Identification of a Stochastic Dynamic Model for Aircraft Flight Attitude Based on Measured Data

Li, Haiquan; Chen, Xiaoqian; Zhang, Jiatu; Wang, Bochen; Peng, Jiahui; Wang, Liang

doi:https://doi.org/10.1155/2023/5429395

International Journal of Aerospace Engineering

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Abstract Introduction Numerical Example Conclusion Data Availability Conflicts of Interest Authors’ Contributions Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2023 | Article ID 5429395 | https://doi.org/10.1155/2023/5429395

Identification of a Stochastic Dynamic Model for Aircraft Flight Attitude Based on Measured Data

Haiquan Li,¹Xiaoqian Chen,²Jiatu Zhang,³Bochen Wang,³Jiahui Peng,³and Liang Wang³

Academic Editor: Jiaying Zhang

Received20 Dec 2022

Revised07 Aug 2023

Accepted29 Sept 2023

Published28 Oct 2023

Abstract

Stochastic disturbances are everywhere. The influence of stochastic factors on the modeling and simulation of aircraft flight dynamics should be considered. Therefore, a stochastic differential equation for aircraft flight attitude is modeled based on the traditional one in this paper. After that, an identification method based on the idea of sparse recognition for unknown parameters and stochastic disturbance is proposed. Finally, a set of measured flight data is used to verify that the identified stochastic model has obvious advantages over the traditional deterministic model when the aircraft is maneuvering in flight. This method can improve the accuracy and reliability of the aircraft flight dynamic model.

1. Introduction

The flight dynamic model of the aircraft plays an important role in aircraft design. It can provide accurate and reliable data for aircraft modification design, flight control system optimization, ground simulation experiment, flight quality, and combat effectiveness evaluation. And it can reduce the number of costly and time-consuming flight tests and shorten the data processing time and the flight test cycle. However, the accuracy of the flight dynamic model is limited by complex aerodynamics. The system identification based on the measured flight data has provided technical support for improving the confidence of the model.

Modeling aircraft flight dynamics using the system identification method began in the 1950s. In 1951, Greenburg [1] and Shinbrot [2] first used the system identification method to study aircraft and proposed the least squares method to determine the stability derivative of aircraft based on flight tests.

Before the 1990s, the identification methods developed mainly include time domain identification and frequency domain identification. And the algorithm is no longer only for single-parameter identification; it has been developed to identify the structure of the model. The basic framework for model identification of aircraft flight dynamics has been gradually developed through the continuous efforts of scholars. This framework is roughly divided into the following three steps. First, the initial value of unknown parameters in the model is estimated. Second, the model structure is identified. Third, unknown parameters in the model are identified. Molusis [3] used the Kalman filter to identify the stability derivatives of helicopters. Kaletka [4] summarized the key technologies to be solved in the identification of the helicopter flight dynamic model system. Guy and Williams [5] identified the MK-50 Sea King helicopter by using the least squares method, the output error method, the maximum likelihood method, and the augmented Kalman filter algorithm, which have been successfully applied to fixed-wing aircraft. Klein and Tischler et al. [6, 7] proposed the frequency domain identification algorithm of the helicopter, which was used for the flight dynamic model identification of the UH-60 helicopter. The development of frequency domain identification has laid a solid foundation for the identification of high-order flight dynamic models.

Since the 1990s, model identification has been mainly based on the high-order flight dynamic model, which has been extended to the nonlinear field at the same time. The identification accuracy and computational efficiency have been greatly improved. Most importantly, model identification has gone beyond the theoretical level to a variety of practical applications. Tischler and Cauffman et al. [8] developed a multiwindow spectrum optimization method to improve the accuracy of spectrum calculation and improve the identification ability of the high-order flight dynamic model. Jategaonkar and Plaetschke [9] carried out the early work of nonlinear model identification for BO-105 helicopter. Zhang et al. [10] proposed an aircraft’s parameter identification algorithm, which optimizes the ML function with the cloud model optimization theory in accordance with the ML estimation principle, thus obtaining the values of the parameters to be identified. Majeed and Vikalp [11] build a neural model of an aircraft from flight data and online estimation of the aerodynamic derivatives from the established neural model. Bagherzadeh [12] provided a novel method that extracts observable flight modes from flight test data and uses them in the identification process by conducting a gray box time domain method. Wu and Chen [13] developed an online system identification method for tiltrotor aircraft flight dynamic modeling by establishing a weighted recursive least squares algorithm.

