Document
3 1176 00156 5234
- NASA Technic a l M e mo ra nd u m so163
NAS A -TM- 8 0163 19800001985
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C O MPARI SO N OF S TABI L ITY AND CO NTRO LP ARAME T ER S
F OR A L I GHT , SINGLE-ENGINE , HIGH -W I NGED A I R CRAFT
US I NGDIFFERENT FL I GHTT E STA N D PA R AMETER
E STI M ATION TECHNI Q UES
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Wi ll i a mT. Su i t a n d Rober t L . C a nnad a y N O _$ O B F " T AIII;NI_gO_I ' _III
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kANGLEY RESEARCH CE ' _TfR LIBRAR Y , IWAS A N ASA _-_AI _/ ]_I::_TOJ' . \. / ]R GIN ,/ ' .
National Aeronautics and Space Administration Langl e y ResearchCenter Hampt o n, Virginia 23665 CO MPARISON OF STABILITY AND C ONTROL P ARAMETERS FOR A LIGHT, S INGLE-ENGINE, HIGH-WINGED AIR C RAFT USING DIFFERENT FLIGHT TEST AND PARAMETER ESTIMATION TECHNIQUE S William T. Suit and Robert L. C annaday SUMMARY Longitudinal and lateral stability and c ontrol parameters were estimated from flight data for a hlgh-wlng, general aviation, airplane using flight data obtained at various flight c onditions within the normal range of the air c raft.
These parameters were estimated using an output error te c hnique (ma x imum likellhood) a nd an equ a tion error te c hnique (linear regression). Longitudinal stati c parameters were also estimated from c limbing, des c ending, and quasi- steady-state flight data. For the lateral ex c itations, four input forms were used involving some combination of rudder and ailerons. The resulting longi- tudinal and lateral parameter estimates were used to c ompute the periods and time-to-damp to one-half amplitude of the various alr c raft modes of motion to determine the sensitivity of these motions to variations in the parameter estimates.
INTRODUCTION The use of simulators for resear c h investigations into air c raft dynami c s is be c oming an in c reasingly important tool of the resear c h engineer. With this in c re a sed use of simulations c omes the dem a nd for greater simulator fidelity, whi c h requires improved mathemati c al models. The aerodynami c s of an air c raft c an be des c ribed mathemati c ally using its stability and c ontrol parameters in a set of equations of motion.
Several te c hniques have been used in the past to estimate stability and c ontrol parameters from flight data in c luding analog mat c hing (refs. I and 2), the time ve c tor method (ref. i), and regression analysis (refs. 3 and 4). In recent years the estimation of parameters from flight data by use of a maximum- likelihood algorithm (ref. 5) has be c ome fairly rou t ine where adequate c omputer fa c ilities are available. In general, the analyses reported in these referen c es were based on the small perturbation equations of motion.
B To date, stability and c ontrol parameters for several low-wlnged light airplanes have been determined from both flight test data and wlnd=tunnel tests (refs. 6, 7 , 8, and 9). For hlgh-wlnged c onfigurations wind tunnel test data are available (ref. I0) but little has been published on the estimated values of stability and c ontrol parameters determined from flight data. Wind tunnel tests usually do not in c lude estimated values for rotary derivatives, whi c h c an be estimated from flight test-data using c urrent estimation techniques.
f The present paper is similar in some respects to the work that was done for the low-wlnged, general aviation aircraft as reported in references 7 and 8, However, in the present study, flight test data were obtained at three different airspeeds corresponding approximately to landing, approach, and cruise " conditions. During the flight test, the aircraft was perturbed from trim conditions using either elevator or rudder and ailerons, and the stability and control parameters were estimated from the resulting data using both an output error method (maximum likelihood of ref. 5), and an equation error method (linear regression of refs. 3 and II). Static longitudinal aerodynamic parameters were also calculated from steady climb, descent, and quasi-steady flight data for comparison with those obtained from perturbation flight data.
This report describes the flight test procedure, presents the results obtained from the perturbation flight tests using the two estimation procedures, and compares values for the longitudinal static parame t ers with values calculated from independent flight tests at two different center-of-gravlty locations.
