A Joint Online Estimation Method for Aircraft Aerodynamic Parameters and Thrust Deviation

Abstract

Direct measurement of engine thrust during aircraft flight remains challenging. Currently, engineering methods yield only coarse approximations of thrust during flight. This limitation significantly impedes aerodynamic parameter identification from powered flight data, particularly undermining the credibility and accuracy of the identification results of aerodynamic force coefficients. To address these challenges inherent in aerodynamic parameter identification from powered flight data, this study proposes a novel joint online estimation method capable of simultaneously estimating system states, unknown aerodynamic parameters, and engine thrust. The algorithm integrates Kalman filters with computationally efficient recursive least squares (RLS) estimators to perform sequential estimation of flight data. This structure provides real‐time access to unmeasurable engine thrust and enhancement of the estimation precision of aerodynamic parameters. The effectiveness of the proposed algorithm was rigorously validated and assessed using both simulation and flight test data from the CAE‐AVM benchmark aircraft model. The method successfully generated valid estimates of engine thrust and aerodynamic parameters from both datasets and exhibited superiority over the EKF and MMAE algorithms. Specifically, for flight test phases including climb, cruise, and descent, the maximum root mean square relative error (RMSE) for thrust estimates was found to be only 17.36%. These results demonstrate the high estimation accuracy of the proposed joint estimation method for both simulation and flight test data and validate its high effectiveness for the identification and processing of aircraft‐powered flight data.

1. Introduction

In the aerospace and aviation industry, parameter identification is essential for identifying a system’s unknown parameters or modeling an unknown system from measurement data. For acquiring in-flight aircraft aerodynamic characteristics, identification technology is often the primary approach. However, its application is constrained by measurement devices and data quality. Data quality, usually defined by the signal-to-noise ratio, describes the proportion of noise and the relevant information about the parameter characteristics. Measurement devices determine which system states can be measured. Poor data quality and the inability to measure critical system states lead to inaccurate identification results, poor performance, or even failures.

Prevailing engine thrust in-flight measurement methods include direct measurement [1] and analytically based or model-based calculation methods [2–5]. NASA attempted to perform direct in-flight thrust measurement using strain gages on a supersonic aircraft [1]. However, the difficulties in reliable sensor calibration and measurement procedures compromised thrust measurement data accuracy and credibility. Analytical or model-based numerical methods rely on ground and flight test data for onboard estimation, leveraging relationships between thrust and measurable correlation quantities to approximate the real value. The estimation accuracy of numerical methods is significantly affected by the choice of correlation quantities and the dataset. Consequently, aerodynamic parameter identification from powered flight data has long been plagued by thrust measurement deviations, especially as most aircraft lack real-time thrust measurement equipment or software. When thrust data is unavailable, alternative strategies are typically employed, such as using passive flight data or carefully designing tests to isolate aerodynamic parameters from thrust effects. Apart from such specially designed cases, the inaccurate thrust reduces the credibility of the identified results of aerodynamic parameters. Thus, a feasible method for identifying aerodynamic parameters under inaccurate thrust measurements is essential for real flight.

Thrust deviation introduces errors into dynamic system models. Research on parameter estimation under model uncertainties primarily focuses on joint estimation of system states, unknown parameters, and noise characteristics. Methods include parallel estimation using dual estimators [6, 7], multiple model adaptive estimation (MMAE) [8], ensemble Kalman filters (EnKF) [9–14], joint Bayesian estimation [15], and variational Bayesian technologies [16–18]. Parallel methods typically use two distinct estimators separately for augmented system states (containing unknown parameters) and the noise covariance matrix. MMAE and EnKF employ multiple homogeneous filters to sample or estimate distributions of unknown parameters. Recently, Bayesian and variational Bayesian techniques have been extensively employed to address process and measurement noise distributions and joint state/parameter estimation in linear and nonlinear systems. Bayesian technologies introduce optimization or variational analysis to approximate posterior probabilities or probability distributions, offering potentially strong information extraction capabilities. However, directly applying existing estimation paradigms to the problem of online thrust estimation presents significant challenges. The primary limitations are

• 1.

  Computational efficiency limitations. Methods that employ multiple filters or incorporate optimization and variational technologies often suffer from high computational complexity. This renders them unsuitable as onboard applications for aircraft due to prohibitive computational latency.

• 2.

