Controlling the Helicopters Turboshaft Engines Free Turbine Speed at Flight Modes

Abstract

The work is devoted to the development of a method for controlling the main rotor speed, which is a key task to be solved in a helicopter flight. A modified block diagram of the circuit for maintaining the helicopters turboshaft engines free turbine speed an electronic PIDcontroller with a neural network tuning of the amplification factor has been developed, which made it possible to automatically adjust the amplification factor and, thereby, reduce the time of the transition process. The use of a dynamic neural network of direct data transmission based on neurons with a radial-basis activation function in the first layer and adalines - neurons with a linear activation function in the second layer is proposed, which made it possible to improve the quality of the transient process in terms of helicopters turboshaft engines turbine rotation frequency, which consists in an increase in performance up to 3 seconds, an increase in statistical accuracy up to ± 0.05 % and the elimination of parameter overshoot. The use of a dynamic neural network of direct data transmission as some functional converter that generates for a set of input and output signals the amplifications factors of the PID-controller made it possible to improve the probability of errors of the 1st and 2nd kind in comparison with the known controllers by 35... 85 %. Helicopters turboshaft engines, free turbine speed, neural network, training, PID-controller, MoMLeT+DS 2023: 5th International Workshop on Modern Machine Learning Technologies and Data Science, June 3, 2023, Lviv, Ukraine.

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Helicopters transient processes, amplification factor, error

1. Introduction

The helicopters turboshaft engines (TE) are a complex thermogas-dynamic system with many features that must be taken into account when designing an automatic control system (ACS). The ACS of a modern aircraft engine performs many functions. These functions are distributed and carried out by a digital controller and hydromechanical actuators. The digital controller performs the main part of the engine control functions. Its main tasks are to control the TE operating modes, maintaining and/or limiting its various parameters, diagnosing and monitoring the state of the TE and ACS elements, and providing service and information functions. The task of the hydromechanical part of the ACS includes the control of the engine mechanization by the commands of the digital controller and the control of the operation of the gas turbine engine according to simplified laws in the event of an electronic system failure [ 1, 2 ].

The quality of control of TE parameters largely depends on the quality of tuning of electronic algorithms. Often in electronic control systems of TE linear controllers of P-, PD-, PI- and PID-type are used. Their popularity is explained by the simplicity of the mathematical description, low cost of implementation and sufficient efficiency. However, as practice shows, within the framework of the linear theory, it is not always possible to tune the PID-controller to ensure the required quality of

2023 Copyright for this paper by its authors. transients in a nonlinear system [ 3, 4 ], which are helicopters TE. Under these conditions, the use of neural network technologies is relevant and promising.

2. Related Works

When controlling complex non-linear objects, such as a helicopters TE, such controllers cannot always provide the required quality of control over TE parameters, stability and robustness of the system under changing operating conditions and failures. In this case, it makes sense to use alternative nonlinear controllers [ 5, 6 ]. For example, it can be a fuzzy logic controller (FLR), which has the property of robustness [ 7, 8 ]. Due to the absence of the need for a strict mathematical description of the object, logic controllers have gained great popularity among developers of electronic systems [ 9, 10 ]. The fuzzy control law obtained as a result of synthesis is non-linear and works well in systems with a high degree of complexity, non-linearities such as a dead zone, hysteresis, when the parameters of the unchanging part of the system deviate from their nominal values and information is lost in case of failures [11, 12]. It is worth noting that Elizaveta Chicherova conducted a number of alternative control methods research that make it possible to increase the stability margins of gas turbine engines ACS and eliminate oscillations using the following controllers: a linear PD controller with a reduced proportional gain, a square-law controller, a variable gain controller, a proportional fuzzy logic controller, a fuzzy logic controller with a proportional gain and a corrective differential link [13, 14], which made it possible to ensure the aperiodic nature of the transient process, to achieve a statistical accuracy of ± 0.2 % and a speed of up to 6 seconds.

However, when using these types of controllers, the overshoot value ranges from 0.1 to 2.2 %. In order to eliminate overcasting, increase statistical accuracy and develop a method for controlling the speed of the free turbine of helicopters TE using neural network technologies, this is an urgent scientific and practical task.

