This research presents the development of an intelligent controller for the helicopter turbosha engines gas temperature, aimed at compensating for the measuring sensor's inertial delays and optimizing transient processes. The aim is to compensate for inertial delays τ ≈ 0.025 seconds and optimize transient processes. The method is based on a double summation circuit with a channel selector, comparing signals from a thermocouple and a gas-generator rotor speed sensor, and an adaptive observer based on Pade approximation and Taylor series expansion provides a prediction of the state at t + τ. The intelligent control law includes a proportional-integral-differential structure with the coefficients γi correction via gradient descent. To refine the delay estimate, a two-layer fully connected multilayer perceptron with a SmoothReLU activation function is implemented trained on flight test data was implemented, which reduced τ to 0.016 seconds (-36 %). This module allows to approximate nonlinear relations between input features and the delay value, which ensures the control signal's timely correction and the system's adaptation to changing operating conditions. Modeling of the system in the Matlab Simulink environment demonstrated a significant improvement in the transient process characteristics: overshoot was reduced from 8.0 to 1.5 %, and the mode establishment time was reduced from 4.2 to 3.3 seconds. The neural network module testing showed high predicting accuracy (99.537 % with losses of 0.511 %), confirmed by the determination coefficient R2 = 0.9717. The neural network use made it possible to reduce the delay value to 0.016 seconds, which corresponds to an improvement of 36 % compared to traditional methods. The obtained results indicate a proposed technique's high potential for improving the helicopter turbosha engines automatic control system's dynamic accuracy and stability.
The aviation industry's evolution is closely related to the helicopter's new types of operation (e.g., Eurocopter AS350, Eurocopter EC145, Eurocopter EC225 Super Puma, etc.), characterized by high speed and long range, which requires increasingly sophisticated automation control systems [1, 2]. The manual and automatic control synthesis, as well as the latter rapid development, forced designers to create not only visual devices for humans but also a sensor set, which signals directly affect the automatic system's subsequent links [3]. The most important characteristic of helicopter turbosha engines (TE) is the gas temperature in front of the compressor turbine, since it significantly determines both the power plant efficiency and its reliability [4, 5]. Thus, these parameters maintaining accurate values is critical to ensuring the helicopter TE's stable operation.
To maintain the set parameters at a fixed throttle position or to change them according to a given law depending on flight conditions and operating modes, the helicopter TE automatic control systems (ACS) are used [6]. The main requirement for modern ACS is compensation for the temperature sensor's inertia so that the measuring devices function without delays [7, 8].
The main requirements for helicopter TE ACS include the high static accuracy to maintain a given range (0.3…0.5%), efficiency with a quick response to control (2…3 seconds), and transient processes close to monotonous, which ensures regulation without drops (2…4%) and with a minimum stabilization time [9, 10]. These strict and contradictory conditions cannot be solved by standard methods, which creates the developing complex task of multifunctional automatic control systems for helicopter TE.
There are known helicopter TE ACS [11–13] that affect a single control parameter, which is the fuel consumption in the combustion chamber, which includes measuring devices for input signals, comparison elements, and an actuator, and the signal from the gas temperature controller directly adjusts the rotor speed controller setting. This scheme's main disadvantage is the reduction in stability reserves and permissible gain factors, which worsens static and dynamic accuracy, and to eliminate this effect, systems with a selector are used, excluding controllers’ joint operation area and thereby improving the system's overall characteristics [14, 15]. The helicopter TE gas temperature control system presented in [16] uses a correction link with a differentiator, multiplier blocks, and adders to compensate for the dynamic error caused by the first temperature sensor inertia. The correction coefficient at the derivative changes based on the current gas flow rate signal, which ensures high measurement and control accuracy. However, the scheme's key disadvantage is that in transient modes, the rotor speed and temperature channel's interaction through the selector is not taken into account, which reduces the inertial delay compensation efficiency.
The helicopter TEs gas temperature control using the controller presented in [17] is carried out through correcting devices for the control channels transfer functions changing. This allows for minimizing overshoot and ensuring stable engine operation in various modes. However, its key disadvantage is related to the inertial delays compensation: since the transfer functions correction depends on the velocity pressure, changes in the system occur with a delay, which can cause a temporary mismatch between the required and actual parameters, especially with sharp changes in engine operating modes.
In [18] the engine temperature controller is presented that uses a double summation scheme, in which the measured temperature signal is compared with the set value, and the correction is carried out by a nonlinear element that compensates for the delay in the compressor turbine blades heating. Due to the inertial link, the rotor speed signal is corrected, which eliminates sharp increases in gas temperature in front of the compressor turbine, thereby improving the transient processes quality. As a result, the fuel consumption changes proportionally to the summing amplifier's output signals, ensuring stable and accurate regulation of the engine operation. This controller's main limitation is the long return time to the original mode, due to the isodromic feedback inertia. The intelligent component introduction will allow dynamically optimizing the control parameters and reducing the system response time to changes in operating conditions.
