Estimation of Systematic Errors of Onboard ...

ISSN 10642307, Journal of Computer and Systems Sciences International, 2011, Vol. 50, No. 1, pp. 130–143. © Pleiades Publishing, Ltd., 2011. Original Russian Text © O.N. Korsun, B.K. Poplavskii, 2011, published in Izvestiya Akademii Nauk. Teoriya i Sistemy Upravleniya, 2011, No. 1, pp. 133–146.

CONTROL SYSTEMS OF MOVING OBJECTS

Estimation of Systematic Errors of Onboard Measurement of Angle of Attack and Sliding Angle Based on Integration of Data of Satellite Navigation System and Identification of Wind Velocity O. N. Korsun and B. K. Poplavskii FSUE Flight Research Institute Named after M.M. Gromov, Zhukovskii, Moscow reg., Russia email: [email protected] Received February 11, 2010; in final form, August 17, 2010

Abstract—A method for estimation of systematic errors of onboard measurement of angle of attack and sliding angle of an aircraft in the course of flight tests using high precision velocity measurements performed by a satellite navigation system is proposed. The main specific feature of the proposed method is that for providing compatibility of measurements of angle of attack and sliding angle sensors and data of the satellite navigation system the identification of wind velocity on the processed leg is used. The operation of the proposed method and the correctness of applied assumptions are proved by processing large amounts of experimental data obtained in the course of flight tests. DOI: 10.1134/S1064230711010126

INTRODUCTION Precise onboard measurements of angles of attack and sliding angles of an aircraft are necessary for the following tasks: (i) satisfaction of operational constraints on the angle of attack and sliding angle in the course of piloting; (ii) application of these angles in the flight control system as feedback signals; (iii) verification and specification of mathematical models of aircraft engines and mathematical mod els of aerodynamic forces and torques using data of flight tests. Results of onboard measurements of the angle of attack and sliding angle in the general case possess essential systematic errors. The main reason is that sensors measure the socalled local angles, rather than true angle of attack and sliding angle, and these local angles essentially depend on specific features of aero dynamic streamline of the aircraft at places of sensor installation. Therefore, in the course of flight tests it is necessary to determine as precisely as possible systematic errors of these sensors, or in other words, the law of calculation of true angle of attack and sliding angle using local values measured by sensors. This requires information on true angle values. Conventionally sensors placed in an unperturbed flow using a rather long outrigger bar temporarily installed in the head part of the aircraft are used for this purpose. However, in this case the errors con nected with the bar geometry and its elastic properties occur, and the account of these properties is not a simple task. Before description of the proposed method, let us present a brief survey of the history and status of this problem. Onboard measurers of the angle of attack and sliding angle became widespread in the 1940s– 1950s. In this period the main problem was clearly formulated (this problem consists in the difference between the true angle of attack and sliding angle and local angles recorded by sensors [1]) and the main method for solution of this problem, application of outrigger bar, was proposed [1, 2]. The achieved level of methodical support mainly preserved until recently [3–6]. In relation with development of inertial and satellite navigation systems, the idea of their application [3, 5] for estimation of values of true angle of attack and sliding angle in the conditions of unperturbed atmosphere [5] was put forward. In the presence of wind [3], it was proposed to use data on wind velocity whose source are meteorological observations. In practice this approach was not popular due to organizational complexity and low accuracy, since wind at the point at which the aircraft is situated can substantially differ from that near the meteorological station or along the weather balloon flight trajectory. 130

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In the recent years, papers devoted to application of satellite navigation systems in combination with other onboard systems for estimation of aircraft motion parameters and atmosphere parameters, includ ing wind velocity, without additional devices (weather balloons, airlogs [1], reference aircrafts [1–3], out rigger measuring bars [1–3]) have been published. In [7] for light aircrafts, which are not equipped with inertial navigation system measuring aircraft position angles (bank, pitch, and course angles) and sensors measuring angle of attack and sliding angle, these parameters were estimated by Kalman filtering of mea surements of satellite navigation system and solution of aircraft motion equations including the data base of aerodynamic coefficients. In [8] the description of onboard system for estimation of wind velocity and other atmospheric parameters is given; this system, along with measurements of satellite navigation sys tem and inertial navigation system, involves data from the system of air signals and the angle of attack and sliding angle sensors. It is assumed that systematic errors of the system of air signals and sensors were determined in the course of earlier tests. In [9] the estimates of wind velocity are made based on the data of satellite navigation system, inertial navigation system, system of air signals, and measurement of the sliding angle. In this study, only high precision data of satellite navigation system (projections of aircraft velocity) and inertial navigation system (aircraft position angles) are used for estimation of systematic errors of mea surement of the angle of attack and sliding angle. Measurements of the system of air signals and the data base of aerodynamic coefficients of the aircraft are not used, because these sources possess large errors, which should be estimated and corrected in the course of flight tests. Outrigger bars are also not necessary, which eliminates the requirement of installation of offnominal accessories on the aircraft. Wind velocity is estimated using parametric identification. Note that similar approach was earlier mentioned by us for estimation of errors of measurement of air velocity [10]. 1. STATEMENT OF THE PROBLEM In this paper, the method for estimation of errors of onboard measurement of angle of attack and slid ing angle using data of flight tests which applies relatively novel high precision measurement tool, satellite navigation system, is proposed. The novelty of the proposed method is that for providing compatibility of measurements of the angle of attack and sliding angle sensors and data of satellite navigation system the wind velocity at the processed leg is identified; in this case, measurements of the system of air signals are not used. The following measurements are the source of information: (i) three projections of absolute velocity of the aircraft measured by the satellite navigation system operating in the differential regime which is considered as the tool for externaltrajectory measurements; (ii) bank, pitch, and course angles which are measured by the inertial navigation system; (iii) angle of attack and sliding angle which are received from sensors of aerodynamic angles. If the errors of the satellite navigation system substantially increase upon aircraft maneuvering, it is rec ommended to perform correction [11] based on measurements of three projections of absolute aircraft velocity using the inertial navigation system. Integration of satellite navigation system recording the aircraft velocity with respect to the Earth and sensors of aerodynamic angles measuring angular position of the aircraft with respect to the air flow is achieved based on estimation of the wind velocity. For this purpose, the assumption on constant wind velocity direction and value on the processed flight leg with a duration of 1–3 min is made. For finding estimates of three projections of wind velocity on the axes of the ground normal coordinate system and the parameters characterizing systematic measurement errors parametric identification is used. The algorithm described in the next section was developed for solution of this problem. The operation of this algorithm and correctness of assumptions were verified by processing data of flight tests. It is known that identification is referred to the class of inverse problems [12, 13]. In the general case, parameter estimates can be biased due to the structural incompleteness of the model, systematic measure ment errors not taken into account, imprecision of assumptions on statistical properties of random mea surement errors. It was shown in [14] that the efficient method for solution of this problem is the compar ison of information received from different sources or processed in different ways. Therefore, operation of the proposed algorithm was verified, first, using rather large amounts of flight data, and second, additional procedures proving the correctness of results obtained using the main algo rithm were considered. JOURNAL OF COMPUTER AND SYSTEMS SCIENCES INTERNATIONAL

