Aerodynamic Parameters of High Performance Aircraft Estimated from Wind Tunnel and Flight Test Data

18-1 NASA/TM..

207993

AERODYNAMIC PARAMETERS OF HIGH PERFORMANCE AIRCRAFT ESTIMATED FROM WIND TUNNEL AND FLIGHT TEST DATA Vladislav Klein The George Washington University Hampton, Virginia 23681 and Patrick C. Murphy NASA Langley Research Center Hampton, Virginia 23681 SUMMARY 1. INTRODUCTION A concept of system identification applied to high performance aircraft is introduced followed by a discussion on the identification methodology. Special The introduction of highly maneuverable and often emphasis is given to model postulation using time inherently unstable aircraft has been presenting new invariant and time dependent aerodynamic parameters, challenges to aircraft identification and parameter model structure determination and parameter estimation estimation. These aircraft can perform rapid large amplitude maneuvers, often extended to the stall and using ordinary least squares and mixed estimation methods. At the same time problems of data collinearity poststall region where nonlinear and unsteady detection and its assessment are discussed. These parts of aerodynamic effects could be pronounced. This introduces methodology are demonstrated in examples using flight a problem of determining how complex the model should data of the X-29A and X-31A aircraft. In the third be. Although a more complex model can be justified for example wind tunnel oscillatory data of the F-16XL more accurate description of airplane motion, it has not been clear in parameter estimation which relationship model are used. A strong dependence of these data on frequency led to the development of models with between model complexity and measurement information unsteady aerodynamic terms in the form of indicial would be the best. If estimates for too many parameters functions. The paper is completed by concluding remarks. are sought from a limited amount of data, a reduced accuracy can be expected or attempts to identify all parameters might fail. The high performance aircraft may NOMENCLATURE also have more control surfaces moved through a flight control system than conventional aircraft. Such a system Only the main symbols are introduced. Other symbols are can introduce a close relationship between the deflections explained in the body of the paper. of various surfaces and at the same time can preclude maneuvers from being suitable for system identification.

mean aerodynamic chord, m These characteristics can be reflected in an inability to b wing span estimate the effectiveness of individual control surfaces lift coefficient CL and to obtain accurate estimates of the remaining Ct, Cm, Cn rolling-, pitching- and yawing-moment parameters. One of the reasons for these problems is coefficient related to the near linear relationship among several reduced frequency variables entering the model for various estimation t characteristics length, m techniques. This near linear relationship is often called roll, pitch and yaw rate, tad/see p,q,r data collinearity [i].

R 2 coefficient of determination V airspeed, m/see In recent years more attention has been given to the ct, 13 angle of attack and sideslip angle, rad analysisofdata obtained from dynamic wind tunnel tests.

control stick deflection rio The frequency and amplitude dependency of oscillatory aileron and rudder deflection, rad 8, 8 and ramp test data led to postulation of models with linear or nonlinear unsteady effects and subsequent canard, flap and strake deflection,rad 8, 8f, 8 parameter estimation in these models [2,3].

thrust vectoring deflection in pitch and yaw, rad As follows from the above mentioned reasons and unknown parameter experience, a successful parameter estimation requires the

18-2

following: where the state vector, x, is comprised of V, co and Euler a) design of an experiment for obtaining data with high angles, and u is the control vector of control surface accuracy, high information content and low collinearity deflections. The outputs, y, are the variables defining among measured inputs and outputs; aircraft responses. The measured outputs, z(i), are corrupted by measurement noise, v(i), and the number of b) determination of model structure which represents an data points is N.

adequate model for the aircraft under test in each maneuver analyzed; Aircraft identification can be defined as follows: Aircraft identification is a determination, from input and output c) introduction of techniques which reduce the adverse measurements, of a structure for F(O) and G(O) and effect of data collinearity on parameter estimates when estimationof unknown parameters, O, in F(O) and G(O).

severe data collinearity could not be avoided by In many practical applications, the muctm'e of F(0) andG(0) experiment design.

is assumed to be known and aircraft identification is reduced to parameter estimation. A general approach to The purpose of this paper is to present a general approach aircraft identification adopted at NASA Langley Research to aircraft identification with emphasis on the above Center is shown in figure I in the form of a block mentioned requirements for successful identification of a diagram. Various steps in the procedure include model high performance aircraft model. The paper starts with an postulation, design of an experiment, data compatibility overview of system identification methodology followed analysis, model structure determination and parameter by a discussion on data collinearity and biased estimation.

estimation combined with collinearity diagnostic, and In examples the flight data from experiments on X-29A model validation.

and X-3 IA aircraft will be used. The forced oscillatory wind tunnel data on F-16XL model were selected for Model Postulation estimation of unsteady aerodynamic parameters. The Model postulation is influenced by the type of selected paper is completed by concluding remarks.

maneuver intended for system identification and by a prior knowledge about aircraft aerodynamics. The 2. AIRCRAFT IDENTIFICATION aerodynamic forces and moments are expressed in the METHODOLOGY form of polynomials or polynomial splines as When system identification is applied to an aircraft the equations governing its motion are postulated and an a-!

experiment is designed for obtaining time histories of the

c.=c.(0)+E%xj (3)

j=l input and output variables. The equations of motion are formed by rigid-body force and moment equations where C, is the aerodynamic coefficient, C,(0) is the value of the coefficient at initial steay-state conditions m_/+mto xV =F o +F r + F(V,o,u,O)

(1)

Id_ + {o x I_ = G (V,{o,u,O) and the xj now represents input and output variables, their combinations and/or spline terms. The postulated model is then used in model structure determination and and by a set of kinematic equations relating the Euler parameter estimation. For a model with unsteady aerodynamics the forces and moments can be formulated (attitude) angles and angular velocities. In eq. (1)m is the in terms of indicial functions [4,5] as mass, I is the inertia matrix, V and co arc the linear and angular velocity vectors, and u is the control vector. The vectors Fc and F r represent the gravity and propulsion t force, F and G aerodynamic force and moment

(4)

C.(t) = C,(O) +fc, (t-*;_(x))T-d-_x _(x)dx

respectively, and 0 is a vector of parameters which specify aerodynamic characteristics of the aircraft.

where _, is a vector of aircraft state and input variables For system identification, the aircraft state equations are completed by the output and measurement equations. The upon which the coefficient C_ depends, C_(I) isa vector complete set of all these equations can be written as of indieial functions whose elements are the responses in C to unit steps in _. The indicial responses, C,_, are i = tlx(t),u(t),0], x(O) =x o functions of elapsed time (t-x) and are continuous single-valued functions of E,(t). The indicial functions (2) y = h[x(t),u(t),0] approach steady-state values with increasing values of the z(i)=y(i)+v(i), i=l,2,...,N argument (t-x). If the indicial response C,_ is only a function of elapsed time, equations (4) is simplified as 18-3 Rather than minimize p(Z 10), it is more convenient to !

minimize the negative logarithm of the likelihood

(5)

c,(t) -- c,(o) *fc._(t- _F d _)dx

function L(0) =p(ZI0), called log-likelihood function, 0 =_n{- _nL(0)} When analytical forms of indicial functions are specified, the aerodynamic model based on equation (4) or (5) can be used in the aircraft equations of motion for stability Substantial simplification to the ML estimation is and control studies involving either nonlinear or linear obtained by assuming no external disturbances to the unsteady aerodynamics, respectively. The resulting system and no measurement errors in input data. Then the equations of motion will be represented by a set of ML estimation is reduced to the output error method with integro-differential equations.

