Neural Network-Based Parameter Estimation and Compensation Control for Time-Delay Servo System of Aeroengine

Abstract

Servo systems are important actuators of aeroengines. The repetitive, reciprocating motion of the servo system leads to significant changes in its time delay and gain characteristics, and degradation increases the uncertainty of these changes. These characteristic variations may have an adverse effect on the dynamic performance of the aeroengine. Therefore, a neural network-based parameter estimation and a multi-loop neural network-based predictive control (ML-NNPC) method for aeroengine inlet guide vane (IGV) servo systems (SVS) were proposed. In this study, the time delay estimation of the servo system was treated as a classification problem, and an SE (squeeze-and-excitation)-GRU (gated recurrent unit) network was proposed to estimate the time delay by using the selected dynamic data of the servo system. The estimated delay was embedded into an online sequential extreme learning machine, and a nonlinear model predictive controller was designed to obtain an optimal control sequence. The compensation control loop was designed to reduce the impact of the model and delay mismatch problems of the control system. The proposed method was applied to the IGV SVS control of a turboshaft engine. The simulation results demonstrate that the time delay is estimated accurately and compensated effectively. Compared to the existing PI and PI with Smith predictor methods, the ML-NNPC method achieves better control performance in the control of both the SVS and the engine rotor speed system. The stability and robustness of the ML-NNPC also show superiority. The results verify the effectiveness of the proposed time delay estimation method and the ML-NNPC method.

1. Introduction

An aeroengine is a complex nonlinear system. To fully leverage the performance of an aeroengine and enhance its dynamic response capabilities, the engine control system is subject to increased standards [1]. As the actuator of the engine, the servo system (SVS) plays an important role in regulating the position of geometric mechanisms, such as fuel metering, guide vane angle, etc., and thus influences the engine’s operating state [2].

During aeroengine operation, the SVS operates reciprocally under conditions of vibration and significant load variations. It may experience performance degradation due to wear, corrosion [3,4,5,6], and load and oil pressure variations [7,8]. If degradation occurs or the displacement direction changes, a mismatch between controller parameters and the changed SVS characteristics may occur, resulting in poorer response properties of the SVS, which affects the engine performance and can even cause safety problems.

The inlet guide vane (IGV) SVS is one of the key actuators of aeroengines, playing an important role in controlling the direction of the airflow [9] as well as improving the efficiency and the aerodynamic stability of the engine. In order to ensure the safe operation of the engine, the attack angle of the airflow into the rotor blades needs to be adapted to the engine speed; therefore, the guide vane needs to be quickly and smoothly adjusted according to the rotor speed [10]. The biggest control challenge facing the IGV SVS is the deterioration of control performance due to parameter variations, especially the variation of delay parameters.

In the servo control system, time delay has a significant impact on its dynamic performance [11]. The Smith predictor (SP) is often used to compensate the performance degradation caused by time delay [11]. However, the SP relies heavily on the accurate modeling of the plant, including a precise estimation of the delay, which limits its application in time-varying delay systems [12]. To solve this problem, some modified SP controllers have been proposed, such as Astrom’s modified SP [13,14] and the gain adaptive SP [15]. The active disturbance rejection controller (ADRC) has also been used together with the SP, which can treat both internal and external uncertainty, including time delay disturbances, and can then mitigate the impact of the parameter-mismatching problem of the SP and enhance the system’s performance [16,17,18,19,20,21,22]. The neural network (NN) can model the nonlinear system accurately, so some adaptive control methods based on NNs have been applied to robotic manipulator systems to mitigate the time delay and other nonlinear characteristics of the system [23,24,25,26,27,28]. However, when the parameters and time delay of the plant vary over a wide range, their effects on system response are coupled. It is difficult for most of the above methods to maintain satisfactory performance. Even model predictive control (MPC) with online horizon optimization capability, which has been widely used in various fields [29,30], will encounter performance degradation when facing delay systems [31,32]. The outputs become even more unpredictable when the delay is larger than the length of the control horizon.

Due to the harmful effects of time delay on SVS control performance, it is essential to accurately estimate these delays and implement appropriate compensation strategies. Some researchers have attempted to identify parameters using frequency-domain methods [33,34], the gradient optimization algorithm [35,36], the least squares method [36], or the expectation maximization algorithm [37], and so on. But most of them are time-consuming and liable to fall into local optimization. The extreme learning machine (ELM) is a fast-learning neural network with strong generalization performance [38,39]. Consequently, it is widely used by researchers for online modeling [40,41,42,43]. But for parameter-coupled time-varying systems, the varying delay is difficult to estimate accurately online along with other varying parameters. Considering the unique response lag characteristic of time-delay systems, an independent time delay estimation method based on a SE (squeeze-and-excitation)-GRU (gated recurrent unit) network is proposed in this paper to realize time delay estimation, treating the estimation problem as a classification problem, and this method can estimate the time delay accurately over a wide range of parameter variations. GRU networks, as simplified long short-term memory (LSTM) networks, need less computational capability and are suitable for time series feature extraction [44]. SE networks contain an attention mechanism, which assigns high weights to important information and low weights to irrelevant information. By utilizing the attention mechanism of the SE network and integrating it with the GRU network, the classification performance of the network can be enhanced [45,46].

