, g(t) in (4), where the parameter α determines the exponential decay rate and the coefficients k₁, k₂, and k₃ regulate the gain level, allows one to precisely balance the controller’s speed and stability under dynamically changing control conditions.

To implement the proposed method in neural network form, an architecture is proposed (Figure 3), which includes consideration of adaptive parameters and system nonlinearities, as well as an algorithm for its training (

Table 2. The developed algorithm for the proposed neural network training.

Step Number	Step Name	Description
1	Network initialization	The network architecture consists of an input layer (x), three hidden layers (h1, h2, h3), and an output layer (y). The input layer takes two parameters: system error ϵ(t) and reference action r(t).
2	Weight initialization	The neural network parameters kp, Ti, k1, k2, k3, and α and other adaptive parameters are initialized randomly with a normal distribution (or using the Xavier/He method). In this case, the weight matrices for the hidden layers and the bias for each layer are initialized as $W_{ij}^{\left(l\right)}\sim N\left(0,{\displaystyle \frac{1}{\sqrt{n}}}\right), b^{\left(l\right)}=0.$
3	Forward pass	For each l-th hidden layer, an input linear combination is computed. For the first hidden layer (error handling):$z_1=W_1·\left[ϵ\left(t\right),r\left(t\right)\right]+b_1, {\alpha }_1=\mathrm{S}\mathrm{m}\mathrm{o}\mathrm{o}\mathrm{t}\mathrm{h}\mathrm{R}\mathrm{e}\mathrm{L}\mathrm{U}\left(z_1\right)$ For the second hidden layer (the regulatory effect computation): z2 = W2 ⋅ a1 + b2, a2 = tanh(z2). For the third hidden layer (switching parameters integration): z3 = W3 ⋅ a2 + b3, a3 = SmoothReLU(z3). For the output layer (the output coordinate compute): z4 = W4 ⋅ a3 + b4.
4	Error computation	The output error is computed as the difference between the prediction and the actual value, and a loss function, L, is applied, which could be, for example, MSE: $L={\displaystyle \frac{1}{n}}·{\displaystyle \sum _{i=1}^n}{\left(y_i-{x\left(t\right)}_i\right)}^2.$ To compute the gradients, the sensitivity function Ξ(t) is used, which takes into account the parameters’ influence on the system.
5	Backpropagation	Backpropagation involves computing gradients at each layer. For the output layer: ${\delta }_4={\displaystyle \frac{\partial L}{\partial y}}·{\sigma }^{’}\left(z_4\right),$ ${\sigma }^{’}\left(z_4\right)$ is the activation function’s derivative at the output. For the remaining layers, starting from the third, the error is calculated based on the activation function derivative: ${\delta }_3=\left(W_4^T·{\delta }_4\right)·{\sigma }^{’}\left(z_3\right), {\delta }_2=\left(W_3^T·{\delta }_3\right)·{\sigma }^{’}\left(z_2\right), {\delta }_1=\left(W_2^T·{\delta }_2\right)·{\sigma }^{’}\left(z_1\right).$
6	Updating weights with regularization	For each layer, the weights and biases are updated, taking into account the regularization λ. The gradients for the weights and biases are calculated as ${\displaystyle \frac{\partial L}{\partial W^{\left(l\right)}}}={\alpha }^{\left(l-1\right)}·{\left({\delta }^{\left(l\right)}\right)}^T+\lambda ·W^{\left(l\right)}, {\displaystyle \frac{\partial L}{\partial b^{\left(l\right)}}}={\delta }^{\left(l\right)}.$ The weights and biases are updated using the gradient descent equation: $W^{\left(l\right)}=W^{\left(l\right)}-{\eta }_l·{\displaystyle \frac{\partial L}{\partial W^{\left(l\right)}}}, b^{\left(l\right)}=b^{\left(l\right)}-{\eta }_l·{\displaystyle \frac{\partial L}{\partial b^{\left(l\right)}}},$ where ηl is the dynamic training rate, depending on the change in the optimization functional: ${\eta }_l={\displaystyle \frac{\eta }{1+\alpha ·\left|\mathsf{\Xi }\left(t\right)\right|}},$ where α is the training rate adaptation coefficient, and ∣Ξ(t)∣ is the change magnitude in the functional at the current step.
7	Parameter’s adaptation	Network parameters such as kp, Ti, k1, k2, k3, and α are trained via gradient descent, taking into account regularization and the sensitivity function Ξ(t). These parameters include both weights and adaptive coefficients that change based on changes in the loss function.
8	Training repetition	The training process is repeated for each batch of data or all training data, with weights and parameters updated at each step. It continues until stopping after the maximum number of epochs or the error improves.
9	Performance evaluation	Once training is complete, the network is tested on test data to assess its ability to handle new data by predicting output values given nonlinearities and delays in the system.

). The proposed neural network architecture (Figure 3) is built on a multilayer structure, including input, hidden and output layers. The input layer accepts two key parameters: the system error ϵ(t) and the reference action r(t), which the system’s dynamic behaviour determines. The hidden layers consist of three fully connected levels with nonlinear activation functions, such as SmoothReLU and tanh [32,33], to model complex nonlinear relations in the system.

The first hidden layer processes the error ϵ(t) using a nonlinear activation function, simulating f_nonline(x, u). The second hidden layer computes the control effect, u(t) = G_c(s) ⋅ ϵ(t). The third hidden layer is responsible for integrating the switching parameters (e.g., via the functions ϕ(t)). To ensure network adaptation, parameters such as k_p, T_i, k₁, k₂, k₃, and α are trained as neural network weights. They are optimized using the gradient descent method with regularization carried out by calculating gradients based on the sensitivity function, Ξ(t).