The helicopter TEs intelligent gas temperature controllers integrate modern automatic control methods, including correction links with differentiators, nonlinear elements, and double summation schemes, which allow achieving high accuracy and stability of operation [19, 20]. They ensure overshoot minimization and fast system response, which is critically important for modern helicopters with high-speed and long-range characteristics. However, these systems key disadvantage is insufficient compensation for the measuring sensors inertial delays, which leads to the parameters temporary mismatch in transient modes. Some schemes, for example, [21, 22], aimed at regulating fuel consumption, experience a decrease in stability margins and permissible gain factors, which negatively affects the static and dynamic accuracy of regulation.
Thus, a goal of this paper is to develop the helicopter TEs gas temperature intelligent controller, compensating for the measuring sensors inertial delays and optimizing transient processes. To achieve the goal the following objectives are formed: (i) developing the intelligent method for regulating gas temperature, (ii) designing the adaptive algorithms and corrective links with differentiators implementation to control parameters dynamic adjustment for eliminating the time discrepancy between the required and actual values. Finally, it’s expected to increase the system's static and dynamic accuracy and improve its stability, as well as reduce the engine stabilization time.
The proposed controller (Figure 1) is based on a double summation circuit [18]. It is based on a comparative analysis of signals received from thermocouples and a gas-generator rotor speed sensor [23], using an inertial link, a comparison element, and a nonlinear unit for adjusting the fuel supply. This ensures compensation for the delay in heating up the turbine and the temperature regime stabilization. Under normal conditions, the signals from the speed sensor compensate for each other, and when the regime changes abruptly, a correcting pulse appears, leading to a change in fuel consumption proportional to the largest deviation in the summing amplifiers. Under transient conditions, the signal delay through the inertial link leads to the different signal appearance in the comparison element, which is then amplified through the summing devices. The control channel selector selects the largest received deviations, and the nonlinear element with an exponential or parabolic characteristic reduces the specified temperature setting, compensating for the delay in heating up the compressor turbine blades.
The proposed scheme for the helicopter TE gas temperature regulating integrates adaptive algorithms and correcting links with differentiators, which allows for dynamic optimization of control parameters and time discrepancies elimination between required and actual values. Adaptive algorithms analyze the transient processes current dynamics and correct the gain factors and inertial links time constants in real time, thus providing a more accurate and timely response to changes in the engine operating mode. The integration of differentiators into measuring circuits allows for predicting trends in temperature conditions, compensating for the delay in turbine warm-up, and a nonlinear element with an exponential characteristic synchronizes system responses, minimizing sharp transient fluctuations. The result is an ACS of adapting to changing operating conditions, increasing the helicopter TE reliability and efficiency.
The proposed controller features include the adaptive control module introduction, which is connected between the summing amplifiers and the control channel selector. This module performs the transmission coefficients real-time correction and the inertial links time constants, responding to the transient processes dynamics and eliminating the time discrepancy between the specified and actual parameters. The adaptive differentiator introduction integrated into the measuring circuit between the thermocouple unit and the first summing amplifier facilitates preliminary signal processing, predicting temperature change trends, and compensating for the delay in heating the compressor turbine blades. The proposed controller also includes a transient mode monitoring unit connected to the actuator to form feedback and dynamically optimize the control algorithms.
Thus, the scientific novelty consists in obtaining further development of the helicopter TE gas temperature controller according to the double summation scheme, which, due to the adaptive algorithm’s introduction integrated with corrective links equipped with differentiators, allows control parameters, real-time optimization, and inertial delays compensation of measuring sensors. This ensures synchronization between the required and actual temperature modes, significantly increasing the helicopter TE accuracy, stability, and efficiency.
Based on [17, 18, 23, 24], it is assumed that the gas temperature dynamics is described by the following equation with a delay of the form:
T˙ (t )=a⋅ T (t )+b⋅ u (t τ )+d (t ) , where T(t) is the gas temperature at time t, u(t) is the control action (for example, change in fuel consumption), a and b are the system coefficients, τ is the measuring inertial delay or actuator links, and d(t) is the external disturbance.
The delayed system transfer function based on the Padé approximation [25] for exponential delay
T^˙ (t )=a⋅ T^ (t )+b⋅ u (t )+ K obs⋅ (T (t ) T^ (t )) , where T^ is the temperature estimate and Kobs is the observation coefficient (can be chosen as a vector for more complex models, for example, as in [26, 27]).