the cost function Design of Experiment N The most important part of the experiment design is the selection of input forms. It has been recognized that the j=l Evr(i)R__v(i) +NQn IRI (6) 2 i-I shape of an input signal could influence the accuracy of estimated parameters from flight measurement. Attempts for obtaining parameter estimates with high accuracy led where v(i) are the residuals, v(i)--z(i)-y(i,0), and R is many researchers to the development of an optimal input.

the measurement noise covariance matrix. Experience One of the latest techniques for optimal input design is shows that a suitable technique for minimization of(6) is discussed in [6].

the Modified Newton-Raphson method. Using this technique, the step size, A0, for parameter estimates is Data Compatibility Analysis given by In practice, often the measured response data, even after careful handling, can still contain bias errors. In order to verify data accuracy, a compatibility check can be applied AO =M -I aQnL(0) (7) to the measured aircraft responses. This check includes dO aircraft state estimation, based on known kinematics and the available sensor measurements, estimation of unknown bias errors and a comparison of reconstructed where M is the Fisher information matrix containing responses with those measured. The state equations are products and sums of the first order partials of the log- formed by kinematic relationships and the parameter likelihood function. The expression for the information vector usually contains constant offsets and scale factor matrix is errors. The estimation techniques are similar to those used in estimation of states and aerodynamic parameters.

Methods of Parameter Estimation

(8)

Model structure determination and parameter estimation combined with collinearity diagnostic form the principal pan of the identification procedure. From the postulated The information matrix provides also a lower bound on model and measured data the model structure can be parameter covariances, i.e.

determined as explained later. When the model structure is known the parameter estimation can follow. In aeronautical application three methods, the maximum Coy(O) = Z{($ - O)(O -o)'r} > M -' (9) likelihood (ML), linear regression (LR) and extended Kalman filter (EKF), are the basic techniques.

The inverse of M is called the Cramer-Rao lower bound The ML estimates are obtained by maximizing the and the ML estimates approach it asymptotically as N conditional probability of measurement increases. Expression (8) is valid if the measurement Z=[z(I),z(2) ..... z(n)] r given a value of 0, i.e.

noise is random, Gaussian and white. Numerous analyses from measured flight data showed, however, that these

6=n_xp(Z]O)

residuals can be far from being white.

18-4 for white noise and as The residuals very often contain some deterministic components. The result is colored residuals leading to Cov(6) = new expression for parameter covariance matrix [7] in the (16) form

Coy(6) --

(10) M-I[_ (OY(i)]TR-' _ _(i-j)R- --_ N I 0Y(i)]M-I

L"' k-"_-J ,-'

for colored noise where ¢r2 is the variance of e(i) 1 N-K . .

_tc_(k) = _--_==_==_ i_=, eO)eO+k) = _,_(-k) (17) where 91vv(i-j) is the autocorrelation matrix for the output residual vector. It is estimated from and N-k _t (k)=__ k _ v(i)v(i+k)X=_,_(_k) (11) e(i) = y(i) - x(i) r _ (18) -- i-I Further simplification to the ML method is obtained by The linear regression is a widely used estimation method assuming that both the input and states variables are for the following reasons: measured without errors. This assumption leads to an equation error method which can be formulated as a a) it is a simple non-iterative method for parameter linear regression applied to aerodynamic model equation estimation; (3). When the states, and inputs in this equation are replaced by measured values the equivalent of the general b) the LS estimates serve as nominal starting values for regression equation the ML and EKF methods; y(i) =0o+ 01xt(i ) ..... 0a_t Xu_, (i) + e(i) (12) c) linear regression can be applied to data generated by partitioning an ensemble of data from repeated measurements with respect to one or more variables [8]; is obtained. In this equation, y now represents a d) linear regression can be extended to a technique for dependent variable, x_ to x,., are the regressors and e is model structure determination, e.g.stepwise regression; the equation error. When the regression equations are expressed as e) formulation of a regression problem can be used for investigation of near-linear dependence (eollinearity) YfX0+e (13) among measured state and input variables and for the development of biased estimation techniques for dealing with highly collinear data (see Section 3).

the least squares (LS) parameter estimates are obtained as For the development of the EKF algorithm the state vector is augmented by a parameter vector, x = [x r i 0"r]'r.

=(XrX)-IXTy (14) The vector x, and its covariance matrix are estimated from measurements by minimizing the cost function and their covariance matrix as J = E {[x. (i Ii - 1) -x. (i)] "r ix. (i li - I) -x. (i)]} Coy (6) = o2(XTX) -' (I 5) Because of inherent feed-back in the algorithm, the EKF can be easily applied to an unstable system for which the use of the output error method might be difficult or even impossible. On the other hand, the main disadvantage of the EKF method is that the initial conditions and the covariance matrices of process and measurement noise must be specified at the start of the estimation process.

18-5 Model Structure Determination 3. DATA COLLLINEARITY AND BIASED ESTIMATION A major problem in system identification is the selection, from measured data, of an adequate (parsimonious) As pointed out in the Introduction the augmentation of model. An adequate model is considered to be a model high performance aircraft very often introduces near which sufficiently fits the data, facilitates the successful linear relationships among the input and output variables estimation of unknown parameters whose existence can (data collinearity). When linear regression is used in data be substantiated, and has good prediction capabilities. For analysis the collinearity results in an ill-conditioned XrX model structure determination a method of stepwise matrix in expression (14) for LS parameter estimates.

regression is being used [9]. The determination of a Because of that the collinearity can cause computational model for the aerodynamic coefficients includes three problems and reduce the accuracy of estimates. Three steps: postulation of terms that might enter the model, procedures for detection of collinearity are recommended selection of an adequate model, and validation of the in [101 and applied to flight data. They are: model selected. After postulating the aerodynamic model equations, significant terms among the candidate variables a) examination of the correlation matrix are determined and the corresponding parameters X'rX ' and its inverse, where the matrix X" is formed by estimated. At every step of the stepwise regression, the centered and scaled regressors; variables incorporated into the model in previous stages and a new variable entering the model are reexamined for b) eigenvalue analysis of the xrx matrix their significance. Experience shows, however, that the or singular value decomposition of the X matrix; model based only on the statistical significance of individual parameters in the model can still include too c) parameter variance decomposition into many terms and may have poor prediction capabilities. a sum of components, each corresponding to one and Several criteria for the selection of an adequate model only one oftbe eigenvalues of the XrX matrix or singular values of the X matrix.

have been,therefore, introduced. The most often used are: a) The computed values of F-statistic, The parameter variance decomposition approach for given as the ratio of regression mean square to residual detecting collinearity was proposed in [ I ]. It follows from mean square. Heuristically, the model with the maximum the covariance matrix of parameter estimates 0 which can F-values is the "best" one for a given set of data.

be also obtained as b) The value of the coefficient of Cov (0) = o2TA-IT T (20) determination, R 2, which can be interpreted as measuring the proportion of the variation explained by the terms where A is a diagonal matrix whose elements are the other than 00 in the model.

eigenvalues of xrx and T is a matrix whose columns are the eigenvectors of XrX.

c) The prediction sum of squares PRESS defined for the k* subset of model parameters as The variance of each parameter is equal to PRESS = (21) (19) lq {y(i)- _,[i[ x(1) ..... x(i-1),x(i+l) ..... x(N)lk}2 i-I where tjk are the elements of eigenvector tj associated with kj. Eq. (21) decomposes the variance of each For the model to be a good predictor the value of PRESS parameter into a sum of components, each corresponding should be minimal.