To model the time delay characteristics of the IGV SVS, the estimated delay is directly embedded into the online sequential extreme learning machine (OS-ELM) input layer. Then, the OS-ELM with a delay layer (DOS-ELM) is employed to capture the characteristics of IGV SVS, which can effectively mitigate the coupling between the delay and other varying parameters. Meanwhile, when the delay layer is removed from DOS-ELM, the remaining online sequential extreme learning machine (OS-ELM) can be used as a delay-free nominal model, and the output of the IGV SVS can be predicted with the OS-ELM. Based on the classical Smith predictive compensation structure, the delay that is ignored in the nominal control loop can be compensated by an additional compensation control loop. Then, a multi-loop neural network-based predictive control structure is developed, which makes the system robust even under large parameter variation conditions, including time delay.

The main contributions of this paper are as follows: (1) A time delay estimation method based on SE-GRU network is proposed and applied to IGV SVS, which independently separates the delay estimation problem from the system parameter identification and treats the estimation problem as a classification problem. It effectively solves the difficulty of delay estimation under multi-parameter coupling conditions, and estimates the time delay accurately. (2) A nonlinear model predictive control (NMPC) based time-delay system control with a special predictive model is proposed. The model is derived from the OS-ELM, which is trained with a DOS-ELM structure and formed the nominal model in the Smith predictor control configuration, while solving the problem of unpredictable output of a large-delay system. (3) A time delay compensation control loop is introduced, which can compensate for the bias brought by the time delay-free optimization of the MPC and form a multi-loop neural network predictive control (ML-NNPC) structure. This structure effectively inhibits the performance deterioration caused by changing the SVS parameters of IGV and ensures the stable operation of the engine by providing a high-performance compensation control for widely varying time delays.

The structure of this paper is as follows: Section 2 briefly describes the typical structure of the inlet guide vane servo systems of aeroengine; then, the delay characteristic is analyzed according to the test data of the IGV SVS. Section 3 presents the parameter estimation methods and the multi-loop neural network-based predictive and compensation control structure. Section 4 validates the parameter estimation method and the multi-loop control method proposed in this paper. Section 5 provides the conclusions of the study.

2. Introduction of the Inlet Guide Vane Servo System of an Aeroengine

A typical aeroengine inlet guide vane (IGV) servo control system is shown in Figure 1. The key components include an electro-hydraulic servo valve (EHSV), an actuating cylinder, and a servo controller [3]. The servo system receives commands from the engine full authority digital engine control (FADEC) system, compares them with the linear variable displacement transducer (LVDT) signal, and computes the tracking error. The controller generates a current command for the torque motor of the EHSV in accordance with the error and actuates the slide valve of the EHSV to supply oil at different pressures to both sides of the actuating cylinder. Subsequently, the IGV is driven to track the control command until the steady-state error is minimized to zero.

Considering the effect of time delay, the traditional single-loop IGV servo control system diagram of aeroengine is shown in Figure 2. In the figure, r represents the command signal, α is the feedback signal of the IGV angle converted from the displacement measured by the LVDT, and I is the controller’s output current.

Research has been conducted on the IGV SVS of a turboshaft engine, and the experimental data obtained are shown in Figure 3.

The sampling time T_s of the SVS is 5 ms. During the total 180-s test, the system command performs seven upward steps and seven downward steps. During the first upward step, the system experiences the longest delay of 23 T_s. This may be due to the time required to build up pressure and overcome inertia and damping effects during the initial movement of the system. Similarly, during the first downward step, the delay is also relatively long at 12 T_s. This could be attributed to the sudden reverse movement causing the system to need to overcome inertia and friction effects once again. During the normal unidirectional movement of the IGV SVS, the time delay is within 5 T_s.

The seventh- and eighth-step responses are shown in Figure 4.

In Figure 4, the delay of the seventh-step response is 3 T_s. However, the eighth-step response exhibits a significantly greater delay of 12 T_s. The delay of the servo system varies considerably during operation. This substantial variation inevitably affects the performance of the servo system. Usually, when the servo system parameters do not change significantly, the typical PI control can meet the requirements. However, when the servo system undergoes reciprocal operations and performance degradation, its characteristic parameters—especially the delay—will change significantly. Due to the significant variations in both the time delay and the gain of the SVS, the typical PI control system cannot maintain a satisfactory performance consistently. To solve this problem, improvements in time delay estimation and control methods are necessary.

3. Neural Network-Based Predictive Control and Compensation

A nonlinear model predictive control method is adopted here for the main controller design of the guide vane SVS. In order to obtain an accurate predictive model, the SE-GRU-based delay estimation method is firstly presented, and the estimated delay is passed to the DOS-ELM to capture the characteristic of the IGV SVS.

3.1. SE-GRU Network Based Time Delay Estimation

Time delays are a commonly encountered problem in engineering systems, and lots of studies have been carried out to address this problem. Most of them treat it as a parameter identification problem and solve it by using recursive or iterative algorithms.