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Satellite navigation system in differential regime

Velocity projections in ground normal coordinate system

Air velocity projections in ground normal coordinate system

Air velocity projections in connected coordinate system

Vxg Vyg,

Vxg_a, Vyg_a,

Vx_a Vy_a,

Vzg

Vzg_a

Vz_a

Complex processing of velocity measurements

Wind model

Vxg_W Vyg_W,

Strapdown Inertial Navigation System

Direction cosines according to GOST 2005880

Vzg_W

Estimate of angle of attack and sliding angle

Algorithm o f parametric dentification

Error Sensors estimates of aerodynamic angles for sensors (onboard measurements of aerodynamic of angle of attack angles and wind and sliding angle) velocity projection

Euler angles

Fig. 1. Structural diagram of formation of the algorithm for estimation of systematic measurement errors for angle of attack and sliding angle.

2. ALGORITHM FOR ESTIMATION OF SYSTEMATIC ERRORS OF ONBOARD MEASUREMENT OF ANGLE OF ATTACK AND SLIDING ANGLE For solution of the problem of estimation of systematic measurement errors of the angle of attack and sliding angle, an algorithm of integrated processing of flight information whose structural diagram is shown in Fig. 1 is proposed. Let us describe the algorithm operation according to the diagram in Fig. 1. The velocity measured by the satellite navigation system is specified in the block of integrated information processing, using the velocity values whose source is the inertial navigation system. The description of the processing algorithm can be found in [11]. It is recommended to perform this stage if the aircraft maneuvering and the change of composition of the “constellation” of satellites connected with it result in a substantial growth of errors of the satellite navigation system [11]. The calculated projections of the aircraft velocity on the ground normal coordinate system after sum ming with corresponding projections of the wind velocity (received from the block of wind model) provide cal culation of the projection of the air aircraft velocity on the axes of the ground normal coordinate system. Using the matrix of direction cosines [15], we find the projections of air velocity in the connected coordinate system; in this case, the bank, pitch, and course angles are received from the inertial navigation system. For identification of systematic measurement errors for the angle of attack and sliding angle, estimates of these angles obtained with account of the values of projections of air velocity on the axes of the con nected coordinate system are used. These estimates are compared with the values of the angle of attack and sliding angle recorded in the flight in the block of parametric identification, which makes it possible to determine three projections of the wind velocity and the parameters of systematic measurement errors. The description of the method of identification is given in the next section. Then let us present the mathematical formulation of the algorithm described above. For realization of parametric identification, it is necessary to specify the models of the object, observations, and the vector of identified parameters. Let us form the object model. The equations for projections of air velocity of the aircraft in the ground normal coordinate system have the form

V xg _ a(ti ) = V xg _ sns(ti ) + V xg _W , V yg _ a(ti ) = V yg _ sns(ti ) + V yg _W , Vzg _ a(ti ) = Vzg _ sns(ti ) + Vzg _W , JOURNAL OF COMPUTER AND SYSTEMS SCIENCES INTERNATIONAL

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where V xg _ sns (t i ),V yg _ sns (t i ),V zg _ sns (t i ) are the values of projections of the aircraft velocity on the axes of the normal ground system measured by the satellite navigation system; and V xg _ W ,V yg _ W ,V zg _ W are the unknown values of projections of the wind velocity on the axes of the normal ground system which should be determined. It is assumed that they are constant on the processed flight leg. Obviously, the absolute value of the air velocity vector is

Va(t i ) = V xg2 _ a(t i ) + V yg2 _ a(t i ) + V zg2 _ a(t i ).