To compensate for the delay τ, the state prediction at time t + τ is used using a Taylor series expansion of the form:
To compensate for the measurement delay, an adaptive observer is used that estimates the system state. Let the observer have the form:
In practical implementation, one can limit oneself to the first or second term of the expansion, as shown, for example, in [16, 18]. To estimate the temperature's second derivative, a difference scheme of the form is used: is: from where the system transfer function has the form:
Based on the adaptive observer mathematical model (
The coefficients ki(t) are adjusted in real time using adaptive laws, for example, according to the gradient descent scheme: k˙ i (t )= γi⋅ e (t )⋅ ϕi (t ) , (i = 1, 2, 3, … e(t) = Tref(t) − T(t). e τ⋅ t ≈
τ 1 ⋅ s 2 τ 1+ ⋅ s 2
,
G ( s )=
b⋅ (1
τ
2
(
2
T (t ) 2⋅ T (t T^¨ (t ) ≈ Δ t )+T (t 2⋅ Δ t ) Δ t 2 . where γi > 0 are the adaptation rates, ϕi(t) are the signal functions depending on the system's current state, t for example, ϕ1 (t )=T ref (t ) T^ (t + τ ), ϕ2 (t )=T˙ ref (t ) and ϕ2 (t )=∫ (T ref ( s ) T^ ( s + τ )) ds. 0 To determine the optimal parameters, the quality functional can be minimized:
t+T p
J = ∫ (Q⋅ (T ( s ) T ref ( s ))2+ R⋅ (u ( s ))2) ds , (
t where Q, R > 0 are the weighting coefficients, Tp is the predict horizon.
To improve the compensation accuracy for inertial delays, it is proposed to implement a neural
network module [28–30] that corrects the delay estimate:
τ^ ( s )=τ 0+ f NN (φ (t )) ,
(
Thus, the scientific novelty of the developed intelligent method lies in the adaptive observer with state prediction implementation and a neural network module for compensating for inertial delays, which allows for the control parameters real-time optimization.
Within the developed method framework, theorem 1, “On adaptive stability and convergence of a closed system of intelligent gas temperature control with inertial delays compensation,” is formulated. According to the developed intelligent method, the system dynamics is given by the equation T˙ (t )=a⋅ T (t )+b⋅ u (t τ )+d (t ) provided that the disturbances d(t) are bounded and the adaptive observer has the form T^˙ (t )=a⋅ T^ (t )+b⋅ u (t )+ K obs⋅ (T (t ) T^ (t )), in this case, the control action t is determined by the law u (t )=k1 (t )⋅ (T ref (t ) T^ (t + τ ))+ k2 (t )⋅ T˙ ref (t )+k3 (t )⋅ ∫ ¿ ¿ 0
T^ ( s + τ )) ds, and the adaptive coefficients change as k˙ i (t )= γ i⋅ e (t )⋅ ϕi (t ) , (i = 1, 2, 3, …). If there exists a Lyapunov function of the form V (t )= 1 ⋅ (e (t ))2+∑3 1 ⋅ (ki (t ) ki¿ (t ))2, where ki¿ are 2 i=1 2⋅ γi the coefficient’s optimal values, and if for some α, β > 0 the inequality holds V˙ (t ) ≤ α ⋅ V (t )+ β ⋅‖d (t )‖2, then the closed system is uniformly bounded in finite time, and the tracking error e(t) = Tref(t) − T(t) asymptotically tends to an arbitrarily small neighborhood of zero, provided that the perturbations d(t) are sufficiently small.
Proof of Theorem 1.
V (t )= 1 ⋅ ( e (t ))2+∑3 1 ⋅ ( ~ki (t ))2 , (
2 i=1 2⋅ γi where
~ e(t) = Tref(t) − T(t), ki (t )=ki (t ) ki¿ (t ) (i = 1, 2, 3, …), and ki¿ are the coefficient’s optimal (constant) values, and γi > 0 are the adaptation constants.
3 1 ~ ~ V˙ (t )=e (t )⋅ e˙ (t )+∑ ⋅ ki (t )⋅ k˙ i (t ) ,
i=1 γi
Since ki¿ are constant, then ki¿ (t )=k˙ i¿. Taking into account the adaptation law (
i=1
Since the gas temperature dynamics is given by equation (
3
e˙ (t )= ∑ ki¿⋅ e (t )+δ (t ) , (
i=1 where δ(t) combines model errors, delay compensation and external disturbances, and ϕi(t) are signal functions that depend on the system’s state.
