# Identification and Modeling Method of Longitudinal Stall Aerodynamic Parameters of Civil Aircraft Based on Improved Kirchhoff Stall Aerodynamic Model

^{1}

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## Abstract

**:**

## 1. Introduction

_{DX}(1 − X) and pitching moment correction term C

_{mX}(1 − X).

_{mX}(1 − X) is added to describe the nonlinear pitching moment. However, in fact, the magnitude and aerodynamic center of the lift generated by the wing change simultaneously during stall, and the change of the pitching moment characteristics, are the superposition of the two effects above, rather than only related to 1 – X. Obviously, it is difficult to describe the nonlinear variation of the pitching moment during stall with the added C

_{mX}(1 – X) correction term only.

_{cr}for switching between the extended stall model and the pre-stall model is defined. The unknown aerodynamic parameters representing steady aerodynamics at small AOA in the stall aerodynamic model are replaced with the known parameters of the pre-stall model at α

_{cr}. Then, the theoretical model given in reference [27] is analyzed. The unsteady pitching moment coefficient terms, in the form of ODE, which are difficult to identify and model, are transformed into high-order correction terms that can easily be identified and modeled. Thus, the aerodynamic parameter identification method based on an improved stall aerodynamic model is developed.

## 2. Improvement of Stall Aerodynamic Model and Identification of Aerodynamic Parameters

#### 2.1. Determination of Critical Angle of Attack for Model Switching and Steady Aerodynamic Parameters

_{cr}). After such improvement, when the flow on the upper surface of the wing is not separated at all, that is, X≈1, the aerodynamic characteristics of the stall aerodynamic model are almost the same as those of the pre-stall model, and the two models can be switched smoothly. When the AOA increases to the point where the upper surface of the wing begins to generate flow separation, i.e., X < 1, the linear and steady aerodynamic characteristics of the aircraft are still regarded as unchanged, while the nonlinear and unsteady aerodynamic increments are all described by X-related correction terms.

_{cr}should be steady, and the X of α

_{cr}should be as close to 1 as possible. Under different change rates of AOA, the value of X is not the same even if the AOA is same, so α

_{cr}is not a fixed value, which needs to be determined according to the degree of flow separation, that is, the value of X. When determining α

_{cr}according to the magnitude of X, it is necessary to determine the threshold of X firstly; that is, when the value of X is less than the threshold, flow separation is considered to lead to non-negligible aerodynamic nonlinearity, and the calculation error of X and the measurement error of AOA should also be taken into account. Generally, the X-threshold range of the aircraft’s stall buffeting that the pilot can notice is about 0.7~0.9 [29], that is, the aerodynamic force has begun to show clear nonlinear characteristics at this time, so the X corresponding to α

_{cr}should be greater than 0.9 at least. Further considering the calculation error of X and the measurement error of AOA (usually greater than 1°), the selection of the critical AOA requires a certain margin, so α

_{cr}is determined as the AOA corresponding to X = 0.95 multiplied by 0.8.

_{cr}can be obtained:

_{cr}, e is the Oswald factor, Λ is the aspect ratio and C

_{DX}and C

_{mX}are the empirical correction coefficients of the drag and pitching moment, respectively. Subscripts wb, t and δ

_{e}, respectively, represent the wing-body, horizontal tail and elevator. ΔC

_{Lt}, ΔC

_{Lδe}, ΔC

_{mt}and ΔC

_{mδe}represent the lift force and pitch moment coefficient increment generated by the horizontal tail and elevator, respectively (relative to the trim deflection angle of the horizontal tail and elevator at α

_{cr}):

_{t}and ε

_{t}are the local AOA and downwash angle, respectively, at the horizontal tail. ε

_{0}is the downwash angle corresponding to zero lift AOA. After integrating Equation (2), we can get:

_{m}

_{αt}∂ε

_{t}/∂X(1 − X) and C

_{m}

_{δe}∂ε

_{t}/∂X(1 − X) in the Equation are similar to the form of C

_{mX}(1 − X) in Equation (1), ∂ε

_{t}/∂X and C

_{mX}are collinear. Therefore, ∂ε

_{t}/∂X and C

_{mX}can be combined into C

_{mX}for identification, rather than the identification of the two parameters separately.

_{1}is the time constant of the unsteady separation process and represents the transient characteristics of the aerodynamic force. τ

_{2}is the time constant of flow separation hysteresis, which represents the hysteresis effect caused by flow separation and reattachment. The separation time of flow is mainly related to the separation characteristics of flow and the flight speed of aircraft. $\overline{c}$/V, the ratio of MAC to airspeed, is used as the units of τ

_{1}and τ

_{2}to avoid the change of time constant caused by the difference of airspeed, so as to reflect the separation characteristics of flow more directly. X

_{0}(α) is the flow separation point under steady flow, which represents the change of the separation point coordinates with AOA in steady flow. X

_{0}is mainly determined by aircraft airfoil and flap/slat configuration, and can be approximated by the hyperbolic-tangent function [28].

_{1}represents the stall characteristics of airfoil and mainly affects the slope of the X-curve at different AOA and change rates of AOA, and α* is the AOA when the flow separation point X

_{0}is equal to 0.5. A complete X expression is formed by combining Equations (4) and (5). The four constants τ

_{1}, τ

_{2}, a

_{1}and α* can describe the change characteristics of X under different stall degrees when the aircraft configuration and Ma are constant.

#### 2.2. The Modification of the Pitching Moment Model and Improved Stall aerodynamic Model

_{mb}(α) is the pitching moment coefficient of the fuselage; C

_{mw}

_{0}is the constant pitching moment coefficient of the wing. C

_{Lw}is the lift coefficient of the wing; X

_{w}(α) is the coordinate of the wing pressure center relative to the gravity center position in MAC fraction. C

_{Lαt}(α) is the lift curve slope of the horizontal tail; α

_{t}is the local AOA at the horizontal tail, which is the same as Equation (2). C

_{mq}(α) is the pitch damping derivative varying with the AOA.