to one and only one of the n singular values laj. In (21) Model Validation the singular values appear in denominator, so one or more small singular values can substantially increase the Model validation is the last step in the identification variance of 0j. This means that an unusually high process and should be applied regardless of the proportion of the variance of two or more coefficients for complexity of the estimation method. The resulting model the same small singular value can provide evidence that must demonstrate that its parameters have physically the corresponding near dependency is causing problems.

reasonable values and acceptable accuracy, and that the Introducing model is a good predictor. For those reasons the parameter estimates are compared with any information available about aircraft aerodynamics. Prediction Cjk - -'S and CJ = _1 Cjk _tj capabilities of the model are checked on a set of data not used in the identification process.

the j,k variance-decomposition proportion as the 18-6 proportion of the variance ofthejth regression coefficient For known o 2 the application of least squares to this associated with the kth components of its decomposition model results in the mixed estimator in (21) is given as _jk t_ME =(XTX+ATWqA)-I (xTy+ATWqd) (24) (22) x •=--, j,k = 1,2,...,n I_ ®j It is shown in [10] that the addition of prior information Since two or more regressors are required to create near to the ordinary regression results in reduction of dependency, then two or more variances will be adversely parameter variance. In real application of the ME the a affected by high variance-decomposition proportions priorivalues are not known exactly therefore the resulting associated with a single singular value. Variance- estimator is biased [10].

decomposition proportions greater than 0.5 are recommended in [I] as a guidance for possible 4. EXAMPLES collinearity problems. It is also suggested that the columns of X should be scaled to unit length but not In the following three examples the measured data from centered. Thus the role of the bias term in near-linear experiments on the X-29A, X-31A and F-16XL aircraft dependencies can be diagnosed.

will be used. These examples will demonstrate mainly specific problems related to identification of high If the collinearity diagnostic reveals a serious problem performance aircraft, in addition, variations of some some way of dealing with it should be chosen. Additional parameters with the angle of attack or Math number, and data can be selected, the experiment can b¢ redesigned, their correlation with wind tunnel data and flight results the model can be respecified or different techniques from from scale model will be also shown.

the ordinary LS procedure can be used. Several estimation techniques which can be applied to data with 4.1 X-29A Aircraft severe collinearity have been developed [11]. Their The test vehicle is a single engine, single seat fighter-type development drops the requirement that the estimator of 0 research aircraft with forward-swept wings. The aircraft be unbiased. The new estimator, however, should have has highly relaxed static longitudinal stability in subsonic smaller variance than the LS estimator. By allowing the and transonic regimes and near-neutral stability in small amount of bias the parameter variance can be made supersonic regimes. For longitudinal control, deflections of canard, wing flap (flaperon), and fuselage strake are small such that the mean square error of 0 is less than used. The lateral control is provided by the rudder and the variance of the unbiased LS estimator. Techniques asymmetric deflection of flaperon. In addition to manual with this property belong to a class of biased estimation control of the aircraft, the concept of remotely augmented methods. Two of the techniques, the mixed estimation vehicle (RAV) could be used for the excitation of aircraft and rank reduction regression, have been introduced in responses. The RAV arrangement employs a ground [10] and used in flight data analysis. The experience with the rank reduction regression showed some difficulties in computer to augment the onboard control system. This its application to aerodynamic model equations, mainly capability is used to introduce a command to the control stick (pitch stick or roll stick command), rudder pedal, or because of small number of unknown parameters in these individual control surfaces. The RAV commands, usually equations. For that reason, only the mixed estimation will be introduced.

a pulse or doublet, are summed onto the already existing commands in order to independently move flaps, strake, canard, rudder, or differential flap. More about the RAV The mixed estimation (ME) is a procedure which uses system can be found in [12]. A drawing of the aircraft is prior information on parameters to augment measured presented in figure 2. A more detailed description of the data directly instead of through a prior distribution.

Mixed estimation includes the usual regression model aircraft and its control system is contained in [13].

given by eq. (! 3) and the additional assumption that a set The following three sets of data were available for of prior conditions on 0 can be written as estimation of aircraft parameters: d = A0 + _ (23) 1. longitudinal maneuvers excited by a pilot at Mach numbers from 0.5 to 1.4, 2. longitudinal and lateral maneuvers with computer In this equation, A is a matrix of known constants and_ generated inputs (RAV experiment) at Mach numbers is a vector of random variables with E(_)=0 and from 0.6 to 1.3, E(_ T) =o2W, where W is a known weighing matrix.

Combining (13) and (23) the mixed model is obtained.

3. low speed lateral maneuvers initiated by a pilot at the angles of attack between 80 to 500 .

18-7 for accurate estimation of C remains small.

During data analysis several problems associated with raq inherent instability, high augmentation and sometimes insufficient excitation of maneuvers had to be addressed.

The estimates of two parameters Cm_and Cm_c which Among them were: contribute the most to the pitching moment are plotted against the Mach number in figures 5 and 6 respectively 1. parameter estimation of an unstable vehicle, and compared with the wind tunnel results. The estimates were obtained from data with pilot input using the least 2. data collinearity and its diagnostics, squares and mixed estimation, and from RAV experiment using only the least squares method. The wind tunnel 3. adverse effect of data collinearity on parameter values of the strake effectiveness were used as a priori identifiability and accuracy.

values. The accuracies of the a priori values were determined from repeated test in two different wind Because of these problems, a linear regression and mixed tunnels. As can be seen from these figures, the accuracy estimation were used in data analysis.

of parameters was improved either by applying the mixed estimation to data with pilot input or by using data from Time histories of a typical longitudinal pilot and the RAV experiment. Similar conclusion can be drawn computer-generated input are presented in figures 3 and from average standard errors in estimated parameters.

4. From figure 3 close correlation among all open-loop These values are shown in table VI together with the fit inputs is obvious. The change in data cotlinearity caused error for the coefficient C m.

by replacing the pilot by computer-generated input as a sequence of commanded flap, strake, canard and stick The measured data from all low speed lateral maneuvers deflections is demonstrated in figure 4. The collinearity were assembled into one set with 51,200 data points was assessed by comparing correlations between which was then partitioned into 42 one-degree- a subsets regressors, and corresponding, condition indexes and and 1 three-degree-ot subset. A half of the selected lateral parameter variance proportions. The condition index is maneuvers was analyzed as individual maneuvers using defined as the ratio of the maximum eigenvalue of the stepwise regression. The possibility of data collinearity in information matrix xTx tO one of the remaining measured data was investigated by procedures explained eigenvalues.

in the previous examples. Application of stepwise regression to partitioned data resulted in models for the The correlation matrices for the two sets of data are given lateral aerodynamic coefficients and least squares in tables 1 and II, respectively. The data with the pilot estimates of parameters in these models. For the data input show correlation between subsets with a<40 °, models with linear stability and Stand qc-/2Vand 8c and 8. The RAV data do not exhibit control derivatives were adequate. For data at ct>40 °.

any significant correlation between regressors. The some nonlinear and longitudinal control terms were condition indexes and parameter variance proportions are selected by the stepwise regression. These additional given in tables III and IV. In this example the maximum terms did not provide any comprehensive information condition index (condition number) for the set of data about aerodynamic nonlinearities or effects of with pilot input is 174, in the second case 14, thus longitudinal control setting on the lateral aerodynamic indicating reduced spread of eigenvalues where the RAV coefficients.