In this paper, the time delay of the IGV system is taken as an integer multiple of the sampling time, and the range of the time delay variation due to degradation is known. Therefore, the time delay estimation problem can be considered as a classification problem based on the dynamic response characteristics of the system. A combination of an SE network and a GRU network named SE-GRU Net was proposed in this paper to classify the time delay. GRU can handle time series tasks with a significantly short training process. SE-Net can select important features with an attention mechanism. It can enhance the reinforcement and suppression effects of the network during the training process [47], which, in turn, improves the network’s classification ability.

The configuration of the GRU unit is shown in Figure 5.

The output of the GRU network can be calculated as follows:

$$\left\{\begin{array}{l}{\mathit{r}}_{\mathrm{t}}=\sigma ({\mathit{W}}_{\mathrm{r}}\cdot [{\mathit{h}}_{\mathrm{t}-1},{\mathit{x}}_{\mathrm{t}}]+{\mathit{b}}_{\mathrm{r}})\\ {\tilde{\mathit{h}}}_{\mathrm{t}}=\tanh ({\mathit{W}}_{\mathrm{h}}\cdot [{\mathit{r}}_{\mathrm{t}}\odot {\mathit{h}}_{\mathrm{t}-1},{\mathit{x}}_{\mathrm{t}}]+{\mathit{b}}_{\mathrm{h}})\\ {\mathit{z}}_{\mathrm{t}}=\sigma ({\mathit{W}}_{\mathrm{z}}\cdot [{\mathit{h}}_{\mathrm{t}-1},{\mathit{x}}_{\mathrm{t}}]+{\mathit{b}}_{\mathrm{z}})\\ {\mathit{h}}_{\mathrm{t}}=(1-{\mathit{z}}_{\mathrm{t}})\odot {\mathit{h}}_{\mathrm{t}-1}+{\mathit{z}}_{\mathrm{t}}\odot {\tilde{\mathit{h}}}_{\mathrm{t}}\end{array}\right.$$

(1)

where ⊙ represents the Hadamard product and r_t represents the reset gate, which controls the amount of the previous state h_t−1 entering the GRU unit. x_t represents the input vector. W_r and b_r are the weight and bias of the reset gate, and σ represents the Sigmoid activation function. z_t represents the update gate vector, while W_z and b_z are the weight and bias of the update gate. h_t represents output state. $\tilde{\mathbf{h}}$ is the candidate state vector.

The SE-GRU network proposed in this paper is shown in Figure 6.

In Figure 6, the fully connected layers in the SE-Block represent the squeeze-and-excitation layer, respectively. The input data sequence is passed through the GRU1 network for the initial feature extraction. The outputs of GRU1 are then weighted at the multiplication layer according to their importance as evaluated by the SE-Block. This weighted sequence is subsequently passed to the GRU2 network for further feature extraction and the last unit of GRU2 stores the final information of the sequence for classification purposes; thus, the output of the last unit is passed to the fully connected layer that follows it [48].

Here, the delay is expressed as an integer multiple of the sampling time, i.e., the number of delay variation is finite (13 in this paper), so the network uses a Softmax layer to obtain the classified outputs of time delays.

The input sequence used to classify the time delay is

$$\left[\begin{array}{l}y\left(k-n_{\mathrm{g}}+1\right),y\left(k-n_{\mathrm{g}}+2\right),\dots ,y\left(k\right)\\ \Delta y\left(k-n_{\mathrm{g}}+1\right),\Delta y\left(k-n_{\mathrm{g}}+2\right),\dots ,\Delta y\left(k\right)\end{array}\right]$$

(2)

where Δy(k) = y(k) – y(k − 1), n_g represents the length of the sequence, and k − n_g + 1 is the moment when the system command r starts to change, i.e., when Flag_r = 0. The output y and Δy can reflect the response time delay relative to the command change.

The training and test data are gathered by iterating all delay cases of the IGV SVS through simulation, and the SE-GRU net is trained offline. After that, it can be used to classify the time delay online.

3.2. Neural Network-Based Model Predictive Control

3.2.1. The DOS-ELM Model

NMPC is a model-based control algorithm that obtains optimal control sequences by minimizing an objective function [49]. The DOS-ELM with a delay layer was used to model the IGV SVS online, and the OS-ELM with the same parameters but without the delay layer was used to construct the predictive model.

The input–output model of the IGV SVS can be described in NAR (nonlinear autoregressive) model format [50]:

$$y_{\mathrm{m}}\left(k\right)=G_{\mathrm{m}}\left(y\left(k-1\right),\cdots ,y\left(k-n_{\mathrm{y}}\right),I\left(k\right),\cdots ,I\left(k-n_{\mathrm{I}}\right)\right)$$

(3)

where n_y and n_I are the orders of y and I.

The DOS-ELM neural network was used to model Equation (3) for its fast online learning ability. The structures of DOS-ELM and the OS-ELM model used in this paper are shown in Figure 7. The two models have the same inputs, weights, and the same active function.

In Figure 7, G_m,d is the DOS-ELM model with a time delay layer, G_m is the OS-ELM model without a time delay layer, and the parameters of G_m are directly taken from G_m,d. τ_m is the estimated time delay. W are the input weights, and β are adjustable output weights.