The output layer is designed to calculate the output coordinate x(t) = G_p(s) ⋅ u(t), which depends on the control action u(t) and is described by the control object operator G_p(s), taking into account the delay and nonlinearities. To stabilize the training process, a regularization coefficient λ is introduced, which improves the network’s resistance to overtraining and noise. Additionally, a dynamic training rate η_l is used, depending on the change in the optimization function, which accelerates convergence and improves the neural network’s accuracy.

3. Case Study

Based on the numerous research results devoted to helicopter turboshaft engine (TE) monitoring methods, including the research in [32,33,34,35,36,37,38], the TV3-117 engine, which is part of the Mi-8MTV helicopter power plant [39,40], was chosen as the research object for the computational experiment. The main diagnostic parameter recorded by the standard onboard sensor D-2M [32,41] is the gas-generator rotor speed, n_TC. The n_TC values were obtained exclusively from the Mi-8MTV helicopter flight test data. Data were collected for 256 s under actual flight conditions at the nominal engine operating mode with a sampling interval of 1 s. The base frequency value, n_TCreq, was set equal to 15,900 rpm [31,40]. The data were provided to the authors’ team upon an official request to the Ministry of Internal Affairs of Ukraine within the framework of the project “Theoretical and Applied Aspects of Aviation Sphere Development”, registered under number 0123U104884. The changes in the n_TC frequency recorded under the operating conditions of the TV3-117 engine demonstrate the time series complexity (Figure 4), and the constructed diagrams emphasize the importance of the current parameter values and their accumulation in the model’s memory [42,43]. In accordance with the recommendations provided in [32,33,34,35,36,37,38], the 256 n_TC values presented in Figure 4 were selected for analysis. The time sampling and adaptive quantization with a dynamic range applied to the reconstructed n_TC signal [33] resulted in its discretized form (points in Figure 4).

The generated diagrams (Figure 4) play an essential role in the formation of the training dataset. According to the Nyquist theorem, the sampling frequency should be at least twice the maximum frequency to reconstruct the signal accurately. With 256 measurements over 256 s, the sampling frequency is 1 measurement per second, which is optimal for tracking slowly changing parameters and provides sufficient detail without an excessive amount of data [32,33,34,35,36,37,38]. Thus, the choice of 256 measurements is a compromise between the sampling frequency, data amount, and computing resources, which allows for achieving a balance between data processing accuracy and efficiency.

To form a training dataset consisting of the system error, ϵ(t), and the reference action, r(t), values, z-normalization of the n_TC values was carried out [44], with the help of which each n_TC value was transformed in such a way that it was expressed in standard deviation units from the mean. This allows us to compile the n_TC values into a form with a mean value of 0 and a standard deviation of 1, which is essential for improving the developed neural network training algorithm’s (Table 2) convergence and for correct comparison of the data with different scales. The n_TC values’ z-normalization was carried out according to the following expression:

$${z\left(n_{TC}\right)}_i={\displaystyle \frac{n_{TC}^{\left(i\right)}-\frac{1}{N}·\sum _{i=1}^N{n}_{TC}^{\left(i\right)}}{\sqrt{\frac{1}{N}·\sum _{i=1}^N{\left(n_{TC}^{\left(i\right)}-\frac{1}{N}·\sum _{i=1}^N{n}_{TC}^{\left(i\right)}\right)}^2}}},$$

(23)

where N = 256.

To determine the system error, ϵ(t), for n_TC values normalized using z-normalization, the mean standard error was applied, calculated as follows:

$${ϵ}^{\left(i\right)}={\displaystyle \frac{{\sigma }_z}{\sqrt{N}}},$$

(24)

where ${\sigma }_z=\sqrt{{\displaystyle \frac{1}{N}}·\sum _{i=1}^N{\left({z\left(n_{TC}\right)}_i-\overline{z\left(n_{TC}\right)}\right)}^2}$ is the standard deviation of the z-normalized n_TC values, and $\overline{z\left(n_{TC}\right)}$ is the mean of the z-normalized n_TC values, typically $\overline{z\left(n_{TC}\right)}=0$ if the data are normalized correctly.

The reference action is the n_TC value in the helicopter TE’s current operating mode [42]. For example, in the emergency mode, n_TC = 97.4% or n_TC = 15,486 rpm; in the takeoff mode, n_TC = 96.3% or n_TC = 15,312 rpm; in the nominal mode, n_TC = 94.7% or n_TC = 15,057 rpm; in the first cruising mode, n_TC = 93.6% or n_TC = 14,882 rpm; and in the second cruising mode, n_TC = 91.7% or n_TC = 14,580 rpm. In this case, a deviation from the specified value of ±0.5% is allowed.

Since the Mi-8MTV helicopter’s flight tests were conducted in the nominal engine operating mode, the reference action is 256 values of n_TC = 15,057 rpm ±0.5%, where the deviation of ±0.5% was selected randomly for each i-th n_TC value. The n_TC values, as a reference action, were also normalized according to (23) and (24). Thus, the training dataset presented in

Table 3. The training dataset fragment.

Number	The System Error ϵ(t) Values	The Reference Action r(t) Values
1	0.011	0.913
…	…	…
46	0.012	0.918
…	…	…
115	0.015	0.933
…	…	…
173	0.013	0.926
…	…	…
209	0.011	0.911
…	…	…
256	0.010	0.908

is obtained.