Simmons [14] established a mathematical model of the propulsion system by using the flight data of isolated propellers collected on the flight envelope line of aircraft. Ayyad et al. [15] proposed a new comprehensive method, DNN-MRFT, to complete the real-time identification and tuning of multirotor UAV. Rohr et al. [16] completed the modeling, system identification, and nonlinear model predictive control design of the longitudinal full envelope speed control of a small hybrid UAV. Ivler et al. [17] extracted the flight dynamic model of a hovering six-wing UAV through the state-space system identification method. Verma and Peyada [18] estimated and optimized unknown variables of the aircraft dynamic model by combining the recursive mechanism learning machine network and the Gauss-Newton method. Finally, the maximum likelihood method was used to verify estimated parameters and demonstrate the effectiveness of this method. Avcolu [19] identified and verified unknown aircraft aerodynamic parameters through an improved subspace identification method, and the identification results were satisfactory. Cao et al. [20] proposed system identification method based on interpretable machine learning for unknown aircraft dynamics. These model recognition methods are based on a definite model.

However, random noise is widely existed in practical problems and has an important effect on the real system [21–25]. Stochastic disturbance will affect simulation accuracy and efficiency and increase fragmentation, proximity, and shape complexity of simulation results [26, 27]. With the further improvement of the accuracy of aircraft flight dynamic model, it is necessary to consider the influence of stochastic factors on flight dynamics. In the study of infectious disease system identification, Tutsoy et al. [28] extended the SpID-N model as in the form of multi-input-multi-output structure by adding the multidimensional unknown uncertainties and proposed identification method of the system. For flight dynamic research, the calculation shows that the stochastic factors have a significant impact on the flight posture, and the stochastic response of the simulation model considering the uncertainty factors can more accurately simulate the flight state in the real environment than the deterministic response. Moreover, the existing deterministic identification methods cannot fully meet these requirements; stochastic identification methods came into being [29–32]. The main purpose of this paper is to identify the stochastic model of flight attitude dynamics based on the sparse identification algorithm and measured flight data.

The paper is organized as follows. In Section 2, we refine the deterministic model and obtain the stochastic model we studied. In Section 3, the identification procedure for the stochastic model obtained in Section 2 is fully showed. In Section 4, measured data of a particular type of aircraft is used as an example to validate the effectiveness of the method and the accuracy of the identified stochastic model. The concluding remarks are given in Section 5.

2. The Establishment of the Stochastic Model of Aircraft Flight Dynamics

2.1. The Traditional Deterministic Flight Attitude Dynamic Model

The traditional deterministic aircraft flight attitude dynamic model is a three-dimensional nonlinear motion equation, which is established under ideal assumptions. The deterministic model has good applicability when the aircraft is not subjected to any stochastic disturbance. However, the aircraft is inevitably affected by various stochastic loads, and this deterministic model is not enough to accurately describe the dynamic behavior of the aircraft in the actual flight. It can be used as prior information to provide some reference for the establishment of the stochastic model. The traditional deterministic model equation is shown in [32] where , , and are the angular velocity of the body shafting, respectively;, , and are the moment of inertia of the aircraft on the body shafting, respectively; , , and are the projection of the combined torque on the body shafting; and are the wingspan and reference area, respectively; is the mean aerodynamic chord; is the air density; and are the angle of attack and sideslip, respectively; , , and are the aileron angle, rudder angle, and elevator angle, respectively; , , and are manipulation derivatives; and are the statically stable derivatives; , , and are the derivatives of the damping moment and the derivatives of the washout time difference; and is the velocity of the aircraft. For any variable in this paper, the operator stands for .

This paper mainly studies axisymmetric aircraft, and it can be considered that the aircraft has no asymmetry between error and shape, that is, . In actual analysis, the influence of , , and on can be ignored. Similarly, the influence of and on and influence of and on can be ignored; thus, Eq. (2) can be simplified to Eq. (3). A more detailed derivation is given in Ref. [33].

Among the above variables, aircraft ontology parameters, flight status parameters, flight environment data, and flight status parameters can be measured in real time by sensors and other equipment in the flight experiment process, while the aircraft ontology parameters and flight environment are set in the aircraft design process.

2.2. The Establishment of a Stochastic Flight Attitude Dynamic Model

In the previous section, the traditional model (Eq. (1)) of aircraft attitude dynamics is presented. A stochastic model is obtained by adding a random term to the torque term of the traditional model. where , , and are represented by Eq. (2).

Insight 1: the uncertainties in the actual situation are complex and diverse, and a stochastic model is obtained by adding a random term to the torque term of the deterministic model.