The DOS-ELM network is initialized offline and trained online. The output weight is updated as [51]:

$$\mathit{\beta }\left(k+1\right)=\mathit{\beta }\left(k\right)+\left[\mathit{y}\left(k+1\right)-\mathit{\beta }\left(k\right)\mathit{H}\left(k+1\right)\right]{\mathit{H}}^T\left(k+1\right){\mathit{P}}_{\mathrm{ELM}}\left(k+1\right)$$

(4)

$$\mathit{H}\left(k+1\right)={\displaystyle \frac{1}{1+\exp \left\{-\mathit{W}\cdot {\left[I\left(k-{\tau }_{\mathrm{m}}\right),\dots ,I\left(k-n_{\mathrm{I}}-{\tau }_{\mathrm{m}}\right),y\left(k-1\right),\cdots ,y\left(k-n_{\mathrm{y}}\right)\right]}^{\mathrm{T}}-\mathit{b}\right\}}}$$

(5)

where P_ELM is an intermediate matrix used for updating the output weights, H are the hidden layer outputs, and b are the hidden layer bias. The estimated delay τ_m is used to obtain the hidden layer output H.

$${\mathit{P}}_{\mathrm{ELM}}\left(0\right)={({\delta }^2\mathit{I}+{\mathit{H}}_0{\mathit{H}}_0^T)}^{-1}$$

(6)

$${\mathit{P}}_{\mathrm{ELM}}\left(k+1\right)={\mathit{P}}_{\mathrm{ELM}}\left(k\right)-{\mathit{P}}_{\mathrm{ELM}}\left(k\right)\mathit{H}\left(k+1\right){\left[I+{\mathit{H}}^T\left(k+1\right){\mathit{P}}_{\mathrm{ELM}}\left(k\right)\mathit{H}\left(k+1\right)\right]}^{-1}{\mathit{H}}^T\left(k+1\right){\mathit{P}}_{\mathrm{ELM}}\left(k\right)$$

(7)

3.2.2. The Nonlinear Predictive Model

The nonlinear predictive control was adopted in this paper. The predictive model is built based on the delay-free OS-ELM model and the structure and parameters of the OS-ELM model are derived from the DOS-ELM model, i.e., the OS-ELM model G_m is updated with the DOS-ELM online during the operation. The NMPC thus developed a delay-free model-based optimal control.

Multi-step prediction is achieved using the OS-ELM model G_m. The actual inputs and outputs of the IGV SVS are denoted by y and I, respectively. y_m is the output of model G_m, and I_m is the predicted input. When OS-ELM input layer signals are available from the SVS system, the model utilizes actual signals as inputs. When prediction of future outputs is required, the predictive inputs I_m and the outputs of the predictive model y_m are also used as inputs to the OS-ELM. Then, the predictive model denoted by G_m,p can then be expressed in the format of a nonlinear autoregressive moving average with exogenous inputs (NARMAX) model as follows [50]:

$$\begin{array}{l}{y}_{\mathrm{m}}\left(k\right)=G_{\mathrm{m}}\left(y\left(k-1\right),\cdots ,y\left(k-n_{\mathrm{y}}\right),I\left(k\right),\cdots ,I\left(k-n_{\mathrm{I}}\right)\right)\\ y_{\mathrm{m}}\left(k+1\right)=G_{\mathrm{m}}\left(y_{\mathrm{m}}\left(k\right),y\left(k-1\right),\cdots ,y\left(k-n_{\mathrm{y}}+1\right),I_{\mathrm{m}}\left(k+1\right),I\left(k\right),\cdots ,I\left(k-n_{\mathrm{I}}+1\right)\right)\\ ⋮\\ y_{\mathrm{m}}\left(k+n_{\mathrm{p}}\right)=\left\{\begin{array}{l}{G}_{\mathrm{m}}\left(y_{\mathrm{m}}\left(k+n_{\mathrm{p}}-1\right),\cdots ,y_{\mathrm{m}}\left(k\right),y\left(k-1\right),\cdots ,y\left(k-n_{\mathrm{y}}+n_{\mathrm{p}}\right),I_{\mathrm{m}}\left(k+n_{\mathrm{p}}\right),\cdots ,I_{\mathrm{m}}\left(k+1\right),I\left(k\right),\cdots ,I\left(k-n_{\mathrm{I}}+n_{\mathrm{p}}\right)\right),if \left(n_{\mathrm{I}}>n_{\mathrm{p}},n_y>n_{\mathrm{p}}\right)\\ G_{\mathrm{m}}\left(y_{\mathrm{m}}\left(k+n_{\mathrm{p}}-1\right),\cdots ,y_{\mathrm{m}}\left(k-n_{\mathrm{y}}+n_{\mathrm{p}}\right),I_{\mathrm{m}}\left(k+n_{\mathrm{p}}\right),\cdots ,I_{\mathrm{m}}\left(k-n_{\mathrm{I}}+n_{\mathrm{p}}\right)\right),else\end{array}\right.\end{array}$$

(8)

where n_p is the predictive horizon.