(2.2)

The projections of air velocity in the connected coordinate system are calculated by multiplication of projections (2.1) by the transition matrix from the ground normal to the connected coordinate system according to [15],

cos ψ cos y sin y − sin ψ cos ϑ ⎡V x _ a ⎤ ⎡ ⎤ ⎡V xg _ a ⎤ ⎥. ⎢V ⎥ = ⎢sin ψ sin y − cos ψ sin ϑ cos y cos ϑ cos y cos ψ sin y + sin ψ sin ϑ cos y ⎥ ⎢V ⎢ y_a⎥ ⎢ ⎥ ⎢ yg _ a ⎥ ⎣⎢V z _ a ⎦⎥ ⎢⎣sin ψ cos y + cos ψ sin ϑ sin y − cos ϑ sin y cos ψ cos y − sin ψ sin ϑ sin y ⎦⎥ ⎢⎣V zg _ a ⎥⎦

(2.3)

In processing the values of bank, pitch, and yaw angle are taken from the results of measurements of the inertial navigation system. In modern inertial navigation systems, measurement errors for pitch and bank angle are of order of angular seconds; therefore, in the considered problem they can be neglected. The yaw angle ψ is charac terized by slowly varying error, which can reach tenth fractions of a degree. Such an error cannot be neglected. Let us make the following assumption: since the duration of pro cessed legs does not exceed 1–3 min, the measurement error for the yaw angle can be assumed an unknown constant Cψ = const. This constant is included in the vector of identified parameters. In order to identify multiplicative component of systematic measurement error for the angle of attack and take into account its nonlinear character, we introduce, for example, the following parameters repre senting the coefficients of calibration characteristic for different angles of attack: K α1 for α ≤ α1, α1 = 15 ; Kα for α > α1. Obviously, the structure of nonlinearity description can change depending on a particular problem. For taking into account the multiplicative component of systematic measurement error for the sliding angle, we use the parameter Kβ representing the coefficient of inclination of the calibration char acteristic of the sliding angle sensor. Based on projections (2.3) of air velocity on connected axes, we write the equations for estimates of the angle of attack and sliding angle [4],

⎛V (t )⎞ α true(t i ) = −arctan ⎜ y _ α i ⎟ + Cα, ⎝V x _ α(t i )⎠

(2.4)

⎛V (t )⎞ β true(t i ) = − arcsin ⎜ z _ α i ⎟ + Cβ, ⎝ Vα(t i ) ⎠

where α true(ti ), βtrue(ti ) are estimates of true angle of attack and sliding angle (here the term “true” means that it is not the local angle); Cα, Cβ are the constant (additive) components of systematic measurement errors for the angle of attack and sliding angle. Thus, the object model is determined by Eqs. (2.1)–(2.4). The model of observations takes the form

z1(t i ) = K α1α true(t i ) + ξ α(t i ), if α ≤ α1, z1(t i ) = K α1α1 + K α(α true(t i ) − α1) + ξα(t i ), if α > α1, z 2(t i ) = K ββ true(t i ) + ξβ(t i ),

(2.5)

where ξ α (t i ), ξ β(t i ) are the measurement noise described as the sequences of independent normally distrib uted random variables with zero mathematical expectation and constant variance. The vector of identified parameters has the following structure:

αT = [C α Cβ V xg _ W V yg _ W V zg _ W C ψ K α K α1 K β ] . JOURNAL OF COMPUTER AND SYSTEMS SCIENCES INTERNATIONAL

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Thus, the estimates of three components of the wind velocity, the constant component of the measure ment error for the yaw angle, and the estimates of parameters characterizing additive and multiplicative system atic measurement errors for the angle of attack and sliding angle are determined using identification. 3. CHOICE OF PARAMETRIC IDENTIFICATION METHOD At present, the number of papers on system identification is infinitely large, and list [12–14, 16–38] is incomplete. Among these papers, we note monographs of general character [13, 16–23], monographs on aircraft identification [12, 28–30], proceedings of international conferences on identification [24–27], papers in which new identification methods were proposed [14, 31–37], textbook [38]. However, the experience of identification of aircrafts and their systems accumulated during the recent decades makes it possible to choose an appropriate method for solution of the problem formulated above. First of all, it is not necessary to apply structural and structural–parametric identification [12, 20, 22, 28, 37], since the object model and the model of observations are determined to several unknown parameters. Fre quency [22, 28] and frequency–time [33, 35] parametric methods can be eliminated, since they yield advantages if it is possible to separate most informative frequency range, which takes place, for example, in identification of aeroelasticity models. This feature is absent in the examined problem. The linear trans formation method [36] yields noticeable effect only if the form of noise is known. Methods based on mul tiple regression and the classical least squares method [23, 16, 20, 22] should also be eliminated from con sideration, since they are applicable to linear or parameterlinear systems, while the problem formulated in Section 2 is essentially nonlinear. Thus, methods of adjustable model [12, 16, 22, 30, 31] and the family of algorithms of stochastic nonlinear filtering, including of Kalman type [12, 17–20, 29, 34] remain among methods widely used for identification of aircrafts and their systems. Stochastic filtering, including Kalman filtering, has indisputable advantage in the presence of object noise, i.e., random process which is the input signal. This situation takes place in aircraft identification first of all in study of flights in con ditions of atmospheric turbulence [29]. Estimation of systematic errors of the angle of attack and sliding angle sensors in flight tests is usually performed using the results of flights performed in the absence of tur bulence in order to eliminate the additional uncertainty factor. Another advantage of filtering methods, which is the possibility of real time data processing, is also insignificant for the solved problem, since cal culations are performed during afterflight processing. Therefore, it is not necessary to use stochastic non linear filtering and it is possible to choose simpler methods of adjustable model. These methods rather well suppress measurement noise present in the considered problem due to averaging over the whole leg. Let us consider, for example, algorithm [31] widely used in foreign practice of aircraft identification [30]. Unfortunately, in [31] working formulas are given without derivation which complicates understand ing of their properties. Moreover, the name “maximum likelihood estimation” is terminologically poor, since the specific feature of the method [31] is not the principle of maximal likelihood, which is actually reduced to common functional of the least squares method, but the method of adjustment of the model parameters. Therefore, let us derive the formulas of the method and analyze its basic properties. It is assumed [31] that the object model and the model of observations are determined as