  Deterministic problem constraints. Methods based on Bayesian theory are primarily designed to infer the statistical characteristics of random variables. Our problem, in contrast, concerns the estimation of a deterministic parameter—engine thrust. The strong correlation between thrust and system states (e.g., acceleration and velocity) leads to an ill-posed mathematical formulation, where Bayesian estimators—reliant on stochastic assumptions—typically struggle to provide a precise deterministic solution.

Consequently, there exists a compelling need for an estimation strategy that simultaneously addresses computational efficiency and the deterministic, ill-posed nature of the online thrust estimation problem. Kalman filters (KFs) and recursive least squares (RLS), commonly used in resource-constrained environments like onboard flight computers, offer real-time capability and versatility. As an exact closed-form solution for recursive Bayesian estimation under linear Gaussian assumptions, KF is a minimum variance estimator. For nonlinear systems, the extended KF (EKF) developed by researchers, such as Jazwinski [19], Gelb [20], and Maybeck [21] since the 1970s, serves as a numerical approximation for nonlinear system states. The EKF has been widely implemented in online aircraft parameter estimation [22, 23], online fault detection [24], and parameter estimation–based adaptive/reconfigurable control [25–27]. The RLS estimator [28] has also found extensive applications in various system parameter estimations. Due to its recursive nature, RLS is particularly suitable for real-time applications in digital signal processing, communication, and control systems. By estimating unknown system parameters, RLS facilitates system modeling. For “gray-box” problems with known model structures, RLS directly estimates parameters in aircraft aerodynamic characteristics modeling [29, 30]. For nonlinear systems lacking accurate models, pseudolinear or auxiliary models are typically employed to fit input–output relationships, for example, using RLS with Wiener models [31], output–error moving average (OEMA) models [32], or neuro-fuzzy models [33]. To address challenges related to convergence rate, tracking accuracy, misadjustment, robustness, and stability in RLS parameter estimation, numerous improvements in RLS algorithms have been proposed. These enhancements include introducing QR decomposition to improve stability [34], incorporating correction terms to eliminate estimation bias [35], using a regularized counterpart to enhance performance [36], and implementing dual-loop structures or forgetting factors to enhance tracking capability for time-varying parameters [37–40]. Given the real-time capability, simplicity, and versatility of KFs and RLS, these methods have found extensive application in aircraft system identification and online parameter estimation. Researchers from Delft University of Technology, including Mulder et al. and Chu et al., proposed the Delft two-step method [41–43], combining KFs with RLS for online aircraft aerodynamic model identification. This method demonstrated robustness, stability, and adaptability through long-term engineering application. NASA subsequently adopted this method in its “Learn-to-Fly” project for online acquisition of aircraft aerodynamic models and parameters [44, 45]. While these classical estimation techniques, such as KF, EKF, and RLS, have been extensively studied and proven to be computationally efficient for real-time onboard applications in aerospace systems, they are not directly applicable to the deterministic and ill-posed problem of online thrust estimation. The KF and its nonlinear extension, the EKF, are inherently designed for state estimation under stochastic uncertainty, making them ill-suited for precisely determining a deterministic parameter. Similarly, while RLS is a powerful tool for parameter identification, its standard formulation often struggles with the ill-posed nature of this specific problem, where the lack of persistent excitation or high parameter–state correlation can lead to observability issues and unreliable convergence. Therefore, there is an urgent need to develop a new estimation strategy that combines these proven methods to overcome the mathematical challenges of this thrust online estimation issue.

Motivated by achieving real-time estimation of engine thrust and aerodynamic parameters on resource-constrained onboard computing equipment, this paper proposes an estimator integrating EKF and RLS, which have been successfully applied in system control and parameter estimation. This approach enables joint online estimation of system states, aerodynamic parameters, and engine thrust. Its effectiveness is verified through an aircraft’s simulation and flight test data. This work contributes to enhancing system modeling, parameter identification, and data processing capabilities for aircraft-powered flight.

The paper is organized as follows: First, the “Materials and Methods” section introduces the system model, EKF, RLS, the joint estimation method, and the method’s convergence analysis. Next, the “Results and Discussion” section introduces the AVM aircraft model and the validation results based on simulation and flight test data. Finally, the paper concludes with the “Conclusions” section.