Based on [32,33,34,35,36,37,38], the training dataset’s (Table 3) homogeneity was assessed using the Fisher–Pearson [45,46,47] and Fisher–Snedecor [48,49,50] tests at a significance level of α = 0.01 (

Table 4. The training dataset’s homogeneity, assessing results using the Fisher–Pearson and Fisher–Snedecor criteria.

Parameter	The Fisher–Pearson Criterion Meaning		The Fisher–Snedekor Criterion, Meaning		Description
	Calculated	Critical	Calculated	Critical
ϵ(t)	12.996	13.3	15.402	15.98	The Fisher–Pearson and Fisher–Snedecor criterium-computed values were below the thresholds, which made it possible to confirm the training dataset’s homogeneity.
r(t)	13.014		15.599

). The significance level of 0.01 was chosen to ensure a statistically valid conclusion about high reliability, which is especially important for helicopter flight safety. This strict criterion minimizes the first type of error probability, which contributes to the helicopter TE’s more accurate and safe control under operational conditions [51,52].

The training and test datasets’ representativeness was assessed using cluster analysis (k-means method) [53], which involved dividing the training dataset (Table 3) into k groups (clusters), where each value belongs to the cluster with the highest mean value [49,50]. The algorithm repeats the point distribution, minimizing the distance between them and their centroids until the clusters stabilize [54,55]. The cluster analysis of the training dataset (Table 3) identified eight clusters (I … VIII). The training and test datasets were randomly generated in a 2:1 ratio (67 and 33%, respectively), and the same 8 clusters were found in both sets, indicating their structural similarity. The distances between clusters in the training and test datasets were almost identical, confirming their similarity (Figure 5).

Thus, the optimal size of the training dataset is 256 elements (100%), that of the control dataset is 172 elements (67% of the training dataset size), and that of the test dataset is 84 elements (33% of the training dataset size).

The conducted performance test results include the computation of the H(J) matrix and the transient process diagrams at the automatic parametric optimization algorithm’s initial and final points (Figure 6, Figure 7, Figure 8 and Figure 9) with the above-described reference action. The research object’s (the TV3-117 engine) parameters are as follows: $T_{ob}^{\left(1\right)}$ = 1.0, $T_{ob}^{\left(2\right)}$ = 4.0, k_ob = 1.0, and τ_ob = 5.0 [32,33,34,35,36,37,38].

Figure 6 demonstrates that the automatic parametric optimization-developed algorithm of the system with controller (5) ensures ${\overline{\theta }}^{*}$ computation from the initial values ${\overline{\theta }}^0$, at which point the transient process in the automatic system has both oscillatory and monotonic characteristics.

Table 5. The automatic parametric optimization algorithm results.

Number	${\mathit{\theta }}_1^0$	${\mathit{\theta }}_2^0$	${\mathit{\theta }}_1^{\mathit{*}}$	${\mathit{\theta }}_2^{\mathit{*}}$	J0	J*	H(J)	det(H(J))
1	0.1	0.01	0.253	0.016	38.325	20.974	$\left[\begin{array}{cc}1.049& 0.889\\ 0.659& 0.569\end{array}\right]$	0.011
2	0.25	0.025	0.347	0.016	38.261	20.659	$\left[\begin{array}{cc}1.013& 0.805\\ 0.611& 0.554\end{array}\right]$	0.069
3	0.35	0.001	0.415	0.012	59.939	19.993	$\left[\begin{array}{cc}1.001& 0.738\\ 0.602& 0.519\end{array}\right]$	0.075

shows that the matrix H(J) is positive-definite at the algorithm’s endpoints ${\overline{\theta }}^{*}$. Figure 6, Figure 7, Figure 8 and Figure 9 illustrate the algorithm’s (10) convergence patterns. Figure 6 shows that the proposed method’s application allowed the transient process time for the parameter n_TC to be reduced by 2 times, while the n_TC values were set at the level of ≈1 (i.e., n_TC values were restored to the initial ones). The search-free method’s [25,26,27,28] application also allows for a reduction in the transient process time for parameter n_TC by 2 times. However, the n_TC values were set at the level of ≈ 0.5 (i.e., n_TC values were restored to half of the initial ones). The n_TC = 0.5 · n_TC values used in the real helicopter TE’s operation conditions are impossible.

Figure 7 illustrates the change in the two components of the gradient of the optimization function J during the automatic parametric optimization algorithm’s operation. Figure 7 displays the changes in trajectories in the corresponding gradient components for each of the initial parameters (“set 1” is the black curve, “set 2” is the blue curve, and “set 3” is the green curve) in three sets. Figure 7 shows that, as the iterations progress, the gradient component values gradually decrease, indicating that the algorithm converges to the optimization criterion’s minimum point. The comparison of the curves for parameters with different sets (indicated in Table 5) allows us to estimate the algorithm’s convergence sensitivity to the choice of initial values.

Figure 8 shows the changes’ trajectories in the controller’s adjustable parameter values (e.g., parameters θ₁, θ₂) during the iterative optimization process. Each curve (black, blue, and green) corresponds to one of the three sets of initial parameters, which allows us to trace how the parameters are gradually adjusted and approach the optimal values, ensuring an improvement in the system’s characteristics. According to Figure 8, the parameter’s observed convergence confirms that the selected update method (based on gradient descent with a dynamic training coefficient) effectively tunes the controller.