3. Methods and Materials

The main task of the automatic control system of helicopters TE is to maintain the rotational speed of the main rotor nнв. This task is accomplished by controlling the free turbine speed nFT through the required fuel flow rate GT. The value of the required fuel consumption is formed from the gas generator rotor r.p.m. nTC and its derivative nTC_req. Depending on the engine operating mode, the value of the required derivative of the gas generator rotor r.p.m. is formed by various circuits. For example, in idle mode, the value of nTC_req is determined by the circuit for maintaining the required speed of the turbocharger rotor, and at flight mode, by the circuit for maintaining the free turbine speed. Since the helicopter TE can be operated at flight mode for a significant part of the time, the choice of the structure and parameters of the electronic controller determines the dynamic quality of helicopter TE control and its resource.

The mathematical model of the regulated object – the helicopter TE (in this work, the deviations of the state variables are considered) according to [15] has the form:  dnTC  nTC  k11GT ;

1 dt  dnFT  nFT  k21GT  k22nTC ;

2 dt where GT – fuel consumption, τ1 and τ2 – time constants, k11, k21, k22 – amplification factors.

The analysis of the object of regulation – the helicopter TE according to [15] is presented in the form of a series connection of dynamic links with transfer functions:

WTC  nTC  k11 ;

GT  1 p  1 WFT  nFT  1 k21    k22 .

nTC  k22WTC   2 p 1 

The mathematical model of the fuel dispenser with a direct drive from an electromechanical converter according to [15, 16] after dry friction linearization is represented as: (1) (2)

GT   kGT ; where J – rotor inertia moment, α – rotor rotation angle, kv, ki, kG – coefficients of viscous friction, T torque and fuel consumption, i – control current.

Thus, the transfer function of the fuel dispenser is presented in the following form [15]: WFD  ki  kGT .

p   J  p  kv  For the internal fuel consumption control loop, a proportional-differential control law is adopted: i WPD 

 kP  kD  p;

GTgiven  GT where GTgiven – fuel consumption setpoint, kP and kD – proportional and differential gains.

According to [15, 17], to take into account the delay of the digital control unit, a link of pure delay by 1.5·T is introduced, where T – sampling period in time: (3) k  kP1  i ;

p  kP2; (4) (5) (6) (7) (8) (9) (10) (11) (12) at the same time, for the transfer function of a closed internal fuel consumption control loop, the following expression is obtained: where W0  Wdelay WPD WFD .

According to [15], as a control law nFT, a proportional-integral control law with a transfer function is adopted:

WPI  WP 

i

GT _ set nFT _ req  nFT where nFT_reg – set value of the free turbine speed, kP1 and ki – gains of the proportional and integral components.

To exclude the sequential inclusion of two integral components for an additional internal control loop nTC, a proportional control law is adopted:

nTC _ set  nTC where nTC_set – set value of the gas generator rotor r.p.m., kP2 – amplification factor.

According to [15], the value of the gain is chosen from the condition of ensuring the required accuracy of the implementation of the law dnTC  f U  in all modes of engine operation.

dt

The transfer function of the open control loop nFT with independent operation of the controller is presented in the following form:

W1  WPI WP WGT WTC WFT .

Similarly, the open-loop transfer function nFT when operated in series with an nTC controller is: Wdelay  e1.5 pT ; WGT 

W0 ; 1  W0 W2  W1 W4;

1 W4  . where

1  WP WGT WTC

In connection with the foregoing, an improved typical circuit for maintaining the free turbine speed of helicopters TE with a linear PID controller is proposed (fig. 1).

nFT_max+

nFT nFT kp . nFT kI kd + + + . nTC_req

The error of the mismatch between the current and required value of the free turbine speed ΔnFT, and the derivative of the current free turbine speed nFT , is fed to the loop input, and the output signal is the value of the required derivative of the turbocharger rotor speed. The expression describing the circuit for maintaining the free turbine speed with a linear PID-controller has the form:

The gains of the proportional kp and differential kd links are determined from the equations: nTC _ req  nFT  kp  nFT  kd . k p  k nFT   kstat _ p  GT _ stat nTC   a13 nTC ;

kd  kstat _ d  FT  nTC ; where GT _ stat nTC  – characteristic of the static fuel consumption, a13 nTC  – coefficient of engine linear dynamic model in terms of fuel consumption,  FT nTC  – time constant of the free turbine rotor.