The NMPC control structure is shown in Figure 8.

To obtain optimal control performance, the future control sequence needs to be determined by optimizing the objective function. The objective function J used in this paper is

$$\begin{array}{l}J=E\left\{{\displaystyle \sum _{j=0}^{n_{\mathrm{p}}}{\left[y_{\mathrm{m}}\left(k+j\right)-y_{\mathrm{r}}\left(k+j\right)\right]}^2+{\displaystyle \sum _{j=0}^{n_{\mathrm{p}}}{\left[\gamma \Delta I_{\mathrm{m}}(k+j)\right]}^2}}\right\}\\ \mathrm{s}.\mathrm{t} \begin{array}{c}{I}_{\min }\le I_{\mathrm{m}}\left(k+j\right)\le I_{\min }\\ y_{\min }\le y_{\mathrm{m}}\left(k+j\right)\le y_{\min }\end{array}\\ \Delta I_{\mathrm{m}}=I_{\mathrm{m}}(k+j)-I_{\mathrm{m}}(k+j-1)\end{array}$$

(9)

where y_m(k + j) and y_r(k + j) correspond to the output of G_m,p and the reference trajectory of the system at the future moment k + j. γ is the control weighting coefficient. I_min and I_max are the minimum and maximum limiting values of the control current, and y_min and y_max are the minimum and maximum limiting values of the output displacement.

By minimizing the objective function in Equation (9) at each control period, a predictive control sequence [I_m(k), I_m(k + 1),…, I_m(k + n_p − 1)] can be obtained. This sequence enables the future predicted output sequence to reach the target value. To ensure a smooth transition of y_m, the reference trajectory is set as follows:

$$\left\{\begin{array}{l}{y}_{\mathrm{r}}\left(k\right)=y\left(k\right)\\ y_{\mathrm{r}}\left(k+j\right)=a_{\mathrm{f}}{y}_{\mathrm{r}}\left(k+j-1\right)+(1-a_{\mathrm{f}})r\left(k+n_{\mathrm{p}}\right), j=1,2,\cdots ,n_{\mathrm{p}}\end{array}\right.$$

(10)

where a_f is flexible coefficient, and 0 ≤ a_f < 1.

In k-th step, only the I_m(k) is implemented, i.e., I(k) = I_m(k). The Levenberg–Marquardt (LM) algorithm is used to solve the nonlinear optimization problem involved in predictive control [52]. To expedite the iteration speed and ensure real-time performance, the maximum iteration of the LM algorithm is set to 30 and the convergence tolerance is set to 10⁻⁴.

3.3. Structure of the Multi-Loop Neural Network-Based Predictive Compensation Control System

Although NMPC can achieve satisfactory control results in most cases, its performance may be degraded due to the varying time delay and gain of the servo system. Moreover, the outputs become even unpredictable when the delay exceeds the length of the control horizon. The Smith predictor is a widely utilized solution for addressing control issues associated with systems exhibiting a delay, commonly encountered in engineering applications [53]. Its original structural configuration is shown in Figure 9a, and its equivalent transformation structure is shown in Figure 9b.

The Smith predictor can achieve satisfactory performance when the nominal model and estimated delay are accurate, meaning that the nominal model can represent G_p and the estimated delay τ_m matches the actual delay τ_p. However, the performance will deteriorate significantly once parameter mismatch occurs. Therefore, a multi-loop neural network-based predictive and compensation control (ML-NNPC) method is proposed in this paper, which consists of an NMPC (instead of PI control) loop and a delay compensation loop.

The control structure of ML-NNPC, shown in Figure 10, is designed to realize parameter estimation and delay compensation for the servo system. Compared to the Smith predictor structure, the controller freedom is increased by incorporating an independent G_c2 controller.

In Figure 10, τ_m is the time delay estimated by the SE-GRU net. G_m,d is the onboard DOS-ELM model of the servo system trained online, its parameters are updated in real time. G_m,p is the delay-free predictive model of NMPC.

NMPC has relatively poor control performance for systems with time delays. To address this, the proposed two-degrees-of-freedom control structure divides the predictive control and delay compensation into two control loops. The NMPC controller in Loop 1 generates I₁(k) to control the predictive model output tracking the reference signal. The controller G_c2 in Loop 2 generates I₂(k) to drive the physical system output tracking the DOS-ELM model output, thus compensating the influence of time delay.

The output signal of ML-NNPC is

$$I(k)=I_1(k)+I_2(k)$$

(11)

The control logic of ML-NNPC is illustrated in Figure 11. In dynamic state, the input-output characteristic reflects the time delay information. So, the quasi-steady-state judgement is performed according to the variation of r(k). When Flag_r = 0, the system is in a dynamic state, and the dynamic data series for delay estimation is stored. Then, the SE-GRU module is activated to estimate the delay τ_m when the number of data stored reaches n_g.