2. Materials and Methods

2.1. System Model Under Thrust Deviation

The nonlinear dynamics model of an aircraft under inaccurate thrust measurement can be expressed as 1 $\begin{array}{c}\stackrel{.}{\mathbf{x}}=f\left(\mathbf{x},\mathbf{u},\mathbf{\theta },\mathbf{d},\mathbf{\nu }\right),\mathbf{y}=g\left(\mathbf{x},\mathbf{u},\mathbf{\theta },\mathbf{d},\mathbf{w}\right),\end{array}$ where x represents the system state vector; y is the output vector; f and g denote nonlinear state and output functions, respectively; u is the control input vector; θ represents system unknown parameters; d is the thrust measurement deviation; and ν and w are the process and measurement noises. The noises are typically modeled as Gaussian random processes with known prior characteristics, characterized by covariance matrices R_v and R_w, respectively.

For this nonlinear system, where input–output measurement data (u, y) are available, the parameters requiring estimation include the system states x, the unknown parameters θ, and the thrust deviation d. As Equation (1) indicates, physical interconnections exist among these three types of unknowns. Consequently, simply applying standard joint estimation often leads to ill-posed problems. To address this challenge, a specifically designed estimator is developed in this paper.

2.2. EKF

The EKF is a classic two-step approach comprising prediction and update steps. For the nonlinear system in Equation (1), EKF estimates the state variable x. The superscripts “^~” and “^∧” are utilized to distinguish the predicted and updated values, respectively. The state predictions and covariance matrix at step k are calculated from the state at step k − 1: 2 $\begin{array}{c}\tilde{\mathbf{x}}\left(k\right)=\stackrel{\wedge }{\mathbf{x}}\left(k-1\right)+{\int }_{t\left(k-1\right)}^{t\left(k\right)}f\left(\stackrel{\wedge }{\mathbf{x}}\left(t\right),\overline{\mathbf{u}}\left(k\right)\right)dt,\tilde{\mathbf{P}}\left(k\right)=\Phi \left(k\right)\stackrel{\wedge }{\mathbf{P}}\left(k-1\right){\Phi }^T\left(k\right)+R_v,\end{array}$ where $\overline{\mathit{u}}\left(k\right)$ denotes the mean input values from step k − 1 to k, Φ is the state transition matrix, P is the state covariance matrix, and t is the time.

The state updates based on output data can be written as 3 $\begin{array}{c}\tilde{\mathbf{y}}\left(k\right)=g\left(\tilde{\mathbf{x}}\left(k\right),\mathbf{u}\left(k\right)\right),K\left(k\right)=\tilde{\mathbf{P}}\left(k\right)C^T\left(k\right){\left[C\left(k\right)\tilde{\mathbf{P}}\left(k\right)C^T\left(k\right)+R_w\right]}^{-1},\end{array}$ where C is the linearized observation matrix, which can be calculated via central difference, K is the gain matrix, I is the identity matrix, and z represents the output observations. The integral term in Equation (2) can be solved by the fourth-order Runge–Kutta algorithm, and the transition matrix Φ is computed using the Padé approximation or the Taylor series expansion.

2.3. RLS

Under the assumption of white noise, least squares estimation is unbiased. When the system model is linear, as shown in Equation (4), the unknown parameters θ can be estimated recursively: 4 $\begin{array}{}\mathbf{z}\left(t\right)=H\left(t\right)\mathbf{\theta }\left(t\right)+\mathbf{\nu }\left(t\right),\end{array}$ 5 $\begin{array}{c}\stackrel{\wedge }{\mathbf{\theta }}\left(k\right)=\stackrel{\wedge }{\mathbf{\theta }}\left(k-1\right)+\Delta \stackrel{\wedge }{\mathbf{\theta }}\left(k\right),\Delta \stackrel{\wedge }{\mathbf{\theta }}\left(k\right)=K\left(k\right)\left[\mathbf{z}\left(k\right)-H\left(k\right)\stackrel{\wedge }{\mathbf{\theta }}\left(k-1\right)\right],\end{array}$ where H is the linearized matrix and ν represents Gaussian white measurement noise.