y '(t) = f (y(t), a, u(t)),

(3.1)

z(ti ) = h(y(ti ), a, u(t i )) + η(t i ),

(3.2)

where y(t), u(t) are the vectors of output and input signals with the dimensions n and m, respectively, z(t i ) is the observation vector with the dimension r, η(ti ) is the observation noise representing the vector normal ran dom sequence of the type of white noise with zero mathematical expectation and known covariance matrix R(ti ); and a is the vector of unknown parameters which should be identified. It is assumed that u(t ) is the known function of time. The initial conditions y(t 0 ) are either assumed known or are included in the vector of esti mated parameters. Observation noise is interpreted as normal and independent random vector variables. Therefore, their combined probability distribution density is equal to the product of densities for each time instant t i , i = 1, N . In these conditions the likelihood function takes the form N ⎧⎪ ⎫⎪ p(z(N ) a) = A exp ⎨− 1 (z(t i ) − h(yˆ(t i ), a, u(t i )))T R −1(t i )(z(t i ) − h(yˆ(t i )a, u(t i )))⎬ ⎪⎩ 2 i = 1 ⎪⎭

∑

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where N

A=

∏ ( R(t )

−1

i

(2π) −r )1/2

i =1

is the constant independent of the parameter vector; yˆ(t i ) is the estimate of the state vector obtained from numerical solution of Eq. (3.1) for fixed parameter vector. It was shown in [23] that the maximum of the likelihood function for these assumptions on the noise properties results in the unbiased effective estimates. Due to the monotonic character of logarithmic func tion the maximum of (3.3) with respect to the parameters a corresponds to the minimum of the logarithm, which results in the criterion N

J (a) =

∑((z(t ) −h(yˆ(t ), a, u(t )))

T

i

i

i

R −1(t i )((z(t i ) − h(yˆ(t i ), a, u(t i ))).

(3.4)

i =1

It can easily be noted that (3.4) represents the criterion of the least squares method with the matrix of weighting coefficients R(t i )−1 . Therefore, the specific feature of the maximum likelihood method [31] is not in the method of choosing criterion that was used in the name of the method, but in the form of rep resentation of object model and model of measurements (3.1), (3.2) and applied numerical optimization method. For minimization of (3.4) in [31] it is proposed to use one of the modifications of the classical Newton method [40], −1

⎛ d 2 J (ak )⎞ dJ (ak ) (3.5) aˆk + 1 = ak − ⎜ . 2 ⎟ ⎝ dak ⎠ dak For elucidation of the meaning of identification, we calculate the first and second order derivatives for the parameter vector included in (3.5) with account of (3.1), (3.2). Let us introduce the notation zˆ(t i , a) = h(yˆ(t i ), a, u(t i )). Then (3.4) can be written as N

J (a) =

∑(z(t ) − zˆ(t , a))

T

i

i

R −1(t i )(z(t i ) − zˆ(t i , a)),

(3.6)

i =1

where zˆ(t i , a) is the estimate of the observation vector obtained by numerical integration of equations for the object and observations (3.1), (3.2) for t ∈ [t 0, t N ] at η(t i ) = 0, i = 1, N . For calculation of derivatives of scalar functional (3.6) with respect to the vector a with the dimension p we use the following vector–matrix relations in which it is assumed that the corresponding scalar prod ucts exist, T

d ( x A x) = 2 A x, dx where x is the vector with the dimension r, A is the symmetric matrix with the dimension r × r,

(

)

dJ (x(a)) da

( p × 1)

( ) (dJdx )

= dx da

(3.7)

T

( p × r)

(3.8)

, (r × 1)

where x(a) is the vector function with the dimension r of the vector argument a with the dimension p; J(x(a)) is the scalar function of the vector argument x and at the same time complex function of the vector argument a. In (3.8) dimensions are indicated in brackets. Relations (3.7) and (3.8) can easily be proved, for example, by writing scalar elements in explicit form. Let us assume that

ε(t i, a) = z(t i ) − zˆ(t i , a). JOURNAL OF COMPUTER AND SYSTEMS SCIENCES INTERNATIONAL

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Then (3.6) can be represented as N

J (a) =

∑ ε (t , a) R T

−1

i

(t i ) ε(t i , a).