_{Lw}consists of the steady linear lift C

_{Lw}

_{1}and the unsteady nonlinear lift C

_{Lw}

_{2}[27]:

_{Lw}

_{2}can be expressed by ODE [27]:

_{1}(α) is the characteristic time varying with the AOA, and describes the transient characteristic of C

_{Lw}

_{2}. C

_{Lw}

_{20}(α) is the steady dependency between the nonlinear lift force and the AOA.

_{w}(α) is also represented by ODE [27]:

_{2}(α) is the characteristic time varying with the AOA, and describes the transient characteristic of X

_{w}. X

_{w}

_{0}(α) is the corresponding steady dependency of the pressure center of the wing upon AOA.

_{t}includes the influence of the downwash angle ε

_{t}, which can also be expressed by ODE [27]:

_{3}(α) is the characteristic time varying with the AOA, and describes the transient characteristic of ε

_{t}. ε

_{t}

_{0}(α) is the steady dependency of the downwash angle upon AOA.

- The C
_{mcr}and C_{mαwb}_{_cr}(α − α_{cr}) of model (1) represent the linear pitching moment of the wing-body, and can represent the linear components of C_{mw}_{0}and C_{mb}(α) of model (6), but the nonlinear components of C_{mb}(α) are missing. Considering that the nonlinear pitching moment characteristics of the fuselage, which is a conventional body of revolution, are not very significant [27], this part of the pitching moment can be compensated by C_{mX}(1 − X). - In model (1), ΔC
_{mt}represents the pitching moment of the horizontal tail. It can be seen from Equation (2) that ε_{t}in model (1) includes the term (1 − X)∂ε_{t}/∂X of the downwash angle varying with X, which is similar to Equation (10) and can characterize the influence of downwash on the pitching moment of the horizontal tail. The C_{Lα}of model (1) is constant and cannot represent the change of the slope of the lift curve of the horizontal tail. Therefore, ΔC_{mt}in model (1) cannot accurately represent the $-{l}_{ht}{\overline{S}}_{t}{C}_{L\alpha t}(\alpha ){\alpha}_{t}$ in model (6). Considering that the maximum AOA in civil aircraft flight tests is generally less than 25°, the local AOA at the horizontal tail is not large due to the effect of the flow downwash of the wing; hence, the change of C_{Lαt}(α) is slight. Meanwhile, the pitch moment of the horizontal tail is the main component of the pitch damping moment of the aircraft. Therefore, the ΔC_{mt}error caused by C_{Lαt}(α) can be attributed to the pitch damping moment error. Thus, the model correction terms could be simplified without another separate correction of ΔC_{mt}. - The ${C}_{mq\_cr}\overline{q}$ and ${C}_{m\dot{\alpha}\_cr}\overline{\dot{\alpha}}$ in model (1) can only represent the pitch damping moment coefficient at α
_{cr}. When the AOA increases, the value of C_{mq}(α) will change greatly compared with that of C_{mq_cr}[3], which leads to easily noticed errors of the stall pitch damping moment characteristics. - C
_{mX}(1 − X) in model (1) can represent the pitching moment C_{Lw}_{1}X_{w}(α) generated by a linear aerodynamic force with the change of aerodynamic center position, but it lacks the pitching moment term specifically corresponding to C_{Lw}_{2}X_{w}(α), that is, the nonlinear pitching moment generated by the simultaneous change of lift force and aerodynamic center position.

_{Lw}

_{2}X

_{w}(α). The forms of Equations (8) and (9) are similar to those of Equation (4), both of which are related to flow separation. Therefore, it can be considered that C

_{Lw}

_{2}and X

_{w}(α) are positively correlated with (1 − X), namely:

_{Lw}

_{2}with X

_{w}(α):

_{t}is the local velocity at the horizontal tail, and Δα

_{tq}and Δα

_{t}

_{ε}are the changes of local AOA at the horizontal tail, caused by the change of pitch angle velocity and downwash angle, respectively. l

_{ht}is the distance from the aerodynamic center of the horizontal tail to the gravity center of the aircraft. α(t) and α(t − τ) are the AOAs without and with the lag of wash, respectively. The additional lift force ΔL

_{t_ro}multiplied by l

_{ht}is the increment of the pitch damping moment and damping moment of the lag of wash.

^{2}S$\overline{c}$ is the increment of the pitch damping moment coefficient and damping moment coefficient of the lag of wash:

_{Lαt}, local velocity V

_{t}at the horizontal tail, downwash rate ∂ε/∂α and the lag time of downwash τ. C

_{Lαt}, V

_{t}and ∂ε/∂α are all related to flow separation, and τ is determined by V

_{t}. Similarly, C

_{Lαt}, ∂ε/∂α are positively correlated with (1 − X), and the ratio of V

_{t}to V is also positively correlated with (1 − X):

_{ht}and K

_{5}(1 − X)$\dot{\alpha}$l

_{ht}have similar effects, they can be combined, then Equation (17) is transformed into:

_{mX}

_{2}(1 − X)

^{2}is proposed to describe the pitching moment generated by the simultaneous change of the lift force and the aerodynamic center position. The third order correction term C

_{mX}

_{3}(1 − X)

^{3}is proposed to describe the variation of the pitch damping moment due to the variation of AOA and pitch velocity. Since X is close to 1, 1 − X is close to 0 in the steady flight state, and the order of magnitude of (1 − X)

^{2}and (1 − X)

^{3}is small in this case, and the two higher order correction terms have no significant impact on the fitting in the steady flight state.

_{mX}

_{2}(1 − X)

^{2}and C

_{mX}

_{3}(1 − X)

^{3}to the pitching moment model, an improved stall aerodynamic model is established:

_{cr}, and are no longer iterated as unknowns. This not only greatly reduces the calculation amount, but also reduces the identification errors of the X characteristic parameters, correction coefficients and other parameters caused by incompatible steady aerodynamic parameters, which can effectively improve the reliability of the identification results.