V˙ (t )=e (t )⋅ ( 3 ∑ ki¿⋅ e (t )+δ (t )) i=1 3 ~ ∑ ki (t )⋅ e (t )⋅ ϕi (t ) . i=1
V˙ (t )= (e (t ))2⋅ ∑3 ki¿⋅ ϕi (t ) e (t )⋅ ∑3 ~ki (t )⋅ ϕi (t )+e (t )⋅ δ (t ) .
i=1 i=1 3 3 ∑ ( ki¿+ ~ki (t ))⋅ ϕi (t )=∑ ki (t )⋅ ϕi (t ) , i=1 i=1 3 and with the ϕi(t) and ki(t) correct choice it is assumed that the total term ∑ ki (t )⋅ ϕi (t ) is a positive i=1 definite function, that is, there exists λ > 0 such that
3 ∑ ki (t )⋅ ϕi (t ) ≥ λ>0.
i=1
Thus, the assessment can be written as:
(
t →∞ α
Hence, the final assessment was received: Denoting α = λ and β = λ⋅ (e (t ))2+ λ ⋅ (e (t ))2+ 1 ⋅ (δ (t ))2= 2 2⋅ λ λ ⋅ (e (t ))2+ 1 ⋅ (δ (t ))2 . 2 2⋅ λ
⋅ (e (t ))2, this means that the error e(t) asymptotically approaches a zero neighborhood whose size is determined by the quantity ¿ ‖δ (t )‖. Provided that the disturbances t ≥ 0 and delay compensation errors are sufficiently small, this neighborhood can be made arbitrarily small.
Thus, it is proved that the closed system with the chosen adaptive control law and the coefficient adaptation law is uniformly finitely determined, and the tracking error e(t) = Tref(t) − T(t) asymptotically tends to an arbitrarily small neighborhood of zero. This means which proves the theorem on adaptive stability and convergence.
Based on [31–33], we propose to use a deep fully connected (feedforward) neural network to approximate the function fNN(φ(t)) (Figure 2). The neural network architecture is defined by an input layer of dimension n, where φ(t) ∈ ℝⁿ, followed by one or more hidden layers with nonlinear activation functions; in particular, for the two-layer MLP architecture example [34], the first hidden layer contains m1 neurons, and the second contains m2 neurons, a er which the output layer, consisting of one neuron with linear activation, forms the correction value.
In the general case, for a neural network with L layers (excluding the input), the input is initialized as a(0) = φ(t). For each hidden layer l = 1, …, L − 1, the calculation is performed: where W(l) is the weight matrix, b(l) is the bias vector, and σ(l)(•) is the activation function (e.g. ReLU,
z(l)=W (l)⋅ a(l 1)+b(l) ,
a(l)=σ (l)⋅ ( z(l)) ,
z(l)=W (L)⋅ a(L 1)+b(L) ,
f NN ( φ (t ))=W (L)⋅ a(L)= z( L) ,
τcorr(t) = τ0 + z(L).
(
T^ (t + τ corr (t ))=T^ (t )+ τ corr (t )⋅ T^˙ (t )+
⋅ T^¨ (t )⋅ ( τ corr (t ))2+.. .
Thus, the neural network module for correcting the delay estimate is implemented according to the
scheme τcorr(t) = τ0 + fNN(φ(t)), where the function fNN(φ(t)) is approximated by a neural network
constructed according to the scheme (
Θ ← Θ
η⋅ ∇ Θ L (Θ ) , δ(L)= ∂∂zL(L) =a(L) ( τ actual τ 0) .
δ(l)=W (l+1)T ⊙ σ ' (l) ( z(l)) , W (l)=W (l)
η⋅ δ(l)( a(l 1))T , b(l)=b(l) η⋅ δ(l) .
Analy cal expression
z(
1 N 2 L
L (Θ )= 2⋅ N ⋅ ∑i=1 (τ actual (t i) (τ 0+ f NN ( φ (t i)))) + λ⋅ ∑l=1 (W (l))22 , where λ > 0 is the regularization coefficient [36].
where η > 0 is the training rate.
At the same time, gradients are calculated at each layer using backpropagation. For example, for the last layer, the error at the output layer is calculated as:
For each previous layer l = L − 1, L − 2, …, 1 the following is determined: where ⊙ denotes element-wise multiplication and σ ' (l) is the activation function derivative.
u (t )=k1 (t )⋅ e (t )+ k 2 (t )⋅ e˙ (t )+ k 3 (t )⋅ ∫ e (t ) dt , then, taking into account the delay correction, it can be supplemented as follows. For example, the parameter τcorr(t) can be taken into account when predicting the state through the expansion in a
It is accepted that, in general, the proposed intelligent gas temperature controller (Figure 3) is a twochannel controller: the gas temperature in front of the compressor turbine regulating channel and the gas generator rotor speed regulating channel [23, 38] (the free turbine rotor speed regulating channel [39] is not taken into account).