Compared to EKF, RLS lacks a prediction step; parameter estimates are updated incrementally using a feedback matrix and innovation residuals. Incorporating a forgetting factor λ into the RLS algorithm adjusts the algorithm’s tracking ability for time-varying parameters. With the forgetting factor, the gain and parameter covariance matrices in Equation (5) become 6 $\begin{array}{c}K\left(k\right)=\stackrel{\wedge }{\mathbf{P}}\left(k-1\right)H^T\left(k\right){\left[\lambda I+H\left(k\right)\stackrel{\wedge }{\mathbf{P}}\left(k-1\right)H^T\left(k\right)\right]}^{-1},\stackrel{\wedge }{\mathbf{P}}\left(k\right)=\frac{1}{\lambda }\left[I-K\left(k\right)H\left(k\right)\right]\stackrel{\wedge }{\mathbf{P}}\left(k-1\right).\end{array}$

For the forgetting factor RLS (FRLS) method, a forgetting factor close to 1 results in small overshoot and good stability but poor tracking ability. Conversely, a smaller forgetting factor enhances tracking ability but increases overshoot and reduces the stability of the algorithm.

2.4. Thrust Deviation Correction–Based Online Aerodynamic Parameter Estimation Method

The engine thrust deviation correction approach leverages discrepancies in multisource measurement data. Specifically, the aircraft position, velocity, angular rate, and attitude angles are measured by GPS, pitot tubes, and gyroscopes, while the overload is usually provided by accelerometers. These measured states are fused with overload to estimate thrust deviations. Then, the thrust estimates subsequently refine the aerodynamic parameter estimation results. As shown in Figure 1, the algorithm framework integrates one EKF, one FRLS estimator, and one conventional RLS estimator. Considering that engine thrust during flight is typically time-varying while aircraft aerodynamic derivatives are generally constant, the FRLS algorithm is utilized to estimate the thrust, while the conventional RLS algorithm jointly estimates the aerodynamic derivatives. This combined approach yields a more realistic estimation. To ensure algorithm stability and maintain continuity in aerodynamic parameter estimation, a convergence judgment mechanism for the thrust ratio coefficient C_F is implemented. Corrections are applied only after confirming the convergence of FRLS estimates.

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The computational workflow of the algorithm is as follows:

• 1.

  Initialization: Set initial parameters for EKF, FRLS, and RLS estimators, including initial values, covariance matrices, noise covariance matrices, window length, forgetting factor, and other relevant parameters.

• 2.

  State estimation: Augment the state variables x with the wind-axis aerodynamic coefficients $C_a^w$. Using the EKF filter, estimate the augmented state values and covariance matrix at step k based on step k − 1 estimates and Equations (2) and (3).

• 3.

  Thrust ratio and aerodynamic coefficient estimation: Input the k-step state estimates and overload measurement data into Equation(7). Use the FRLS estimator to estimate the body-axis aerodynamic coefficients $C_a^b$ and thrust ratio coefficient C_F. $C_a^b$ components (C_x, C_y, and C_z) are projected in the body coordinate system, where m is the vehicle mass, q_∞ is dynamic pressure, and S is the reference area. Body-axis accelerations (a_x, a_y, and a_z) are calculated directly from the overloads. 7 $\begin{array}{c}\left[q_{\infty }SF\right]\left[\begin{array}{c}{C}_x\\ C_F\end{array}\right]=m{a}_x,q_{\infty }S{C}_y=m{a}_y,\end{array}$

• 4.

  Coordinate transformation: Convert the body-axis aerodynamic coefficients to wind-axis coefficients $C_{ac}^w$ using a transformation matrix. Calculate the corrected thrust estimates F_c using the estimated C_F.

• 5.

  Convergence judgment: Determine the thrust ratio convergence time within a given window. Calculate the mean and standard deviation of the thrust ratio estimates within this window. If both statistics fall below predefined thresholds, mark the window end-time as the convergence moment.

• 6.

  Aerodynamic derivative estimation: If the thrust ratio has not converged, perform RLS estimation using the measured thrust and uncorrected $C_a^w$. If the thrust ratio has converged, use the corrected F_c and $C_{ac}^w$ in the RLS estimator from the convergence moment onward to obtain the k-step aerodynamic derivative estimates.

• 7.

  Repeat Steps 2~6 until all data are processed.