#### 2.3. Identification of Aerodynamic Parameters Based on Improved Stall aerodynamic Model

**θ**to be identified includes four correction coefficients, C

_{DX}, C

_{mX}

_{1}, C

_{mX}

_{2}and C

_{mX}

_{3}, and four X characteristic parameters, a

_{1}, α*, τ

_{1}, τ

_{2}; e and ∂ε

_{t}/∂α. It is difficult to obtain the dynamic data needed to identify τ

_{1}by a quasi-steady stall maneuver, so it is impossible to obtain accurate τ

_{1}, and the lack of identifiability of τ

_{1}may affect the estimation of τ

_{2}. Therefore, in order to obtain more accurate τ

_{2}, identification should start with a simplified model with τ

_{1}removed. Special attention should be paid to the identification of C

_{mX}

_{1}, C

_{mX}

_{2}and C

_{mX3}: when flow separation is not clear (such as X > 0.8), 1 − X, (1 − X)

^{2}, (1 − X)

^{3}have obvious collinearity, and accurate identification results of C

_{mX}

_{2}and C

_{mX}

_{3}cannot be obtained at this time. In order to obtain more accurate estimates, the collinearity of 1 − X, (1 − X)

^{2}, (1 − X)

^{3}should be assessed through VIF, and C

_{mX}

_{2}and C

_{mX}

_{3}should be removed from the model when significant collinearity exists among the three. For data with easily noticed flow separation (such as X < 0.6), C

_{mX}

_{2}and C

_{mX}

_{3}should be retained.

_{cr}, which no longer need to be identified.

_{L}, C

_{D}and C

_{m}of model (19) constitute the output vector

**Y**, and the measurement data of C

_{L}, C

_{D}and C

_{m}of aircraft constitute the observation vector

**Z**. The measurement data of C

_{L}, C

_{D}and C

_{m}are calculated by acceleration and angular acceleration flight test data.

## 3. Aerodynamic Modeling Method for Longitudinal Stall Process

_{DX}, C

_{mX}

_{1}, C

_{mX}

_{2}and C

_{mX}

_{3}; and the third is the modeling of e and ∂ε

_{t}/∂α. Because the identification results of the correction coefficients and e and ∂ε

_{t}/∂α vary with the stall degree 1 − X, X modeling needs to be completed firstly, and then the modeling of the correction coefficient and e and ∂ε

_{t}/∂α with the change of 1 − X can be completed.

#### 3.1. Modeling Method of X

_{1}, α*, τ

_{1}and τ

_{2}. a

_{1}, α* and τ

_{2}are mainly determined by airfoil and wing configuration. τ

_{1}is mainly determined by the characteristics of the flow field at infinity and the trailing edge flow characteristics of the wing-body, i.e., τ

_{1}is mainly related to Ma and is independent of the airfoil and wing configuration [28,31]. Therefore, a

_{1}, α*, τ

_{1}and τ

_{2}are constants that do not change with the stall degree when the configuration is the same and Ma is similar. Through the change of X, the existing stall aerodynamic model can well describe the quasi-steady stall lift characteristics of aircraft without adding additional correction terms. This means that a set of optimal a

_{1}, α*, τ

_{1}, τ

_{2}can characterize the lift stall characteristics at different stall degrees. That is, it is necessary to obtain a set of optimal a

_{1}, α*, τ

_{1}, τ

_{2}by using flight data of different stall degrees.

_{1}, α*, τ

_{1}, τ

_{2}, in the quasi-steady stall process, except the stall and recovery stage, the change rate of the AOA is small, and the flow separation point changes slowly, that is, in most of the data, dX/dt ≈ 0, $\dot{\alpha}$ ≈ 0, and X and $\dot{\alpha}$ have strong changes only in the stall and recovery stage. Since τ

_{1}is the coefficient of dX/dt and τ

_{2}is the coefficient of $\dot{\alpha}$, most flight test data of a quasi-steady stall cannot fully reflect the effects of τ

_{1}and τ

_{2}on X characteristics. Therefore, when the quasi-steady stall data is used to identify a

_{1}and α*, the impact of data measurement errors and atmospheric disturbance is small, and the identification results are relatively accurate. On the contrary, the identification results of τ

_{1}and τ

_{2}will be significantly affected, and the accuracy of the identification results is relatively low. Therefore, a

_{1}and α* with higher accuracy values are first determined here.

_{1}and α* do not vary with the change of stall degree, the identification results obtained through different data are theoretically the same. However, the identification results of a

_{1}and α* obtained through different data are not exactly the same due to the impact of measurement error, atmospheric disturbance and other factors, and distribute in a certain area. The statistical test method can be used to determine the value of the identification results. Ideally, a

_{1}and α* identification results obtained through flight test data with different stall degrees should only be affected by random errors and show normal distribution. If the identification results do not meet the normal distribution, it indicates that the identification results are not accurate, and some identification results that are too deviated should be eliminated. The K-S one-sample test is a test method to determine whether the overall sample of the identification results shows a certain distribution through a set of observed values. In this paper, the one-sample K-S test is used to test whether the discrete identification results show normal distribution.

_{1}and α* when the configuration is the same and Ma is similar. In a quasi-steady stall, the steady separation characteristics of the flow, that is, the change characteristic of X when $\dot{\alpha}$ ≈ 0 and dX/dt ≈ 0, are determined by a

_{1}and α*. Therefore, the test of the distribution characteristics of the two identification results is also carried out in no order simultaneously. In practice, the values of a

_{1}and α* in the X model can be taken as the mean value of the identification results according to the actual fitting of the aircraft flight dynamic model to the flight test data.