Figure 9 demonstrates how the value of the optimization criterion J changes as the algorithm’s iterations are performed. Figure 9 shows that the J value decreases with each iteration and converges to a minimum in about 20 iterations, indicating the algorithm’s rapid convergence. In comparison, in the “search-free” method [25,26,27,28], the criterion’s optimization is achieved in 24 iterations, which emphasizes the proposed method’s advantage (the number of iterations reducing by about 1.2 times). This result confirms the effectiveness of the approach used: a rapid decrease in the J value demonstrates that the algorithm successfully minimizes the optimization function, thereby improving the control characteristics.

According to Table 5, the matrix H(J) is positively defined, which confirms, together with the technique given in [25], that the optimal parameters for controller (5) were found by the automatic parametric optimization-generated algorithm, based on the minimum value of criterion (6).

The presented stability analysis method was carried out under conditions where the object parameters take pre-specified values: $T_{ob}^{\left(1\right)}$ = 1.0, $T_{ob}^{\left(2\right)}$ = 4.0, k_ob = 1.0, and τ_ob = 5.0.

The controlled object G_p(p) operator takes into account the system’s inertia, delay and possible nonlinearities. For these parameters, the object is characterized by a significant delay (${\displaystyle \frac{{\tau }_{ob}}{T_{ob}^{\left(m\right)}}}>1$), which increases the control and raises the complexity of the requirements for the controller’s algorithm. The modified PI controller with the switching function $g\left(t,\overline{\theta }\right)$ (4) is adaptive and allows for taking into account the transient process’s features. In the stability analysis, the choice of parameters k₁, k₂, k₃, and α plays an important role since they determine the controller’s response to error deviations and its change rate. The first- and second-order sensitivity functions (ξ_i(t) and ξ_ij(t)) show how a change in the controller parameters (θ₁, θ₂) affects the system’s dynamics. For the given object parameters, the sensitivity function takes into account a significant delay and inertia, which leads to the need to adjust the training rate, η_l, in the gradient procedure (10). Sylvester’s condition (16) is used to check the Hessian matrix’s positive definiteness, which is necessary to confirm that the minimum optimization criterion J was found. For this object, given its parameters, the Hessian matrix determinant and its diagonal elements require additional regularization (λ > 0) (15) to compensate for the significant delay effect. The switching function, $g\left(t,\overline{\theta }\right)$, significantly affects the transient processes’ stability. For $\epsilon \left(t,\overline{\theta }\right)·\dot{\epsilon }\left(t,\overline{\theta }\right)\ge 0$, the system demonstrates smoother control due to a decrease in the gain. However, for $\epsilon \left(t,\overline{\theta }\right)·\dot{\epsilon }\left(t,\overline{\theta }\right)<0$, the presence of exponential damping, e^{−α ·t}, can lead to oscillations if the value chosen for the parameter α is too small. The optimization functional J (6) reflects the balance between minimizing the error and control costs. For these parameters of the object, the weighting factors a₁ and a₂ must be selected in such a way as to take into account the delay’s influence on the control action, u(t).

Based on the above, it was found that the method is stable for the given parameters $T_{ob}^{\left(1\right)}$ = 1.0, $T_{ob}^{\left(2\right)}$ = 4.0, k_ob = 1.0, and τ_ob = 5.0, provided that the controller’s parameters (k_p, T_i, k₁, k₂, k₃, α) are selected correctly. The significant delay of the object (${\displaystyle \frac{{\tau }_{ob}}{T_{ob}^{\left(m\right)}}}>1$) is levelled out by the use of regularization and an adaptive training rate. The Hessian matrix’s positive definiteness confirms the system’s stability. To further confirm the stability, simulation modelling was carried out using the second-order sensitivity functions according to the developed algorithm (

Table 6. Developed algorithm for conducting simulation modelling.

Modelling Step Number	Modelling Step Name	Description
1	Setting system parameters	$T_{ob}^{\left(1\right)}$ = 1.0, $T_{ob}^{\left(2\right)}$ = 4.0, kob = 1.0, τob = 5.0.
		Controller parameters: θ1 = 0.5, θ2 = 0.1.
		Reference action: r(t) = sin(t).
2	System modelling	The system block diagram creation, including the controller Gc(p, $\overline{\theta }$) and the object Gp(p).
		Adding blocks to compute ξi(t) and ξij(t).
3	Sensitivity computation	The implementation of the numerical calculation of the derivatives ${\displaystyle \frac{\partial u\left(t\right)}{\partial {\theta }_i}}$ and ${\displaystyle \frac{{\partial }^{2}u\left(t\right)}{\partial {\theta }_i\partial {\theta }_j}}$.
		The finite difference methods or analytical solution application.
4	The optimality criterion computation	$J=\underset{0}{\overset{\infty }{\int }}\left(a_1·{\epsilon }^2\left(t,\overline{\theta }\right)+a_2·u^2\left(t\right)\right)dt,$ where a1 = 1.0 and a2 = 0.5.
5	Analysis of results	The computation of the first- and second-order sensitivity functions.
		The gradient ${\nabla }_{\overline{\theta }}J$ and Hessian matrix H(J) computation.
		Testing Sylvester’s criterion for H(J) and optimizing the $\overline{\theta }$ parameters.

).

The resulting diagram (Figure 10) for the sensitivity functions demonstrates the system parameters’ influence on its behaviour time dynamics.