system was used. The variance proportions in table !II for the largest condition index show strong collinearity The estimated parameters (stability and control among the bias term, canard and strake effectiveness. The derivatives) from 43 subsets were plotted against the same quantities in table IV indicate only a possibility of angle of attack and fitted by quadratic polynomial splines.

collinearity between the bias term and canard effectiveness. In addition to fitted splines, the 20-confidence limits on the mean were computed as Table V demonstrates a possibility of estimating parameters in the regression equation for the pitching- (25) moment coefficient with sufficient accuracy. Included are _(_) ± _ _/xT(XTx')-Ix the increments in coefficient of determination, AR z, and t-statistics, t'. The values of AR z represent the amount of information in the data explained by the individual terms In (25), 0 is the mean value of a parameter given by the in the model, t-statistic can be considered as a measure of significance of individual parameters. The data with the fitted spline, s is the standard error of 0 estimated from the residuals, x is a vector of regressors. As an example, pilot input revealed that C,,_8 c is the highly influential term in the pitching-moment equation, that there is a estimated values of the parameter C,p, fitted spline, and limited possibility for accurate estimates of parameters 2a-confidence limits are shown in figure 7.

Cmsf and Cm6 ,, and that the significance of theCmqq_/2V Stepwise regression analysis of data from single term is almost zero. The RAV experiment improved the maneuvers showed that linear models for the aerodynamic identiflability of parameters C,_, Cm6 f and Cm6 ,, still coefficients were adequate within the angle-of attack maintaining the C,k5 as a dominant term. The chance range from 80 to 400 . For angles of attack greater than 40 o 18-8 models for the lateral-force and yawing-moment boom on the full scale aira_ Positive yaw damping foret>30 ° coefficient included some of the nonlinear terms is predicted by flight data whereas the wind tunnel results show low yaw damping over the whole range of angles [3z, _3 ct[3, [38cor 8_. As for the partitioned data, there of attack. In figure 9 three rolling moment parameters are was no consistent information about parameters associated with these nonlinear terms. included. The parameter Ct_ estimated from flight data agrees, in general, with wind tunnel predicitions. The roll The least squares parameter estimates at low and damping decreases above ct--15 °, from wind tunnel moderate angles of attack were close to results from above ct--25 °. Positive values of C_ from flight are partitioned data. For ct>20 °, however, increased scatter about three times higher than those from wind tunnel. As in the estimates and their deviation from the previous mentioned in [15] and [16] the forebody aerodynamics results using partitioned data were observed. This dominates the roll-damping parameter at high angles of inconsistency was caused by data collinearity detected attack and causes the unstable damping. Therefore the mostly among the variables pb/2V, 8and 8 r. In applying differences between flight and wind tunnel results maybe the mixed estimation technique, the a priori values were attributed to different forebody aerodynamics in these two selected for parameters which were affected by test conditions. The parameter Ct6 . is estimated with high cotlinearity. Their changes had only a small effect on consistency and agrees well with wind tunnel data for the aerodynamic coefficients. The a priori values and their angles of attack between 120 and 40 o . Some differences uncertainty were set to the mean values and their standard exists outside this interval. More about the parameters can errors obtained from partitioned data. An example of data be found in [17].

collinearity is given in table VII where the eigervalues of the information matrix, condition indexes and parameter The results from this example lead to the following variance decompositions are included. The damaging conclusions: effect of data collinearity on parameters associated with 1. A collinearity diagnostic and a comparison of the variables pb/2V, 8and 8 can be expected. Table parameter estimates from flight data using the RAV VIII presents the parameter estimates and their standard errors in the rolling-moment equation using stepwise experiment with those from flight data using pilot input regression and, mixed estimation. In the last column of revealed that the computer-generated deflections of individual control surfacescan substantially decrease data table VIII the parameter estimates from partitioned data are given. Large changes in the least square estimates of collinearity. A simple least squares technique can be used in estimation of all parameters in the model which also Cfp, C¢_and Ct_ r are visible during the model means estimation of the effectiveness of all controls used.

development. The final estimates of the three parameters are also quite different from those obtained from 2. The experiment providing the data for the analysis was partitioned data. On the other hand, the parameters from not properly designed. The selected input forms resulted the mixed estimation technique are close to the result in very small excitation of response variables. As a result from partitioned data. These estimates also have lower some parameters were not identifiable and the standard errors than the least squares results. The decrease identifiability of several remaining parameters was in the fit error, s(Cr), and squared correlation coefficient, significantly reduced. These identifiability problems were R 2, are not substantial (see last two rows in table VIII).

apparent from the diagnostic of selected maneuvers.

Selected parameters obtained from partitined data and 3. For low speed lateral maneuvers linear aerodynamic single maneuvers are plotted against the angle of attack models determined by applying stepwise regression in figures 8 and 9, and compared with wind tunnel data.

techniques to partitioned data and single maneuvers were The 20 - confidence limits were omitted in these figures.

found to be adequate for the angles of attack less than However, the minimum and maximum values for standard 400 . For angles of attack greater than 400 , nonlinear and errors of the estimated parameters from partitioned data longitudinal control terms entered the models selected by are given in table IX. Both figures indicate no significant the estimation technique. These terms, however, did not differences between the two sets of flight results. Large provide any comprehensive information about differences, however exist between flight and wind tunnel aerodynamic nonlinearities and the effect of longitudinal results in figure 8. Flight data exhibit a sudden increase control on lateral aerodynamic coefficients. Because of data coilinearity detected in single maneuvers, the data in parameter values of C:O and C,_ at angles of attack from these maneuvers were reanalyzed by using mixed around 400 to 450 . As indicated by wind tunnel estimation. No significant differences existed between investigations in [14], an increase of lateral force and parameters estimated from partitioned data and those yawing moment due to sideslip is caused by the forebody vortex assymetry. This assymetry can produce a sideforce estimated from single maneuvers.

which moves the nose into the sideslip and thus enhances 4.2. X-31A Aircraft directional stability. The same effect was observed in [15] during wind tunnel testing of the scaled model. The The X-31A is a single engine, single seat fighter-type present differences between flight and wind tunnel results can be caused by different Reynolds numbers in the two research aircraft with delta wing and canard. The aircraft is inherently unstable in subsonic regimes and is experiements (0.68 x l06 in the wind tunnel; approx. 6 x controllable by a digital control system. For longitudinal l06 during the flight test) and by the effect of the nose 18-9

control, canard and symmetrical flaperon deflections are

Lateral parameters:

used. The lateral control isprovided bytherudder and

differential flaperon. In addition to theaerodynamic

In maneuvers with pilot inputs, high augmentation

control surfaces, athree-vane thrust vectoring system is

prevented sufficient excitation ofsideslipangle and lateral

mounted around the engine exhaust nozzle. This system

acceleration, especially at angles of attack above 30 °, and

allows thrust deflection up to 150 andis used for

introduced high pairwise correlation between variables p

augmentation ofpitch and yaw control during low speed

andpost-stall flights. A drawing of the aircraft is

and r, 5 and 8, 5 and 8q_. The effect of limited presented in figure 10. excitation and high correlation was reflected in low accuracy of parameter estimates and, in many cases, Maneuvers from eight flights were selected for inability to estimate some parameters at all. In the aerodynamic model structure determination and parameter rolling-moment equation the terms with 5 and _ were estimation. These maneuvers were initiated from trim predominant, explaining about 750 variation in the data conditions at altitudes between 6,000 to 9,000 meters and of the 87% for the complete model. The main influential angles of attack between 10 ° and 70 °. Two types of input terms in the yawing-moment equation were those with were used, the pilot inputs and inputs generated by a either 5 (for low a) or 8q_ in maneuvers at high angles flutter test box installed on one of the aircraft. The pilot of attack. Their contribution represented about 85 to 96% inputs were either pitch command or yaw and roll from the overall value of 98% for the model with all commands in the form of a single doublet or a terms included.

combination of doublets. The flutter test box allowed separate excitation of all aerodynamic control surfaces.