_{1}and α*, the modeling of the transient characteristics and hysteresis characteristics of X when $\dot{\alpha}$ ≠ 0 and dX/dt ≠ 0, namely the determination of values of τ

_{1}and τ

_{2}, should be further completed. It is necessary to extract the data segments that can fully characterize the transient stall characteristics and aerodynamic hysteresis effect of aircraft, that is, the data of aircraft entering stall with strong buffeting until it recovers from stall, to determine τ

_{1}and τ

_{2}. However, one data segment is not enough to determine τ

_{1}and τ

_{2}because these data segments account for a small proportion of the whole quasi-steady stall data (usually the length of these data segments is only a few seconds), so it is necessary to extract multiple (at least two) lift coefficient observation data segments with significant unsteady separation and hysteresis. In addition, in order to eliminate the impact of the changes of a

_{1}and α* values on the results of τ

_{1}and τ

_{2}, a

_{1}and α* should be fixed as known quantities after the exact values of a

_{1}and α* are determined. Then, τ

_{1}and τ

_{2}are optimized by the optimization function (such as fmincon in Matlab), so that the lift model can fit all the extracted observation data segments of the lift coefficient. Finally, a set of optimal τ

_{1}and τ

_{2}are used to characterize the unsteady separation characteristics of flow at different stall degrees.

#### 3.2. Modeling Method of Correction Coefficients

_{DX}and pitching moment correction coefficients C

_{mX}

_{1}, C

_{mX}

_{2}and C

_{mX}

_{3}. The correction terms include the correction of the stall aerodynamic force or moment coefficients of different components such as wing-body, horizontal tail and elevator. On the other hand, they also include the correction of different types of derivatives, such as static derivative, dynamic derivative and control derivative. Therefore, the values of correction terms are different under different stalling degrees. The reference [1] holds that the correction coefficient varies proportionally with flow separation. Considering that the aerodynamic characteristics of stall show clear nonlinear characteristics with the change of the flow separation, this proportional model will have nonnegligible errors in the stall and recovery stages. Therefore, knots spline functions are needed for modeling. The general form of the m-order knots spline function of a single variable is [23]:

_{i}is the knot of spline function. X is used as the independent variable to represent the stall degree. First, 1, X, X

^{2}, …, X

^{m}are taken as the candidate set of independent variables, and then the significance test is used to determine whether there is an evident correlation between each independent variable and the identification results of the correction coefficients at different stall degrees. Then, the independent variables with significant correlations with the identification results of each correction coefficient are determined, and the correction coefficient model is established by fitting the identification results through Equation (23). The results of this paper show that the best fitting can be obtained when m is set to three. Finally, according to the difference between the mathematical simulation results of the constructed aircraft motion model and the test flight data, the parameters of each coefficient C

_{i}and D

_{i}in Equation (23) can be adjusted to complete the correction coefficient modeling.

#### 3.3. Modeling of e and ∂ε_{t}/∂α

_{t}/∂α mainly affect the accuracy of the drag and pitching moment models, but have little influence on the lift model. Different from the stall characteristic parameters in X and the correction terms of the drag and pitching moment, e and ∂ε

_{t}/∂α also have significant contributions to C

_{D}and C

_{m}when the flow is completely unseparated (i.e., X = 1). In addition, the nonlinear characteristics of the drag model are mainly characterized by ${C}_{L}^{2}$ and C

_{DX}; the nonlinear characteristics of the pitching moment model are mainly represented by C

_{mX}

_{1}, C

_{mX}

_{2}, C

_{mX}

_{3}. Hence, the identification results of e and ∂ε

_{t}/∂α do not contain evident nonlinear characteristics of the drag and pitching moment models. Therefore, in order to ensure the accuracy of the drag and pitching moment models before stall occurs and simplify the model structure as much as possible, e and ∂ε

_{t}/∂α of different stall degrees are mainly regarded as constants independent of X.

- a
_{1}and α* are identified by the whole quasi-steady stall data. Then, it is checked whether the identification results of stall maneuver data with different stall degrees show normal distribution. If these identification results show the normal distribution, the values of a_{1}and α* are determined near the mean value of the normal distribution. If so, a_{1}and α* are taken as the mean of normal distribution. If the normal distribution is not followed, the identification results that are too discrete are eliminated and the mean value of the normal distribution is reset. The remaining identification results are tested to determine the values of a_{1}and α*. - The flight test data segments of at least two groups of aircraft from deep stall to stall recovery are extracted, and the τ
_{1}and τ_{2}optimization are carried out to make the lift model fit all the extracted observation data segments well. Then, X modeling is carried out using a_{1}, α*, τ_{1}and τ_{2}. - After X modeling is completed, knots splines with 1, X, X
^{2}and X^{3}as independent variables are used to carry out the modeling of C_{DX}, C_{mX}_{1}, C_{mX}_{2}and C_{mX}_{3}under different stall degrees. e and ∂ε_{t}/∂α are determined as constants.

## 4. Stall Aerodynamic Parameter Identification and Modeling Example

#### 4.1. Identification of Stall Aerodynamic Parameter

^{2}and a wingspan of 35 m. Because the deflection of the slat can reduce the adverse pressure gradient of the upper airfoil, eliminate the separation vortex of the upper airfoil, delay the separation of flow and increase the maximum lift coefficient and the critical AOA, the flight test data with more obvious aerodynamic nonlinear and hysteresis characteristics at a larger AOA could be obtained. Hence, the flap deflection angle of 25° and the slat deflection angle of 21° configuration (Configuration 3 of the test aircraft) is used as an example to discuss the identification and modeling methods of aerodynamic parameters. In order to obtain the aerodynamic characteristics of the aircraft at different stall degrees, the quasi-steady stall maneuver flight tests with different maximum AOA were carried out. The aerodynamic parameters are identified using the stall maneuver flight test data of Configuration 3 with different stall degrees at the same Ma. There are 15 sets of evident stall flight test data with a flight altitude of 4500~5500 m, Ma = 0.275~0.30 and a maximum AOA of 15.4°–21.8°.

_{1}, α*, τ

_{1}, τ

_{2}, C

_{DX}, C

_{mX}

_{1}, C

_{mX}

_{2}, C

_{mX}

_{3}, e and ∂ε

_{t}/∂α are given as 20, 20°, 15 c/V, 5 c/V, 0.1, 0, 0, 0, 0.75 and −0.3, respectively. Then, iterating the unknown parameters to minimize the cost function (22). The identification results of stall aerodynamic parameters are shown in Table 1.