The first-order sensitivity function (ξ₁(t) and ξ₂(t)) diagrams show two curves: the red curve, ξ₁(t), is the system’s sensitivity to the first controller parameter, θ₁, and the blue curve, ξ₂(t), is the system’s sensitivity to the second parameter, θ₂. At the beginning of the process (t ≈ 0), the ξ₁(t) and ξ₂(t) values can be large, indicating the significant influence of parameters θ₁ and θ₂ on the system dynamics at the initial moment in time. As time increases (t → ∞), both functions tend to zero, meaning a decrease in the parameters’ influence on the system’s output in the steady-state mode. The ξ₁(t) values are higher than ξ₂(t), which may indicate that the parameter θ₁ had a more substantial influence on the system’s dynamics, and the time delay and curve amplitudes differ due to each parameter’s different contributions.

The second-order sensitivity function (ξ₁₁(t), ξ₁₂(t), ξ₂₂(t)) diagrams show three curves: The red curve, ξ₁₁(t), is the sensitivity’s second derivative with respect to θ₁; it shows how θ₁ influences changes over time. The green curve, ξ₁₂(t), is the second-order joint sensitivity between the parameters θ₁ and θ₂; it reflects the parameter interactions. The blue curve, ξ₂₂(t), is the sensitivity’s second derivative with respect to θ₂; it shows how the influence of θ₂ changes over time. The second-order functions ξ₁₁(t) and ξ₂₂(t) have high values at the beginning (t ≈ 0) and then quickly decrease, demonstrating a decrease in the parameters’ influence in the steady-state mode. The joint sensitivity, ξ₁₂(t), has a smaller amplitude, which may indicate a weak relationship between the parameters θ₁ and θ₂. At the same time, the ξ₁₁(t) value is higher than ξ₂₂(t), confirming the greater significance of θ₁ compared to θ₂. The ξ₁₂(t) values remain low, indicating weak cross-dependence between the parameters.

Thus, the diagrams show that the parameters θ₁ and θ₂ have the most significant influence on the system’s dynamics at the initial moment in time. At the same time, the parameter θ₁ has a more substantial impact on the system’s behaviour than θ₂, both in the first-order and second-order sensitivity terms. In turn, the mutual influence of the parameter ξ₁₂(t) is insignificant, which simplifies the optimization issue since the parameters can be selected almost independently. As a result, it is proven that the output coordinate _x(_t) stabilizes at optimal q^*, and the criterion J is minimized at q^* = [0.45, 0.12].

When conducting a computational experiment using the developed neural network (Figure 3) and the developed algorithm for its training (Table 2), a maximum accuracy of 0.993 was achieved on both the training and validation datasets (Figure 11a) with 160 training epochs. At the same time, the loss function drops from 2.5 to 0.5% on both the training and validation datasets (Figure 11b) with 160 training epochs.

The obtained data confirm that the proposed neural network (Figure 3), which trains 160 epochs using the developed training algorithm (Table 3), is sufficient to achieve the minimum value of the error function (MSE_min = 0.005). For comparison, similar experiments were performed using standard training algorithms for this neural network (Figure 3), the results of which are presented in

Table 7. Comparative analysis results for the proposed neural network training.

Number	Training Algorithm	Achieved MSEmin Value	Minimum Number of Epochs	Conclusions
1	Developed algorithm	0.0050	160	The most efficient learning algorithm is characterized by the minimum MSEmin value achieved in the minimum number of training epochs.
2	Traditional backpropagation algorithm	0.0093	220	The minimum MSEmin value achieved exceeds the developed algorithm’s similar indicator by 1.86 times, with an increase in the optimal number of training epochs by 1.38 times.
3	Genetic algorithm	0.0108	240	The minimum MSEmin value achieved exceeds the developed algorithm’s similar indicator by 2.16 times, with an increase in the optimal number of training epochs by 1.5 times.
4	Modified inverse gradient descending method [51]	0.0163	250	The minimum MSEmin value achieved exceeds the developed algorithm’s similar indicator by 3.26 times, with an increase in the optimal number of training epochs by 1.56 times.
5	Traditional inverse gradient descending method	0.0237	330	The minimum MSEmin value achieved exceeds the developed algorithm’s similar indicator by 4.74 times with an increase in the optimal number of training epochs by 2.07 times.
6	Hybrid algorithm	0.0301	380	The minimum MSEmin value achieved exceeds the developed algorithm’s similar indicator by 6.02 times, with an increase in the optimal number of training epochs by 2.38 times.

.

The comparative analysis results showed that the developed training algorithm is the most effective, achieving the error function’s minimum value (MSE_min = 0.005) for the minimum number training epochs (160). Compared with traditional and alternative algorithms, the minimum values of MSE_min achieved by them exceed the developed algorithm’s similar indicator by 1.86 to 6.02 times, while the optimal number of training epochs increases by 1.38…2.38 times.

The developed method for neural network implementation using the developed neural network (Figure 3) comparison was made with other well-known neural network architectures—the traditional RBF network [20,56], Jordan neural network [21,57], Elman neural network [22,58], Hopfield neural network [23,59], and Hamming neural network [24,60]—according to traditional quality metrics: accuracy, precision, recall, F1-score, and AUC-ROC, as well as neural network training time (

Table 8. Comparative analysis results.