2.5. Convergence Analysis

The parameter differences between step k and step k − 1 are defined as 8 $\begin{array}{}\Delta \stackrel{\wedge }{\mathbf{\theta }}\left(k\right)=\stackrel{\wedge }{\mathbf{\theta }}\left(k\right)-\stackrel{\wedge }{\mathbf{\theta }}\left(k-1\right).\end{array}$

Assuming the changes in the measurements z, the linearized matrix H, and the gain matrix K between adjacent steps are negligible, rearranging Equation (8) with (5) and (6) yields 9 $\begin{array}{}\Delta \stackrel{\wedge }{\mathbf{\theta }}\left(k+1\right)=K\left(k+1\right)\left[\mathbf{z}-H\stackrel{\wedge }{\mathbf{\theta }}\left(k\right)\right]=K\left(k+1\right)\left[\mathbf{z}-H\stackrel{\wedge }{\mathbf{\theta }}\left(k-1\right)-H\stackrel{\wedge }{\mathbf{\theta }}\left(k\right)+H\stackrel{\wedge }{\mathbf{\theta }}\left(k-1\right)\right]=K\left(k+1\right)\left[\mathbf{z}-H\stackrel{\wedge }{\mathbf{\theta }}\left(k-1\right)\right]-K\left(k+1\right)H\left[\stackrel{\wedge }{\mathbf{\theta }}\left(k\right)-\stackrel{\wedge }{\mathbf{\theta }}\left(k-1\right)\right]=\Delta \stackrel{\wedge }{\mathbf{\theta }}\left(k\right)+\left[\lambda \stackrel{\wedge }{\mathbf{P}}\left(k\right){\stackrel{\wedge }{\mathbf{P}}}^{-1}\left(k-1\right)-I\right]\Delta \stackrel{\wedge }{\mathbf{\theta }}\left(k\right)=\lambda \stackrel{\wedge }{\mathbf{P}}\left(k\right){\stackrel{\wedge }{\mathbf{P}}}^{-1}\left(k-1\right)\Delta \stackrel{\wedge }{\mathbf{\theta }}\left(k\right).\end{array}$

The matrix P is usually positive definite. Parameter estimates converge when the inequality holds: 10 $\begin{array}{}\left|\lambda \times \text{eig}\left(\stackrel{\wedge }{\mathbf{P}}\left(k\right){\stackrel{\wedge }{\mathbf{P}}}^{-1}\left(k-1\right)\right)\right|<1,\end{array}$ where eig(∙) represents the eigenvalues of a matrix and |∙| calculates the absolute value. Substituting the iterative expression for matrix P (Equation(6)) into Equation (10) yields 11 $\begin{array}{}\left|\text{eig}\left(I-K\left(k\right)H\left(k\right)\right)\right|<1.\end{array}$

Parameter estimates converge when Inequality (11) holds.

Define the predicted errors at k-step in FRLS as 12 $\begin{array}{}\stackrel{\wedge }{\mathbf{e}}\left(k\right)=\mathbf{z}-H\stackrel{\wedge }{\mathbf{\theta }}\left(k\right).\end{array}$

Assuming negligible changes in the measurements z and the linearized matrix H between adjacent steps, the difference in predicted errors between step k and step k − 1 is 13 $\begin{array}{}\stackrel{\wedge }{\mathbf{e}}\left(k\right)-\stackrel{\wedge }{\mathbf{e}}\left(k-1\right)=-H\left[\stackrel{\wedge }{\mathbf{\theta }}\left(k\right)-\stackrel{\wedge }{\mathbf{\theta }}\left(k-1\right)\right]=-HK\left(k\right)\stackrel{\wedge }{\mathbf{e}}\left(k-1\right).\end{array}$

Substituting Equation (6) into (13) yields 14 $\begin{array}{}\stackrel{\wedge }{\mathbf{e}}\left(k\right)=\left[I-HK\left(k\right)\right]\stackrel{\wedge }{\mathbf{e}}\left(k-1\right)=\left\{I-H\stackrel{\wedge }{\mathbf{P}}\left(k-1\right)H^T{\left[\lambda I+H\stackrel{\wedge }{\mathbf{P}}\left(k-1\right)H^T\right]}^{-1}\right\}\stackrel{\wedge }{\mathbf{e}}\left(k-1\right)={\left[I+{\lambda }^{-1}H\stackrel{\wedge }{\mathbf{P}}\left(k-1\right)H^T\right]}^{-1}\stackrel{\wedge }{\mathbf{e}}\left(k-1\right).\end{array}$

If the forgetting factor λ is a positive real number, the matrix P is positive definite, and H is bounded, then it yields 15 $\begin{array}{c}{\lambda }^{-1}H\stackrel{\wedge }{\mathbf{P}}\left(k-1\right)H^T>0,{\left[I+{\lambda }^{-1}H\stackrel{\wedge }{\mathbf{P}}\left(k-1\right)H^T\right]}^{-1}<1.\end{array}$

If Inequality (15) holds, the predicted errors converge to 0 in finite time. Thus, the proposed FRLS estimator converges in both parameter estimates and predicted errors if and only if Inequalities (11) and (15) hold. When λ = 1 and these two inequalities hold, the RLS estimator also converges in both parameters and errors.