_{min}< 0.82, the identification results of a

_{1}, α*, e and ∂ε

_{t}/∂α are concentrated. This means that although the degrees of stall during the flight test are different, the identified stall characteristics of the aircraft, the characteristics of drag generated by lift and the downwash characteristics at the horizontal tail are similar. The standard deviations of τ

_{2}identification results are large. It should be pointed out that due to the strong buffeting in the stall and recovery, the acceleration data at this stage contains strong noise generated by buffeting. Hence, the C

_{L}hysteresis curve calculated by acceleration also contains strong noise. As a result, the standard deviation of τ

_{2}is excessively large (the estimation of τ

_{2}is mainly determined by a hysteresis curve, and the quasi-steady data with low noise has no evident influence on the estimate of τ

_{2}). However, the buffeting is approximated as white noise and has little impact on the mean of the estimated results. Therefore, the distribution range of estimated τ

_{2}obtained from tests 6 to 15 is closer to the real standard deviation of τ

_{2}, namely, s(τ

_{2}) ≈ 2.39 c/V.

_{mx}

_{1}is retained in the model, and C

_{mx}

_{2}and C

_{mx}

_{3}are removed. When α

_{max}is greater than 18.5° and X

_{min}is less than 0.82, C

_{mx}

_{2}and C

_{mx}

_{3}are retained. Taking the 11th group of data as an example, the fitting of the identification model on C

_{L}and C

_{D}observation data is shown in Figure 3.

_{L}and C

_{D}contain relatively strong noise. This is mainly because the aircraft is accompanied by strong buffeting when it is stalling, so the recorded overload data contains the disturbance generated by buffeting, and the observed C

_{L}and C

_{D}values calculated from the overload data contain strong noise. The improved stall aerodynamic model can accurately fit the C

_{L}and C

_{D}observation data at the same time, so the identification results of a

_{1}, α*, C

_{DX}and e, which are directly related to the fitting of C

_{L}and C

_{D}observation data, are reliable.

_{m}observation data is shown in Figure 4.

_{m}observation data with the AOA and time. In Figure 4a, C

_{m}observation data changed in the order of arrow direction and serial number. The pitch angular acceleration required to calculate C

_{m}observation data is obtained by smoothing the angular velocity data and differentiating it. Detailed processing methods of angular velocity data can be referred to Reference [32]. Therefore, C

_{m}observation data does not contain significant noise.

_{m}observation data is approximately 0, and the C

_{m}curves of the unimproved pitching moment model and the improved model are almost coincident, so only the comparison of the fitting curves after 40 s is given. In the quasi-steady stall flight test, the aircraft speed slowly decreases and the AOA slowly increases. Before entering the deep stall, the aircraft maintains a steady state. Therefore, there are more data points in the quasi-steady flight test, and relatively few data points at the state of deep stall and stall recovery. In Figure 4a, it can be seen that the curves of stages ①, ② and ③ are thick, and the curves of stages ④~⑦ are thin. In Figure 4b, it can be seen that the curve before 55 s is thick and after 55 s is thin.

_{m}observation data of aircraft can be fitted by using the unimproved pitching moment model. When the AOA increases to the point where obvious stall buffeting occurs (i.e., stages ② and ③ in Figure 4a and 45 s~60 s in Figure 4b), the fitting error of C

_{m}increases significantly. The error is more apparent during the stall recovery (i.e., the stages ④~⑦ in Figure 4a, and the stage after 60 s in Figure 4b). This means that C

_{mX}obtained from the unimproved pitching moment model cannot accurately describe the characteristics of the aircraft stall pitching moment. The error of C

_{m}will lead to the error of pitch angle velocity, which will further lead to a rapid increase of the error of pitch angle, airspeed, AOA and other state variables over time. Compared with C

_{L}and C

_{D}, C

_{m}error has more clear adverse effects on the accuracy of simulation calculation.

_{mX}

_{2}(1 − X)

^{2}is used to characterize the increment of the pitching moment resulting from the simultaneous change of aerodynamic force and aerodynamic center. C

_{mX}

_{3}(1 − X)

^{3}is used to characterize the nonlinear increment of the pitch damping moment due to the change of AOA and pitch velocity. The fitting of the improved pitching moment model on C

_{m}observation data is obviously improved. The maximum C

_{m}fitting error decreases from 0.027 to 0.013 when entering stall (i.e., stages ② and ③ in Figure 4a and 45 s~60 s in Figure 4b). The maximum C

_{m}fitting error is reduced from 0.086 to 0.034 when recovering stall (stages ④~⑦ in Figure 4a and the stage after 60 s in Figure 4b). The two high-order pitching moment correction terms have little effect on the fitting during steady flight (stage ① in Figure 4a and the stage before 45 s in Figure 4b). It shows that the improvement of the pitching moment model proposed in this paper makes sense. The identification results of ∂ε

_{t}/∂α, C

_{mX}

_{1}, C

_{mX}

_{2}and C

_{mX}

_{3}obtained by using the improved pitching moment model can more accurately characterize the characteristics of the aircraft’s stall pitching moment.

#### 4.2. Aerodynamic Modeling of Stall Process

#### 4.2.1. Modeling of X

_{1}in Table 1 show the normal distribution of x~N(22.5,10

^{2}). The identification results of α* follow the normal distribution of x~N(20,0.4

^{2}). Therefore, the steady separation characteristics of quasi-steady stall maneuvers with different stall degrees can be characterized by the same set of a

_{1}and α*. Based on the analysis above, a

_{1}and α* can be set as 22.5 and 20.0°, respectively.

_{1}and α*, two sets of flight test data segments in the interval from the beginning of evident buffeting to the complete recovery of stall (maximum angles of attack are 20.6° and 21.8°, the duration from stall to recovery is 8.2 s and 9.5 s, respectively) are selected for the optimization of τ

_{1}and τ

_{2}. The resulting τ

_{1}and τ

_{2}are 11.93 c/V and 6.66 c/V, respectively. Combined with a

_{1}and α* already established, X modeling can be completed. In order to validate the optimization results, the optimized τ

_{1}and τ

_{2}are substituted into the lift model to fit another flight test data segment with different hysteresis characteristics and a maximum AOA of 21.1°. The optimization results and validation results are shown in Figure 5.