Neural Network Architecture	Metrics for Evaluating Methods					Training Time
	Accuracy	Precision	Recall	F1-Score	AUC-ROC
Developed neural network	0.993	0.991	1.0	0.995	0.825	1 min 12 s
Traditional RBF network [20,56]	0.992	0.989	1.0	0.994	0.811	2 min 25 s
Jordan neural network [21,57]	0.993	0.989	1.0	0.994	0.781	4 min 17 s
Elman neural network [22,58]	0.993	0.989	1.0	0.994	0.780	4 min 12 s
Hopfield neural network [23,59]	0.837	0.832	0.835	0.833	0.635	0 min 47 s
Hamming neural network [24,60]	0.811	0.806	0.831	0.819	0.602	0 min 33 s

). These metrics are calculated according to the following expressions [61,62,63]:

$$\begin{gathered} Accuracy={\displaystyle \frac{1}{N}}·\sum _{i=1}^{N}1\left(n_{TCi}={\widehat{n}}_{TCi}\right),Precision={\displaystyle \frac{TP}{TP+FP}}, Recall={\displaystyle \frac{TP}{TP+FN}}, \\ F1-score={\displaystyle \frac{2·Precision·Recall}{Precision+Recall}}, AUC-ROC=\underset{0}{\overset{1}{\int }}TPR·\left({FPR}^{-1}\left(t\right)\right)dt. \end{gathered}$$

(25)

According to the comparative analysis data (Table 8), the developed neural network is trained significantly faster and shows improved quality indicators. Thus, the training time is only 1 min 12 s (72 s), which is approximately 2 times faster than that of the traditional RBF network, whose training time reaches 2 min 25 s (145 s) [20,56]. In comparison with the Jordan and Elman architectures, the training time is reduced by approximately 3.5 times (72 s vs. 257 and 252 s, respectively) [21,22,57,58]. Moreover, in terms of metrics such as accuracy and F1-score, the developed network shows an about 18…22% improvement compared to the Hopfield (accuracy 0.837, F1-score 0.833) and Hamming (accuracy 0.811, F1-score 0.819) architectures [23,24,59,60]. These quantitative improvements indicate that the proposed architecture combines high accuracy and significantly reduced training time, making it more effective for real-time applications.

4. Discussion

4.1. Generalization of Results

The developed method for studying processes in ACSs (see Figure 2) is based on the operator approach used for modelling both the control object and the controller. The processes in the automatic control system are described by the system of equations given in (1), which determines the relationship between the control error, ε(t, $\overline{\theta }$), the reference action, r(t), the control action, u(t), the output coordinate, x(t), and the controller, G_c(s, $\overline{\theta }$), and control object, G_p(s), operators. The control object operator, presented in the form of (2), includes parameters that allow for describing the most complex object dynamics, including the static gain, k_ob; time constants $T_{ob}^{\left(1\right)}$ and $T_{ob}^{\left(2\right)}$; the delay time, τ_ob; and the nonlinear part, f_nonline(x, u). A modification of the traditional PI controller is used for the controller, presented in the form of (3). It includes the switching function $g\left(t,\overline{\theta }\right)$, which describes the controller’s adaptation to the current operating conditions. This function is regulated by the parameters k₁, k₂, k₃, and α and is determined according to (4). The control action, u(t), is determined by expression (5), which depends on the error ε(t, $\overline{\theta }$), parameters θ₁ and θ₂, and the switching function state $g\left(t,\epsilon \left(t,\overline{\theta }\right),\dot{\epsilon }\left(t,\overline{\theta }\right)\right)$. The functional expression in (6), including the weighting coefficients a₁ and a₂, which take into account the control errors and the control action contributions, is selected as the optimization criterion for adjusting the controller’s parameters. Expression (7) is used to calculate the sensitivity to the parameters $\overline{\theta }$, and the update in the parameters’ gradient in the optimization process is determined by expression (10). The conditions for stopping the optimization process are formed on the basis of the analysis of changes in the optimality criterion, as indicated in expressions (12) and (13). The process completion is determined by fulfilling the conditions presented in expression (14). To check the criterion’s minimum, the Hessian matrix H(J) is calculated, whose expression is given in (15). The matrix’s positive definiteness condition is estimated using Sylvester’s criterion, presented in expression (16). The developed method differs from the search-free method [25,26,27,28] by the integration of the nonlinear components into the control object operator, the parameterization of the dynamic training rate η_l, the regularization coefficient λ used to stabilize the optimization, and improved stopping criteria taking into account the permissible error and the number of changes in the gradient sign.

The developed method is implemented as a neural network with a specified architecture (see Figure 3), taking into account the system’s dynamic features and its behavioural nonlinearities. The basis of this architecture is a multilayer structure, where the input layer receives the system error ϵ(t) and the reference action r(t) parameters, and the hidden layers with nonlinear activation functions (SmoothReLU, tanh) model complex relations. A unique feature is the first hidden layer used for processing ϵ(t) with the function f_nonline(x, u), the second layer for calculating the regulatory action u(t) = G_c(s) ⋅ ϵ(t), and the third layer for integrating the switching parameters through the functions ϕ(t). Adaptive network parameters such as k_p, T_i, k₁, k₂, k₃, and α are trained using the gradient descent method with regularization, where the coefficient λ stabilizes the training, preventing overfitting and noise. The dynamic training rate η_l, depending on the change in the optimization functional, ensures convergence acceleration and increased model accuracy, and the output layer calculates the coordinate x(t), described by the control object operator G_p(s), which allows for taking into account delays and system nonlinearities.

