1. Introduction
Despite the increasing prevalence of advanced beyond-visual-range missiles, close-range dogfights remain a critical aspect of aerial combat. High agility and maneuverability are still the indispensable key features of next-generation fighter aircraft. The aerodynamic characteristics of aircraft at high angles of attack exhibit highly nonlinear and unsteady characteristics due to phenomena such as flow separation and vortex shedding. The traditional database-based interpolation method is no longer sufficient for the precise simulation and control of post-stall maneuvers. Therefore, the establishment of accurate aerodynamic models is crucial for aircraft dynamics investigation, stability analysis, and control system design at high angles of attack. Past work has focused on mathematical models. These models establish a mathematical relationship between aerodynamic loads and flight states (such as velocity, angle of attack, and sideslip angle) based on unsteady flow phenomena and physical principles, such as the dynamic derivative model [
1,
2], the state-space model [
3,
4], and the indicial function model [
5,
6]. While these models often possess explicit physical interpretations, their accuracy is constrained by the level of understanding of physical phenomena and the degree of mathematical simplification involved.
In recent years, with the rapid advancement of artificial intelligence, machine learning models have found widespread application in unsteady aerodynamic modeling. Machine learning models, also known as black-box models, bypass the need for explicating complex physical mechanisms. With a sufficient amount of training data, these models possess powerful nonlinear fitting capabilities to establish the relationship between flight states and aerodynamic loads. Models such as support vector machines [
7,
8], random forest [
9], and neural networks [
10,
11,
12] have been employed for this purpose. Furthermore, recurrent neural networks improve accuracy by capturing the time lag effects of unsteady aerodynamics [
13,
14,
15]. In addition to flight state data, the geometric representation of the airfoils is used as an input feature for the deep neural network to obtain the aerodynamic parameters [
16,
17]. While machine learning models can fit nonlinear relationships effectively, they cannot provide a reasonable explanation for how inputs are used to make predictions.
If the explanation of physical relationships between flight states and aerodynamic loads could be applied to model training, it would enable the model to possess the interpretability of traditional mathematical models and the powerful nonlinear fitting capabilities of black-box models simultaneously. Currently, there are two approaches to exploring the combination of traditional mathematical models and black-box models.
One approach involves using physics-informed neural networks (PINNs) [
18]. The physics equations are incorporated into the loss function of a neural network to constrain the model while training, thereby ensuring outputs follow known physical laws [
19]. Zhao et al. [
20] proposed an identification method of aerodynamic models using a physics neural network that incorporates the attitude dynamics of an aircraft. Li et al. [
21] utilized a PINN model to predict parameters of the state-space model using neural networks instead of predicting aerodynamics directly. While this method enhances the neural network’s extrapolation capabilities, it remains fundamentally a state-space model, with no significant improvement in interpolation accuracy.
Another approach is using the fusion model, which attempts to integrate traditional mathematical models into machine learning models [
22]. The combination of models with different accuracy is also known as the multi-fidelity method. Low-fidelity models’ predictions, which are assumed to have similar trends to the high-fidelity models’ predictions, are used to provide additional information. Wang [
23] and Li, et al. [
24] combined dynamic derivative models with black-box models for unsteady aerodynamic prediction. These are used to compute low- and high-fidelity outputs, respectively. Finally, the least squares method is used to merge the outputs. The fusion model exhibited superior generality when compared to black-box models. However, the effectiveness of the least squares method depends on the correlation between models and the assumptions about the error distribution. If the models are highly correlated, the least squares method may not provide a significant advantage. Zhang et al. [
25] presents an innovative aerodynamic modeling method using heterogeneous data and physical feature embedding, significantly improving prediction accuracy while reducing training data needs. The complexity of implementation, reliance on high-quality data, need for further real-world validation, and demand for substantial computational resources may be potential challenges of this approach.
This paper proposes a weighted feature fusion model (WFFM) based on the state-space model and long short-term memory network (LSTM) to predict nonlinear unsteady aerodynamics. The main contributions of this paper can be summarized as follows:
- (1)
An architecture of an aerodynamic model is proposed, which combines the physics model and black-box model, exhibiting high accuracy in both interpolation and extrapolation tests.