SYMBOLS The aerodynamic parameters are referenced to a system of body a x es with the orlg_n at the airplane center of gravity, which is loca t ed at 28.8 percent c, and with orientation .of body axes as shown in figure I, which also shows the direction of positive forces, moments, displacements, angles, and linear and angular velocities.
ax, ay, aZ acceleration measured along X, Y, and Z body axes, respectively, g units b wing span, m wing mean geometric chord, m FX, Fy, F Z force along X, Y, and Z body axes, respectively, N g acceleration due to gravity, m / see 2 IX , _, IZ moment of inertia about X, Y, and Z body axes respectively, kg_m 2 IXZ product of inertia, kg-m 2 5t distance from airplane center of gravity to center of pressure of horizontal tail, m MX, My, M Z rolling, pitching, and yawing moments, respectively, N-m m mass, kg p roll rate, tad / see q pit c h rate, rad / se c
dynam ic pre ssur e, N / m 2
r y aw rate, r ad / s e c R est i mate of error c ovarlan c e ma t r ix S wing area, m T thrust, N "- , u, v, w velo c ity along X, Y, and Z body axes, respe c tively, m / see u', v', w' velo c ity c omponent along X, Y, and Z body axes, respe c tively, at angle-of-atta c k sensor on w l ng-tlp boom, m / se e U c ontrol ve c tor V a i rplane total velo c ity, m / se e X i matrl x, o f measured states and i nput variables X, Y, Z body c oordinate axes through a i rplane c enter of grav i ty x(1) state ve c tor
y(1 ) outputve c to r
x, y , z x- , y- , a n d z -c oo rd i n a te s , r e s pec ti v e ly , o f t h e s e ns o rs o n wl n g - tip bo om relat iv e to air pl ane c e n t er o f g ravi t y, m zi meas u remen t ve c t or angle of a t t a c k, ra d 8 angle of sideslip, rad 6a left aileron defle c t i on minus right a i leron defle c tion, rad • 5e s tabilator defle c tion, pos i tive tra i l i ng edge d own, tad 6r rudder defle c tion, po s itive trailing edge lef t , t ad e angle between thrust ax i s and a i rp l ane X bo d y axis, po s i t iv e for thrust up, tad \
8 p i t ch a ngle, t ad
parametervector 9 air density, kg / m 3 measurement noise vector roll angle, tad perturbation in parameter vector CL lift coefficient, gm / qS 0 C rolling-moment coefficient, Mx / qSb C pitching-moment coefficient, My / qS_ m Cn yawing-moment coefficient, My / qS_ CT thrust coefficient, T / qS (used in some publications as T')c CX axlal-forcecoefficient, Fx / qS Cy side-forcecoefficien t, Fy / qS CZ norma-l- l oree coefficien t ,Fz / qS _C CL 5C L 5C C =_ C - CL - rb _8 _ C p 5P b _r 5 2-_ 2V 5C 5Cm 5C _C C - 5 C - C m
c - q_ m = _- i -
_Sa _Sa _Sr _Sr mq _ 2V
_c Bc _c _cn
C - m C _ m C - n. C - m. _ m6e A8 e n pb n pb 5 2V P 5 2V r 5 2V 5C _Cn 5Cn 5CT_ _ n C - C = CT =_-- Cn_ _B nsa _8a nsr _ 8cx 8Cy _Cy CX - _ CX' = CX + CT cos € Cyp = _ Cy - rb
_ _ _ _ 2 v r ._-_
• 5Cy _Cy 5c z _cz Cy8 = _- Cy - CZ =-----= C Z - _ 6r 56r q _2"_
5c z 5c T
C_ =C Z +C T sin_ C - CT - 6_ _ _ _6e _6e Subscripts: c computed k index m measured o coefficient at trimmed conditions t trimmed conditions Superscripts: -i inverse matrix T transposematrix M m e a sured qu a n t i t y o nominal evalua t ion A estimated value A dot over a symbol signifies a derivative with respect to time.