A neural network training algorithm (see Table 2) is developed that includes a multilayer architecture with adaptive parameters and takes into account the system’s nonlinear dynamic features. In the first stage, the network architecture and its parameters (weights and biases) are initialized using the normal distribution or methods such as the Xavier or He methods. In the forward pass stage, the input parameters ϵ(t) and r(t) are sequentially processed through three hidden layers: the first layer computes the nonlinear error processing using SmoothReLU, the second layer models the control action with the tanh function, and the third layer integrates the switching parameters via SmoothReLU. The output layer forms the coordinate x(t) defined by the control object operator. The error is calculated using a loss function, for example, MSE, and the gradients are computed taking into account the sensitivity function, Ξ(t), reflecting the parameters’ effect on the system. In the backpropagation stage, the gradients are updated for all layers using regularization, λ, and a dynamic learning rate, η_l, depending on the change in the optimization function. Adaptive parameters such as k_p, T_i, k₁, k₂, k₃, and α are optimized, taking into account the error changes. The training process is repeated until the specified criteria, such as the minimum error or the maximum number of epochs, are achieved, and the algorithm’s accuracy is evaluated on test data, demonstrating the neural network’s ability to effectively handle nonlinearities and system delays.

To conduct the computational experiment, it was noted that the choice of the TV3-117 engine as the research object was based on its widespread use in the Mi-8MTV helicopter power plant, and the key diagnostic parameter of the gas-generator rotor speed, n_TC (see Figure 4), recorded by the standard D-2M sensor, was used to form the training dataset (see Table 3). The use of z-normalization made it possible to standardize the data, improving the neural network training algorithm’s convergence (see Table 4), and cluster analysis (see Figure 5) confirmed the representativeness of the training and test datasets. This approach ensures the statistically substantiated reliability of the dataset and error minimization, which are critically important for improving helicopter operation safety.

This research demonstrated the effectiveness of the proposed method for automatic parametric optimization for the TV3-117 engine control system. When analyzing transient processes (see Figure 6, Figure 7, Figure 8 and Figure 9 and Table 5), it provided a twofold reduction in the n_TC parameter transition time to a level of ≈1, which confirms its suitability for actual operating conditions. Unlike the search-free method [25,26,27,28], where n_TC is restored only to a level of ≈ 0.5, the developed method provides higher-quality control, which makes it practically applicable in aviation technology.

Sensitivity analysis (ξ₁, ξ₂, ξ₁₁, ξ₂₂, ξ₁₂) showed (see Figure 10 and Table 6) that the parameters θ₁ and θ₂ have the most significant influence at the initial time, with θ₁ dominating, and their interaction is insignificant, which simplifies the tuning of the controller. The iteration process confirmed the positive definiteness of the Hessian matrix, and the method’s convergence was observed to be faster compared to analogues (1.36-fold reduction in iterations). The application of the developed neural network and training algorithm showed a maximum accuracy of 0.993 at 160 epochs (see Figure 11a), ensuring the error function minimization to MSE_min = 0.005 (see Figure 11b). These results confirm the reliability and stability of the proposed method for complex objects with significant delays and inertia.

The comparative analysis (see Table 7) results of the proposed neural network training algorithm and alternative approaches’ efficiency confirmed its superiority. The developed algorithm achieved the mean square error minimum value (MSE_min = 0.005) for the minimum number of training epochs (160), which is significantly better compared to traditional error backpropagation and other methods (exceeding MSE_min by 1.86 to 6.02 times and increasing the number of training epochs by 1.38 to 2.38 times).

The comparison of neural network architectures showed that the developed network has the highest accuracy (0.993), recall (1.0) and F1-measure (0.995) while taking only 1 min and 12 s to train (see Table 8). Alternative networks, such as RBF, Jordan and Elman, have comparable accuracy but require significantly more time to train (from 2 to 4 min). Hopfield and Hamming networks showed substantially lower accuracy values (0.837 and 0.811, respectively), which limits their use for tasks requiring high accuracy.

4.2. Limitations and Further Research

The developed method’s key limitations are its dependence on the control object’s model parameters (such as the static gain, time constants, and nonlinear components) and precise identification, which can reduce the modelling and control accuracy in the event of errors in these parameters [64,65]. The method also requires careful adjustment of the controller’s adaptation parameters and optimization coefficients, which complicates its implementation in practical conditions [66]. Nonlinear components in the control object model may not be applicable to systems with more complex dynamics, and the dependence on the training data (noise, sensor errors) quality can significantly worsen the results [67,68,69]. High computational complexity, requirements for the neural network architecture, and the need for complex optimization of the stopping criteria also limit the method’s applicability, especially in actual conditions with limited computing resources [70].

In addition, there is an overfitting risk when using small or nonuniform datasets, and modelling systems with considerable delays may be inaccurate. The method showed high efficiency for a specific engine type (TV3-117), but its generalizability to other kinds of control systems requires additional adaptation. The algorithm’s implementation in real systems requires powerful computing resources [71,72,73], which may be problematic for existing systems with limited power.

A roadmap for future research to address the developed method’s limitations is presented in

Table 9. The proposed roadmap for future research.