- (2)
A new method for weighting data is proposed. To reduce the impact of the state-space model error, the feature standardization layer and weighting layer, which is implemented using a single neuron and an activation function, are introduced.
- (3)
Two mappings are established and fused by LSTM. One is the mapping from flight states to aerodynamic loads, and the other is the mapping from low-fidelity data to high-fidelity data.
- (4)
To test the model, the proposed model is used to predict aerodynamic loads at high-angles-of-attack oscillations. Furthermore, the model is applied to a flight simulation of the F-16 with different control inputs to evaluate the generalization capability.
The paper is organized as follows. In
Section 2, a brief introduction to the state-space method and neural network approach in aerodynamic modeling is given, and the structure of the WFFM based on the state-space model and LSTM is discussed in detail. In
Section 3.1, the WFFM is used to predict the pitching moment coefficient at a high-angle-of-attack oscillation data obtained from wind tunnel experiments using a wire-driven parallel robot with eight wires (WDPR-8) and a traditional tail support to verify the proposed aerodynamic model. The results are then compared with three other models. In
Section 3.2, the WFFM is applied to flight simulation and tested with different control inputs to validate the robustness and generalization of the models. Finally,
Section 4 presents conclusions.
2. Modeling Methods
2.1. State-Space Method
Aircraft exhibit unsteady characteristics at high angles of attack, with airflow separation being a primary cause of aerodynamic time delays. To address this issue, Goma et al. proposed a state-space modeling method by introducing internal state-space variables into traditional aerodynamic derivative models [
26]. The internal state-space variable is defined as the nondimensional coordinates of the airflow separation point, formulated as
. Here,
represents the distance between the position of the separation point and the leading edge of the airfoil, and
represents the chord length. The range of values of the airflow separation point
is
. Introducing the airflow separation point allows the state-space model to depend not only on the instantaneous state variables but also on the physical mechanisms of airflow separation and attachment. The aerodynamic force and moment coefficients can be expressed as:
where
represents the aerodynamics coefficients,
represents the static component of aerodynamic coefficients,
represents the dynamic component of aerodynamic coefficients, and
represents the effect of control surface deflection on aerodynamic coefficients. In the state-space models, these coefficients are expanded in their Taylor series, using the first derivatives only and truncating higher-order terms, which may lead to insufficient accuracy. For example, considering the aerodynamic coefficient
along the body axis, its dynamic aerodynamic coefficient
is typically approximated by the first five terms of its Taylor series expansion:
In this formula, all coefficients of the expansion terms in the dynamic aerodynamic coefficients are approximated using a quadratic polynomial model. For example,
can be expressed as follows:
where
,
, and
are unknown parameters within the model, which are determined using parameter identification techniques.
2.2. Neural Network Approach
Neural networks are powerful machine learning tools capable of learning complex nonlinear relationships from data. They have found widespread applications in fields such as image recognition, natural language processing, and time-series prediction. A neural network consists of multiple layers of neurons, including an input layer, one or more hidden layers, and an output layer. All the neurons connected by links take in some data and use it to perform specific operations and generate output through an activation function.
Aerodynamic loads could be regarded as a function of the instantaneous values of the aircraft’s motion state variables [
27]. In general, aerodynamic force and moment coefficients
can be expressed as:
where
is a vector of flight states.
Assuming the considered flight states include angle of attack
, sideslip angle
, pitch rate
, altitude
, and Mach number
, then
. A typical neural network used to predict aerodynamic loads is shown in
Figure 1. By training a neural network on an aerodynamic dataset, it is possible to fit the mapping between flight states and aerodynamic loads, thereby achieving aerodynamic force and moment coefficients prediction.
The nonlinear and unsteady aerodynamics exhibit time lag effects. The unsteady aerodynamic forces and moments not only depend on the instantaneous states but also their time histories. Consequently, aerodynamic coefficients
can be further modeled as a function of flight states over a continuous period:
where
denote the flight states corresponding to the time step from
to
.