The coefficient is given by the following system of equations:   nFT , if nFT  1;  k  nFT   1  nFT 1, if 1  nFT  3; (15)  2  2   nFT  4  , if nFT  3;  7

The static coefficient of the proportional link is kstat_p = 0.35, while that of the differential link is kstat_d = 0.05 [13, 14].

Since the main task solved in the helicopter flight mode is to maintain the main rotor speed, it is advisable to apply the neural network setting of the coefficients kp, ki and kd, which will eventually allow dynamic maintenance of the main rotor speed.

It is assumed that, in general, the circuit for maintaining the speed of the free turbine of the gas turbine engine of helicopters with a linear electronic controller is described by the following equation:

X  At  X  B t u  f t ; (16) where х = (х1 ... хn)T – state vector of the system under the action of control u = (u1, u2, u3)T, – components of which are constants under the constraints |ui| < 1, i = 1, 2, 3. The elements of the matrices A(t), B(t) and the vector f(t) will be considered real, continuous functions for t  [0, T], their dimensions are (n × n), (n × 3) and (n × 1), respectively. The matrix B(t) is determined by a predetermined real and  b1 t  

 continuous at t  [0, T] vector b t    ...  in the form:  bn t  

 t  B t    b t ,  b   d ,b t  .

 0  (13) (14) (17)

To obtain an equation describing the search for the coefficients kp, ki and kd, under which the quality of the control system will be the best or will satisfy the specified requirements, the assumptions are made that kp = const, ki = const and kd = const. It is assumed that the vector function b(t) satisfies the inequality a et  bt   a2et for some positive λ and a1 < a2. Then, due to the choice of control u, 1 it is necessary to achieve exponential stability of the original system. It is accepted that t bt   B1 t ;  b  d  B2 t ; b t   B3 t ; (18)

0 where the vectors B1(t), B2(t), B3(t) make up the matrix B(t).

Initially, the coefficients u, considered as control, are assumed to be constant. However, for further research, a new control u t  is introduced, which will be a function of time, that is, u  u t  x.

In order for the program motion x t,u t  together with the control u t  to set the product constant u t  x t,u t  , system (16) is represented as:

Let be Z t,u  – matrix of the fundamental system of solutions corresponding to the homogeneous system (20) for f t   0 . Then for Z t,u  will be fulfilled (19) (20) (24) (21) in this case, Z 1 t,u  – inverse matrix Z t,u  .

It follows that the general solution in the Cauchy form of system (21) with initial data х0 = х(0) will be written in the form: t x t,u  Z t,u  x0  Z t,u   Z 1  ,u  f  d . (22) 0

It should be noted that the choice of control is subject to restrictions caused by the operating conditions of helicopters TE at flight mode. Therefore, it is necessary to take into account this possibility and add admissibility conditions for the values of the coefficients:

T 3 2  ui t dt  ; (23) 0 i1 where φ – positive constant, T – length of the time interval on which the program control law will operate in the form of fixed coefficients u t  . Let’s introduce the following functional: J u   u t   x2 t,u ;  3  x   At    Bi t ui t   x  f t .

 i1 

 3  Z t,u    At    Bi t ui t   Z t,u ;

 i1  where x t,u  – solution of the Cauchy task (22) determined by the functional law of coefficients u t  during the time interval [0, T].

Let us assume that there is an optimal control [18, 19] in the form u 0  u10 t ,u2 t ,u3 t  that 0 0 T gives the functional (24) the minimum possible value, while simultaneously fulfilling the condition: T 3 2  ui0 t  dt .