3. Results and Discussion

3.1. AVM Standard Aircraft Model

The CAE-AVM (Chinese Aeronautical Establishment–Aerodynamic Validation Model) [46, 47], developed by the Chinese Aeronautical Establishment, is an aerodynamic standard model based on a large long-range business aircraft with a cruise Mach number of 0.85. Figure 2 shows the configuration of the aircraft, while Table 1 lists the geometric and physical parameters of the aircraft. Using wind tunnel test data at Mach 0.2, the aerodynamic stability and control derivatives at this Mach number (see Table 2) were fitted via the polynomial form in Equation (16). 16 $\begin{array}{c}{C}_D=C_{D0}+C_{D\alpha }\alpha +C_{D{\alpha }^2}{\alpha }^2,C_L=C_{L0}+C_{L\alpha }\alpha +C_{L\stackrel{.}{\alpha }}\frac{c}{2V}\stackrel{.}{\alpha }+C_{Lq}\frac{c}{2V}q+C_{L\delta e}{\delta }_e,\end{array}$ where C_D is the aircraft drag coefficient, C_L is the lift coefficient, C_m is the pitch moment coefficient, α is the Angle of attack, q is the pitch angular velocity, δ_e is the elevator deflection angle, and V is the aircraft velocity.

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Table 1 Mass and inertial characteristics of AVM aircraft.

Aircraft physical parameter	Values
Mass m	160 kg
Wing reference area S	1.391 m2
Wing span l	3.53 m
Mean aerodynamic chord c	0.5151 m
Moment of inertia Ixx	0.853 kg × m2
Moment of inertia Iyy	41.797 kg × m2
Moment of inertia Izz	40.945 kg × m2
Product of inertia Ixz	1.573 kg × m2

Table 2 Longitudinal aerodynamic derivatives of AVM aircraft.

Aerodynamic derivatives	Values
C D0	0.0320
CDα	−0.1277
$C_{D{\alpha }^2}$	3.001
C L0	0.1516
CLα	5.4786
$C_{L\stackrel{.}{\alpha }}$	−42.6
CLq	40.2
CLδe	0.2336
C m0	0.0805
Cmα	−1.5836
$C_{m\stackrel{.}{\alpha }}$	−6.1
Cmq	−11.5
Cmδe	−1.0188

3.2. Validation by Flight Simulation Data

Based on the AVM parameters and aerodynamic model, a six-degree-of-freedom (6-DOF) flight dynamics model in the body coordinate system was established. Flight dynamics simulations generated aircraft simulation flight data for algorithm validation. A longitudinal-level flight simulation case was developed. The initial flight states and trim parameters are listed in Table 3, and other unspecified initial attitude angles and angular velocities were set to 0. When the simulation started, a 2-s period, 3° amplitude “3211” elevator deflection signal was introduced as a perturbation. Gaussian white noises were added to the longitudinal state variables, with standard deviations determined by the resolution of typical aircraft measurement equipment (see Table 4).

Table 3 Initial level flight states and trim parameters.

Flight state parameters	Values
Initial velocity V0	Mach 0.2
Initial height h0	100 m
Initial pitch angle φ0	2.5°
Trim angle of attack αtrim	2.5°
Trim elevator angle δetrim	−0.55°
Trim axial thrust Pxtrim	99.7475 N

Table 4 Noise standard deviations of longitudinal output measurements.

Longitudinal state measurements	Noise standard deviations
Axial velocity u	3.91e − 2 m/s
Normal velocity w	3.91e − 2 m/s
Pitch angular velocity q	2.91e − 2 rad/s
Pitch angle φ	5.11e − 4 rad
Height h	0.1 m
Velocity V	3.91e − 2 m/s
Angle of attack α	5.24e − 5 rad
Axial acceleration ax	1e − 3 m/s2
Normal acceleration az	1e − 3 m/s2

Using the proposed algorithm and the AVM longitudinal simulation data, parameter estimation was performed for four thrust measurement deviation scenarios: 0.5, 0.8, 1.2, and 1.5 times the reference trim axial thrust from Table 3. These cases covered both under- and overmeasured thrust. The measured thrust input for this algorithm was defined as the trim thrust multiplied by the above four deviation factors. Unknown parameters were initialized to 0, with algorithm parameter settings listed in Table 5 (where I denotes the identity matrix).