Insight 2: Eq. (4) is a stochastic differential equation. It is worth noting that stochastic terms , , and are completely unknown. Unlike other identification methods that treat random items as Gaussian white noise, we do not make any assumptions about stochastic terms in advance.

Insight 3: on the right-hand side of Eq. (4), except for the stochastic terms , , and, the remaining terms can be obtained by in-flight experiments. The derivative term on the left-hand side of Eq. (4) needs to be solved by some methods, which will be covered in Section 3.

3. The Identification of the Stochastic Model of Aircraft Flight Dynamics

In this section, the actual measurement data of the aircraft and the traditional deterministic flight dynamic model (Eq. (1)) are taken as the priority information, and the process of constructing the flight dynamic stochastic model based on the sparse identification idea and the identification method of the model parameters is proposed.

3.1. TV Algorithm for Solving Derivatives

The derivative in Eq. (10) of the measured data is crucial for the identification of the stochastic model. The central difference method performs well for smooth data without disturbance. But during the actual flight, aircraft may be affected by stochastic factors such as air pressure, temperature, atmospheric density, turbulence, aircraft main parameters, and pilot control, which lead to random fluctuations in the attitude dynamics of the aircraft. Therefore, the derivatives of flight attitude data obtained by using the central difference method have great disturbance, and it is not satisfactory to use these derivatives for model identification.

In order to solve the problem, we can use total variation (TV) method to obtain the relatively smooth derivative values of the measured data. A brief introduction of TV algorithm is as follows.

In 1992, Rudin, Osher and Fatemi [34] first proposed the TV method, which can obtain relatively smooth derivative values without losing the jump information of the original data. The TV method needs to adjust the parameters so that the derivatives have good regularity. The derivative of the function can usually be obtained by solving the following equation:

Here, is the penalty term, is the integral of , and the upper limit is a variable; in other words, the integral result is the expression about . is a regularization parameter that controls the balance of the penalty term and fidelity term. is a data fidelity item, which tries to reduce the differences between and . The data fidelity item is usually the square of norm, that is, .

The TV method solves the derivative of in as the minimum of the function [35].

We assume that (a null hypothesis in the discrete case), and let ; is defined in . Actually, is continuous on , and it exists minimum value due to the compactness of on and the lower semicontinuous of seminorm. Meanwhile, is strictly convex enough to ensure that has a unique minimum . is the integral operator with a bounded kernel, and it is continuous on .

3.2. The Identification Process of the Stochastic Model

The traditional identification method should traverse the functional basis composed of state variables to realize model identification. However, the method of traversing the basis will bring some problems to model identification. For example, going through all the functional basis will greatly increase the time cost of the computation. When the functional basis is less, the accuracy of the model may be insufficient. The model is more complicated when the functional basis is more. The sparse identification method can well balance the accuracy and complexity of the model [36].

Based on the idea of sparse identification, this paper establishes a stochastic model with the traditional deterministic model as the prior information. In this paper, , , , , , , , , , , , , and related combinations in model (1) are taken as the basis. Then, stochastic noise is introduced to build a stochastic flight dynamic identification model with unknown coefficients and unknown noises. The general form of the stochastic model is shown in where , , , , , , , , , , and are the unknown parameters and , , and are the independent stochastic disturbances. The meanings of the remaining parameters are given in Eq. (1).

Next, the stochastic model will be divided into a deterministic part and a stochastic perturbation part for identification, respectively. To facilitate description, the model is written in the following form: where denotes the deterministic part of the stochastic model (7) and is the stochastic perturbation part. In the following, represents the measured data; represents the solution of ; is the difference between and , that is, , which represents the stochastic disturbance contained in the measured data; and is recorded here.

The solution of equation can well represent the dynamic behavior of aircraft flight without stochastic factors. Therefore, we firstly identify the unknown parameters in equation based on the measured data and the least squares method. A brief introduction to the least squares method is as follows.

First, construct the data matrix:

An important step in the sparse identification method is to select the correct model terms and construct data matrix. The general method is to extract the terms as the base according to some physical equations that the system may satisfy. The deterministic flight attitude dynamic model (Eq. (1)) can describe the flight attitude of the aircraft well without disturbance. Thus, it is credible to add the items contained in this deterministic model into the base library.

The equation is split into overdetermined where , , and . Solving Eq. (5) is equivalent to finding , , and , which satisfy

The solution (12) of the system (10) can be simply expressed as Eq. (11) by using the optimal solution in the sense of two-norm. where is the inverse of and is the transpose of .