0 i1 Then any other control from some local proximity to the optimal will have the form: u  u0   u10 t   1 t ,u20 t   2 t ,u30 t   3 t T ; (25) (26) and the admissibility condition must also be satisfied for it, but the inequality will necessarily be true: for all sufficiently small ε. Then, taking into account the individual choice, a vector function ν(t) is accepted that satisfies the requirement and, therefore, under the admissibility condition for non-optimal control u  u account the square under the integral sign in this condition,   and taking into we notice that sign  sign  i t ui0 t  dt . From the need to minimize the functional J u0    for the value T 3 0 i1 of the parameter ε = 0, the following inequalities will necessarily hold: dJ T 3 0

 0 for ε > 0 or   i t ui t  dt  0; ddJ 0T i31 0 (29)

 0 for ε < 0 or   i t ui t  dt  0.

d 0 i1 It is assumed that the structure of the derivative of the functional J(u) has the form 0 dJ u0    T 3

   i u0  i t  dt; d 0 i1 where  i u0  – functions, but depending on the optimal control and on the right-hand sides of system (20).

If we assume that u10 t   0 , but at the same time assume that the controls  2 t   3 t   0 and J u0   J u0  

T 3 0   i t ui t  dt  0 0 i1 , then by direct substitution into (28) where constants  i  0 T  i2 t  dt 0 T 0  ui t  i t  dt , i = 1, 2, 3. (27) (28) (30) (33) T 0  1 t u1 t  dt 0  1 t    u1 t   t  , where is a constant   0T  u10 t  dt

2 0

T 0 we obtain that  u1 t  t  dt , that is, the function ω(t) will be orthogonal to u10 t  . Noting that for 0 each control of the form  1 t   u10 t    t  for any β, we find that any of the inequalities (29), taking into account the choice of the structure in the form (30), for a sufficiently large modulo value of the parameter β will be violated:

T 3 T T T 0   i t  i t dt   1 t  1 t  dt    1 t  t  dt   1 t u1 t  dt  0; (31) 0 i1 0 0 0 but it can only be done under one condition:

T  1 t  t  dt  0. (32) 0 The resulting expressions show that the function ψ1(t) is orthogonal to any function orthogonal to 0 0 u10 t  , that is, we can assume that  1  1  u1 t  . Similarly,  i   i  ui t  for some constants αi, i = 1, 2, 3. Let us express the optimal control u

0 u1 t   i  i t ; 0

from these equations in the form find the derivative of the functional

d solution formula in the Cauchy form, we first derive: 0 0 0 dZ T ,u d   

dZ T ,u  dJ u0   

0    T   Z 1 t,u 0 The resulting expression (33) is a consequence of relations (29) and, in essence, represent the 0 0 0 necessary conditions for the optimality of controls u1 t  , u2 t  , u3 t  on the time interval [0, T].

Let us now formulate the general principle of actually finding the functions  i u0 t  , which determine the method of software adjustment of the gains u according to expression (33). Let us first at ε = 0 and time t = T. To do this, using the general 0    f t  dt  Z T ,u 0     dx T ,u

   d

0

T dZ 1 t,u  0

T  Z 1 t,u 0 d

0  x0Z T ,u 0  x0    d 0 0 f t  dt  x0Z T ,u 0

T      Bi t  i t  dt  Z T ,u 0 0

T 3       Bi t  i t  dt  0 i1   f t  dt  Z T ,u 0

T     Z 1 t,u

T     Bi t  i t  dt  Z T ,u 0 0 t 3   f t dt    Bi   i  d f t dt 

0 i1    T 3 Bi t  i t dt  T Z 1 t,u0  f t dt  0 i1 0 dJ u0   d  2x T ,u0  dx T ,u d

  Z T ,u 0 T t 3  t        Bi   i   d d   Z 1  ,u 0 0 i1  0

0 T t 0 Z T ,u      Z  ,u

0 0 Then, assuming the moment of time t = T, we find 0

   f   d   x0Z T ,u 0  0

   f   d   Bi t  i t  dt.

