A fractional derivative aerodynamic model at high angles of attack

A fractional derivative aerodynamic model is put forward based on Grünwald-Letnikov fractional derivative in this paper. In order to express the hysteresis property of aerodynamic coefficients, a fractional derivative state-space equation is developed from Goman’s differential equation. A fast and high-accuracy calculation method of Grünwald-Letnikov fractional derivative is proposed. Then, the model is demonstrated by several experiments data from literatures. The simulation results show that the presented fractional derivative aerodynamic model can preferably express the nonlinear unsteady aerodynamic characteristics of airfoils and aircrafts. The proposed modelling framework is meaningful to the designation of flight and control system.


Introduction
It is an important and hard task for precisely predicting the nonlinear unsteady aerodynamic forces and moments during aircraft design [1][2].It plays a significant role in flight dynamics analysis, control law design, and flight simulation of aircraft at high angles of attack [3][4][5][6].
So far, a great deal of the investigations on the aerodynamic model in unsteady flow and at high attack angles have been published [7][8][9][10][11].A neural network model for predicting onboard aircraft's aerodynamics is developed by Peterson et al. [12].Systematic discussions on the prevailing unsteady aerodynamic models and the fundamental restrictions of the conventional stability derivative models are presented by Greenwell [13].Watanabe et al. [14] presented an optimization of the near-sonic passenger plane, making use of the CFD data.Brunton et al. proposed a unsteady fluid dynamics model with lowdimensional balanced and the finite-time Lyapunov exponents were utilized to visualize the flow structure [15].Mifsud et al. reduced the unsteady fluid model dimension in literature [16], making use of data mining methods.In summary, the current researches could be divided into two classes: mathematic methods and artificial intelligent methods.The mathematic models contain differential equation model, state-space model, step response model et al, which are given based on the knowledge of unsteady flow phenomenon and mechanism.The intelligent models include fuzzy logic, neural networks et al, avoiding the analysis on the complex flow mechanism, and considering the flow system as a black-box.Different types of models are suitable for different practical demands.In this paper, we mainly focus on the mathematic one.
Goman model is a wide used state-space model in simple form.It is presented assuming that the hysteresis characteristics at high angles are mainly issued by the flow separation and broken vortex.Based on Goman model, many researches are carried out in literatures [17][18][19][20], e.g.Hao et al improved the Goman model by introducing the power item of the angular rate of attack in literature [21].However, it is always assumed that the separation location and the time variable are first-order differential relations.Is this assumption reasonable and accurate?Depending on this question, in this paper, the fraction-al derivative theory is introduced.The relations between the separation location and the time variable are denoted by fractional derivative form, aiming at improving the model accuracy.
In this paper, an aerodynamic model is presented based on Grünwald-Letnikov fractional derivative.The definition of fractional derivative (FD) is firstly introduced.Then the improved model and the highaccuracy calculating method of Grünwald-Letnikov FD are proposed.Afterwards, the parameter identification methods are given.The proposed Grünwald-Letnikov FD aerodynamic model is verified by three types of wind tunnel tests.

Fractional derivative
Fractional derivative is purely a mathematical concept.There are several types of definitions about fractional derivative [22].And literature [22] show that it is reasonable to replace an ordinary integerorder derivative by a FD.
To expand the representation range and improve the accuracy of the models, the fractional derivative theory is employed by many researchers in literatures [23][24][25][26][27] for kinds of physical or mathematic models, and verified to be useful.For example, in the research field of viscoelastic model, considering the experimental data and the prevailing Maxwell model, a generalized Maxwell model is studied in literature [28] by introducing and making use of the fractional derivative theory.It is figured out that the generalized Maxwell model could precisely describe the viscoelastic characteristics of the structures as long as sufficient orders are defined in the fractional derivative part.
Since the successful application in some other research areas, a new aerodynamic model based on the Grünwald-Letnikov fractional derivative would be presented in this paper, to improve the accuracy of the Goman model.The details about Grünwald-Letnikov fractional derivative could be found in literature [22].
The modelling process is started with Goman's model as [31][32][33]: The definitions of the parameters in Eq. ( 1) was given in literature [20].Based on the factional derivative theory, the traditional Goman's model is improved as: where  is the fractional derivative order.Denote the k-th time step as k t .In order to obtain the highaccuracy   k x t , a computing approach of the fractional derivative   k D x t  will be given as follows.
Divide the time interval   0 k t into N subintervals and the length of every subinterval is Submitting Eq. ( 4) into Eq.( 3), we obtain Submitting Eq. ( 5) into Eq.( 2), the following relation Eq. ( 6) could be obtained.
Rewrite Eq. ( 6) as Then we have the following recursion formula , then Eq. ( 8) can be described in the form of matrix shown as Eq. ( 9).
, , ,, , where   The aerodynamic coefficients (e.g.lift (L), drag (D), and pitch moment (M)) are nonlinear functions of the state variable x , the angle of attack  , the angular rate of pitch motion  .Then, we obtain Eq. ( 13) is approximated at   0, 0 where , , i L D m  .And we define that: where , , i L D m  .We define the input vector as Eq. ( 17) and the output vector as Eq. ( 18).

𝜶 𝜶 𝜶 𝜶
In Eq. ( 18), , Therefore, the parameters of the static component to be determined are ), the total number is twenty-three.These twenty-three static parameters can be solved by an optimization problem as Eq. ( 20).
The parameters of the dynamic component to be determined are 1  , 2  , v ,  , 3  Eq.( 20) and Eq. ( 24), the results are sensitive to the initial values.Therefore, the determination process with the randomly created initial values is carried out by ten times and the means are considered as the final results.These operation aims to obtain preferable accuracy and reduce the impact of uncertainty.The proposed model, which bases on fractional derivative theory, is validated by three types of wind tunnel tests in the following section.The detailed algorithm is displayed in Figure 2.