(3.9)

i =1

Since the covariance matrix R(ti) is symmetric, R–1(ti) is also symmetric. Let us find the derivative of (3.9) with respect to the parameter vector with account of (3.7) and (3.8), dJ (a) = da

N

d ε (t i , a) dJ (a) = da d εT (t i , a) i =1

∑

T

N

N

d ε (t i , a) −1 2R (t i ) ε(t i , a) da i =1

∑

T

(3.10)

T

dzˆ (t i , a) −1 R (t i )(z(t i ) − zˆ(t i , a)). da i =1

∑

=−2

Differentiating (3.10), we find the matrix of second derivatives (Hesse matrix), N

N

dzˆT (t i , a) −1 dzˆ(t i , a) d 2 zˆT (t i , a) −1 d 2J (a) =2 −2 R (t i ) R (t i )(z(t i ) − zˆ(t i , a)). 2 2 da da da da i =1 i =1

∑

∑

(3.11)

In [31] it is proposed to estimate the matrix of second derivatives by the following expression: N

dzˆT (t i , a) −1 dzˆ(t i , a) d 2J (a) R (t i ) ≅ 2 . da da da 2 i =1

∑

(3.12)

Exact expression (3.11) shows that the modification used in [31] consists in simple rejection of the term containing the second order derivative. It was pointed out in [40] that the main disadvantage of the Newton method is the requirement of cal culation of second derivatives. It can be seen that formula (3.12) solves this problem in a radical way. The advantages of approximation (3.12) are obvious. First, the amount of calculations essentially reduces, and second, errors connected with calculation of second derivatives are eliminated. In this relation, the question on validity of this approach occurs. Arguments in favor of this variant of the Newton method can be found, for example, in [16], where it is pointed out that the accuracy of cal culation of Hesse matrix is important only near the point of minimum when the optimized function is well approximated by the quadratic function, since the Newton method possesses singlestep convergence for quadratic functions [16, 40]. However, for the considered object and observations models near the point of minimum the term con taining Hesse matrix is close to zero. Indeed, in the neighborhood of minimum a ≈ atrue , where atrue is the true value of parameters. Then

ε(ti, a) = z(ti ) − zˆ(ti , a) ≈ η(t i ) and consequently,

M [ z(t i ) − zˆ(t i , a)] ≈ 0.

(3.13)

Thus, the rejected term in (3.11) is proportional to the estimate of mathematical expectation (3.13) for N observations. According to the conditions of application of the Newton method [40], the functional is twice continuously differentiable, i.e., second derivatives in (3.11) are continuous and bounded. There fore, near the point of minimum the second term in (3.11) is close to zero. At large distances from the point of extremum, the quadratic character of the functional is weakly manifested, and it is not necessary to calculate Hesse matrix with high accuracy. Thus, the rejected term should not essentially influence the recurrent search of extremum. JOURNAL OF COMPUTER AND SYSTEMS SCIENCES INTERNATIONAL

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Upon the algorithm realization the derivatives of estimates in (3.11) are numerically determined for the time instants t i , i = 1, N , using the formulas [31]

dzˆ(ti, a) ⎡∂zˆ(ti , a) ∂zˆ(ti , a) ∂zˆ(ti , a)⎤ =⎢  , ⎥ ∂a2 ∂a p ⎦ (r × p) da ⎣ ∂a1

(3.14)

∂zˆ(ti, a) zˆ(ti, a + ε e j ) − zˆ(t i, a) = , ∂a j ε

where ej is the vector with the dimension p with all zero elements except the jth element which is equal to 1; and ε is the small number usually determined on a level of 0.001–0.2% of the nominal parameter value. The estimates zˆ(t i , a), i = 1, N , are the numerical solution to the object and observations equations for η(t i ) = 0. The identification is finished by the condition aˆk +1 − aˆk < δ aˆk , where δ = 0.05…0.02. The matrix of the weighting coefficients R −1(t i ) is as a rule diagonal, its elements are the variances of residuals between measurements and estimates of the angle of attack and sliding angle. The numerical values are given in Section 4. The analysis of the identification method makes it possible to answer the following question. Object model (3.1) is determined in the form of vector nonlinear differential equation, while in Section 2 the object is described by nonlinear algebraic equations (2.1)–(2.4) which can be combined into one vector nonlinear algebraic equation

y(t i ) = f (y(t i ), a, u(t i )).

(3.15)