3 i1  2   x0Z T ,u0   Z T ,u0  T Z 1 T ,u0  f t  dt  

 0 

It can be seen from (36) that the functions  i t,u0  depend on the index i only through the corresponding function Bi(t). Thus, having determined the type of functions  i t,u0  through the 0 dependence on the optimal control u and the type of system (20), in the future, an algorithm for programmatic adjustment of the gains and will be proposed, which will play the role of a “teacher” T      Bi t  i t  dt  0 (34) (35) (36)

4. Experiment

On a cycle, the neural network receives the setpoint and generates the control coefficients of the PID-controller, which are fed to the PID-controller along with the value of the current feedback error e(k). The PID-controller calculates the control signal according to the expression: u k   u k 1u1 k ek   ek 1  u2 k ek   u3 k ek   2ek 1  e k  2; (37) used for discrete PID-controllers and feeds it to the control object – the helicopters TE. The neural network is trained in real time by feedback error.

Let the entire observation time be divided into time intervals of length Т: [0, Т], [0, 2Т], [2Т, 3Т], ... Within each individual segment [sТ, (s + 1)Т], s = 0, 1, 2, … is used as a “teacher” for a neural network, the method of programmatic adjustment of gain factors, described in this paper.

When synthesizing the controller, a dynamic network of direct data transmission was used, based on neurons with a radial basis activation function in the first layer and adalines – neurons with a linear activation function, in the second layer. At the same time, on test examples, the optimal settings of the neural network were obtained, which provide the smallest overshoot for a given time of the transient process [20]. The following sequences are used as neuroregulator inputs: – reference signal – a master sequence that determines the final state of the object, – controller output –feedback from the controller output, – object error – the difference between the reference signal and the real output of the object, – integrable error – the error accumulated by the controller for the entire time of the object operation, – object output – signal from the object output.

The choice of input sequences is not random. Some of the sequences are intended only for a certain component of the control signal. So, the output of the object and the output of the controller are necessary for the differential component and the adjustment of the parameters of the predictor, which actually implements the differentiation function. The “integrable error” sequence is only necessary for the integral component and affects only it. The remaining incoming sequences affect all neurons of each of the blocks (fig. 2).

nFT_max .

u(t) .

nFT . . nTC_req.

Reference signal Input

Delay line layer Regulator output

Delay line

Error

Delay line Integrable error

Adjustable

storage Object output

Delay line

Layer of non

linear neurons

Layer of

linear neurons Σ u(t)

The ACS structure, which includes a neural network in the role of setting the coefficients, using a PID controller, is schematically shown in fig. 3 (developed on the basis of [21]), in which the neural network plays the role of some functional converter that generates the required coefficients of the PID controller kp, ki, kd for a set of signals nFT_max, ∆nFT, u, nTC _ reg .

nFT_max. +

nFT

Neural

network

ki kp kd nFT .

NFT_max u .

P

. . nTC_req

Let us consider a method for approximate construction of an optimal control based on the application of (33) and assume that uil1  ul i ul  i ul ; (38) where u l  u1l ,u2l ,u3l  – l-th successive approximation, and the constant factors β have the form

T ul t  i t,ul  dt  i ul   0

T  i2 t,ul  dt 0 . The value  ul  is chosen so that  ull1 t  dt  , that is

T 3 2 0 i1  T 2   i2 t,ul  dt   0   T l 2   u t  i t,ul  dt  T T 3ull1 t 2dt  3 T ull1 t 2 dt  2 ul  3  0   i2 t,ul  dt  2 ul   0 i1 i1 0 i1 0   i31  T0 ulT0 ti2ti,ut,luld t dt 2 . Thus,  ul   k  i31  T0 ulT0 ti2ti,ut,luld t dt   . 2 1 of an approximate control to stop one’s choice. Having fixed some approximate control u

Thus, following [21], a sequence is constructed that approximates the optimal control u for the entire observation interval [0, T]. Note that in [21] it is not indicated at which step in the construction l t  , we

As a first approximation u

obtain a time-varying control (in this problem, the time-varying [0, T] coefficients of the PIDcontroller). However, this control will be deprived of the possibility of correction if the value of the control error is exceeded, caused, for example, by fixing an insufficiently large approximation step or by the presence of an unexpected random perturbation of the function f(t) (although it is considered deterministic).

It is assumed that the time T is not the entire observation time of the controlled dynamic process, but only its rather small segment of the algorithmic stepwise sampling (not the time sampling step used in the calculations), that is, the entire observation time will be divided into time segments of length Т: [0, Т], [0, 2Т], [2Т, 3Т], … Within each individual segment [sТ, (s + 1)Т], s = 0, 1, 2, …, we will apply the method of software adjustment of gain coefficients.