U ={U 1 , U 1 ≤ U 2 ,
U 2 , U 1>U 2 , where H1 is the object’s (engine TV3-117) transfer function in the first control channel (the gas 1 generator rotor speed channel) H 1= , H2 is the object’s (engine TV3-117) transfer function in the
T ⋅ s second channel (the gas temperature in front of the compressor turbine channel) H2 = 1, where: where Y10 and Y20 are constant settings.
W contr=
kcontr s⋅ ( τ ⋅ s+1)
, W1 = k1, W2 = k2, W I ( s )=
k1⋅ kcontr s⋅ (T ⋅ s+1)⋅ ( τ ⋅ s +1) , W I ( jω)=
k1⋅ kcontr jω⋅ (T ⋅ jω+1)⋅ ( τ ⋅ jω+1) ¿ k1⋅ kcontr⋅ ( τ +T )2⋅ ω4
2 2 ( τ +T )⋅ ω4+ω2⋅ (1 τ ⋅ T ⋅ ω ) = j⋅ k1⋅ kcontr
2 =¿ ( τ +T )⋅ ω2+ jω⋅ (1 τ ⋅ T ⋅ ω ) k1⋅ kcontr⋅ (1 τ ⋅ T ⋅ ω ) 2
2 2 ( τ +T )⋅ ω4+ω2⋅ (1 τ ⋅ T ⋅ ω ) W I ( s)=
k2⋅ kcontr , s⋅ ( τ ⋅ s+1) W II ( jω)= k2⋅ kcontr τ ⋅ ω2+ jω = k2⋅ kcontr⋅ τ ⋅ ω τ 2⋅ ω4+ω2 2 j⋅ k2⋅ kcontr⋅ ω τ 2⋅ ω4+ω2 , where k1 = 10, k2 = 1, kp = 10, Т = 0.5 second, τ = 0.025 second [37, 38].
W I ( s ) W II ( s)
k1⋅ kcontr k2⋅ kcontr⋅ (T ⋅ s+1) Φ ( s )=
= 2+W I ( s )+W II ( s) 2⋅ (T ⋅ s+1)⋅ (T ⋅ s+1)⋅ s+k1⋅ kcontr+k2⋅ kcontr⋅ (T ⋅ s+1) (41) (42) (43) (44) (45) (46) (47) Φ ( jω)= 2⋅ k 2⋅ k contr⋅ T ⋅ ω2)2+((2+k2⋅ k contr⋅ T )⋅ ω
The Nyquist hodographs constructed for the system’s individual loops with the values k1 = 10, k2 = 1, kcontr = 10, T = 0.5 second, τ = 0.025 second [23, 38] demonstrate that in this automatic control system with a minimum selector, the closed loop WІ(s) is unstable, while the closed loop WІІ(s) is stable (Figure 4).
Nyquist Hodographs 4 3 2 s i x A1 y r a0 n i g a-1 m I -2 -3 -4
0 dB 2 dB
-2 dB 4 dB -4 dB 6 dB -6 dB 10 dB-10 dB
WII(s)
WI(s) -4 -2 0 2 4 Real Axis 6 8 10
The transformed system’s Ф(s) linear link amplitude-phase characteristic is shown in Figure 5. The condition for the oscillation’s occurrence in such a nonlinear system is the equivalent linear part’s Ф(s) hodograph’s intersection point with the complex nonlinearity coefficient’s hodograph. In this case, the latter corresponds to the negative segment of the real axis in the range from –1 to –∞. It follows from this those oscillations with a frequency of ω ≈ 12.227 s–1 (f ≈ 1.96 Hz) can occur in this system.
Therefore, this system is characterized by two differential equations set and an equation that determines the selector switching:
W nTC ( s )=W k 1 ( s )⋅ W TG¿ ( s ) , W T¿G ( s )+W k 1 ( s )⋅ W TG¿ ( s )=1 ,
The system remains stable when the condition is met: Y10 = 1, Y20 < 1 (if Y20 > 1, oscillations occur). In this region, the output variables change exponentially. The second circuit is closed (Figure 3), and Y2 reaches the steady-state value in tcontr = 0.25…0.3 seconds. At the setpoint value Y20 > 1, oscillations occur in the system that do not damp. Their frequency remains unchanged and is f ≈ 1.96 Hz; T = 0.427 seconds. The oscillation’s amplitude increases as the setpoint increases.
To eliminate temperature overload, corrective elements are introduced [40]. The selection condition is determined by the following expression [40, 41]:
U nTC=U T¿G . ε2 = Y20 – Y2 = 0,
ε1 = ε2. (54)
In this research, the correcting link’s Wk1(s) and Wk2(s) transfer functions are determined based on the selection conditions. For the first condition (53):
U nTC=W nTC ( s )⋅ ε1 ,
U TG¿=W TG¿ ( s)+W k 1 ( s )⋅ W T¿G ( s)⋅ s+W k 2 ( s)⋅ W TG¿ ( s )⋅ ε2 .