After obtaining equation with known parameters, we can get the behavior data of aircraft without stochastic factors. Further, the stochastic disturbances contained in the measured data can be calculated by using equation . Note that model (7) is a differential equation, and is the increment of the disturbance part of the stochastic model. It is worth mentioning that the Gaussian white noise is expressed as a formal derivative of the Wiener process. This expression is very common in the field of stochastic dynamical systems. The stochastic process usually does not have strictly mathematical theoretical derivatives; its formal derivative is typically expressed as a noise. In this paper, is the increment of the stochastic response within a sampling time interval , and is a formal derivative of the stochastic process, which can be referred to as noise. To sum up, the relationship between and can be expressed as

After the above steps are performed, the aircraft flight stochastic dynamic model can be identified. In the following, we will demonstrate the effectiveness of our method through the numerical example.

4. Numerical Example

In this section, we will illustrate the identification process in actual application and demonstrate the effectiveness of our method on the basis of actual data from a certain type of aircraft.

4.1. Real Data Preprocessing

Some parameters of this aircraft are shown in Table 1. Next, the stochastic model is identified based on the measured flight data, and the sampling frequency of the measured data here is 32 Hz. The measured data is divided into training data and prediction data. The training data is used to identify the stochastic model, and the prediction data is used to verify the effectiveness of the identified model.

Figure 1 is the time history diagrams of the training data of the angular velocity component of the aircraft. Figure 2 draws the time history diagrams of the training data of the three rudder declination angles according to the measured data. Figure 3 represents the time history diagram of training data for the attack angle and sideslip angle .

(a)

(b)

(c)

(a)

(b)

(c)

(a)

(b)

In addition, according to the actual situation, the rudder deflection angles and of the aircraft during flight should be roughly symmetric about 0, and and should be roughly equal. However, the measured data show different characteristics, which may be caused by measurement errors such as the accuracy of the measuring instrument. In order to make the model identified closer to reality, the rudder deflection angles and were slightly shifted to make them roughly symmetric about 0, and and were equal through . After data processing, the training data used for model identification are shown in Figures 1 and 2.

Finally, it should be noted that when the least squares method is used to identify the parameters of the equation, it is necessary to solve the derivative of angular velocity. For the data without noise, the central difference can be used to solve this problem. However, the measured data contains a lot of random information, and the direct derivation will lead to large fluctuations in the derivative value. Therefore, the TV algorithm mentioned in the previous section is combined to obtain the derivative of , , and .

4.2. The Identification of the Stochastic Model

First, we identify the deterministic part of Eq. (7) based on and the least squares method. The identification results of the equation using the TV algorithm and the least squares method are as follows.

Then, the fourth-order Runge-Kutta method is used to solve the deterministic part Eq. (5). In order to keep consistent with the sampling frequency 32 Hz of the measured data, the fourth-order Runge-Kutta iterative step is selected . Thus, we can obtain solutions of , , and by numerically solving Eq. (14). Then, the difference between the training data , , and and the simulation results , , and can be used to obtain the probability density function of stochastic disturbances , , and , as shown in Figure 4, respectively.

(a)

(b)

(c)

We can carry out numerical simulation of stochastic disturbances , , and by the probability density function of stochastic disturbances. Then, the stochastic model (15) can be obtained by using .

4.3. Numerical Simulation of the Stochastic Model

To verify the accuracy of the identified model, we solve the dynamic response of this model (15). It should be emphasized that the response of the stochastic system is different from the solution of the deterministic model. The response of a stochastic system is a stochastic process (or a stochastic variable), which cannot be directly compared with the measured data. Therefore, we consider extracting the numerical characteristics of the stochastic response, that is, the sample means, and comparing it with the deterministic result and training data to show whether the stochastic system can more accurately describe the actual flight problem.

The comparison diagrams of numerical simulation results of flight dynamic responses , , and with measured data , , and are, respectively, drawn below. The red line represents the solution result of the deterministic model (1). The blue line represents the solution of stochastic model (15). The black line represents measured training data.

It can be seen from Figures 5–7 that and only show small fluctuations, while rolls due to the pilot’s control in the actual flight process. Small fluctuations may be caused by small changes in the environment, such as height, temperature, or wind, and such stochastic factors have relatively little influence on the response of the system. The large fluctuations are caused by the pilot’s large maneuvering action, which has a great influence on the flight dynamic behavior of the aircraft. Therefore, we mainly analyze the solution result of , which contains more complex stochastic information.