1,s

, we take the constant values u2,s1  sT  obtained at the previous step of the algorithm with the help of control u2,s1  sT  at the time sT of the corresponding time interval of the algorithmic discredit [(s – 1)Т, sТ]. In this case, the initial point x0 from (22) will also be given l by the value xsT ,u2,s1  obtained at the boundary time. Thus, at this stage, we define the following approximation:

ui2,s t   u2,s1  sT i u2,s1 sT  i t,u2,s1 sT ; where the functions  i t,u2,s1 sT  , i = 1, 2, 3 are found by expressions (36), taking into account the time shift t = t + sT for all functions included in this expression:

 i t,u2,s1 sT   2xsT ,u2,s1  Z s 1T ,u2,s1 sT   Z s 1T ,u2,s1 sT   s1T Z 1  ,u2,s1  sT , f   d    x  sT ,u2,s1  Z  s  1T ,u2,s1  sT   Z  s  1T ,u2,s1  sT   sT 

s1T    Z 1  ,u2,s1  sT , f   d  Bi t .

sT  Thus, we obtain the final expressions for determining α and βi:

s1T u2,s1  sT  i t,u2,s1  sT  dt i u2,s1  sT   0 . s1T   i2 t,u2,s1  sT  dt 0 (39) (40) (41) (42)

The input parameters of helicopters TE mathematical model are the values of atmospheric parameters (h – flight altitude, TN – temperature, PN – pressure, ρ – air density). The parameters recorded on board of the helicopter (nFT – free turbine rotor speed) reduced to absolute values according to the theory of gas-dynamic similarity developed by Professor Valery Avgustinovich (table 1). We assume in the work that the atmospheric parameters are constant (h – flight altitude, TN – temperature, PN – pressure, ρ – air density) [22].

To form the training and test subsets, cross-validation [23] was used to estimate the values of nFT, the results of which are shown in fig. 4.

In order to establish the representativeness of the training and test samples, a cluster analysis [22, 24] of the initial data was carried out (table 1), during which seven classes were identified (fig. 5, a). After the randomization procedure, the actual training (control) and test samples were selected (in a ratio of 2:1, that is, 6 7% and 33 %). The process of clustering the training (fig. 5, b) and test samples shows that they, like the original sample, contain seven classes each. The distances between the clusters practically coincide in each of the considered samples, therefore, the training and test samples are representative.

An important issue is the assessment of the homogeneity of the training and test samples. To do this, we use the Fisher-Pearson criterion χ2 [22, 25] with r – k –1 degrees of freedom:

 2  min r1 i  mi npnipi   ; (43) where θ – maximum likelihood estimate found from the frequencies m1, …, mr; n – number of elements in the sample; pi(θ) – probabilities of elementary outcomes up to some indeterminate k-dimensional parameter θ.

The specified statistics χ2 allows, under the above assumptions, to test the hypothesis about the representability of sample variances and covariances of factors contained in the statistical model. The area of acceptance of the hypothesis is  2   nm, , where α – significance level of the criterion. The results of calculations according to (43) are given in table 2.

Calculating the value of χ2 from the observed frequencies m1, …, mr (summing line by line the probabilities of the outcomes of each measured value) and comparing it with the critical values of the distribution χ2 with the number of degrees of freedom r – k –1. In this work, with the number of degrees of freedom r – k –1 = 13 and α = 0.05, the random variable χ2 = 0.687 did not exceed the critical value from table 2 is 22.362, which means that the hypothesis of the normal distribution law can be accepted and the samples are homogeneous.

The neural network was trained by the above method for 1000 epochs, the training accuracy characteristic is shown in fig. 6, the steady-state mean square error is 0.382. Fig. 7 shows the results of the neural network training validation test, from which it can be seen that the average value of the gradient is, and the optimal value of the training coefficient does not exceed 1.

