Analysis of the Effects of Numerical Differentiation Methods on the Estimation of Longitudinal Stability and Control Derivatives of the Aircraft Mathematical Model

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Introduction Estimating aircraft stability and control derivatives is relevant in aircraft engineering because these parameters provide crucial information regarding aircraft performance, stability, and control [1; 2]. Stability and control derivatives estimation also plays a significant role in flight testing, aircraft simulation, and control system design. It helps in predicting aircraft behavior during various flight conditions, such as takeoff, landing, and maneuvering. Methods for the estimation of these parameters are important in flight control system design, optimization of aircraft design, improvements in fuel efficiency, enhancement of flight safety, and help ensure the stability and maneuverability of aircraft, and these methods have been proposed in real time in some papers [3-5]. Moreover, aircraft stability and control derivative estimation is essential for simulation, aircraft performance analysis, and optimization, whereas it helps in the determination of the aircraft’s maximum speed, range, payload capacity, and fuel consumption [6]. By understanding these parameters, aircraft operators can make informed decisions on flight planning, route selection, and operational efficiency. To estimate the dynamic stability derivatives, a computational fluid dynamics (CFD)-based force oscillation method was also applied by engineers [7]. Therefore, the accurate estimation of the aerodynamic parameters, aircraft stability, and control derivatives is relevant in the field of aircraft engineering. It is evident that measurement errors affect the accuracy of aerodynamic coefficient estimation [8]. There are various forms of input signals, i.e. the control organs’ deflections, which also have a significant influence on the estimation accuracy [9-11]. It was observed that the application of appropriate methods for signal filtering can ensure an improvement in the accuracy of the estimation of the aerodynamic coefficients depending on the nature of the signals [12-14]. The main purpose of this study is to propose an approach for enhancing the accuracy of parameters estimation by analyzing the effects of numerical differentiation methods on the accuracy of estimation performed by the least-squares method without the application of any methods for filtering noisy signals. The derivative of the pitch rate is used as the output variable or the dependent variable in the least-squares method. Because the output vector represents the actual output values, its range and variations can influence parameter estimation accuracy. 1. Problem statement In aircraft dynamics, the short-period mode of aircraft motion refers to the pitch motion, in which the aircraft rotates around its lateral axis. This motion includes changes in the pitch angle, pitch rate, and airspeed. In this short-period mode, the aircraft experiences rapid pitch angle fluctuations owing to interference such as turbulence or pilot actions. The stability and controllability characteristics of the aircraft in this mode are crucial for flight safety and efficiency. Therefore, to estimate the longitudinal stability and control derivatives of the aircraft mathematical model, the mathematical simulation of the short-period mode of the aircraft motion in a longitudinal channel is performed by applying the aircraft dynamic equations [15]. Because the problem under the current research focuses on the longitudinal stability and control derivatives, only the equations for the angle of attack and pitch rate are used. The elevator deflection was mathematically simulated, as shown in Figure 1. The necessary aerodynamic coefficients are determined to perform a mathematical simulation of the aircraft spatial motion, and they are estimated after the measurements. In this study, the normally distributed random variables with a zero mean and different standard deviation values characterize the measurement noise. The ordinary least-squares method (LSM) was used to estimate the aerodynamic coefficients. The derivative of the pitch rate was used to form the LSM output vector. It is very important to choose the correct numerical differentiation methods so that the accuracy of the estimation can be assured, whereas every numerical differentiation method has its own distinct effectiveness depending on the intensity of the measurement noise. Figure 1. Angle of elevator deflection S o u r c e: made by O.N. Korsun, M.H. Om, S. Goro For the simulation, it was necessary to form the object model and determine the output signals. To form the object model, the angle of attack and pitch rate of the aircraft were simulated using mathematical equations. In this study, integration was performed by applying simpler Euler methods for the angle of attack and pitch rate. Usually, it is convenient and easy to perform simulation and identification in a discrete form; for this reason, the simulation of all the necessary signals is