W nTC ( s )⋅ ε1
W TG¿ ( s )+W k 1 ( s )⋅ W TG¿ ( s )⋅ s +W k 2 ( s )⋅ W T¿G ( s )⋅ ε2 ,
For modeling and calculations in this study, the parameters of the TV3-117 engine, which is the
(0.21⋅ s +1)⋅ xnTC=(0.229⋅ s +1.306 )⋅ xG ,
T where xnTC is the output signal for the turbocharger rotor speed, xT¿G is the output signal for the gas temperature in front of the compressor turbine, xGT is the input signal for fuel consumption.
From (66), (67) the gas generator rotor speed W GnTTC and the gas temperature in front of the compressor turbine W TGG¿T transfer functions are obtained, identical to those in [38]:
W nTC ( s ) W k 1 ( s)⋅ W TG¿ ( s )=1 ,
From (67) and (68) it follows that the gas generator rotor speed controllers W nTC ( s ) and the gas temperature in front of the compressor turbine W TG¿ ( s ) transfer functions have the form:
Then, according to (60), (61), (64), (65), the first and second correction links Wk1(s) and Wk2(s) transfer functions are obtained, respectively, for conditions (53) (I) and (54) (II): W GnTTC ( s )=
.
W nTC ( s )=
W T¿G ( s )=
.
W (kI1) ( s )=
,
W (kI2) ( s )=0.522⋅ s +2.
It is noted that in [38], the analytical expressions describing the first and second correcting links
Thus, in the refined transfer functions Wk1(s) and Wk2(s) (72)–(75) compared to (76)–(77), a decrease in the variable s orders is observed, which allows eliminating high-order terms that increase the (62) (63) (64) (65) (66) (67) (68) (69) (70) (71) (72) (73) (74) (75) (76) (77) system’s dynamic sensitivity. In this case, only those terms are preserved that to the greatest extent determine the phase and amplitude characteristics necessary to compensate for inertial delays. The analysis shows that the dominant low-order terms (e.g., constant and linear in s) provide adequate delay suppression and maintenance of the required transient process, minimizing overshoot and stabilizing the ACS. This approach simplifies the correcting link’s model, reduces the computational load and reduces the high-frequency noise amplification, which significantly increases the adaptability and reliability of the helicopter TE control system.
According to Figure 3, the Matlab Simulink 2014b so ware package has constructed simulation schemes in two versions: without correction links (Figure 6a) and with correction links (Figure 6b). The simulation results are shown in Figure 7. It is noted that the actuator’s (isodromic controller’s) transfer function is adopted, according to [38], in the form:
3⋅ (0.56⋅ s +1) 1.68⋅ s +3 W І C ( s )= = . (78)
s⋅ (0.02⋅ s +1) 0.02⋅ s2+ s 1 0.229s+1.306
Gas generator rotor speed controller transfer Fcn
1 0.522s+3 Gastemperature controller
transfer Fcn Gas generator rotor
speed sensor (normalized values)
Comparison element1 Gas temperature
sensor (normalized values)
Comparison element2 Gas generator rotor
speed sensor (normalized values)
Comparison element1
0.522s+3 0.229s+1.306 Gas generator rotor speed correctionlink 0.522s+2
s+1 Gas temperature correctionlink
Ful adder1
Key
min Minimum selector a 1 0.229s+1.306
Gas generator rotor speed controller transfer Fcn
Comparator < 2.298s+3 0.02s2+s Isodromic controller transfer Fcn min Minimum selector 2.298s+3 0.02s2+s Isodromic controler transfer Fcn 0.229s+1.306
0.21s+1 Gas generator rotor speed transfer Fcn
0.522s+3 0.064s2+0.667s+1 Gas temperature transfer Fcn 0.229s+1.306
0.21s+1 Gas generator rotor speed transfer Fcn
0.522s+3 0.064s2+0.667s+1 Gas temperature transfer Fcn
Scope Scope 1 0.522s+3 Ful adder2 Gastemperaturecontroler
transfer Fcn Gas temperature
sensor (normalized values)
Comparison element2 For the first diagram (Figure 7a) σ 1=
⋅ 100 %=8 %. For the second diagram (Figure 7b) 1.015 1 σ 2= 1 ⋅ 100 %=1.5 %. Thus, the correction links with transfer functions (72)–(75) introduction makes it possible to virtually eliminate overshoot in the helicopter TE gas temperature in front of the compressor turbine control channel (the overshoot value does not exceed 1.5 %).