Table 5 Parameter settings in the joint online estimation method.

Estimators	Parameters	Values
EKF	Process noise covariance matrix Qn	I × 10−6
Measurement noise covariance matrix Rn	I × 10−4
Initial state covariance matrix Px	I
FRLS	Forgetting factor λ	0.98
Initial covariance matrix Pcf	I
Convergence window length Nw	2 s
RLS	Initial covariance matrix Pc	I

Figure 3 shows thrust estimation results for all four deviation scenarios, directly calculated from the estimates of the thrust ratio coefficient C_F. Figure 4 presents the converged C_F estimates with their error bars under a 95% confidence interval. Figures demonstrate the algorithm’s capability to identify the true thrust from deviated measurements. The estimates of thrust and the ratio coefficient C_F converged within 12 s across four scenarios, with stable error bars of C_F estimates after convergence. Figure 5 compares the identified drag stability derivative C_Dα with its reference value. After thrust estimates converged, applying corrected thrust effectively reduced deviations between the estimated C_Dα and its reference. The drag derivative estimates with error bars are shown in Figure 6, which were significant only initially before rapidly converging to near-imperceptible levels. These results and error analysis validate the effectiveness, convergence, and stability of the proposed online joint estimation method under simulation conditions.

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3.3. Validation by Flight Test Data

The performance of the proposed method was further validated using real powered flight data from an AVM flight test. This flight test included takeoff, climb, cruise, descent, and landing phases, with measured parameters encompassing angle of attack, sideslip angle, accelerations, angular rates, attitude angles, airspeed, and position. The aircraft used two identical SW400Pro engines (left and right) during this flight, with recorded engine speeds but no direct thrust measurements. Flight data from 1975 to 2235 s (e.g., climb, cruise, and descent phases) were chosen to validate the proposed method. Figure 7 shows the recorded left and right engine speeds during this chosen period. The ground-based theoretical relationship between engine speed and thrust at altitude 2000 m and Mach 0.1 is presented in Table 6.

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Table 6 Theoretical relationship between engine speed and thrust at altitude 2000 m and Mach 0.1.

Engine speed RPM	Thrust value N
50,000	53
54,000	65
58,400	79
62,800	97
67,500	117
71,800	140
75,900	169
79,800	197
83,000	226
86,000	254
89,500	284
91,900	308
93,800	314

The reasonableness of the estimation results derived from flight data is validated by comparing them with the ground data. However, the inherent existence of the ground-to-flight deviation prevents a direct comparison between the results and ground data. Based on previous research on the physical laws governing this deviation [48], it was found that the ground-to-flight differences in the aircraft’s aerodynamic characteristics can be described by a mathematical expression, rooted in the principles of flow similarity. Based on this foundation, a correction term is applied to the aerodynamic model for this flight data. A simple Expression (17) is formulated that defines the increment for the aerodynamic coefficients as a function of angle of attack or sideslip angle. The constant coefficient k in this expression is determined from the flight data. Finally, this derived correction term is incorporated into the system dynamics model. The estimation algorithm is subsequently applied to process the flight data based on this updated, bias-corrected model. 17 $\begin{array}{c}C=C_{\text{base}}+\Delta C,\Delta C=f\left(\alpha ,\beta ,k\right),\end{array}$ where C denotes the aerodynamic coefficients, C_base is the ground data, ΔC is the increment, and β denotes the sideslip angle.

The effectiveness of the proposed joint online estimation method is demonstrated through comparisons with both the EKF and MMAE algorithms. All three algorithms employ the same modified system model to identify flight test data from the aircraft climb phase. The EKF approach augments the system state with unknown aerodynamic coefficients and thrust for joint estimation, with algorithm parameters consistent with those listed in Table 5. The MMAE estimation follows the standard MMAE process described in Reference [8], here combining multiple UKF estimators to jointly estimate the system state, unknown thrust, and aerodynamic coefficients. Specifically, the thrust ratio factor is treated as a system uncertainty and uniformly partitioned within the interval [0.01, 10] at 0.1 intervals. At each partition point, a UKF estimator is used to estimate the augmented system states with unknown aerodynamic coefficients. The weights corresponding to each partition point are updated based on Bayesian posterior probabilities, and a weighted sum is performed to obtain the estimated thrust and augmented system states at each time step. The parameters of the UKF estimators are set to be consistent with those of the EKF algorithm in Table 5.