A recurrent neural network (RNN) is a type of neural network which uses sequential data or time series data. In contrast to the feedforward neural network, an RNN has feedback connections that enable the network to remember the previous input. Improved network structures such as Long Short-Term Memory (LSTM) and Gated Recurrent Units (GRUs) have resolved the vanishing gradient problem of traditional RNNs. Therefore, by inputting a continuous sequence of flight states into an RNN, it is possible to extract temporal information and improve the accuracy of machine learning models in aerodynamic forces and moments prediction.
Figure 2 illustrates the process of using LSTM to predict aerodynamic loads based on the flight state history.
In summary, neural network models typically achieve high accuracy. However, the prediction process of neural networks is difficult to interpret, making their application in aerodynamic modeling challenging.
2.3. Weighted Feature Fusion Model
This paper introduces a weighted feature fusion model (WFFM) based on both the state-space model and LSTM. The design of the WFFM aims to overcome the difficulty of obtaining explicit physical mechanisms and high accuracy.
State-space models possess explicit physical meanings and describe the physical characteristics of separated flows. However, their limited fitting capability on nonlinear problems leads to low prediction accuracy. The aerodynamic data obtained through this method are often referred to as low-fidelity data. In contrast, the aerodynamic data obtained through methods such as wind tunnel experiments are referred to as high-fidelity data. The key concept behind the WFFM is to introduce physical information from the low-fidelity model into the neural network model. By using high-fidelity data for training, it establishes a mapping from low-fidelity data to high-fidelity data. This method minimizes the impact of errors in the low-fidelity data on prediction accuracy. In this way, the WFFM maintains physical significance and reduces additional errors to improve the accuracy of predictions for aerodynamic forces and moments.
The structure of the WFFM is illustrated in
Figure 3. The WFFM consists of four layers. The first layer is the state-space model layer, which takes flight states
at time
as input and calculates the low-fidelity aerodynamic coefficients
using the state-space model:
where
is the state-space model described in
Section 2.1.
The second layer is the feature standardization layer, which makes the distributions of each feature in the input data have zero means and unit variances. This step normalizes the range of independent variables or features of data, thus improving training convergence speed and prediction accuracy [
28]. The standardization operation is defined as follows:
where
and
represent the population of flight state data and low-fidelity data.
and
are the means of
and
, respectively.
and
are the standard deviations of
and
, respectively.
and
are the standardized data.
The third layer is the feature weighting layer. The process of weighting is used to assign different levels of importance to the various features in a dataset. Since the low-fidelity model output
exhibits similar trends to the high-fidelity data, but with different values, it should be assigned a reduced weight to play a guiding role. The feature weighting layer is implemented using a single neuron and an activation function:
where
and
are the weight vector and bias term for this neuron, adjusted automatically through backpropagation based on the error between the predicted output and the actual high-fidelity output [
29]. The activation function
transforms any input from the range
to a value that lies on the interval
.
and
represent the weighted data.
The fourth layer is the feature fusion layer, which consists of the LSTM model. In this layer, two mappings have been established. One is the mapping from flight states to aerodynamic loads, which is the same as the black-box model. The other is the mapping from low-fidelity data to high-fidelity data, which includes additional physical information. The LSTM model with strong nonlinear fitting capabilities is used to fuse the features:
where
represents the LSTM model, consisting of an LSTM sublayer with 100 hidden units, followed by two fully connected sublayers with 100 and 50 neurons, respectively, and a dropout layer.
and
compose the input vector for the LSTM model at time
(
).
represents the output of the WFFM.
Finally, the error between the model’s output
and the high-fidelity data
is calculated. Based on the chain rule, the backpropagation algorithm calculates the error gradient of the loss function with respect to each parameter of the network. Parameters are adjusted to minimize the difference between the actual output and the desired output. Mean squared error (MSE) is chosen as the loss function:
where
is the number of training samples.
Overall, the weighted feature fusion model predicts aerodynamic forces and moments guided by the state-space model. It leverages the state-space model’s explanatory power for unsteady aerodynamic effects and the powerful nonlinear fitting capabilities of neural networks. In constructing the model, low-fidelity data are used to indicate trends, while high-fidelity data are used to correct these trends. This ensures both high accuracy predictions and generalization of the model.