The circuit for maintaining of helicopters TE free turbine speed based on a neural network (fig. 3) has one caveat: although the neural network controller finds a minimum, this minimum is only a local minimum and it cannot be argued that this is the optimal solution. To avoid choosing a suboptimal local minimum in the objective function, it is required to repeat the optimization process several times and choose the best result. It is possible that by setting different initial values of the PID-controller parameters, different optimal controller parameters will be obtained. In addition, during the training of the neural network, several random perturbations are used during each cycle and the reaction time of the system is taken into account to calculate the objective function and its gradient. This ensures obtaining local optimal coefficients of the PID-controller for various disturbances affecting the system. In addition, by changing the step size and the number of perturbations, the sensitivity of the results increases during the search process for the controller coefficients. The process of searching for controller parameters is terminated when a steady state of the system is determined, which is achieved through the calculation of a linear regression of the most recent estimates and iteration until the step response reaches a steady state value with an error of 1 % (fig. 8).

It should be noted that the adequacy of the model represented by the neural network directly depends on the training process. There are a number of parameters that affect the quality of training: training rate coefficient (assumed 1.5); number of neurons in the hidden layer (assumed 20); length of the delay line of input signals (5 is accepted); number of completed training epochs (assuming 1000 training epochs).

As a criterion for assessing the quality of training, you can use the final total standard deviation for the epoch, which is determined according to the expression:

Eepoch  1 M 1 m yk out  yk 2 ; (44)

M i1  2 k1  where M – number of training sample elements. The training of the neural network continues until one of the stopping criteria is met. For example, the training time will not run out, a certain number of training epochs will pass, or the error per epoch will not exceed the minimum required threshold.

Let’s carry out a number of additional research that determine the influence of training parameters on the quality of the neural network, namely: 1. Influence of the training rate coefficient. 2. Influence of the number of neurons in the hidden layer. 3. Influence of the delay length of input signals. 4. Influence of the number of training epochs passed.

The results of the research are given in table 3–6 and in fig. 9, where: a – diagram that determines the influence of the training rate on the final standard deviation; b – diagram that determines the effect of the number of hidden neurons on the final standard deviation; c – diagram that determines the influence of the length of the delay line on the final standard deviation; d – diagram that determines the effect of the number of epochs passed on the final standard deviation. c d Figure 9: Research results that determine the impact of training parameters on the quality of a neural network Table 3 Influence of the training rate coefficient on the resulting error

Training rate coefficient Final standard deviation 0.4 1.722 1.5 1.471 3.6 1.471 5.0 1.723

The conducted studies, which determine the influence of training parameters on the quality of the neural network, allow us to preliminarily state:

1. At low values of the training rate coefficient, slow convergence is observed and there is a risk of hitting a local minimum, while at the same time, the accuracy of hitting an extremum point increases. With large values, it is impossible to achieve a small error, since each step we skip past the extremum.

2. A larger number of neurons in the hidden layer allows you to more accurately describe the training sample at the cost of computing power. At the same time, there is a danger of overfitting when the neural network reproduces noises and distortions in the training sample and is not able to adequately represent the circuit for maintaining the speed of the free turbine of helicopters TE.

3. A large length of the delay line increases the number of input neurons and, consequently, the computational load. At the same time, a longer delay line describes the dynamic properties of the object better and avoids contradictions in training, when the object can change its behavior depending on past influences.

4. The final error depends logarithmically on the number of epochs passed, as a result of which it is not rational to use a large number of epochs, since the computation time increases exponentially and there is a risk of retraining the neural network.

5. Results

a b c d e f Figure 10: Free turbine frequency rotation transient’sgprocesses resulting diagram

6. Discussions

A comparison was made of the quality of control of the transient response by free turbine frequency rotation and the stability margins provided by each of the considered controllers (Fig. 10). Fig. 12 shows a diagram of the joint arrangement of the transient curves according to the free turbine speed during the operation of various controllers: 1 – linear PD-controller; 2 – PD-controller with reduced kd; 3 – quadratic controller; 4 – PD-controller with a variable amplification factor; 5 – fuzzy logical Pcontroller; 6 – fuzzy logical P-controller with a corrective differential link; 7 – PID-controller developed on the basis of a neural network. Table 7 presents a numerical analysis of the quality of operation of the circuit for maintaining the free turbine speed with various controllers.