performed in a discrete form. The mathematical formulas in discrete form for the angle of attack (1) and pitch rate (2) are as follows: α(t(i+1) )=α( )ti +Δ -t Yαα( )ti +ωz ( )ti + -( Yδδ( )ti ) ; (1) ωz (t(i+1) )=ωz ( )ti +Δt M zα α( )ti + M zωz ωz ( )ti + M zδδ( )ti , (2) where α(t(i+1) ) is the angle of attack for time instant (t(i+1) ) (rad), ωz (t(i+1) ) is the pitch rate for time instant (t(i+1) ) (rad/s), α( )ti is the angle of attack for time instant( )ti , ωz ( )ti is the pitch rate for time instant( )ti , and Y Y M M Mα, φ, zα, zωz , zδ are the aerodynamic parameters. a b Figure 2. Measurement of signals: a is the simulated angle of attack; b is the pitch rate S o u r c e: made by O.N. Korsun, M.H. Om, S. Goro After the simulation of the short-period mode of the aircraft motion in longitudinal motion was performed, it was necessary to proceed with the measurement of the signals (Figure 2) used in the process of parameter estimation. To imitate the measurement mathematically, the normally distributed random variables with zero mean and different standard deviation values were subjected to the measurement noise for every signal. The values of the standard deviations of the measurements are listed in Table 1. It is assumed that the measurements took place under various Table 1 The values of standard deviations subjected to measurement noise Signal, deg Std, deg Std, deg Std, deg Std, deg Std, deg Angle of attack 0.02 0.08 0.2 0.4 0.8 Pitch rate 0.02 0.08 0.2 0.4 0.8 Elevator deflection 0.02 0.08 0.2 0.4 0.8 a b Figure 3. Measurement of signals under the influence of measurement noise: a - Angle of attack; b - Pitch rate S o u r c e: made by O.N. Korsun, M.H. Om, S. Goro intensities of measurement noise (Figure 3); therefore, the effectiveness of the estimation will be analyzed based on the numerical methods applied to approximate the derivatives of the pitch rate under the influence of various levels of noise intensity. In Table 1, Std represents the values of the standard deviations in degrees subjected to measurement noise. 2. Estimation of stability and control derivatives There are many distinct methods for parametric system identification, and each method has its advantages and disadvantages depending on the system itself [16; 17]. These parametric identification methods are also applicable for detecting dynamic errors in on-board measurements of aircraft based on flight data [18]. The least-squares method (1) was used in this study to estimate the aerodynamic coefficients. Evidently, the least-squares method is more effective for linear systems. The mathematical model of an object can be described as: y( )t = a xˆT ( )t , (3) where y( )t is the vector of the output signal, x( )t is the vector of the regressors, and aˆ is the vector of unknown parameters. The observation model for N number of observations can be expressed as z( )t = y( )t +ε( )t , (4) where ε( )t denotes the measurement noise. Therefore, for each of the N discrete time points at which the measurements are available, it can be described in matrix notation as: Y = Xaˆ + ,ε (5) where Y =[y y y1 2 3………. yN]T , ε=[ε ε ε1 2 3………. εN]T are N×1 size vectors and X is the N × p N × p matrix of the independent variables: 1 X = 1 1 x1t( )1 x1t( )2 … x1t( )N x2t( )1 x3t( )1 x2t( )2 x3t( )2 . … ... x2t( )N x3t( )N Unlike the maximum likelihood, in the leastsquares method (6), the parameters to be estimated must arise in the expressions for the means of the observations. When the parameters appear linearly in these expressions, the least-squares estimation problem can be solved in closed form, and it is relatively straightforward to derive the statistical properties for the resulting parameter estimates. The least-squares method is given by T ) T -1 aˆX X X Y, (6) where aˆ is the vector of unknown parameters to be estimated, X is the matrix of the object model, and Y is the vector of the output signal. Only three moment coefficients involved in the angle of attack and pitch rate were estimated, whereas the present study focused on analyzing the effect of numerical differentiation methods on the estimation. The estimation of three moment coefficients using LSM requires the derivative of . the pitch rate ωz for the formation of the output signal vector Y. The object model matrix X is formed by the angle of attack α, pitch rate ωz and elevator deflection δ. Then, the object model matrix, output signal vector and vector of unknown parameters take the following form: 1 X = 1 1 αt( )1 αt( )2 … αt( )N ωzt( )1 δt( )1 ωzt( )2 δt( )2 , … ... ωzt( )N δt( )N dωz( )1 dt = dωdtz( )2 Y . . ˆ = Mzα , a Mzωz T Mzδ . d ωz N( ) dt 3. Numerical differentiation methods for the approximation of derivatives The aerodynamic moments acting on the aircraft in flight are proportional to the derivatives of the corresponding angular velocities of the aircraft, which are usually obtained from the onboard measurements. Therefore, the numerical differentiation methods that ensure the accurate approximation of the derivatives are crucial in aircraft engineering [19]. Generally, differential equations can be solved