The quasi-steady state is judged as follows.

$$Fla{g}_r=\left\{\begin{array}{c}1,\left|r\left(k\right)-r_{\mathrm{avg}}\left(k\right)\right|\le {\epsilon }_{\mathrm{r}}\\ 0,\left|r\left(k\right)-r_{\mathrm{avg}}\left(k\right)\right|>{\epsilon }_{\mathrm{r}}\end{array}\right.$$

(12)

where ε_r represent the quasi-steady-state discrimination thresholds. r_avg are the average values of command r from k − n_g + 1 to k.

4. Simulation and Validation

4.1. Time Delay Estimation

The time delay is estimated by SE-GRU net. The training data for the SE-GRU net are obtained from the simulation results of the transfer function model of the IGV SVS, as shown in Equation (13) [54]. This transfer function was established based on the nonlinear least square method using the experimental data of IGV SVS presented in Figure 3.

$$G_{\mathrm{P}}(s)={\displaystyle \frac{\alpha (s)}{I(s)}}={\displaystyle \frac{K_v}{s\left(T_{v}s+1\right)}}{e}^{-{\tau }_{d}s}$$

(13)

where α is the angle of IGV, α = y, T_v is the time constant of the EHSV, and K_v is the velocity amplification coefficient or open-loop gain of the servo system.

The accuracy of the transfer function model is shown in Figure 12. The servo system response is very fast, and the output of the transfer function model tracks the original experimental data very well, which means the model can represent the dynamic characteristic of the servo system and can be used as the agent model. With the transfer function model, all the possible time delay and gain variations of the IGV system can be simulated. This has greatly compensated for the lack of real experimental data and significantly enriched the training and testing dataset used for the SE-GRU neural network.

The delay of the researched servo system varies from 3T_s to 15T_s, where T_s = 5 ms represents the sampling time. The transfer function model was used to simulate all these conditions to enrich the training dataset apart from real IGV system data, and a total of 22,536 sets of data have been collected to train and test the SE-GRU net. The input sequence used in Equation (2) to classify the time delay is set to n_g = 20, which can cover all the time delay situations and provide sufficient dynamic characteristics. The output sequence α and its single-step variation Δα are gathered as the network input after Flag_r changes from 1 to 0. The network output is the corresponding time delay. The network is trained and the confusion matrix on the testing dataset is shown in Figure 13. The classification error on the same test dataset is compared with the GRU net and the LSTM net in

Table 1. Classification error of three networks.

Network	Training Set Error/%	Testing Set Error/%
SE-GRU	1.51	2.00
GRU	3.85	4.6
LSTM	3.63	4.4

.

From Table 1, we see that the SE-GRU net achieves a much higher classification accuracy on both the training set and the testing set than that of the GRU net and the LSTM net; the accuracy of the GRU net and the LSTM net is also similar.

The accuracy of the delay estimation was further tested on actual SVS data, and the result is shown in Figure 14. The estimation result was updated each time the stored data reached 20 (i.e., 0.005 s × 20 = 0.1 s) after Flag_r = 0, so the estimated delay may lag behind the actual delay 0.1 s but is smaller than the settling time of the dynamic response of the servo system (0.3 s for common). From Figure 14, the time delay estimated by SE-GRU accurately follows the real delay after each command variation. If the time delay remained unchanged, even when there was command variation between 15 s and 20 s, the estimated result also remained unchanged.

4.2. Multi-Loop Neural Network-Based Predictive Compensation Control Validation

The simulation is carried out based on the transfer function model of the servo system in Equation (13), which is very convenient for simulating the delay and gain variations, where G_c1 is the OS-ELM-based NMPC, and G_c2 is the PI controller. The initial delay τ_p = 4 T_s. The parameters of G_c2 were set to match these initial conditions with K_p2 = 3 (mA/°) and K_i2 = 0.001 (mA/°) for G_c2.

4.2.1. Predictive Model Validation

For the controlled servo system, the prediction accuracy of the delay-free network model G_m,p is tested. The OS-ELM in Figure 7 was used to establish the servo system model online. The number of hidden layer nodes of the delay-free network G_m,p is 20, the orders of output n_y = 2, and input n_I = 2. Thus, the input weights W ∈ R^{20 × 5} and hidden layer bias b ∈ R^{20 × 1} are generated randomly. The initial output weights β are calculated by the least squares method offline [38,39,40] and updated online by using the recursive least squares method using Equations (4)–(7).

The prediction of future moments performed by the OS-ELM network used by G_m,p can be expressed as Equation (8). The error between the G_m,p model’s predictive output and the experimental data of the IGV servo system is compared. The results of the prediction are shown in Figure 15, where the “OS-ELM” represents the predictive output of the G_m,p, and “Real Output” represents the output of IGV servo system of the aeroengine.

As can be seen from Figure 15, the maximum predictive error occurs at 31 s, reaching 2.98%. The predictive accuracy of G_m,p shows a great advantage both in the steady state and in the dynamic process. It can be used as a predictive model for the ML-NNPC control structure proposed in this paper.

4.2.2. Control Validation

The simulation result is compared with the Smith PI control and Smith ADRC [55]. As illustrated in Figure 9, the Smith PI has a constant nominal model and constant estimated delay. All controllers are tuned to achieve satisfactory control performance on the non-degraded IGV SVS.