The parameter identifiability was improved, in general, by These inputs were in the form of"3211" multiple signals.

the introduction of a single surface excitation The surface deflection limits in the tests were 7. I ° for the (5 or 5 inputs). The above mentioned correlations were canard, 11.80 for the flaperons and 7.80 for the rudder. To substantially reduced and the amplitude of sideslip angle ensure good excitation of aircraft motion, the longitudinal increased. As a result of that, parameter identifiability in inputs were in some cases preceded by the pilot pitch the rolling-moment equation was improved. However, the doublets and the lateral inputs by pilot roll and yaw problem of accurate parameter estimation in the yawing- doublets. Maneuvers with pilot inputs were flown with moment and lateral-force equations remained, with the thrust vectoring off (ct<30 °) and on. The single surface exception of the thrust-vectoring effectiveness. Four excitation maneuvers were realized only with thrust parameters in the rolling-moment equation vectoring on.

C I = Estimated parameters from flight data were correlated +C. Pb+c. rb +C.- 6 +C.- 6 r (26) with those obtained from wind tunnel data and drop C_°+C'_ 0 'P2V 'r2V ,0, , ,o, model experiment. Static and dynamic wind-tunnel test were conducted at the 30 - by 60 - Foot Tunnel at NASA are presented in figure I I with their 2g-error bounds. In LaRC using a 19 - percent scale model (see [ ! 8] for some this figure the results from wind tunnel and drop model results). For a comparison with flight results presented, tests are also included. The estimates of C_ are the parameters Cep and Ctr were estimated from the consistent with the exception of the region around oscillatory data using the techniques of [19] and [3].

a =25 °. There is significant departure of flight results More about this analysis will be presented in the third from wind tunnel data for the angle of attack between 30 example.

and 45 °. The reason for that has not been explained yet.

Small scatter is also apparent in the estimates of Ct_.

In preparation for X-3 IA flight testing, the unpowered, The flight data demonstrate about 50% reduction in 27-percent dynamically-scaled radio-controlled model of the X-31A was built and tested at NASA LaRC. During aileron effectiveness when compared to wind tunnel the test, the model was attached to a helicopter and lifted values. The estimates of Cfp and C, demonstrate the to an altitude between 1,500 and 3,000 m, and then advantage of single surface excitation over pilot induced released with the helicopter in forward motion. At an maneuvers. In the latter case, high correlation between altitude of about 300 m the flight was terminated by rolling and yawing velocities, and low information deploying a large parachute. The model was controlled by content in the data prevented the estimates of both a pilot on the ground but the model also had the oscillatory parameters. There are no substantial capability to accept preprogrammed surface deflection differences between full-scale flight and wind tunnel commands. This feature was used in generating estimates of these two dynamic parameters. The drop maneuvers for the purpose of parameter estimation.

model parameter values at 30°<a <450 are scattered with During the experiment, only the lateral parameters were values lower than indicated by results from full-scale estimated. They are presented in [20]. In this paper only aircraft test.

estimates of four lateral parameters and two thrust vectoring parameters are presented. More results can be Thrust vectoring effectiveness: found in [21].

The parameter C,,_,, was estimated directly from 18-10 parameters.

maneuvers with the _5 inputs because the adequate model for the pitching moment included only variable 3. The accuracy of parameters was also affected by their a,6 c and 6q_. These three regressors were not low sensitivity and small excitation of response variables, correlated. The maneuvers with 6, inputs provided especially at high angles of attack.

estimates of C-'bc for a<45 ° and C_aqt _ for a>45 °.

4. All the influential parameters were, in general, in From these estimates the thrust vectoring effectiveness agreement with wind tunnel data and results from drop was computed as model tests. Some unexplained differences were, however, observed.

6¢

c=b,_ =(C'.bo-C.b,) 8_

5. The estimated thrust vectoring effectiveness was found (27) or lower than its theoretical values.

. be Cmbqt v = C mbqtv -Cm&e 6. The predictive capabilities of the resulting model determined from flight test data are very good for low amplitude maneuver at low to moderate angles of attack, where the relation 8 / 8q,, was evaluated from measured Some deterioration in prediction was observed at high data and where the values of C=, c were obtained from angles of attack.

maneuvers with pilot input (no thrust vectoring) and8 4.3. F-16XL Aircraft input. The values of the thrust-vectoring effectiveness, C.b _, were estimated from the data with pilot input and For better understanding of aircraft aerodynamics in large single surface excitation. In the first cases, only amplitude maneuvers, NASA LaRC conducted a series of maneuvers for a>35 ° were used because the effect of wind tunnel tests on a model of the F-16XL. A sketch of rudder deflection on the yawing moment was small.

the O. 10 - scale model is shown in figure i 3 together with When necessary the final estimates were corrected for the some basic model dimensions. Tests included effect of the rudder deflection by computing C.6 _ from measurement of aerodynamic forces and moments under static conditions, followed by oscillatory test about all the estimated values of C ".6=, as three body axes and ramp tests in pitch. One of the reasons for the testing was to determine a mathematical 6, (28) model with unsteady aerodynamic terms from oscillatory Cnbrtv = C'n6ttv- 6rt----_ Cnbr data at different angles of attack and frequencies. In this paper only limited results from small amplitude (+ 5°) oscillations in pitch will be shown. More results can be From maneuvers excited by a single surface deflection, found in [3] and upcoming reports.

the parameter C8,_ was estimated directly. Both parameters C=,q_, and C,,_ are plotted against the thrust For the following analysis of the oscillatory data it is coefficient in figure 12 and compared with their assumed that the aerodynamic coefficients are linear theoretical values functions of the angle of attack, pitching velocity and their rates. Then, for example, the increment in the lift coefficient with respect to its mean value can be Cmbqt_ = ----_Cr formulated as (29)

----_C

C"b"_ = b X ACL= (30) CL, Aa+¢C_ _ _ z .

+ _Culq +(_) C_q where xt, is the distance of thrust impact point from airplane center of gravity. So far there has been no then for the harmonic motion explanation for the differences between the theory and the experiment.

ACt_ = it, (C.L= -kZCL¢)sin o t + a^ k(C_,L,+Ci.q)cos _ t From the results presented here and in [22] the following conclusions can be drawn: =a^(Ct,. sin o t + kC--LqCOS to t )

(31)

1. In piloted maneuvers, high correlations between input and output variables were observed. These correlations resulted in low accuracy of estimated parameters and where u A is the amplitude, co the angular frequency and prevented estimation of effectiveness of several control surfaces.

2. Single surface excitation reduced the above mentioned _^kCa_ = a^k(Ca. , + Ct,q) correlations. It was possible to estimate the effectiveness of all controls and increase the accuracy of estimated 18-11 From the experiment, the in-phase and out-of phase

represent the Fourier coefficients. The in-phase and out-

components are usually obtained for different values of of-phase components of Ct(CL and CL.,) can be obtained the angle of attack and reduced frequency while keeping by integrating the time histories of AC L over a selected the amplitude of the oscillations constant. Then the number of cycles. The in-phase and out-of-phase equations (36) and (37) can be generalized as components of C t are plotted against the angle of attack Uji =Ui -aiZuj (38) in figure 14. The figure shows the effect of frequency which is especially strong on the out-of-phase Vii = Vi -ai Zvj (39) components. A comparison of steady and oscillatory data is given in figure 15.