During the research it was established that oscillations with a frequency of f ≈ 1.96 Hz (the delay is 0.025 seconds) can occur in the developed controller. Therefore, it is advisable to conduct a delay dynamic’s research in this frequency vicinity. For this aim, the Mi-8MTV helicopter flight tests results, the power plant of which consists of two TV3-117 engines [42], are used (the data for the le engine are used in the research). In response to an official request sent by the authors to the Ministry of Internal Affairs of Ukraine, information was obtained on the gas temperature in front of the compressor turbine (T G¿) and the gas generator rotor speed (nTC) in the nominal engine operating mode. The request was fulfilled within the research project “Theoretical and Applied Aspects of Aviation Sphere Development” (number 0123U104884) framework. The data was obtained based on the Mi-8MTV helicopter’s flight tests. The experiments were conducted at an altitude of 2500 meters above sea level. The test duration was 320 seconds. The sampling step was 0.25 seconds.
The nTC and T ¿G data obtained during the Mi-8MTV helicopter’s flight tests using the onboard monitoring system were preliminarily cleared of noise interference and abnormal emissions. A er that, they were transformed into time series are the parameter’s sequences ordered by time [43]. To ensure the time series with different scales comparability, the z-normalization procedure was applied: z ( nTC )i=
, z (T G¿)i= n(TiC)
N ⋅ ∑ n(TiC)
i=1 √ ⋅ ∑ (n(TiC) 1 N N i=1
N ∑ n(TiC)) i=1 2
T G¿(i) 1 ⋅ ∑N T ¿G(i)
N i=1 √ 1 ⋅ ∑N (T ¿G(i)
N i=1
N ∑ T ¿G(i)) i=1 2 , (80)
To check the training dataset (Table 2) representativeness, the cluster analysis method (k-means [48]) was used. The training and test datasets were formed by random division. The proportion was 2:1, which is 67 and 33 % (858 and 422 elements, respectively). The training dataset’s (Table 2) clustering revealed 8 groups (classes I...VIII). This indicates the eight clusters identification. This observation confirms the training and test datasets (Figure 9) structure’s similarity. Based on these results, the optimal dataset sizes for the nTC and T ¿G parameters values were established. The training dataset consisted of 1280 elements (100 %). The control dataset consisted of 858 elements (67% of the training dataset). The test dataset consisted of 422 elements (33% of the training dataset). b Figure 9: The parameters nTC and T ¿G values cluster analysis results: (a) training dataset (858 elements); (b) test dataset (433 elements)
The proposed fully connected neural network (see Figure 2), consisting of two hidden layers with 16 and 8 neurons, respectively, was trained using the Keras library [49]. The “time_delay” factor was separately allocated for the predict, and the original dataset was divided into training and test datasets, where the test dataset constituted 33 % of the total amount. SmoothReLU [35] was chosen as the activation function for the hidden layers, and the mean square error (MSE) [35] was used as the optimization criterion. The model was optimized using the Adam algorithm with the training step parameter set as 10i, where i varies from 1 to 4. Each configuration was trained for 10 epochs, a er which the most successful one was selected based on the loss function and predict accuracy indicators, which was then further trained for 100 epochs. The best results were demonstrated by the model configured with the Adam optimizer (training rate 0.0001) and two hidden layers containing 460 and 230 neurons, with SmoothReLU activation [35].
Figures 10 and 11 show the neural network’s accuracy and loss diagrams. The obtained diagrams prove the neural network’s convergence on 100 training epochs, since both the accuracy and loss on the training and test datasets coincide on the 100th training epoch. In this case, the accuracy reaches 0.99537 (99.537 %), and the loss decreases to 0.00511 (0.511 %). It is noted that a er the 100th training epoch, the neural network’s occurs overtraining effect. The neural network’s overfitting effect, observed a er 100 training epochs, is that the model begins to adjust too precisely to the training dataset, including its noise and random deviations, instead of identifying general patterns, which result the data on the training dataset continues to demonstrate high accuracy and low loss, and on the test (validation) set, a deterioration in performance is observed, since the model loses the ability to generalize to new data, having begun to “remember” the training dataset’s specific features, which reduces its practical applicability.
Thus, it was found that further training leads to the neural network’s generalization abilities
L
deterioration. To prevent this effect, early stopping [50, 51] and regularization λ⋅ ∑ (W (l))22 in (
The developed neural network’s predictive assessment ability was carried out on a test dataset, where Figure 12 shows a diagram demonstrating the delay value’s predicted results correspondence to the actual data.
Figure 12: Diagram of predicted delay values vs. reference values
The predict errors distribution’s analysis was also conducted, presented in Figure 13. It follows from the diagram that the developed model demonstrates high accuracy in determining delays in the gas temperature control channel based on the factor’s given set, without an error’s obvious bias in any direction.