Research Title	Aim	Actions	Expected Results
Optimization of model parameters and increased error tolerance [74].	Develop methods for identifying parameters with increased accuracy and error tolerance.	Study more accurate methods for estimating the controlled object parameters using machine learning and adaptive filtering methods.	Reduce the model’s sensitivity to errors in parameters, increasing the modelling and control accuracy.
		Develop algorithms for automatic correction of errors in parameters in real time.
		Develop methods for taking into account uncertainty in model parameters.
Controller adaptation and tuning simplification [75,76].	Simplify the process of tuning controller adaptation parameters and optimization coefficients for practical applications.	Investigate new approaches for the automatic tuning of controller parameters (e.g., using optimization methods such as genetic algorithms or gradient descent).	Simplify the tuning process and increase the method’s flexibility when applied to various control objects.
		Develop universal methods for parameter optimization that can automatically adapt to system changes.
Processing complex nonlinearities [77].	Develop methods for more efficient processing of complex nonlinearities in object dynamics.	Study more complex neural network architectures (e.g., deep neural networks considering multilevel nonlinear dependencies).	Increase the accuracy and universality of the model for systems with different nonlinear dynamics.
		Implement methods considering the dynamic nature of nonlinearities, such as adaptive neural network models.
Robustness to noise and data quality [78].	Improve the method’s robustness to noise and poor data quality.	Develop data preprocessing methods, including improving noise filtering and eliminating sensor errors.	Improve the accuracy of the method by improving the input data quality.
		Use high-reliability data methods, such as data reconstruction or fragmentation methods, to improve the training datasets’ representativeness.
Reducing computational complexity [79].	Optimizing computational processes and reducing the need for computational resources.	Developing more efficient algorithms for adapting the training rate and regularization to speed up computations.	Reduce computational costs while maintaining the high accuracy of the method.
		Investigating the use of hardware accelerators (e.g., neural processors or graphics processors) to optimize computational processes.
Increasing the method generalization ability [80].	Develop methods for adaptation and training transfer to apply the technique to the systems and engines’ new types.	Study methods for training transfer that allow the model to be adapted to different systems and conditions.	Increase the generalization ability of the method and its successful application to different types of engines and systems.
		Development of universal neural network architectures suitable for different types of control objects.
The significant delays in modelling optimization [81].	Develop methods for more accurate modelling of significant delays in the system.	Research and implementation of methods that take into account dynamic delays and significant delays in the system.	Improve model accuracy for systems with long delays.
		Development of prediction methods to take into account long delays timing models’ integration to take into account phase shifts.
Simplifying the implementation of accurate systems [82].	Simplify the implementing process of the method on real systems with limited computing power.	Developing lighter and faster versions of the algorithm for implementation on systems with limited computing power.	Increase the availability and practicality of the method for implementation in real systems with limited resources.
		Investigating the possibilities of integration with existing platforms, such as microcontrollers and specialized neuro-microprocessors.
Preventing overfitting [83].	Develop methods to prevent overfitting on small or heterogeneous training datasets.	Use cross-validation and regularization methods to prevent overfitting.	Reduce risk of overfitting and improve generalization properties of the model.
		Investigate methods for generating training data and expanding the dataset to improve model robustness.

.

The roadmap’s expected results include improvements in the method’s accuracy and reliability, which will allow for its applications to be expanded in various fields [84,85,86], as well as increased computational efficiency and reduced dependence on hardware resources. In addition, we plan to expand the method’s scope, including its adaptation for working with complex dynamic systems with long delays [87,88,89], which will increase the universality and practical value of the approach.

5. Conclusions

A method for researching processes in automatic control systems using an operator approach and a modification of the PI controller has been developed, which allows for the control and regulation of the efficient modelling of object dynamics. The control object operator includes parameters such as static gain and time constants (e.g., $T_{ob}^{\left(1\right)}$ = 0.5, $T_{ob}^{\left(2\right)}$ = 0.3) and a nonlinear part f_nonline(x, u), which makes it possible to take into account various aspects of the system. The PI controller modification, which includes a switching function, adapts the system to changing conditions, increasing the regulation flexibility and accuracy. The algorithm for optimizing the controller’s parameters using gradient descent and regularization (λ = 0.01) stabilizes the training process, minimizing overtraining and data errors.

The developed method’s application to the TV3-117 engine control system as an example demonstrated high efficiency. The algorithm significantly improved transient processes, reducing the gas-generator rotor speed parameter transition time from 2.0 to 1.0 and increasing the control accuracy. At the same time, the neural network, with the use of adaptive parameters (k_p = 1.2, T_i = 0.8, k₁ = 0.5, k₂ = 0.3, k₃ = 0.7, α = 0.2) and a special architecture, ensured the error function’s minimization to the value MSE_min = 0.005 for 160 epochs, which confirms the method’s high efficiency and stability for the current operating conditions. Comparative analysis showed that the proposed method outperforms alternative approaches in terms of accuracy (accuracy = 0.993), convergence rate (reduction in the number of epochs from 380 to 160) and the number of training epochs, which makes it promising for use in aviation and other complex dynamic systems.

The developed method has a number of key limitations, including high dependence on the precise identification of the controlled object’s model parameters, such as static gain, time constants and nonlinear components, which can reduce the accuracy with errors in these parameters. The method also requires careful tuning of the controller’s adaptation parameters and optimization coefficients, which complicates its implementation in practical conditions. High computational complexity and requirements for the neural network architecture limit its applicability, especially under limited computing resource conditions. Despite its effectiveness for the TV3-117 engine control system, the method has risks of overfitting with small or heterogeneous datasets, as well as modelling accuracy problems for systems with significant delays and complex dynamics, which requires additional adaptation for other types of control systems and implementation in real systems with limited power.

Prospects for further research include the further development of the method to improve the model’s accuracy and robustness to errors in the control object parameters, as well as simplifying the controller’s adaptation process using methods for automatic parameter adjustment. Critical areas include improving the processing of complex nonlinearities in the object dynamics using more complicated neural network architectures and increasing the method’s robustness to noise and data errors. We also plan to optimize the computational processes to reduce computational costs and improve the method’s generalizability for applications to various types of systems with significant delays and limited computing resources.