DESCRIPTION OF AIRPL A NE AND DATA SYSTEM The subject airplane was a four-place, externally braced high-wlng, fixed tricycle landing gear, single-englne airplane, as sho w n in figure 2. Its pertinent geometric details and mass characteristics are given in table I. The mass characteristics shown were obtained from manufacturer's data on the subject aircraft. The airplane instrumentation used torecord control-surface movements and airplane responses to these movements was basically like that of the subject aircraft of reference 7.
The instrumentation system measured and recorded on tape the data used in this study. The variables recorded and the range of each sensing instrument is given in table II. The accuracy of these measurements is c onsidered to be 2 or 3 percent of full scale on each instrument. Unlike the data recorded in • reference 7 in which FM and PAM were merged, all the data were recorded on FM (continuous) channels. The advantage of this was that all data channels could : be filtered using an analog filter without introducing time delays in some channels relative to others. These data were digitized, then sampled at 20 points per second and converted to engineering units to obtain the data used in this study. The pitot-static head for measuring velocity (dynamic pressure), and the angle-of-attack and angle-of-sideslip vanes were mounted on a boom located near the left wing tip. The boom extended 3 / 4 chord ahead of the wing leading edge.
FLIGHT TESTS Three types of flight tests were flown to obtain the data used in this report.
(I) Perturbation tests.- These tests consisted of trimming the airplane with power for level flight, idle, or full power and perturbing the trimmed condition with either elevator or rudder and aileron doublets. A typical time history of the input forms used to excite the longitudinal motions is presented in figure 3. The aircraft was trimmed at three indicated airspeeds; 31.6 m / see (61 knots), 40. 7 m / see (78 knots), and 54.2 m / se e (104 knots), whi c h correspond roughly to landing, approach, and cruise, although no flaps were used in these tests.
The lateral perturbation tests consisted of trimming the aircraft at the same three airspeeds as for the longitudinal data, but only trim power for level flight was used. Four different input forms designated A, B, C, a_d D, were used to excite the lateral motions, and typical time histories of the inputs are illustrated in figure 3. All perturbation data were obtained for one center- of-gravity (e.g.) location, 28.8 percent MAC, and test altitudes ranged from about 600 m to 1500 m in relatively smooth air.
The number of test runs made at each condition longitudinally and laterally and for each input is presented in table III.
(2) Steady tests.- These tests consisted of trimming the airplane for a steady full power climb for a particular airspeed followed by setting the power to idle and trimming for an idle power steady descent at the same airspeed and altitude range as the climb. This test was done for the same three airspeeds as above, with no flap deflection, and two e.g. locations 28.8 percent MAC, and 36.5 percent MAC. No analysis for lateral characteristics was performed on these data.
(3) quasi-steady tests.- These tests consisted of performing slow acceleration-deceleration flight test maneuver (ref. 12), starting from trimmed level flight. Power for these tests was left at the trim setting. The initial trin_ned indicated airspeeds were again 31.6 m / sec (61 knots), 40.7 m / sec . (78 knots), and 54.2 m / sec (104 knots). These tests were performed at the same c.g. locations as the steady tests. No flaps were used nor was an analysis of lateral characteristics made for these data.
DATA REDUCTION AND ESTIMATION METHODS The measured flight data included dynamic pressures, angles of at t ack and sideslip, linear accelerations, rotational rates, and control surface movements.
It was necessary to apply corrections to the measured flight data before it could be used for estimating stability and control parameters. Since the angle of attack and sideslip vanes were mounted on a boom located near the left wing tip and extended about 3 / 4 chord ahead of the wing leading edge, corrections for upwash and angular rates were applied to the measured angle-of-attack. Angle- of-sideslip was corrected for angular rates. Details of the angle-of-attack and sideslip corrections are given in reference 7. Airspeed was corrected for position error by applying a correction to the static pressure determined from an airspeed calibration test. The airspeed was also corrected for altitude to obtain trueairspeed, and since the pitot head was located on the boom, angular rates were taken into account to convert airspeed to the aircraft c.g. (ref. 7).
The accelerometer readings were also corrected to the c.g. of the airplane.
Three methods were used to estimate stability and control parameters using the corrected flight test data. These were the ma x imum likelihood technique described in reference 5, a regression parameter estimation technique described in references 3 and II, and an analytical technique described in reference 13.