_{1}, α*, τ

_{1}and τ

_{2}can make model (19) meet the lift characteristics of aircraft at different stall degrees.

#### 4.2.2. Modeling of Correction Coefficient and e, ∂ε_{t}/∂α

_{DX}, C

_{mX}

_{1}, C

_{mX}

_{2}and C

_{mX}

_{3}are listed by taking X

_{min}, the minimum value of X achieved in the flight test, as the abscissa, as shown in Figure 6.

_{min}is large, the flow separation is not obvious, the nonlinear characteristics of aerodynamic force are not significant, and the accuracy of the identification results of the correction coefficient is low. Therefore, only the identification results of a correction coefficient with X

_{min}less than 0.82 are given in Figure 6. As can be seen from Figure 6a, when X

_{min}< 0.8, C

_{DX}identification results have no clear knots change characteristics, so nonlinear functions can be adopted for fitting. Most of the nonlinear components of the drag are characterized by the induced drag C

_{L}

^{2}/(eπΛ). The proportion of nonlinear components described by C

_{DX}(1 − X) is relatively small, and the absolute value of the estimated result of C

_{DX}is smaller (compared with C

_{mX}

_{1}). Although the s(C

_{DX})/C

_{DX}is relatively large, variations in C

_{D}are still acceptable for stall aerodynamic modeling.

_{mX}

_{1}, C

_{mX}

_{2}and C

_{mX}

_{3}under different X

_{min}are quite different, and the knots spline with X as the independent variable can be used to fit such nonlinear change characteristics. For large civil aircraft, it is too dangerous for the flight test when the flow on the upper surface of the wing is completely separated (i.e., X

_{min}is close to 0). Therefore, the minimum X

_{min}achieved in the flight test is 0.16, which is not significantly less than 0.2. Hence, the X knots are initially determined to be 0.6 and 0.4, according to the distribution characteristics of the identification results. Then, the knots and function forms are adjusted according to the fitting of the stall aerodynamic model on the C

_{m}observation data.

- (1)
- C
_{DX}modeling

_{DX}spline function is Equation (25).

- (2)
- Modeling of C
_{mX}_{1}, C_{mX}_{2}and C_{mX}_{3}

- (3)
- Modeling of e and ∂ε
_{t}/∂α

_{t}/∂α under different X

_{min}is small and does not change with X

_{min}, so it can be regarded as constant. Based on the identification results, parameters are adjusted according to the comparison between the flight dynamics simulation and test flight data, and the value of e is finally determined to be 0.785 and the value of ∂ε

_{t}/∂α is 0.347. So far, an aerodynamic model is established for a large civil aircraft configuration 3 at 4500~5500 m altitude and an initial Ma = 0.275~0.30, which can describe the longitudinal stall aerodynamic characteristics of different stall degrees.

#### 4.3. Mathematical Simulation Validation of Quasi-Steady Stall Flight

_{cr}at entering stall is 12.6°, and the corresponding switching time between the pre-stall model and the extended stall model is 26.2 s. In the range of 0 s to 26.2 s, X ≈ 1, the flow on the upper surface of the wing does not separate, and the aerodynamic model of the motion model is the pre-stall aerodynamic model. The responses of the motion model at this stage are almost exactly consistent with the flight test data, which indicates that the pre-stall aerodynamic model has good accuracy and can accurately characterize the aerodynamic characteristics of the aircraft at small AOA.

_{cr}. In the range of 26.2 s~38.8 s, 0.95 ≤ X ≤ 1, it can still be approximated that there is no flow separation, and the aerodynamic characteristics are steady. However, at this time, the aerodynamic model of the motion model has been switched to the extended stall model. At this stage, the responses of the motion model almost coincide with the flight test data, which indicates that the established stall aerodynamic model has high accuracy in the steady aerodynamic region where the flow is not separated.

_{mX}

_{1}(1 − X), C

_{mX}

_{2}(1 − X)

^{2}and C

_{mX}

_{3}(1 − X)

^{3}, the three high-order correction terms for the pitching moment, can well represent the characteristics of the stall pitching moment of the aircraft, which significantly reduces the error, which is caused by pitching moment error, between the responses of the motion model and the flight test data.

## 5. Conclusions

_{cr}of the switch between the pre-stall model and the extended model. Based on the steady aerodynamic theory at small AOA, the values of unknown aerodynamic parameters representing the steady aerodynamic forces/moments in the stall aerodynamic model are determined. In addition, C

_{mX}

_{2}(1 – X)

^{2}and C

_{mX}

_{3}(1 – X)

^{3}, two high-order correction terms, are added to the pitching moment model. An improved stall aerodynamic model is established. Then, the observation values of lift, drag and pitching moment are calculated using the overload and angular acceleration data from the flight test. Finally, based on the improved stall aerodynamic model, the maximum likelihood method is used to identify four correction coefficients, C

_{DX}, C

_{mX}

_{1}, C

_{mX}

_{2}and C

_{mX}

_{3}, four X characteristic parameters a

_{1}, α*, τ

_{1}and τ

_{2}, and e, ∂ε

_{t}/∂α.