Figures 5–7 show the comparison of solution results of the deterministic model, stochastic model, and measured training data. It can be seen that the simulation result of the stochastic model is more precise than that of the deterministic model when the aircraft is maneuvering. However, there are some errors between the results of the stochastic model and deterministic model and the measured training data. After all, the stochastic result is the average value of multiple groups of samples, while the measured data is only one group of samples. It is difficult to ensure that the sample has the same value as the mean, but the mean can well reflect the characteristics of the sample. The error analysis table of the solution results of the stochastic model and deterministic model and training data is given below to observe the advantages of the stochastic model from qualitative and quantitative perspectives. The stochastic model shows good simulation results in the case of aircraft maneuvering flight. The mean square error of the solution results and the measured training data is calculated when .

As can be seen from Table 2, the accuracy of the solution results of the stochastic model is significantly improved compared with that of the deterministic model. From the value of the mean square error, the mean square error of the stochastic model pair is obviously smaller than that of the deterministic model. This shows that the stochastic model has good accuracy from a quantitative point of view.

The above comparison between the solution results of the stochastic model and the measured training data shows the accuracy of the stochastic model and the effectiveness of the algorithm in this paper. To further confirm the robustness of the stochastic model (15), its ability to predict the flight attitude of the same aircraft will be verified below.

4.4. Prediction Results of the Stochastic Model

In this section, the stochastic model (15) will be used to predict the flight attitude of the aircraft in a certain period, and the prediction results of the model will be compared qualitatively and quantitatively with the measured data to further illustrate the practicality of the stochastic model (15).

Figures 8 and 9 show the data required by stochastic model (15) to predict flight attitude , , and . Figures 8(a)–8(c) show the time history diagrams of predicted data representing the rudder angles , , and , respectively, and Figures 9(a) and 9(b) represents the attack angle and sideshow angle , respectively. Then, the stochastic Runge-Kutta method was used to solve the dynamic response of model (15), and the iteration step was still 0.01.

(a)

(b)

(c)

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(b)

The comparison diagram of the predicted results , , and and measured data is drawn below. The red line represents the solution result of the traditional deterministic model (1). The blue line represents the prediction results of the stochastic model (15). The black line represents the measured data.

The results of Figures 10–12 show that the prediction results of the stochastic model fit well with the flight measured data when the aircraft rolls. However, the prediction results of the traditional deterministic model (1) have great errors with the flight measured data. In particular, the comparison is particularly obvious at the larger maneuvering peak in Figure 10. The prediction results of the stochastic model are closer to the measured data, while the solution results of the deterministic model have a large error, which highlights the higher accuracy of the stochastic model when the aircraft does large maneuvers.

Table 3 shows the comparison of the mean square error of the solution results of the deterministic model, the stochastic model, and the measured predicted data. From the numerical perspective, it can be seen that the stochastic model has better prediction ability. From a quantitative point of view, the stochastic model is closer to reality than the deterministic model, which further confirms the effectiveness of the recognition algorithm and the good robustness of the stochastic model.

In general, the stochastic model is more close to the measured data from qualitative and quantitative perspectives. The stochastic model shows more precise prediction results compared with the deterministic model when the aircraft is rolling. Therefore, the model considering stochastic factors is more consistent with the real flight state and can effectively improve the reliability of flight dynamic simulation model.

5. Conclusion

In this paper, we use the idea of sparse identification to establish the stochastic model of flight attitude dynamics. Based on the measured data of aircraft flight attitude dynamics containing stochastic disturbances, the stochastic model is divided into the deterministic part and random disturbance part for identification, respectively, and then, the stochastic model of flight dynamics is obtained by using the least squares method. Furthermore, we compare and analyze the solutions of the stochastic model and the traditional deterministic model with the measured data, respectively. The comparison results show that the calculation results of the stochastic model are in good agreement with the measured data, especially when the aircraft is maneuvering in flight, and the stochastic model has obvious advantages compared with the traditional deterministic model. This demonstrates that the algorithm in this paper is effective for the identification of stochastic models driven by measured data. It also shows that it is necessary to consider the influence of stochastic factors when modeling and simulating aircraft flight dynamics.

Data Availability

Data will be made available by contacting the corresponding author.

Conflicts of Interest

The authors have declared that no competing interests exist.

Authors’ Contributions

All authors contributed equally to this work.

Acknowledgments

This work is supported by the National Natural Science Foundation of China (Grant No. 11972289).

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Copyright © 2023 Haiquan Li et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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