The maximum likelihood technique utilizes the log-likelihood function T R-I N J_) = - 1 / 2 7 Hi _i - _ l°glRl where
..... _e
_i zi _i Yi Yi_o ) _I 0 =8o with zi the measurement vector and Yi the output vector which comes from x = f(x, U, 8, t) and y = g(x, U, 8, t). In the above equation _i is • assumed to have a Guassian distribution and the representation x = f(x, U, 8, t) is assumed to accurately represent the physical system. The unknowns to be estimated are the elements of 8 and R. Minimizing J with respect to R, A I T R = diag _ 7 Hi Hi is obtained. The estimates for the parameters are obtained l from the equation
8 = 8
which results in yielding the parameter estimates A
e =eo+ ,, e
The regression technique utilizes the cost function Jr(Q ) = Z rl - fri (x' U, (9r i=l where r indicates the rth state equation.
The estimates of the unknown parameters are obtained from the equation _J
_9 -0
which results in where the matrix X. includes measured states and output variables (assumed l noise free).
The analytical technique was used to determine the longitudinal static and control parameters. These were estimated using the measured stick-fixed trim curves at two c.g. positions and the measured C L as a function of angle-of- attack curve. CL_ was determined using the slope of the CL versus _ curve.
Cm was determined from the slopes of the stick-fixed trim curves. Cm6 e was determined using the difference between the stick-fixed trim curves for various CL'S. CZ6 e was then calculated from Cm6 e (see relation in appendix).
The application of these techniques is outlined in figure 4. The maximum likelihood and regression techniques were applied to the perturbation data, and the analytical technique of reference 13 was applied to the steady and quasi-steady data.
One criterion used to evaluate the uncertainty of the parameter estimates obtained using the maximum-likelihood technique is the Cramer-Rao bound _ discussed in reference 14. The Cramer-Rao bounds are an estimate of the standard deviations of the parameter estimates but they are too small except for • the ideal case of a perfect mathematical model, infinite data points, and unbiased random noise in the data. Since this is not the case in a practical situation, the Cramer-Rao bounds can be used in a relative sense to determine ' the estimation accuracy.
The effects of power settings on the estimated parameters were determined by comparing the values of the estimated parameters determined using full power data with those determined using idle °power data.
RESULTS AND DISCUSSION Longitudinal The results of applying the various estimation techniques to the longi- tudinal flight data are shown in figure 5. The longitudinal parameters estimated for each longitudinal run are also given in table IV. All of the parameters except Czq and Cz6 e had Cramer-Rao bounds less than 2 percent of the estimated value (table V). These results indicate that the uncertainty in the estimated values was'small. Where repeat runs were available, the longi- tudinal parameter values estimated by both methods agreed to within i0 percent of each other in the majority of the cases. Also, the fit to the flight data using the maximum likelihood method was considered good since the mean-squared- fit error (area between measured and computed time histories) for each of the states was less than I percent of the full-scale range of the instrument used to measure that state. A typical comparison of measured and predicted flight data time histories is shown in figure 6.
The trends of the estimated parameters with CL were consistent with those approximated from reference i0 where comparisons could be made. The values determined by both the maximum likelihood and the linear-regresslon methods generally showed similar trends. The left half of figure 5 shows the parameters estimated from data which were obtained by perturbing the aircraft from trim powered level flight while the right half shows parameters estimated from data taken when the aircraft was perturbed from idle or full powered flight. The curves fared through the points determined using the maximum likelihood estima- tion method represent the estimates of the derivatives over the CL range for which flight data were available. Derivatives determined by other methods were shown for comparison, but the values determined using maximum likelihood were . considered the most reliable.
Cx was positive and increased linearly with CL. This trend seems .
reasonable since Cx_ was approximately proportional to CL (ref. 15). There was a small power effect at the smallest and mlddle CL values tested.
t Cz_ Cz_ was approximately constant with CL, as was e x pected, since CL varies linearly with _ in the _ range covered by the flight tests (fig. 5).
The power effect observed, idle power giving the least negative value, was greatest at the largest CL values. The same trends can be seen in the results for both the maximum likelihood and the regression extracted methods.