analytically; however, significant effort and effective mathematical theory are often required, and the closed form of the solution may be too confusing to be useful. When an analytical solution to a differential equation is not available, it is too difficult to deduce, or it takes on a sophisticated form that is unhelpful to apply, an approximate solution can be considered. There are two approaches to this purpose. The first approach is a semi-analytical methods that consider the use of series, that is, integral equations, perturbation methods, or asymptotic methods for obtaining an approximate solution expressed in terms of simpler functions. The second one is numerical solutions. Discrete numeric values can represent solutions with a certain precision. Currently, these numerical arrays (and their associated tables or graphs) are obtained using computers to provide effective solutions to many problems that were previously impossible to obtain [20]. In this study, to approximate the derivative of the pitch rate, forward difference, backward difference, and central difference methods were applied. The finite difference methods - forward difference, backward difference, and central difference - are used for the numerical approximation of the derivatives of a certain function. The forward difference method approximates the derivative of a function at a particular point by considering the function values at the very point and a nearby point ahead of it. This can be expressed mathematically as follows: ( ) ( f( )x h+ - f( )x ) f ' x = . (7) h The backward difference method considers the function values at that point and the nearby point behind it to approximate the derivatives of a function at a particular point. This can be expressed mathematically as follows: f '( )x = ( f( )x - f( )x h- ). (8) h The central difference method takes the function values at two nearby points, one ahead and one behind it to approximate the derivative of a function at a particular point. This can be expressed mathematically as follows: f '( )x = ( f( )x h+ - f( )x h- ), (9) 2h where f '( )x is the derivative of a function, f( )x is the function that is differentiated, and h is the step size. Moreover, the combination of these three methods “gradient” that is implemented in Python is also utilized for the approximation of the derivative of pitch rate. This combination method “gradient” uses the forward/backward difference methods for approximating the boundary points (first and last points) of the signal and the central difference method is used to calculate the derivatives of interior points of the signal. Poplavsky method was also applied in this study to approximate the derivatives [19]. The approximation polynomials of k degrees were used to estimate the first derivative. This is expressed as follows: S = mj=-mb y tj ( )i , (10) bj = 5 5 (3m(4 +2 -6m1)3m m- +(3m+ 21))(j4-m7 32(-1m)2(2+ -m3m+3)1) j3 , (11) h m where h is the sampling interval, and m is the sliding interval size. First, the estimation of aerodynamic parameters using LSM was performed under noise-free conditions to clarify how the method functions without noise. Table 2 presents the results. The estimation was repeatedly executed 15 times and the average of the relative errors of the estimates was calculated. Relative errors of estimates given by numerical differentiation methods under noise%free condition Method Relative error Mzα , % Relative error Mzωz , % Relative error Mzδ , % Forward Difference 1.90588286e12 8.38218384e12 3.89965837e12 Backward Difference 56.54558069 167.07475111 90.06760292 Central Difference 28.27279035 83.53737555 45.03380146 Combination of Methods 1.89378435 1.36827312 0.96303547 Combination of Results 1.89414061 1.36895486 0.96301928 Poplavsky Method_1 8.40614113 28.35047731 14.42287996 Poplavsky Method_2 1.89026931 1.33060582 0.97035183 Table 2 After obtaining the results calculated by the forward difference method, backward difference method, central difference method, and Poplavsky method, two approaches are proposed considering the efficiency and weaknesses of these methods. The first one is Poplavsky methods-2, where the sample points are cut off along the left and right edges of the dataset in accordance with the size of the sliding window used in the Poplavsky method. The second approach considered a combination of the results obtained by separately using three finite difference methods. As shown in Table 2, the forward difference method demonstrated superior performance compared to the other methods under noise-free conditions. Combination of methods “gradient”, Combination of results and Poplavsky method also demonstrate their effectiveness in providing satisfactory estimates. The backward difference method is generally less accurate, and when it is used for the output vector, which is crucial in estimating parameters, it provides less favorable estimates. The results provided by the central difference method are less favorable because this method does not accurately approximate the derivatives at the boun-dary points. The results show that Poplavsky method also provides gratifying accuracy in the estimation of the parameters under noise-free conditions. 