Controller type Linear PD-controller PD-controller with reduced kd Quadratic controller PD-controller with a variable amplification factor Fuzzy logical P-controller Fuzzy logical P-controller with a corrective differential link PID-controller developed on the basis of a neural network

Static accuracy, %

Speed, seconds

Transition process nature

Throw value, % ± 0.2 ± 0.3 ± 0.2 ± 0.2 ± 0.1 ± 0.2 ± 0.05

From fig. 12 and table 7 it follows that the best quality of the transient process in terms of the free turbine speed is provided by the PID-controller developed on the basis of a neural network. The transient response during operation of this controller differs from others in its higher speed (about 3.5 seconds) and the absence of oscillations (curve 7). The remaining controllers are inferior in terms of providing the required quality of control. According to [13, 14], the transient response in terms of free turbine rotational speed, obtained with the operation of a quadratic controller, does not differ in quality from the characteristic obtained with the operation of the original linear PD-controller (curves 1 and 3). The process is characterized by the presence of fluctuations and overshoot of about 4 %. The oscillation amplitude is 0.2 %. The transition process time is about 20 seconds. When analyzing the characteristics of transients obtained during the operation of a PD-controller with a reduced proportional gain, a variable gain controller, and a P-type fuzzy logic controller, it can be seen that these characteristics differ slightly from each other. The transient process has an aperiodic character (curves 2, 4, 5 and 6) and almost the same speed (for curve 2, the transition process time is 15 seconds, for curves 4, 5 – 16.5 seconds, for curve 6 – 6 seconds).

A comparative analysis of the accuracy provided by each of the considered controllers (Fig. 12) is given in table 8, which shows the probabilities of errors of the 1st and 2nd kind in determining the optimal parameter nFT. Probability of error in determining

Type 1st error Type 2nd error

Controller type Linear PD-controller PD-controller with reduced kd Quadratic controller PD-controller with a variable amplification factor Fuzzy logical P-controller Fuzzy logical P-controller with a corrective differential link PID-controller developed on the basis of a neural network 1.95 1.74 1.46 1.32 1.08 0.97 0.58 1.42 1.21 1.03 0.95 0.77 0.64 0.22

As can be seen from table 8, the use of a PID-controller developed on the basis of a neural network provides an improvement in the probability of errors of the 1st and 2nd kind compared to the controllers developed in [13, 14] by 35...85 %. 7. Conclusions

1. The method of maintaining the helicopter rotor speed has been further developed, which, through the use of a PID-controller developed on the basis of a dynamic neural network of direct data transmission based on neurons with a radial basis activation function in the first layer and adalines – neurons with a linear activation function, in the second layer, made it possible to improve the quality of the transient process in terms of helicopter aircraft turboshaft engine free turbine rotation frequency, which consists in increasing the speed up to 3 seconds, increasing the statistical accuracy up to ± 0.05 % and eliminating the overshoot of parameters.

2. The circuit for maintaining the helicopter aircraft turboshaft engine free turbine speed has been improved, which, due to the use of a PID-controller with neural network tuning of the gain factors, made it possible to automatically adjust the amplification factor and, thereby, reduce the time of the transition process.

3. The neural network training method based on the method of programmatic gain adjustment, developed by Professor Volodymyr Zubov, was further developed, which, by integrating direct data transmission into a dynamic neural network based on neurons with a radial-basis activation function in the first layer and adalines – neurons with a linear activation function, in the second layer, made it possible to reduce the error of its training for the problem of researching the transient process in terms of helicopter aircraft turboshaft engine free turbine rotation frequency to 0.005 %.

4. The structure of the PID-controller with an auto-tuning unit based on a neural network has been improved, in which, due to the use of a dynamic neural network of direct data transmission based on neurons with a radial basis activation function in the first layer and adalines – neurons with a linear activation function, in the second layer, as a functional converter that generates the desired amplification factor of the PID-controller for a set of input and output signals, made it possible to improve the probability of errors of the 1st and 2nd kind in comparison with the known controllers by 35 ... 85 %.

8. References

References

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