_{1}and α* of stall maneuver data with different stall degrees are tested to check whether they show the given normal distribution, and the values of a

_{1}and α* are determined. Then, at least two groups of flight data segments from deep stall to stall recovery are extracted, and the lift model is made to fit the hysteresis curves of these data segments through the optimization of τ

_{1}and τ

_{2}. The optimal solution is the values of τ

_{1}and τ

_{2}, and the X modeling is established by using a

_{1}, α*, τ

_{1}and τ

_{2}. Finally, knots splines with 1, X, X

^{2}and X

^{3}as independent variables are used to model correction terms under different stall degrees. The Oswald factor and the downwash rate at the horizontal tail are set as constants.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Notation

AOA | angle of attack |

MAC | mean aerodynamic chord |

ODE | ordinary differential equation |

DOF | degree of freedom |

α | angle of attack (°) |

V | airspeed (m/s) |

q | pitch velocity (°/s) |

$\overline{q}$ | dimensionless pitch velocity |

$\overline{\dot{\alpha}}$ | dimensionless change rate of AOA |

$\dot{\alpha}$ | change rate of AOA (°/s^{2}) |

$\overline{Q}$ | dynamic pressure (Pa) |

S | reference wing area (m^{2}) |

${S}_{t}$ | wing area of horizontal tail (m^{2}) |

$\overline{c}$ | mean aerodynamic chord length (m) |

δ_{t} | horizontal tail deflection angle (°) |

δ_{e} | elevator deflection angle (°) |

e | Oswald factor |

Λ | aspect ratio |

X | relative position of the ideal flow separation point of the upper surface of the wing on the mean aerodynamic chord |

α_{t} | local AOA at horizontal tail (°) |

ε_{t} | downwash angle at horizontal tail (°) |

ε_{0} | downwash angle corresponding to zero lift AOA (°) |

τ_{1} | time constant of unsteady separation process ($\overline{c}$/V) |

τ_{2} | time constant of flow separation hysteresis ($\overline{c}$/V) |

a_{1} | stall characteristics parameter of airfoil |

α* | the AOA when flow separation point is equal to 0.5 (°) |

∂ε_{t}/∂α | derivative of downwash angle with respect to AOA |

∂ε_{t}/∂X | derivative of downwash angle with respect to X |

C_{L} | lift coefficient |

C_{D} | drag coefficient |

C_{m} | pitching moment coefficient |

C_{Lαwb} | lift curve slope of wing-body |

C_{DX} | empirical correction coefficient of drag |

C_{mX} | empirical correction coefficient of pitching moment |

C_{mX}_{1} C_{mX}_{2} C_{mX}_{3} | 1-order, 2-order, 3-order pitching moment correction term |

C_{mα} | pitching static stability derivative |

C_{Lq} | lift derivative due to pitching |

${C}_{L\dot{\alpha}}$ | lift derivative due to $\dot{\alpha}$ |

C_{mq} | pitching damping derivative |

${C}_{m\dot{\alpha}}$ | pitching damping derivative of lag of wash |

C_{Lαt} | lift derivative of horizontal tail |

C_{mδe} | pitch control derivative of elevator |

C_{mδt} | pitch control derivative of horizontal tail |

C_{mb} | pitching moment coefficient of fuselage |

C_{mw}_{0} | zero lift pitching moment coefficient of wing |

C_{Lw} | lift coefficient of wing |

X_{w} | the coordinate of wing pressure center relative to gravity center position in MAC fraction |

l_{ht} | the distance from the aerodynamic center of the horizontal tail to the gravity center of the aircraft |

V_{t} | local airspeed at horizontal tail |

a_{x} | longitudinal acceleration (m/s^{2}) |

T | thrust (N) |

I_{xx} | moment of inertia with respect to roll axis (kg × m^{2}) |

I_{yy} | moment of inertia with respect to pitch axis (kg × m^{2}) |

I_{zz} | moment of inertia with respect to yaw axis (kg × m^{2}) |

I_{xz} | product of inertia (kg × m^{2}) |

## Appendix A

**Figure A1.**Model predictive capability validation. (

**a**) Airspeed; (

**b**) Angle of attack; (

**c**) Pitch angular velocity; (

**d**) Pitch angle; (

**e**) z−axis overload; (

**f**) x−axis overload; (

**g**) Change rate of AOA; (

**h**) Change of X.

**Figure A2.**Flight test data of main control variables. (

**a**) Deflection angle of elevator; (

**b**) High-pressure rotor speed of engine.

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**Figure 3.**Fitting of the improved stall aerodynamic model to the observation data (α

_{max}= 21.1°). (

**a**) C

_{L}~α; (

**b**) C

_{D}~α.

**Figure 4.**Fitting of unimproved/improved stall aerodynamic model to the pitching moment observation data. (

**a**) C

_{m}−α; (

**b**) C

_{m}−t.

**Figure 5.**Lift coefficient fitting after optimization of stall aerodynamic parameters. (

**a**) C

_{L}hysteresis curve for optimization of τ

_{1}and τ

_{2}(α

_{max}= 21.8°); (

**b**) C

_{L}hysteresis curve for optimization of τ

_{1}and τ

_{2}(α

_{max}= 20.6°); (

**c**) C

_{L}hysteresis curve for validation of τ

_{1}and τ

_{2}(α

_{max}= 21.1°).

**Figure 6.**Identification results of correction coefficients. (

**a**) C

_{DX}−X

_{min}; (

**b**) C

_{mX}

_{1}−X

_{min}; (

**c**) C

_{mX}

_{2}−X

_{min}; (

**d**) C

_{mX}

_{3}−X

_{min}.

**Figure 8.**Spline curves of correction coefficients of pitching moment. (

**a**) C

_{mX}

_{1}−X; (

**b**) C

_{mX}

_{2}−X; (

**c**) C

_{mX}

_{3}−X.

**Figure 9.**The comparison of response of motion model and flight test data. (

**a**) Airspeed; (

**b**) Angle of attack; (

**c**) Pitch angular velocity; (

**d**) Pitch angle; (

**e**) z−axis overload; (

**f**) x−axis overload; (

**g**) Change rate of AOA; (

**h**) Change of X.

**Figure 10.**Flight test data of main control variables. (

**a**) Deflection angle of elevator; (

**b**) High-pressure rotor speed of engine.

Number | α_{max}(°) | X_{min} | a_{1} | s (a_{1}) | α* (°) | s (α*) (°) | τ_{2}(c/V) | s (τ_{2})(c/V) | C_{DX} | s (C_{DX}) | C_{mX}_{1} |
---|---|---|---|---|---|---|---|---|---|---|---|

1 | 15.4 | 0.98 | 61.85 | 29.13 | 16.28 | 2.12 | 0.00 | 1156 | 1.08 | 0.0298 | 0.75 |