Figure 8 compares the theoretical thrust during the climb phase with the thrust estimation results from the three algorithms. Since the thrust values estimated by both the EKF and MMAE deviate significantly from the theoretical values, the common logarithm of the estimated thrust values is used on the vertical axis for better visualization. The results indicate that for this ill-posed estimation problem, neither the simple KF nor the complex filtering algorithm with multiple parallel estimators can achieve accurate thrust estimates. In contrast, the proposed joint online estimation method effectively addresses this issue. In the subsequent context, further validation of the proposed joint online estimation method for thrust estimation is provided using different flight phases of the same flight test data.

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Using the trim thrust from Table 3 as the known input, the algorithm jointly estimated total thrust and aerodynamic coefficients during climb, cruise, and descent phases, with results shown in Figures 9, 10, and 11. The algorithm adopts the same parameter settings from Table 5. The algorithm gives results of the whole engine thrust and aerodynamic parameters. The estimated results exhibit comparability with the ground data by adding the correction term. The estimation thrust results followed the ground theoretical variation trends across all phases, with about 20 N absolute differences. The root mean square relative errors between the estimates and ground theoretical variations were 3.01%, 6.71%, and 17.36% for the three phases, respectively. Due to the variations in thrust magnitude during the climb, cruise, and descent phases—with the climb phase exhibiting the highest thrust (averaging around 450 N) and the descent phase significantly reduced (averaging approximately 110 N)—the relative error under a fixed absolute deviation is minimized during climb and maximized during descent. The estimated aerodynamic drag coefficients approximated ground data. Accurate thrust estimates effectively corrected drag coefficient deviations from the scenario using only trim thrust. The correction effect for aerodynamic drag coefficients during descent was less pronounced, where the theoretical thrust closely matched the trim thrust. Nevertheless, the corrected drag coefficients during descent still better followed the variation trends of ground data than the uncorrected values. Comparative analysis reveals that the algorithm provides effective drag coefficient corrections when thrust estimation errors are small, enabling accurate aerodynamic characteristics identification in real flight.

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The proposed joint estimation algorithm successfully derived valid thrust and aerodynamic coefficient estimates for the AVM aircraft under three typical flight conditions, demonstrating its engineering applicability. During flight test data identification, a simple proportional correction term—with a constant proportionality coefficient determined by the overall ground-to-flight deviations of the whole flight test data—was added to the aircraft model. However, such correction terms only compensated for average ground-to-flight deviations, resulting in an approximate 20 N absolute bias in thrust estimates across three phases. This limitation correlates with the assumption and correction method employed for model uncertainties.

4. Conclusions

This study addresses the issue of low accuracy in aerodynamic parameter identification caused by inaccurate or unavailable thrust measurements during aircraft-powered flight. A joint online estimation method is proposed to simultaneously estimate flight states, unknown aerodynamic parameters, and engine thrust. The effectiveness of this method is verified through both simulation data and actual flight test data from a standard model aircraft. Evaluation results demonstrate that the proposed joint estimation method achieves converged thrust estimates under various thrust deviation ratios in simulation cases. The aerodynamic parameter estimates corrected by thrust estimates progressively approach reference values. Validation using real flight test data further confirms the method’s capability and superiority in estimating unmeasurable thrust and correcting the drag coefficient than other estimation algorithms.

The proposed joint estimation method integrates classical KFs with RLS, ensuring high computational efficiency. Validation through simulation and flight test data confirms its feasibility for online identification of engine thrust and aerodynamic parameters in powered aircraft. This method can provide critical data for verifying aircraft propulsion/aerodynamic performance and help design aircraft guidance/control systems. While the method achieves satisfactory estimation accuracy for simulation data, real flight applications are still limited by lower-quality measurement data, model bias, and reasonably revealing the ground-to-flight deviation law. Future work will focus on analyzing deviation sources, enhancing the algorithm framework with uncertainty-handling mechanisms or better compensation strategies for model/data discrepancies, and conducting rigorous evaluation and refinement using diverse aircraft simulation and real flight datasets to improve the method’s robustness and stability.

Data Availability Statement

The data that support the findings of this study are available from the corresponding author upon reasonable request.

Conflicts of Interest

The authors declare no conflicts of interest.

Funding

No funding was received for this manuscript.

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