Formulas (3.10), (3.11), (3.14) show that for application of the method it is necessary to obtain the esti mates zˆ(t i , a) = h(yˆ(t i ), a, u(t i )), and it is unimportant which equation is used for finding yˆ(t i ), differential or algebraic one. Thus, it is necessary to solve (3.15) instead of (3.1). The other identification formulas do not change. The verification of the considered algorithm using simulation showed that this algorithm converges to the point of minimum as a rule after 4–8 iterations for errors of initial estimates of up to 50%. 4. RESULTS OF ESTIMATION OF SYSTEMATIC MEASUREMENT ERRORS FOR THE ANGLE OF ATTACK The algorithm described in the previous section was used for processing a number of flight legs in the course of flight tests of one of modern aircrafts. For estimation of measurement errors of the angle of attack 21 flight legs were used, maneuvers “deceleration” and “spiral turn” were performed on these legs. The values of mean square deviations of the angle of attack and sliding angle calculated in the model and measured on board were 0.25–0.35°. This testifies good agreement of the model and the object and proves the correctness of assumptions for the wind velocity. The example of comparison of calculated and mea sured values of the angle of attack for the “deceleration” maneuver is shown in Fig. 2. The plot shows the values of the angle of attack α meas measured in the flight, the values of the angle of attack α est estimated using the proposed algorithm, and the values of the number M on this leg. The lefthand coordinate axis shows the scale for the angle of attack, the righthand axis, that for the number M. The estimates of the angle of attack α est were obtained by numerical solution of Eqs. (2.1)–(2.5) for zero observation noise  ξ α (t i ), ξ β(t i ), i.e., it was assumed that α est = z1(t i , a). In the course of trial processing of several legs, it was established that the coefficient K α1 characterizing the inclination of the calibration characteristic of the angle of attack sensor at α < 15° is equal to 1 with high accuracy. Therefore, hereinafter, this parameters was assumed equal to unity, K α1 = 1, and it was eliminated from the set of identified parameters. The analysis of the estimates Cα demonstrated the presence of constant error with the average value C α = − 1.1 independent of the angle of attack and the number M. In this case, the mean square deviation of estimates was 0.32°. For estimation of the coefficient Kα characterizing the multiplicative error for angles of attack larger than 15° an average value of 1.08 was obtained. In this case, the mean square deviation of errors was 0.045. For additional verification of the presence of nonlinear multiplicative error for the leg shown in Fig. 2 the angle of attack α meas measured in the flight as a function of estimates α meas.lin (conditionally called lin JOURNAL OF COMPUTER AND SYSTEMS SCIENCES INTERNATIONAL

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KORSUN, POPLAVSKII αmeas, αest, deg 35

M 0.70

30 0.66 25 0.62 20 15

0.58

10 0.54 5 0.50 0 −5

0

12

24

36

48

60

72

84

96

108

120

t, s

0.46

Fig. 2. Measured and estimated angle of attack and M upon execution of “spiral turn” maneuver: – αmeas,—αest, … М.

αmeas, deg 35

y=x

30 25 20 15 10 5 0 −5 −10 −4

2

8

14

20

26

32 αest.lin, deg

Fig. 3. Measured angle of attack as a function of estimated angle of attack. JOURNAL OF COMPUTER AND SYSTEMS SCIENCES INTERNATIONAL

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139

y=x

30 25 20 15 10 5 0 −5 −10 −4

2

8

14

20

26

32 αest.lin, deg

Fig. 4. Measured angle of attack as a function of estimated angle of attack after correction of measurement delay for angle of attack.

ear) found by numerical solution of Eqs. (2.1)–(2.4) at zero observation noise was plotted. Since the non linearity of (2.5) was not taken into account upon calculation of α meas.lin , this plot should pictorially rep resent the actual calibration characteristic. The result is shown in Fig. 3; in this figure the ideal calibration characteristic y = x with the inclination coefficient equal to unity is also shown for convenience of analysis. The growth of the inclination coeffi cient at α > 15°, which agrees with the found quantitative estimates of the coefficient Kα, can be clearly seen in this plot. Figure 3 also clearly demonstrates the hysteresis in the measured angle of attack, due to which the experimental calibration characteristic shown in the plot is an ambiguous function. The most probable reason is dynamic errors of onboard measurements of the angle of attack, which occur due to the applica tion of smoothing lowfrequency filters or delays introduced by the onboard registration system. Figure 4 shows the result obtained after correction of measurement delay by 1/16 (in this case the flight data registration frequency was 16 Hz). It can be seen that the hysteresis is much smaller, and the calibration characteristic approaches an unambiguous function, which agrees with its physical meaning. Thus, the proposed method makes it possible to estimate dynamic measurement error for the angle of attack. In conclusion we note that the values of the true angle of attack calculated on board according to the following dependence were used above as the measured values α meas :

α true = f (α local ), where α true is the true angle of attack (here, the term “true” means the difference from the local angle of attack); and α local is the local angle of attack registered by the sensor of aerodynamic angles. The dependence α true = f (α local ) was determined earlier using the temporary outrigger bar. Thus, the obtained error estimates characterize the accuracy of the conventional method for calibration of angle of attack sensors. JOURNAL OF COMPUTER AND SYSTEMS SCIENCES INTERNATIONAL

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KORSUN, POPLAVSKII βmeas, βest, deg 18

12

6

0

−6

−12

−18

0

19

36

58.3

80

101

117

135

155

t, s

Fig. 5. Measured and estimated sliding angles without correction of multiplicative error: βmeas,—βest.

5. RESULTS OF ESTIMATION OF SYSTEMATIC MEASUREMENT ERRORS FOR THE SLIDING ANGLE In this section, the estimation of systematic measurement error for the sliding angle is considered. The model of measurement errors was taken in the form β meas = C β + K ββ est ,

(5.1)

where β meas is the value of the true sliding angle measured in the flight (calculated according to the depen dence of the true angle on the local one realized in the onboard computer); βalg is the estimate of the true sliding angle obtained using the developed algorithm. For estimation of the measurement error for the sliding angle data on 21 flight legs mentioned above were processed. Additional 10 flight legs on which throttles were applied were also used. The average esti mate of the constant error Cβ was C β = − 0.7  , and the mean square deviation of these estimates was 0.35°. The estimates of the coefficient of multiplicative error Kβ had mean square deviations of 0.03, and the average value was 0.87. Figure 5 shows the difference of amplitudes of measured sliding angles with respect to the values esti mated using the algorithm before the correction of the multiplicative error of the coefficient Kβ (constant component Cβ is eliminated) for the regime in which throttles are applied. Figure 6 shows the result of cor rection according to Eq. (5.1) with the estimates of average values C β = − 0.7  and K β = 0.87 . It can be seen from this figure that after the correction the degree of convergence of the measured and estimated sliding angles is high. 6. VERIFICATION OF THE PRESENCE OF ADDITIVE MEASUREMENT ERROR FOR THE ANGLE OF ATTACK USING A SIMPLIFIED METHOD The proposed algorithm uses the data of different onboard systems and the procedure of parametric identification. For additional verification of reliability, we describe the simple method for estimation of the constant component of measurement error for the angle of attack. JOURNAL OF COMPUTER AND SYSTEMS SCIENCES INTERNATIONAL