These equation define Model I where for the lift A strong dependence of wind tunnel oscillatory data on coefficient frequency lead to the development of models with unsteady aerodynamic terms, see e.g. [20] and [3]. As an

ui =C_ (ai) vi : CL, l(ai)

example, the model for the lift increment was formulated in [20] as ACL= t t (32)

fct.. (t-_)d a d.r+ _ fc_ (t-x)--d q(_)dx

o V o dr i = 1,2,...,n for where Cry(t) and Ct4(t) are the indicial functions. j = 1,2,...,m For obtaining the model with limited number of In equations (38) and (39) there are, in general, 3n + I parameters, it was assumed that the effect of q(t) on the unknown parameters: u_,v_,_ and x,. They can be lift could be neglected and the indicial functionCt_(t) estimated from experimental data _ and v_ by can be expressed as minimizing the cost function Ct.,t (t) = a (1 _e-b,t) + c (33) m It J,=,._, ,_,{[ _j,-(ut-a, zei)]_ } +[ vit-(v,-a,z_)]2 } Considering the above mentioned assumptions, equation In formulating airplane equations of motion it might be (33) is simplified as more convenient to obtain expressions for u, v, and a as AC L = a function of the angle of attack rather than their discrete values. For that reason the previous model was ), d _ C__ (,o) q(t) C_ (*o) a (t) -a e -b'(t'O. a(_)d*+_ reformulated as Model II defined by the following equations: (34) ujl = u( a i) - a( cti) zuj (40) where CL,(**) and Cl_(O_) are the rates of change of C L with ct and q in steady flow. The steady form of _ji = v(ctl) -a(ai)z_j (41) equation (32) for harmonic changes in ct is identical to that of equation (31), that is The form of expressions for u(cti), v(ai) and a(_ti) can be either specified from the variation of estimated A CL(t) = aA_q., sin _0t + aAk__q.qcos o t (35) parameters in Model I or the form can be postulated in terms of polynomials and/or polynomial splines. In the second case adequate models for u(ct), v(c0 and a(c0 can However, with the indicial function of equation (33), the be determined from measured data by a stepwise expressions for Ct_ and _St4 have the form regression. The previously estimated value of pararneterxj can be used as an a priori value thus simplifying the =CL_(oo)-a _k---_2 (36) parameter estimation procedure using Model I1 and the 1 +x_k 2 cost function m It xt (37) i-1 - _=Cta(**) -a 1 + x_k2

• /

where x, =V/b t I is the nondimensional time constant.

Both estimation procedures were applied to oscillatory 18-12 conditions were available. By repeating the identification data at four frequencies. The data for frequency k = 0.19 using data at different trim conditions, an extended model were used to demonstrate model prediction capabilities.

could be constructed. However, for obtaining a global The parameters Ca._ (_o), Ct_(_o) and a in both models are model from flight data a dedicated experiment would be plotted in figure 16. The agreement between both sets of needed. In this experiment modeling and optimal input results is very good. The parameter Ct.a(®) agree well design would play the major role. One of the possibilities for an extension of traditional aerodynamic model is the with that from static data. Low accuracy in C_(oo) could result from small number of data points and small inclusion of unsteady effects. Some initial work has sensitivity of this parameter. The parameter a indicated already been done mostly using data from dynamic wind smooth variation of unsteady term with the angle of tunnel testing. A strong dependency of oscillatory data on attack and the largest effect of unsteady terms on the frequency led to identified models with time varying terms in the form of indicial functions. Based on these coefficient at ct around 40 °. The estimated value of%" l experiences the extension to flight data will follow.

was 17.2+1.0, which means b_--2.71:t:0.16(sec). This result indicates that the time constant associated with the References unsteady effect is about 0.4 sec. The predicted components from model eq.(40) and (41) for k =0.19are Belsley, D.A.; Kuh E. and Welsh, R.E.: "Regression presented in figure 17 together with the corresponding Diagnostics: Identifying Influential Data and measured values. This figure demonstrates that Model !I Sources of Collinearity", John Wiley & Sons, Inc., is a good predictor. Similar result was obtained for Model 1980.

I.

2. Goman, M. and Khrabrov, A.: "State-Space As follows from this example, a strong dependence of Representation of Aerodynamic Characteristicsofan wind tunnel oscillatory data on frequency led to the Aircraft at High Angles of Attack", Journal of development of model with unsteady aerodynamic terms Aircraft, Vol. 31, No. 5, 1994.

in the form of indicial functions. These functions were postulated as simple exponentials where the unknown 3.

Klein, Vladislav; Murphy, Patrick C.: Curry, parameters included aerodynamic derivatives, the Timothy J. and Brandon, Jay M.: "Analysis of Wind exponent and multiplication term.

Tunnel Longitudinal Static and Oscillatory Data of the F- 16XL Aircraft", NASA/TM-97-206276, 1997.

5. CONCLUDING REMARKS 4.

Tobak, Murray and Shift, Lewis B.: "On the System identification applied to aircraft has proven to be Formulation of the Aerodynamic Characteristics in a powerful tool for aircraft modeling based on Aircraft Dynamics", NASA TR R-456, 1976.

experimental data. In recent years the introduction of highly maneuverable, inherently unstable and highly 5.

Klein, Vladislav and Noderer, Keith D.: "Modeling augmented aircraft has required high sophistication in of Aircraft Unsteady Aerodynamic Characteristics.

aircraft identification methodology. At present this Part l-Postulated Models", NASA TM 109120, methodology includes basic steps: model postulation, 1994.

experiment design, data compatibility analysis, model structure determination and parameter estimation 6.

Morelli, Eugene A.: "Advances in Experiment combined with coil inearity and identifiability diagnostics, Design for High Performance Aircraft", in "System and model validation.

Identification for Integrated Aircraft Development and Flight Testing", in RTA Conference The examples presented demonstrated the need for the Proceedings, Paper No. 8, 5-7, Madrid, Spain, May, introduction of an on-board computer generated input 1998.

signals. This system and the optimal input design procedure improve identifiability of aerodynamic models Morelli, E. A. and Klein, V.: "Accuracy of and accuracy of estimated parameters. The various Aerodynamic Model Parameters Estimated From modifications of the ordinary least square method proved Flight Test Data", Journal Guidance, Control and to be basic tools for model structure determination, Dynamics, Vol. 20, No. I, 1997, pp. 74-80.

parameter estimation and data collinearity assessment.

The resulting parameters can be used as nominal values 8.

Batterson, J. G. and Klein, V.: "Partitioning of in the application of two parameter estimation techniques Flight Test Data for Aerodynamic Modeling of used in aircraft application, the maximum likelihood Aircraft at High Angles of Attack", Journal of method (usually in the form of output error method) and Aircraft, Vol. 26, No. 4, 1987, pp. 334-339.

the extended Kalman filter method. For meaningful information about the identified model, the parameter 9.

Klein, V.: Batterson, J. G. and Murphy, P. C.: estimates should be completed by their covariances thus "Determination of Airplane Model Structure From indicating the accuracy of parameters and their Flight Data by Using Modified Stepwise correlation.

Regression", NASA TP-1916, 1981.