4. Results and discussion For better accuracy and reliability of the estimates of the stability and control derivatives, the estimation was performed 15 times. For each execution of the program, whereas the normally distributed random variables with zero mean and several values of standard deviation characterized the measurement noise, the relative errors of the estimates were calculated. Subsequently, the average values of the relative errors of the estimates according to the repeated execution of the estimation are obtained. The results obtained by the five methods and two approaches are presented in Tables 3-9 and their graphical presentation is shown in Figures (4-6). Table 3 Relative errors of estimates given by forward difference method Std, deg Relative error Mzα , % Relative error Mzωz , % Relative error Mzδ , % 0.02 0.15205137 0.86688247 0.29560156 0.08 0.66371885 11.94143302 3.2670063 0.2 3.55255766 69.70950238 18.22990503 0.4 19.66459307 258.66536699 63.01107033 0.8 116.90595768 724.5247589 132.21844664 Table 4 Relative errors of estimates given by backward difference method Std, deg Relative error Mzα , % Relative error Mzωz , % Relative error Mzδ , % 0.02 56.46582798 167.64983046 90.18434052 0.08 57.33883809 181.78023716 95.01045211 0.2 58.34679468 250.40740951 116.16826948 0.4 54.42187212 459.89022414 175.76298856 0.8 18.73253971 960.14423628 275.98508977 Table 5 Relative errors of estimates given by central difference method Std, deg Relative error Mzα , % Relative error Mzωz , % Relative error Mzδ , % 0.02 28.30462418 83.64873361 45.08821755 0.08 28.66114439 83.85506646 45.40073055 0.2 30.98583624 89.82494974 48.79572234 0.4 37.10499183 104.01791904 57.41742689 0.8 48.81781812 117.835604 69.74591019 Table 6 Relative errors of estimates given by combination of three methods “gradient” Std, deg Relative error Mzα , % Relative error Mzωz , % Relative error Mzδ , % 0.02 1.7906732 1.53301677 0.83952767 0.08 1.37399282 3.27782994 1.37076629 0.2 3.40201398 15.52870946 6.96162448 0.4 13.13329098 39.18691594 20.89832966 0.8 36.63975232 85.35310955 51.29733977 Table 7 Relative errors of estimates given by combination of results given by finite difference methods Std, deg Relative error Mzα , % Relative error Mzωz , % Relative error Mzδ , % 0.02 1.84662819 1.36845403 0.93535781 0.08 1.44177607 3.93460101 1.52677994 0.2 2.89059324 14.78572012 6.41968791 0.4 11.46850026 35.22754859 18.59589663 0.8 35.05152409 87.77718717 51.02816458 Table 8 Relative errors of estimates given by Poplavsky method_1 Std, deg Relative error Mzα , % Relative error Mzωz , % Relative error Mzδ , % 0.02 8.44056384 28.78966508 14.56942007 0.08 8.18407016 29.35411495 14.57706342 0.2 11.86598703 39.18062433 19.97106469 0.4 22.38493718 65.62501345 35.51506445 0.8 41.11094292 101.28620295 59.36737503 Table 9 Relative errors of estimates given by Poplavsky method_2 Std, deg Relative error Mzα , % Relative error Mzωz , % Relative error Mzδ , % 0.02 1.76466215 1.89400133 0.71172912 0.08 1.73815893 7.24275427 3.36221295 0.2 12.51837471 27.97205869 17.2391036 0.4 49.55022288 65.28447612 54.77609201 0.8 79.72952628 90.19302059 83.77492038 In the presented Tables 3-9, the notation “Std” stands for the values of the standard deviations that characterized the measurement noise for the simulation of the measurement of the aircraft performance signals using the mathematical equations of flight dynamics. The relative errors in the estimates of the pitch moment coefficient with respect to the angle of attack, which is a static longitudinal stability derivative, are shown in Figure 4. The relative errors in the estimates of the pitch moment coefficient with respect to the pitch rate, which is a dynamic longitudinal stability derivative, are shown in Figure 5. Figure 4. Relative errors of the estimates of pitch moment coefficient with respect to angle of attack ܯ௭஑ S o u r c e: made by O.N. Korsun, M.H. Om, S. Goro Figure 5. Relative errors of the estimates of pitch moment coefficient with respect to pitch rate ܯ௭ன೥ S o u r c e: made by O.N. Korsun, M.H. Om, S. Goro Figure 6. Relative errors of the estimates of pitch moment coefficient with respect to elevator deflection ܯ௭ஔ S o u r c e: made by O.N. Korsun, M.H. Om, S. Goro The relative errors in the estimates of the pitch moment coefficient with respect to the elevator deflection, which is the longitudinal control derivative, are shown in Figure 6. In Figures 4-6, the X axes of each figure represent the values of standard deviations in degrees that characterize the measurement noise, while the Y axes represent the relative errors of the estimates of the longitudinal stability and control derivatives calculated by the least-squares method, which requires the derivatives of the pitch rate. According to the results presented in Tables 3-9 and Figures 4-6, it can be observed that the backward difference method is less effective for estimating coefficients. The forward difference method ensures a good and satisfactory result when the noise intensity is relatively low; however, its