To assess the predicted delay values with the observed data correspondence, the determination coefficient and its adjusted version were calculated [52]. The obtained results R2 = 0.9717 and adjusted R2 = 0.9720 indicate a significant degree of relations between the neural network’s predicts and the reference data.
To improve the predicted value’s accuracy, a confidence interval construction technique is used, which aim is to take into account uncertainties arising from errors in data collection, errors in reference values, or random noise generated by a neural network with a reliability given level. Since there is no strictly mathematically sound algorithm for determining such intervals for neural networks, a quantile approach is proposed: the interval boundaries for the 95% reliability level [53] are set based on the quantiles of 0.025 and 0.975, which leads to the interval [–1.162; 1.077], covering forecast errors in 95% of the model cases (Figure 14).
Considering that 95 % of the neural network errors fall within this interval, we can conclude that for any predicted value of τ the following confidence interval is valid: [τ – 1.162; τ + 1.077].
Figure 15 shows the dependence of the delay values on the frequency in the range from 1.9 to 2.1 Hz, with special attention paid to the 1.96 Hz point, where pronounced oscillations are recorded, which may indicate the system’s resonance effects or specific dynamic features. The neural network use in this context has a positive effect on reducing the delay, since it is able to model complex nonlinear relations between system parameters and accurately predict optimal control modes, which ensures the control signal’s timely correction. Due to the neural network’s adaptability, it is possible to achieve a faster system response (the delay values using the neural network did not exceed 0.016 seconds, which is 36 % higher compared to the case without using a neural network), the operation stabilization in critical frequency ranges and, as a result, a significant reduction in delay.
In this research, the helicopter TE’s intelligent gas temperature controller (see Figure 1) was developed based on a double summation scheme, which allows for the measuring sensors inertial delays compensation and transient processes optimization in real time. Its special feature is the integration of adaptive algorithms with a differentiator and a neural network module, providing dynamic correction of control parameters and high control accuracy (up to 99.5 %).
A method has been developed based on a mathematical model of delayed dynamics (
A neural network module for delay estimation correction (see Figure 2) has been developed, which
is implemented using a deep fully connected neural network, which architecture is specified by (
The neural network module’s testing results demonstrate high training accuracy: the accuracy and loss function diagrams (Figures 10 and 11) confirm the model’s convergence on the training and test datasets, and the predicted and reference delay value’s correspondence diagram (Figure 12) indicates small predicting errors, which is further confirmed by the errors distribution on the histogram (Figure 13). In addition, the delay value’s dynamics analysis in the resonant frequency vicinity (Figure 15) shows a 36 % reduction in delay, which indicates a significant improvement in the system’s adaptive capabilities.
However, despite the positive dynamics and results achieved, the research has some limitations:
1. The modeling and testing results were obtained on the experimental data limited set basis
(Figures 7 and 8), which may reduce the methodology applicability in conditions other than the test
dataset.
2. The neural network module’s overtraining effect is observed (Figures 10 and 11), which limits
its ability to generalize and may negatively affect the delay prediction in real operating conditions.
3. The approximate compensation methods use, such as Taylor series expansion (
The research demonstrates that the innovative adaptive control methods development requires technological improvements while simultaneously complying with regulatory and ethical standards when applying it on board a helicopter, taking into account the rights and responsibilities of human operators [60].
The helicopter TE’s intelligent temperature controller has been developed that uses a double summation scheme with an adaptive observer and correction links, which provides effective compensation for the measuring sensor’s inertial delays.
The neural network module’s implementation for delay estimation correction allows the control parameter’s dynamic adaptation in real time, which is a significant improvement compared to traditional approaches.
Simulation showed a reduction in overshoot from 8.0 to 1.5 % and a reduction in the transient process time from 4.2 to 3.3 seconds, and the neural network module’s testing demonstrated a forecasting accuracy of 99.537 % (losses is 0.511 %) with a determination coefficient of R2 = 0.9717 and a reduction in delay to 0.016 seconds (an improvement of 36 %).
In the future, authors are going to explore the experimental base and testing conditions expansion, including additional tests in various operating modes [61] and the data integration [62] from various helicopter platforms to reduce the specific conditions influence. They also plan to develop robust neural network modules using regularization methods [62], ensemble models [63], and adaptive algorithms [64], as well as refine the mathematical models for compensating for inertial delays by integrating adaptive self-tuning algorithms [65, 66] in real time.
The research was carried out with the support of the research projects “Theoretical and Applied Aspects of Development of the Aviation Sphere” of the Ministry of Internal Affairs of Ukraine (number 0123U104884) and “Information system development for automatic detection of misinformation sources and inauthentic behaviour of chat users” (number 187/0012 from 1/08/2024, 2023.04/0012).
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