In many applications of system identification methodology, only data from small excursions from trim 18-13 16.

10. Grafton, Sue B.; Chambers, Joseph R. and Coe, Paul

Klein, V.:"Two Biased Estimation Techniques in

L. Jr.: "Wind-Tunnel Free-Flight Investigation of a

Linear Regression. Application to Aircraft", NASA

TM-100570, 1988. Model of a Spin-Resistant Fighter Configuration", NASA TN D-7716, 1974.

II.

Hocking, R. R.; Speed, M D. and Lynn, M. J.: "A 17.

Klein, Vladislav; Cobley, Brent R. and Noderer, Class of Biased Estimators in Linear Regression", Technometrcs, Vol. 18, No. 4, Nov. 1976, pp. 425- Keith D.; "Lateral Aerodynamic Parameters of the 437. X-29 Aircraft Estimated From Flight Data at Moderate to High Angles of Attack", NASA TM- 12. 104155, 1991.

Mackal, D. A.; Picket, M. D.; Shilling, L. J. and Wagner, C. A.: "The NASA Integrated Test Facility 18. Croom, M. A.; Fratelo, D. J.; O'Rourke, M. J. and and its Impact on Flight Research", AIAA Paper No. 88-2095, 1988. Trilling, T. W.: "Dynamic Model Testing of the X- 31A Configuration for High-Angle-of Attack Flight 13. Dynamic Research", AIAA Paper 93-3674-CP, Gera, Joseph: "Dynamic and Controls Flight Testing 1992.

of the X-29A Airplane", NASA TM-86803, 1986.

19.

14. Brandon, J. M. and Nguyen, L.T.: "Experimental Klein, Vladislav and Noderer, Keith D.: "Modeling of Aircraft Unsteady Aerodynamic Characteristics.

Study of Effects of Forebody Geometry on High Part 2-Parameters Estimated From Wind Tunnel Angle of Attack Static and Dynamic Stability", Data", NASA TM- I i 01666 !, 1995.16 AIAA Paper No. 86-0331, 1986.

15.

Murri, Daniel G.; Nguyen, Luat T. and Grafton, Sue 20.

Klein, V. and Noderer, K. D.: "Aerodynamic B.: "Wind-Tunnel Free-Flight Investigation of A Model of a Forward-Swept-Wing Fighter Parameters of the X-31 Drop Model Estimated From Configuration", NASA TP-2230, 1984. Flight Data at High Angles of Attack", AIAA Paper 92-4357-CP, 1992.

21.

Klein, Vladislav and Nguyen, Luong V.: "Aerodynamic and Thrust Vectoring Parameters of the X-3 IA Aircraft Estimated From Flight Data at Moderate to High Angles of Attack", in "High- Angle-of Attack Technology Conference", NASA Langley Research Center, September 17-19, 1996, Hampton, VA.

Table I. XTX matrix in correlation Table 1I. XTX matrix in correlation

form. Pilot input. form. RAV experiment.

IX q_ 6, 6c

a q_ s, & so

2V 2v

I.O00 0.151 0.753 1.000 0.275 0.673 0.711 -0.795 0.328 -0.772 1.000 0.318 1.000 0.345 -0.341 -0.979 -0.375 -0.089 1.000 0.643 -0.928 1.000 0.098 -0.578 1.000 -0.844 1.000 -0.303 1.000 1.000 18-14 Table HI. Collinearity diagnostic. Pilot input.

Condition Variance proportions (scaled regressors) index _C Eigenvalue 2V 3.3102 1 0.00130 0.0092 0.4806 0.0017 0.0006 0.0001 1.2781 3 0.0099 0.0001 0.0397 0.0720 0.0058 0.0001 1.0333 3 0.0130i 0.2572 0. ! 604 0.0000 0.0342 0.0002 0.2613 ! 3 0.0504 0.1140 0.0991 0.0762 0.1887 0.0245 0.0981 34 0.1220 0.6 !94 0.2143 0.0041 0.7123 0.0548 174 0.8176 0.0001 0.0061 0.8460 0.0585 0.9204 0.0190 Table IV. Collinearity diagnostic for RAV experiment.

Condition Variance proportions (scaled reffressors) index Eigenvalue Ct

8f

2V 2.5320 0.0003 0.0017 0.8594 0.0003 0.00(30 0.0018 1.4339 2 0.0905 0.0002 0.0049 0.0008 0.0132 0.1901 1.0028 3 0.0156 0.2543 0.0098 0. ! 481 0.1685 0.2565 0.4745 5 0.2173 0.0355 0.0171 0.0178 0.4886 0.0172 0.3798 7 0.0460 0.6463 0.0220 0.4605 0.0094 0.0506 0.3724 0.4720 0.1771 14 0.6302 0.0621 0.0867 0.7147 Table VI. Average standard errors of estimated parameters.

Table V. Identifiability diagnostic.

Parameter Parameter RAV RAV Pilot input Pilot input LS LS ME AR 2, % It*l AR 2, % It*l 0.029 0.058 0.054 28.0 9.09 64.8 5.06

s(C..)

c..

(0.014) (0.075) (0.020) 1.08 13.7 0.01 0.0 0.017 0.090 0.048

s(C..)

(0.018 (0.037) (0.020) 12.3 C_, 75.74 99.1 91.49 0.0033 0.0059 0.0060

s(C= )

9.2 (0.0011) (0.0026) (0.0026) C=¢ 18.35 46.4 0.94 Note: figures in parentheses are standard errors.

9.2 C_, 4.46 42.7 1.00 Table VII. Collinearity diagnostic.

Condition Variance proportions (scaled and centered regressors)

index rb

Eigenvalue 2V 2V 0.002 0.013 2.387 1.0 0.000 0.005 0.007 0.084 0.0O6 1.849 1.3 0.085 0.000 0.004 0.528 0.022 0.537 4.4 O. t67 0.001 0.001 0.002 0.207 0.209 I 1.4 0.540 0.000 0.079 0.384 0.752 0.017 !40.4 0. 208 0.994 0.910 18-15 Table VIII. Identifiability diagnostic and parameter estimates.

Mixed* Partitioned

Parameter

Stepwise Regression n=6 estimation data n=4 n=5 n=2 n=3 -0.to20 -0.0172 -0.0031 -0.0036 -0.0014 -0.0028 -0.190 -0.208 -0.254 -0.221 -0.223 -0.272 -0.246 C/, (0.0057) (0.0056) (0.OO87) (0.OO64) (O.OO80) (o.oo7a) (0.toY0) 2.3 0.84 0.62 1.4 0.3 C/, (0.050) (0.046) (0.13) (0.12) (0.15) -3.3 -1.2 -1.04 c,, (0.22) (0.095) (0.26) 0.064 0.064 0.08 i 0.106 0.049 C/a,.

(0.0046) (0.OO35) (0.0041) (0.OO55) (0.OO50) 0.015 0.01 0.03 I 0.054 'fa, (O.oolO) (0.0021) (0.0O47) (0.OO53) 0.0035 0.0040 0.0043 0.0040 0.0050 0.0045

s(C0

88.0 89.7 91.7 86.8 87.6 R2[%1 83.4 * a priori value Ct. = 0.010 + 0.0021 Note: figures in parentheses are standard errors Table IX. Standard errors of estimated parameters from partitioned data.

Parameter Standard Error min max Cr J 0.045 0.15 Cr " 0.42 1.1 Cr ' 0.78 2.3 Cr,. 0.026 0.060 Cr, 0.016 0.053 C/_ 0.0076 0.016 Figure 1. Block diagram of aircraft identification.