2 | 15.0 | 0.87 | 184.49 | 24.84 | 15.07 | 2.86 | −0.00 | 3517 | 0.25 | 0.0594 | 0.82 |

3 | 17.9 | 0.88 | 21.20 | 11.75 | 19.79 | 1.30 | −1.97 | 91.81 | 0.56 | 0.0046 | 0.33 |

4 | 17.0 | 0.95 | 33.88 | 13.69 | 17.82 | 2.44 | 2.18 | 377.8 | 0.69 | 0.0141 | 0.41 |

5 | 17.3 | 0.93 | 40.85 | 12.10 | 17.09 | 1.27 | −0.79 | 253.3 | 0.14 | 0.0155 | 0.32 |

6 | 18.8 | 0.75 | 22.87 | 12.11 | 20.11 | 0.64 | 2.18 | 16.93 | 0.14 | 0.0037 | 0.25 |

7 | 19.4 | 0.63 | 26.50 | 11.90 | 20.19 | 0.52 | 1.90 | 10.39 | 0.11 | 0.0731 | 0.42 |

8 | 19.3 | 0.67 | 29.56 | 12.14 | 20.09 | 0.75 | 0.00 | 13.29 | 0.12 | 0.0886 | 0.37 |

9 | 19.5 | 0.60 | 22.28 | 12.18 | 20.02 | 0.74 | 5.65 | 18.45 | 0.16 | 0.0399 | 0.63 |

10 | 18.5 | 0.82 | 17.37 | 10.62 | 20.46 | 0.23 | 2.24 | 11.80 | 0.18 | 0.0437 | 0.32 |

11 | 21.1 | 0.27 | 21.93 | 9.36 | 19.93 | 0.96 | 8.35 | 1.64 | 0.14 | 0.0092 | 0.25 |

12 | 20.6 | 0.39 | 19.56 | 8.11 | 20.07 | 0.87 | 4.85 | 3.07 | 0.16 | 0.0682 | 0.79 |

13 | 20.6 | 0.36 | 16.86 | 12.54 | 19.79 | 0.18 | 2.59 | 19.89 | 0.16 | 0.0309 | 0.60 |

14 | 21.7 | 0.16 | 27.33 | 8.73 | 19.78 | 0.38 | 3.28 | 0.30 | 0.17 | 0.0141 | 0.55 |

15 | 21.8 | 0.16 | 20.62 | 10.76 | 19.42 | 0.32 | 1.73 | 9.02 | 0.20 | 0.0655 | 0.39 |

Number | s (C_{mX}_{1}) | C_{mX}_{2} | s (C_{mX}_{2}) | C_{mX}_{3} | s (C_{mX}_{3}) | e | s (e) | ∂ε_{t}/∂α | s (∂ε_{t}/∂α) | ||

1 | 0.021 | - | - | - | - | 0.81 | 0.046 | 0.385 | 0.002 | ||

2 | 0.026 | - | - | - | - | 0.80 | 0.017 | 0.36 | 0.002 | ||

3 | 0.009 | - | - | - | - | 0.81 | 0.016 | 0.35 | 0.002 | ||

4 | 0.009 | - | - | - | - | 0.81 | 0.026 | 0.36 | 0.002 | ||

5 | 0.009 | - | - | - | - | 0.79 | 0.022 | 0.40 | 0.002 | ||

6 | 0.028 | −2.32 | 0.39 | 2.81 | 1.38 | 0.79 | 0.011 | 0.40 | 0.001 | ||

7 | 0.023 | −1.71 | 0.21 | 2.32 | 0.49 | 0.81 | 0.016 | 0.38 | 0.001 | ||

8 | 0.024 | −1.33 | 0.25 | 1.83 | 0.67 | 0.80 | 0.012 | 0.38 | 0.001 | ||

9 | 0.027 | −3.48 | 0.24 | 4.49 | 0.52 | 0.83 | 0.014 | 0.36 | 0.002 | ||

10 | 0.055 | −2.67 | 0.97 | 1.85 | 4.65 | 0.80 | 0.011 | 0.40 | 0.003 | ||

11 | 0.023 | −0.06 | 0.086 | −0.91 | 0.086 | 0.77 | 0.009 | 0.37 | 0.001 | ||

12 | 0.019 | −3.66 | 0.087 | 3.57 | 0.110 | 0.78 | 0.017 | 0.37 | 0.001 | ||

13 | 0.017 | −3.15 | 0.080 | 3.17 | 0.100 | 0.79 | 0.011 | 0.34 | 0.003 | ||

14 | 0.018 | −2.10 | 0.061 | 1.35 | 0.055 | 0.80 | 0.008 | 0.36 | 0.001 | ||

15 | 0.020 | −2.16 | 0.073 | 1.85 | 0.069 | 0.78 | 0.011 | 0.34 | 0.002 |

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## Share and Cite

**MDPI and ACS Style**

Wang, L.; Zhao, R.; Xu, K.; Zhang, Y.; Yue, T. Identification and Modeling Method of Longitudinal Stall Aerodynamic Parameters of Civil Aircraft Based on Improved Kirchhoff Stall Aerodynamic Model. *Aerospace* **2023**, *10*, 333.
https://doi.org/10.3390/aerospace10040333

**AMA Style**

Wang L, Zhao R, Xu K, Zhang Y, Yue T. Identification and Modeling Method of Longitudinal Stall Aerodynamic Parameters of Civil Aircraft Based on Improved Kirchhoff Stall Aerodynamic Model. *Aerospace*. 2023; 10(4):333.
https://doi.org/10.3390/aerospace10040333

**Chicago/Turabian Style**

Wang, Lixin, Rong Zhao, Ke Xu, Yi Zhang, and Ting Yue. 2023. "Identification and Modeling Method of Longitudinal Stall Aerodynamic Parameters of Civil Aircraft Based on Improved Kirchhoff Stall Aerodynamic Model" *Aerospace* 10, no. 4: 333.
https://doi.org/10.3390/aerospace10040333