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βmeas, βest, deg 18

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6

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−6

−12

−18

0

22

45

68

94

115

141

159

183

t, s

Fig. 6. Measured and estimated sliding angles after correction of multiplicative error: βmeas,—βest.

It is known that in the case of horizontal flight at constant velocity (regime “plateau”) the pitch angle is equal to the true angle of attack,

ϑ = α true. At small angles of attack the value α meas represents the sum of the angle of attack measured in the flight and the additive error,

α meas = α true + Cα. Then the estimate of additive error can be calculated, (6.1) C α = α meas − ϑ. In this simple estimate the wind velocity was not taken into account, its influence was eliminated due to averaging over the set of legs. 39 “plateaus” were processed using the data on flight experiments. The error Cα was calculated using formula (6.1); and the average value was –0.89°, and the mean square deviation, 0.45°. Comparing this average value with the estimates Cα given in Section 3 (C α = − 1.1) we can see that the simple method described in this section proves the correctness of estimates of the constant measurement error for the angle of attack obtained using the algorithm presented in Section 2, which employs data of the satellite navigation system and wind identification. The statistical verification of the hypothesis of equality of mathematical expectations of two samplings [41] for average values of compared estimates of –1.1 and –0.89°, corresponding mean square deviations of 0.32 and 0.45°, sampling volumes of 21 and 39 also did not demonstrate any difference between these estimates on a significance level of 0.05. CONCLUSIONS The performance of the proposed method for estimation of systematic measurement errors for the angle of attack and sliding angle and the correctness of assumptions has been confirmed by processing essential amount of experimental data accumulated in the course of flight tests. JOURNAL OF COMPUTER AND SYSTEMS SCIENCES INTERNATIONAL

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The developed method makes it possible to estimate additive and multiplicative systematic measure ment errors for the angle of attack and sliding angle. The reliability of obtained results has been proved by additional verification based on graphical analysis and comparison with partial estimates calculated using a simpler method in which assumptions of the main algorithm were not applied. The efficient algorithm of parametric identification of nonlinear systems was described in detail and its properties were analyzed; this may be of interest for solution of identification problems in other applied fields. ACKNOWLEDGMENTS This work was supported by the Russian Foundation for Basic Research, project no. 090800887. REFERENCES 1. V. S. Vedrov and M. A. Taits, Flight Testing of Aircrafts (Oborongiz, Moscow, 1951) [in Russian]. 2. W. Gracey, Summary of Methods of Measuring Angle of Attack on Aircraft. NACA Technical Note 4351 (NACA, Washington, 1958). 3. E. A. Íàering, Airdata Measurement and Calibration. NASA Technical Memorandum 104316 (Dryden Flight Research Center, Edwards, California, 1995). 4. K. K. Vasil’chenko, V. A. Leonov, I. M. Pashkovskii, et al., Aircraft Flight Testing (Mashinostroenie, Moscow, 1993) [in Russian]. 5. V. M. Soldatkin, Methods and Means for Measuring FV Aerodynamic Angles (Izd. KGTU, Kazan, 2001) [in Rus sian]. 6. G. I. Klyuev, V. N. Derevyankin, V. M. Soldatkin, et al., Meters of FV Aerodynamic Parameters (Izd. UlGTU, Ul’yanovsk, 2005) [in Russian]. 7. K. P. A. Lievens, J. A. Mulder, and P. Chu, “Single GPS Antenna Attitude Determination of a Fixed Wing Air craft Aided with Aircraft Aerodynamics,” AIAA Guidance, Navigation and Control Conference and Exhibit (SanFrancisco, California, 2005). 8. T. P. Bui, Meteorological Measurement Systems (MMS) (NASA Ames Research Center, 2006). 9. S. G. Pushkov, E. G. Kharin, V. R. Kozhurin, et al., “Technology for Determination of Aerodynamic Errors of APSs and Air Parameters in Flight Tests of Aircrafts Using Satellite Tools of Trajectory Measurements,” Prob lemy bezopasnosti poletov, no. 7, 12–19 (2006). 10. O. N. Korsun, A. V. Zinov’ev, O. P. Lysyuk, et al., “Algorithm for Estimation of Constant Component of Mea surement Error for Air Velocity with Account of Wind Velocity,” Vestn. Kompyutern. Informats. Tekhnol., No. 9, 2–6 (2008). 11. O. N. Korsun, A. V. Zinov’ev, O. P. Lysyuk, and O. N. Grebnev, “Esimation of Measurement Errors for Velocity and Altitude by Satellite Navigation System at Aircraft Maneuvering,” Problemy Bezopasnosti Poletov, No. 10, 49–58 (2008). 12. K. K. Vasil’chenko, Yu. A. Kochetkov, V. A. Leonov, et al., Structural Identification of Mathematical Model of Air craft Motion (Mashinostroenie, Moscow, 1993) [in Russian]. 13. A. N. Tikhonov and V. Ya. Arsenin, Solutions of Ill Posed Problems (Nauka, Moscow, 1986) [in Russian]. 14. O. N. Korsun, “Principles of Parametric Identification of Mathematical Models of Aircrafts Using Flight Test Data,” Mekhatron. Avtomat., Upravlen., No. 06, 2–7 (2008). 15. GOST 2005880. Aircraft Dynamics in Atmosphere. Terms, Definitions and Symbols (Izd. Standartov, Moscow, 1981) [in Russian]. 16. L. Ljung, System Identification Theory for the User (Prentice Hall, 1987; Nauka, Moscow, 1991) [in Russian]. 17. E. Seidzh and Dzh. Mels, Estimation Theory and its Application in Communication and Control (McGrawHill, New York, 1972; Svyaz’, Moscow, 1976). 18. E. Sage and J. Mels, Management Systems’ Identification (Nauka, Moscow, 1974) [in Russian]. 19. Modern Identification Methods, ed. by P. Eickhoff (Mir, Moscow, 1986) [in Russian]. 20. Handbook on the Theory of Automatic Control, ed. by A. A. Krasovskii (Nauka, Moscow, 1987) [in Russian]. 21. Ya. Z. Tsypkin, Foundations of the Information Identification Theory (Nauka, Moscow, 1984) [in Russian]. 22. P. Eickhoff, Priinciples of Identification of Control Systems (Mir, Moscow, 1975) [in Russian]. 23. Yu. V. Linnik, Method of Least Squares and Principles of the Theory of Observations (Fizmatgiz, Moscow, 1962; Pergamon Press, Oxford, 1961). 24. Proceedings of 7th IFAC Symposium on Identification and System Parameter Estimation (IFAC, York, 1985). 25. Proceedings of 8th IFAC Symposium on Identification and System Parameter Estimation (IFAC, Beijing, 1988). JOURNAL OF COMPUTER AND SYSTEMS SCIENCES INTERNATIONAL