C/, 0.040 0.11 Ct" 0.063 O. 18 Iflap C/,. 0.0037 0.01 I C/. 0.002 ! 0.0044 C. n 0.0064 0.020 C,, 0.054 0.12 flap C,,, 0.086 0.31 C,,,. 0.0029 0.0070 C.. 0.0013 0.0051 Figure 2. Drawing of X-29A aircraft.

18-16 tO ¢

.% .,.%

OS +If.

oeg qC.

0 E m = -.05 ........ I......... I......... I ......... I......... I ......... I -.TO i :.

/ -

..'- / .. 8 5 • ii. ; _ " olg IV" "_; " "" :: '

' E :: 'v"

"8 c.

de9 "4 1 "'O /::'" O t 2 3 • $ :. SIC .,,,,1, ..... ,,,I,,,,,,,,_1 ...... ,,,I 8 ."

Figure 4 Time histories of measured input variables.

R.AV experiment.

. !/,,\

&f.

ae_ oo $ ,,_ o

• -.\ °

1.5 2 ........ I......... l; ........ I......... h ........ I ......... I o o o o C. l 0.5 8 S •

:: ..'- /

. - ,,,..;.

(,m

-a _-- . X

C • ° • i . a . , . , . i -0.$ -12 -,,,,_',,,,,,,,,,,,,,_ _!r, .h , ,..:,:.....,I,,,,,_::t_ 0 T 2 3 4 5 6 2 (b) [. $OC 1.5

o%

Figure 3. Time histories of measured input variables.

C. I

Pilot input.

0.5 o6 -- -0.5 I I __ I _

.o2[ / k .,.,..._ 2

"I0 i I I I I I I . I, ........ I ......... I ......... I Ca" ,% & • , °, ., • ;. • • : .. _.,

/

" o'.6 ' o:, " ; t:2" ,'.4

./." . 3 " / _ : • oe9

M

: V

v • S- "- " " Figyre 5. Static-stability parametr estimated from : V "_.: flight data with pilot input by using (a) least •7 _........ I, ........I.,,.,,,,l ......... I .........l.........I......... I ......... h. ......I,,,,.,,.I squares, (b) mixed estimation, (c) from RAV 3 v 2 ) a | experiment.

', 14_ 18-17 1.6 1.4 "I ° Riih, I : '" -- Wind tunnclJ -- !_llrlillotteddlla 0 0 o o o o o 0 ImaI amp. mamluver ----1 _'_0 _ _ -- 1.2 o

I I ..es _I- _I

o Cy_ I C% 0.8

.... F-T--i...I--o I

-2 0.6 o 0.5 0.4 Q ; I l I i I l I :. \ 0.3 !.6 Co) Cna 1.4 0.1 • 8 1.2 • e o ° -0.1 C.. I I u 0.8 0,6

, /

I * l I I i I , I

o o_ /e

0.4 Cn t

oI !%o_?

i I_o

1.6 (c) o _ _cg._r...j.___l... 1...........

t.4 ., I I 'I' i I

0 10 20 30 40 50 1.2 ,',, deg C% 1 Figure 8. Comparison of lateral-force and yawing- 0.8 moment parameters estimated from flight and wind- 0.6 tunnel experiment.

, I , * I

%.4

M 0.2

I

Figure 6. Canard effectiveness estimated from flight

J

data with pilot input by using (a) least squares, Cl I!

Co) mixed estimation, (c) from RAV experiment.

-0.2 -0.4 1.5 _._ o, olg_q.l i _ =

L' _-_ _

0 parameter Eslim,,tes Cj P 0.5 -- ,F_._timzted mean

..... .-r ......... i

_o .'_ 'f I Io._,.

I i

.5 - -- - 20 confidence limits

t-,g--_ i i I

-0.5 on the mean "°-,_0 ..

O.Z5 _'P. o .°J""°="'°-x_"., --.5 Cnp • __b ""-o_''"" " _'-._ 0.15 o _.

-I CIl,.

O.OS J -1.5

' ' o ' 50

10 20 3 40

I I T'I I'i

_, deg

-0.05 a., deg Figure 7. Roll-rate parameter estimated from partitioned data.

Figure 9. Comparison of rolling-moment parameters estimated from flight data and wind-tunnel experiment.

18-18 X DIMENSIONS Span 3.7 m Length 13.2m Heillh¢ 4.6 m 0.S .... i .... i .... i .... i .... i .... i .... _ ....

0.6 0.4 C t 0.2 P Figure I0. Drawing of X-31A aircraft.

-0.2 .... i .... , .... i .... i .... t .... i .... i ....

°4o....

a, dog -0.05 • o I .... ! .... i .... ! .... ! .... i .... s .... i ....

-0.1 C 1 -0.15 P

_/_ f_°"

-0.2

0.5 O

q

f -0.25 -0.5 u, i t J i t -0.3 0 10 20 30 40 50 60 70 80 O_, deg 0 .... i .... i .... ! .... i .... i .... i .... i .... "I.5 .... I .... I .... I .... I .... J .... I .... I ....

0 I0 20 30 40 50 60 70 80 o_, deg -0.05 Figure 11. Comparison of rolling-moment parameters estimated from full-scale aircraft and drop model data, -0.I and wind-tunel experiment.

Class -0.15 -0.2 i • " * i ! i i .......

-0.25 ........................

0 I 0 20 30 40 50 60 70 80 ct, deg 18-19 2 F--'_\_,._. " ?1 ---.-- _,.10o -0,5 : \ X.\\_ i I _ k,,._7

2.5

Cm6qlv- I - 1.5 -1: .... i .... i .... i .... i ....

-2 0.2 0.4 0.6 0.8 I ! .2

% 15

o • _._...,...,...,...,..., ...,...

_Lq 10 _ _ i i -0.2

\T_'-, t

"%. ""4_

-o.4

C -o.6

n&rw -5 20 30 40 S0 60 70

Io_ _'"",''' I\

-0.8

cr,deg

-! I --_M_l I \

Figure 14. Variation of in-'phase and out-of-phase .... .....

components with angle of attack for different -1.2 0.2 0,4 0.6 0.8 I 1.2 1.4 1.6 values of reduced frequency.

CT

Figure 12. Comparison of estimated thrust- vectoring

effectiveness and its theoretical values.

1.5

CL 0.5

o ,,/ i

"0.5 , . . t . , . t , . . t , , . t . - . t . - • -20 0 20 40 60 80 1 O0 a,deg Figure 15. Lift coefficient obtained from static and oscillatory data for k=0.190.

I f 1.45 1 51

,jJ

Figure 13. Drawing of F-16XL model.

18-20

c4(-) o

-I -2 I I I : -3 0.6 ............ !.... f..,.! ........

,,i i i ....

0.4

J

cL ' (.)o.2

/, _: i

i// i" 1

I , I J I , I I !

-0.2 !

,,,,I .... I .... ! .... I ....

-0.4 0.5

......... l .... I .... I .... I.... _.... ' .... •

I _ , -0.5 J L _ .j -1

i_i i :./ i

! i " !

(l -1.5

t., i

-2 -2.5 30 400_,deg 50 80 70 -3 : ! i : -3.5 0 10 20 30 _q 50 60 70 80 Figure 17. Measured and predicted in-phase and out- (Z, (1¢g of-phase components of lift coefficient. Model IL k=O. 190.

Figure 16. Estimated parameters of lift components.