Initial guess
Calculating the static aerodynamic coefficients by Eq.( 19) and the proposed model

Computing the optimization objective function in Eq.(20)
Is it satisfy the termination condition?

Prepare the wind tunnel experimental data
Calculating the dynamic aerodynamic coefficients by Eq.( 23) and the proposed model Computing the optimization objective function in Eq.( 24) Is it satisfy the termination condition?

Model verification
In this section, the adopted data to validate the FD model are the wind tunnel measurements of NACA 0015 airfoil [36], F18 aircraft [37], and F18 HARV [3,38].The improved Goman model in literature [21] is denoted as Hao model.The Goman model, the Hao model and the proposed model in this paper (denoted as FD model) are compared by F18 HARV data.NLSM (the Nonlinear Least Square Method) is employed to solve the optimization problems of Eq. ( 20) and Eq. ( 24).The optimization procedure is carried out by function "lsqcurvefit" encoded in Matlab.

NACA 0015 airfoil
The measurements of NACA 0015 airfoil were supplied by Jumper et al. [36].In Jumper et al.'s wind tunnel tests, we have After parameter identification, the parameter values of NACA 0015 airfoil (Jumper) are illustrated in Table 1.It can be showed in Figure 3 that the simulated data could well agree with the experimental data for static It can be seen from Figure 4 that for dynamic coefficients, the linear part could be well predicted, while the predictions of the nonlinear part are slightly worse.Nevertheless, the trends of the lift coefficients in nonlinear part related with attack angles are successfully caught.

F18 aircraft
The measurements of F18 aircraft are selected from Brandon's publication [37].The test condition of the angle of attack is provided as follows After parameter identification, the parameter values of F18 aircraft (Brandon) are illustrated in Table 2.As is depicted in Figure 5 and Figure 6, the FD model performs well in describing both static and dynamic coefficients.The simulated results agree well with Brandon's wind tunnel experiments with constant angular rates of attack (57.8 degree/s, 115.0 degree/s, 172.8 degree/s, -57.8 degree/s, and -115.0 degree/s).In addition, the area surrounded by the hysteresis curve increases with the increment of the angular rate of attack.That is to say the surrounded area of 115.0 degree/s    is larger than that of 57.8 degree/s    .

F18 HARV configuration
The experimental data of F18 HARV is from literature [38].The attack angle motion in the wind tunnel tests is controlled by     32.5 30 cos 30 sin After parameter identification, the parameter values of F18 HARV are illustrated in Table 3.

FMIA-2023
1.7671 10 0.01115 0.006402 1.8591 10 +0.003894 0.01736   As is seen in Figure 9 and Figure 12, all three models catch the dynamic nonlinear trends with high attack angles.The HAO model performs better than the Goman model, e.g. the predictions at about 28 degree and 40 degree attack angle in Figure 9.The FD model proposed in this paper perform best, agreeing better than any of the other one, e.g. the predictions among attack angles of 10 to 24 degree in Figure 11, the predictions of the peek feature in Figure 12 (among attack angles of 50 to 60 degree), and the predictions among attack angles of 10 to 24 degree in Figure 12.
In a summary, the FD model has been validated by three types of wind tunnel tests.The present Grünwald-Letnikov FD model for modelling the aerodynamics could represent the complex aerodynamic properties of an airfoil and an aircraft.It may own a wide usage, especially for the flight simulation and control system design.The investigation of the pre-sent modelling framework gives an effective approach of describing the complex aerodynamics.

Conclusions
Based on Grünwald-Letnikov fractional derivative, a generalized aerodynamic model is put forward in the present investigation.A fast and high accuracy calculating method of the model is given and the FD model are validated by the wind tunnel experimental data.The results indicate that the FD model can preferably express the selected cases.The proposed fractional derivative model of aerodynamics owns a wide usage, especially for the flight simulation and control system design.The new model-ling approach of the aerodynamics has a good engineering prospect.
the inverse matrix of k A exists, Eq. (9) can be transformed to    ,    ,  , , , ,  *(12)The variable x at k t is the last element of k x , i.e.  , k N x t.The calculated results of the dynamic separation position x for varying fractional derivative order 0.5,1.0,1.5are shown in Figure1.It can be seen that the fractional derivative order  has apparent influence on the dynamic curve of the separation position x at larger or smaller angles of attack.

Figure 1 .
Figure 1.Dynamic separation position for varying fractional derivative order γ number of the dynamic parameters is thirty-one.The dynamic parameters are

Figure 5 .
Figure 5. Static lift coefficient of F18 aircraft Figure 6.Dynamic lift coefficients of F18 aircraft

2 9Figure 7 .
Figure 7. Static coefficients of F18 HARV Figure 8. Dynamic coefficients of F18 HARVAs is seen in Figure7and Figure8, the FD model can describe the characteristics of Lin and Hu's data which were measured at harmonic oscillations with a large amplitude.The peak feature at static

3. 4 .
Model comparisonThe comparisons of predicting the lift and drag dynamic characteristics of F18 HARV among the Goman model, the HAO model and the FD model are depicted in Figure9and Figure10, separately.

Figure 9 .Figure 10 .Figure 11 .Figure 12 .
Figure 9. Comparisons of dynamic lift coefficient of F18 HARV Two kinds of parameters should be obtained in the FD model.The one is static and the other one is dynamic.To obtain the static component, the aerodynamic coefficients are rewritten by

Table 1 .
Parameter values of NACA 0015

Table 2 .
Parameter values of Brandon's data Variable Value

Table 3 .
Identified parameters of Lin and Hu's data Variable Value