Based on the IGV SVS experimental data of the turboshaft engine shown in Figure 3, and considering the impact of measurement and process noise, Gaussian colored noise was applied to the control input I, and white noise was applied to the feedback signal in the simulation.

During the simulation, the parameters of the IGV SVS varied according to the characteristic of IGV SVS shown in Figure 3, i.e., the delay increases when the direction of IGV SVS movement changes. During this process, the system has to overcome the influence of inertia and friction effects, which result in significant variations in gain and time delay. Therefore, the changes in gain and time delay during reverse movement occurred at 5 s. It is worth mentioning that the effectiveness of the proposed control method can be verified through parameter variation simulations conducted either at increasing or decreasing movement process. The simulation result is shown in Figure 16.

In Figure 16, the initial delay τ_p = 4 T_s. Then, τ_p increases to 12 T_s and the gain of the servo system increases to 1.3 K_v at 5 s. Before time reaches 5 s, the system can be considered as not degraded. In this case, the dynamic performance of the ML-NNPC is superior to the Smith PI and Smith ADRC in terms of both response speed and convergence capabilities. The response of the Smith ADRC is slowest before time reaches 5 s. After time exceeds 5 s, τ_p increased to 12 T_s and the gain of the plant increased to 1.3 K_v; the responses are obviously faster. Overshoots occur in both Smith ADRC and Smith PI control systems. During the decrease process around time = 6 s, the overshoot of Smith ADRC is 12.9% and the overshoot of Smith PI control reaches 22.8%. In contrast, the ML-NNPC system ensures a fast response without overshoot, which demonstrates the advantage of the proposed control method.

The delay change and its estimation result for the full response process are shown in Figure 17.

In Figure 17, the initial time delay is 4 T_s and it increases to 12 T_s after 5 s. After going through a dynamic process, the estimation is completed, and accurate estimation results are achieved for subsequent times without estimation errors.

4.2.3. Controller Stability Validation

The stability of the control system is one of the key issues to be considered in the controller design. In order to validate the stability of the ML-NNPC controller proposed in this paper, the gain margin and phase margin of the controller are measured by frequency sweeping. The bode diagrams of the worst case, i.e., τ_p = 15 T_s and the open-loop gain increase to 1.3 K_v, are shown in Figure 18.

In this case, the gain margin and phase margin of the ML-NNPC controller compared with those of the PI controller and the Smith PI controller are shown in

Table 2. Comparison of frequency characteristics.

Controller	Gain Margin (dB)	Cutoff Frequency (rad/s)	Phase Margin (deg)	Crossover Frequency (rad/s)
ML-NNPC	8.67	28.5	68.9	9.07
Smith PI	4.39	22.1	35.2	13.3
PI	3.95	21.4	32.9	13.6

. The results in Table 2 were obtained by frequency sweeping and calculated as follows.

In Table 2, Gain margin is calculated as

$$GM=-20\log \left|G(j{\omega }_{\mathrm{x}})\right|$$

(14)

where ω_x is the crossover frequency, i.e., Im[G(jω_x)] = 0. G(jω) is the frequency description of the transfer function G(s). G(s) is the equivalent open-loop transfer function of the system.

$$G(s)={\displaystyle \frac{\varphi (s)}{1-\varphi (s)}}$$

(15)

where ϕ(s) is the closed-loop transfer function established by the nonlinear least squares method based on the data obtained from the closed-loop simulation of the corresponding control system.

The phase margin is calculated as:

$$r={180}^{\circ }+\angle G(j{\omega }_{\mathrm{c}})$$

(16)

where ω_c is the cutoff frequency, i.e., |G(jω_c)| = 1.

From the data presented in Table 2, it is evident that compared to the PI controller and Smith PI controller, the ML-NNPC proposed in this paper exhibits better gain margin and phase margin performance in the worst case. The gain margin of ML-NNPC is about twice that of the Smith PI, while also exhibiting a higher cutoff frequency. This indicates the system will achieve a faster response property with a larger gain margin. Additionally, the crossover frequency of ML-NNPC is less than 70% of the Smith PI control system, but with a greater phase margin. This suggests that even if the actual system crossover frequency is slightly larger, perhaps due to the neglect of high-frequency characteristics in the modeling process of IGV SVS, the ML-NNPC system will experience a lower phase margin reduction than the PI and Smith PI controllers. This is because the smaller crossover frequency corresponds to a slower rate of decrease in the phase-frequency characteristic as the frequency increases compared to the high-frequency crossover frequency of the other controllers.

The frequency response of the Smith ADRC closed-loop control system has also been evaluated and illustrated in Figure 19. The figure reveals that the system exhibits a fatal deficiency in high-frequency stability when subjected to a wide range of parameter variations.

The results demonstrate that the ML-NNPC system achieves a faster response time, along with increased gain and phase margins, which means the system achieves enhanced stability and robustness properties.

4.3. Validation on Turboshaft Engine Control

The ML-NNPC system is used in the compressor IGV control of a bi-rotor turboshaft engine [56]. By controlling the angle of IGV SVS, the operating characteristics of the compressor can be adjusted to the optimal state, thus increasing the efficiency and surge margin (SM) of the compressor [57,58].