ability deteriorates with an increase in the noise intensity. The central difference method provides moderate and stable accuracy of the estimation at every level of noise intensity, but it provides a less accurate approximation of the derivatives at the boundary points, resulting in a significant deterioration in the accuracy of the estimation. Poplavsky method also provides a good accuracy of the estimates, and it can be observed that this method is robust and stable despite changes in the noise intensity. The combination of three finite difference methods “gradient” ensures gratifying and stable accuracy for every level of noise intensity. The forward difference method is efficient for approximating derivatives at the left boundary, and the backward difference method is effective at the right boundary. The central difference method provides better results than the other two methods, but it is observed that this method is much more suitable for finding the derivatives of the interior points. Based on these advantages and disadvantages, a combination of three finite-difference methods was proposed. The three combined methods were observed to be more efficient in handling different levels of noise intensity throughout the entire dataset, as they approximated the derivatives at the boundary points using the forward and backward difference methods, which are generally effective at the boundary points. It also potentially allows for a more accurate approximation of the derivative considering the specific characteristics of noise in different locations. Nevertheless, in the combination of the three methods, the central difference method still struggles at the second and second last points of the processing time because these two points become the first and last points at the left and right boundaries, where the method has weakness. The combination of the results obtained by separately approximating the derivatives using three finite-difference methods compensates for this weakness and ensures the enhancement of the accuracy of parameter estimation. Poplavsky method also struggles at the left and right boundaries, but after cutting off the edges according to the size of the sliding window, it becomes more efficient under a low intensity of noise. However, the efficiency of this approach deteriorates with an increase in the intensity of the measurement noise. Conclusion This study comprehensively analyzed the effects of numerical differentiation methods: the forward difference method, backward difference method, central difference method, combination of finite difference methods, and Poplavsky method on the estimation of the longitudinal stability and control derivatives of the mathematical model of the motion of the aircraft. Moreover, two proposed approaches take into account the combination of the results separately obtained by three finite difference methods and Poplavsky method-2, where the unnecessary edges of the dataset at the boundaries are cut off. The numerical differentiation methods are used to approximate the derivatives of the pitch rate that is necessary for forming the output vector in the least-squares method. It is important to note that the choice of method depends on specific tasks and requirements. All of these numerical differentiation methods may have advantages depending on the task. The combination of the results separately obtained by the three finite difference methods can be especially useful in practical applications where noise is present, for example, in scientific experiments, data analysis, or signal processing. This method also has the ability to reduce the effects of noise and provides reliable estimates of derivatives, even at high noise levels. Based on the results obtained in this work, it can also be noted that every single point of the dataset is crucial for the parameter estimation, as the first and last points, that is, boundary points, significantly affect the accuracy of estimation. This advantage of the proposed approach enhances the accuracy and reliability of numerical differentiation methods in various scientific research fields.

About the authors

Oleg N. Korsun

State Scientific Research Institute of Aviation Systems; Moscow Aviation Institute (National Research University)

Author for correspondence.
Email: marmotto@rambler.ru
ORCID iD: 0000-0003-3926-1024
SPIN-code: 2472-6853

Doctor of Technical Sciences, Head of the Scientific and Educational Center, State Scientific Research Institute of Aviation Systems; Professor, Department of Design and Certification of Aircraft Engineering, Moscow Aviation Institute

Moscow, Russia

Moung Htang Om

Moscow Aviation Institute (National Research University)

Email: mounghtangom50@gmail.com
ORCID iD: 0000-0002-7770-2962

Ph.D., Post-doctoral Candidate, Department of Design and Certification of Aircraft Engineering

Moscow, Russia

Sekou Goro

Moscow Aviation Institute (National Research University)

Email: gorosekoi@gmail.com
ORCID iD: 0000-0002-4397-9429

Postgraduate Student, Department of Design and Certification of Aircraft Engineering

Moscow, Russia

References