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26. Proceedings of II International Congress “System Identification and Control Problems (FICPRO’03) (Tra peznikov Institute of Control Sciences, Russian Academy of Sciences, Moscow, 2003) [in Russian]. 27. Proceedings of International ScientificTechnical Conference “Identification of Flying Vehicles, Power Instal lations, and Aviation Equipment” (Gromov Flight Research Institute, Zhukovskii, 1994) [in Russian]. 28. L. M. Berestov, B. K. Poplavskii, and L. Ya. Miroshnichenko, Frequency Methods for Identification of Aircraft (Mashinostroenie, Moscow, 1985) [in Russian]. 29. A. S. Belotserkovskii, B. O. Kachanov, Yu. B. Kulifeev, and V. I. Morozov, Creation and Application of Mathe matic Models of Aircraft (Nauka, Moscow (1984) [in Russian]. 30. V. Klein, “Estimation of Aircraft Aerodynamic Parameters from Flight Data,” Prog. Aerospace Sñi, No. 26, 1–77 (1989). 31. R. E. Main and K. W. Iliff, Identification of Dynamic Systems: Theory and Formulation (NASA Reference Pub lication, 1138, 1985). 32. O. N. Korsun, “Identification of the Longitudinal Motion of a Statically Unstable Aircraft Based on the Parallel Model,” Izv. Ross. Akad. Nauk, Teor. Sist. Upr., No. 3, 34–39 (2001) [Comp. Syst. Sci. 40 (3), 385–390 (2001)]. 33. B. O. Kachanov, “Method of Spectral–Time Identification of Dynamic Systems,” Izv. Akad. Nauk SSSR, Tekh. Kibern., No. 3, 147–152 (1993). 34. Yu. B. Kulifeev, “Discrete–Continuous Method for Identification of Continuous Systems,” Izv. Akad. Nauk SSSR, Mekhanika Tverdogo Tela, No. 5, 47–55 (1981). 35. O. N. Korsun, “Algorithm of Identification of Dynamic Systems with Functional in the Frequency Domain,” Avtom. Telemekh., No. 5, 111–121 (2003). 36. V. A. Leonov and B. K. Poplavskii, “Method of Linear Transformations of Identification of Dynamic Systems,” Izv. Akad. Nauk SSSR, Tekh. Kibern., No. 2, 21–28 (1990). 37. I. N. Beloglazov, “Multidimensional Analysis in Problems of Education Qualimetry,” Izv. Ross. Akad. Nauk, Teor. Sist. Upr., No. 6, 39–52 (2006) [Comp. Syst. Sci. 45 (6), 881–893 (2006)]. 38. A. A. Alekseev, Yu. A. Korablev, and M. Yu. Shestopalov, Identification and Diagnostics of Systems (Akademiya, Moscow, 2009) [in Russian]. 39. Yu. P. Dobrolenskii, Flight Dynamics in a Disturbed Atmosphere (Mashinostroenie, Moscow, 1969) [in Russian]. 40. N. N. Moiseev, Yu. P. Ivanilov, and E. M. Stolyarova, Optimization Methods (Nauka, Moscow, 1978) [in Rus sian]. 41. Handbook on Applied Statistics, ed. by E. Lloyd and W. Lederman, vol. 1, (Fizika i Statistika, Moscow, 1989) [in Russian].

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