The control objective of a turboshaft engine in maximum continuous operation is to maintain a constant power turbine speed N_p = 100%. In this case, the IGV command is scheduled according to the compressor-corrected speed N_c,cor. In our simulation, the N_p speed changes between the ground idle speed N_p = 85% and the maximum continuous state N_p = 100%. The IGV command α_r is scheduled according to the power turbine speed command N_pr to obtain a favorable IGV condition for the power turbine acceleration, i.e., α_r = f(N_pr). The control configuration is shown in Figure 20. The fuel flow command W_fr is calculated by the N_p controller in a closed loop. When the servo system undergoes degradation, the response of the servo control loop deteriorates, which in turn affects the engine’s dynamic performance.

The N_p controller is designed based on the generalized predictive control (GPC) method [59]. The prediction horizon of the GPC controller is set to 10, and the control horizon is set to 2. It controls the engine power turbine speed N_p to track the command N_pr by adjusting the fuel flow. The servo system of the IGV adopts the ML-NNPC configuration, and the fuel servo system is represented by a typical inertial model. Simulation is carried out on the component-level model (CLM) of the turboshaft engine. The CLM is built based on the engine’s operating principles and is the most widely used agent model. The sampling time of CLM is 25 ms, which is 5 times faster than that of the IGV SVS model. As a result, during the simulation, the servo system model runs five times per CLM sampling period.

The simulation replicated the acceleration and deceleration process of a turboshaft engine, starting from its maximum continuous state, decelerating to idle, and then accelerating back to the maximum continuous state from idle [60]. During the simulation, the power turbine speed command N_pr reduced from 100% to 85% and then accelerated back to 100%.

The control command N_pr is shown in Figure 21a. The command α_r is shown in Figure 21b.

Assuming the initial delay of the IGV servo system is 4 T_s, and changes to 12 T_s at time = 2 s. Additionally, the servo loop gain increased by a factor of 1.3 after time = 10 s. The responses of the guide vane, power turbine speed, and compressor surge margin (SMC) are shown in Figure 22.

From Figure 22a, it can be seen that at time = 5 s, as the power turbine speed N_pr decreases, the IGV command α_r increases. In the IGV control loop, the overshoots of Smith ADRC and Smith PI are 5.99% and 9.46% respectively. At this point, the delay τ_p = 12 T_s, which is significantly different from the controller design state. Mismatch between the controller and controlled object occurs. However, for the ML-NNPC, the delay is estimated online and the NN model is also trained online, so the mismatch hardly occurs. Meanwhile, the compensation control loop also can compensate the mismatch. After 10 s, when the gain increases, it can be observed that due to the model and controller mismatch, the IGV overshoots of the Smith ADRC and Smith PI are 17.87% and 26.89%, respectively. The ML-NNPC completes delay estimation at the first dynamic process and achieves much better responses in both the IGV increase and decrease process compared to the other methods. There is nearly no IGV overshoot during the entire dynamic process.

From Figure 22b,c, it can be observed that during the deceleration process of N_p, the responses of the Smith ADRC are the fastest. However, during the N_p acceleration process, the large overshoot of IGV caused a deviation of the IGV output from its command, which brings a decrease in the compressor surge margin SMC under Smith ADRC and Smith PI control systems. Due to the proximity of the SMC to the surge boundary, the safety of the aeroengine may also be compromised. Acceleration performance of the engine is deteriorated that the N_p response of Smith ADRC is slower than that of the ML-NNPC, and the N_p overshoot of Smith PI increased to 4.91%. The ML-NNPC system, on the other hand, shows no overshoot in the N_p speed acceleration process, and the settling time is 2.9 s, which shows the ML-NNPC control system can better cooperate with the fuel control system and effectively enhance the engine response performance. It verifies the advantages of the proposed method in dealing with system parameter variations.

5. Conclusions

The IGV servo system of an aeroengine encounters large parameter variations, such as time delay and gain variations, and the variations will affect the performance of the servo system and the engine. The proposed ML-NNPC has demonstrated remarkable effectiveness in mitigating the effect of these uncertainties and shows advantages in maintaining robust performance and ensuring ample stability margins.

The time delay estimation can be treated as a classification problem and estimated independently by a classification neural network, which can achieve high precision under multi-parameter coupling conditions. The SE-GRU network was adopted and has shown effectiveness in time delay estimation.

The combination of DOS-ELM and OS-ELM makes the NMPC feasible for a large delay system. The dual-loop NMPC improves the response performance of the system under larger parameter variation conditions and enhances the system’s stability and robustness.

The proposed parameter estimation and ML-NNPC methods were validated through simulations at different parameters variation situations. The results demonstrate that the parameter estimation achieves high accuracy for both the simulation dataset and the physical system dataset. The multi-loop compensation control method outperforms the widely used PI and Smith PI control methods in dynamic responses for both the servo system and the engine rotor speed control system.

The parameter estimation and control method proposed in this paper are also valuable for other nonlinear systems. It can be applied to robust control, degradation monitoring, time delay estimation, disturbance compensation control, and so on. We will further research its application in an engine fuel control system and nozzle servo system, etc.