Values of Cz_ for full power, trim power for level flight, and idle power were calculated by the techniques of reference 13 using the quasl-steady and steady measurements and the values of Cz_ are shown on figure 5. The values of Cz_ determined from the steady measurements taken at full power and trim power were about the same, and these values of Cz_ were more negative than those extracted from the'perturbation tests. The Cz_ values calculated by the methods of reference 13 from the steady measurements taken at idle power, were approximately the same as the values estimated from the perturbation test data taken at idle power.
The power effect on the estimated values of Cz_ from steady measurements was about 20 percent for this parameter. The power effect on the values estimated using maximum likelihood was about 12 percent. As C L was decreased, the difference in the estimated values at Cz_ for full and idle power decreased for both the perturbation and steady measurements. However, the values of Cz_ determined from the steady measurements indicated a larger power effect for all CL values tested.
Reference i0 describes wind tunnel tests on a hlgh-winged, slngle-engine airplane. Although the configuration tested in the wind tunnel was not exactly the same as the configuration of the airplane discussed in this report, the two aircraft a re somewhat similar. The data in reference i0 indicates that -C z (CL_ _ -Cz ) becomes greater as power increases. With the throttle in the idle position in flight, thrust was approximately zero, so this case was considered to be comparable with the T_ = 0 ease of reference i0. The thrust coefficient for the full power flight condition was approximately the same as the T'c = 0.26 case of reference i0. The -Cz_ of reference I0 for a T c'= 0.26 is about 14 percent greater than the -Cz_ value for T'c = O.
I0 A similar c hange was noted for the -Cz_ estimated from flight data using the maximum likelihood te c hnique. No reqson has been found for the differences in power effe c ts for the values of Cz_ determined from the perturbation and • steady measurements. However, the power effe c ts seen in the results from the perturbation tests are of the same magnitude as those seen in the wind tunnel test results, so the effe c t of power on Cz_ for the maximum likelihood estimated results were c onsidered reasonable.
C Zq Th e magnitu de s e stimat ed for C Zq a nd its t rend with C L d iff ere d gr e atly f o r th e ma x imum lik e lih o o d a nd th e re gr e ssion e stimatio n m e tho ds. S in ce th e parameter had a Cramer-Rao bound at least three times greater than the bound for any of the other estimated parameters Czq was not c onsidered well determined. Therefore, the estimated value, regardless of its magnitude, did not signifi c antly affe c t the c alculated motions of the air c raft. This c an be seen by examining the period and time to damp to half amplitude of the short period mode (table Vl).
While the a c tual values of all the longitudinal parameters estimated by the maximum likelihood anj linear regression methods are different, both sets give reasonable fits to the flight data. For representative sets of flight data the periods and times to damp to half amplitude for the short period mode are shown as table VI. The quantities in the table were c al c ulated using the sets of parameters determined by examining a parti c ular set of data with both the maximum likelihood and regression methods. The differen c e between the periods and times to half were around I0 per c ent in most c ases so that even though the individual derivatives in the mathemati c al model des c ribing the air c raft were different the resulting motions were similar.
Cz6e The values estima t edfor Cz6e using both t he maximum likelihoodand the regressiontechniqueswere similar even though differentmathematicalmodels were used during t he es t imationprocedure. The maximum likelihoodmathematical model used a cons t raintequationwhich calculatedvalues of Cz6e from the o es t im at edv a lue of Cm6e (see refs. 6 a nd 7 ). There were n o cons t r a in t sin t he regressionmathematicalmodel. For the full and idle power cases the magnitudes of t he estimatedpar a meterswere about the same and therewere no obvious differencesin t he trends wi t h CL between t he two estimationmethods. The trend wi t h CL seemed somewhat differentfor t he two estimation m ethods when the t rim power case was examined,but t he magnitudesof the parame t erswere similar (see fig. 5).
Ii The estimated values of Cm_ be c ame more negative with in c reasing CL (fig. 5). The trends for Cm_ were the same for both extraction methods, but the values determined by the maximum likelihood method were more negative. The results showed a definite power effect on Cm_ for the higher CL values, increasing the power tended to make the values of Cn_ less negative.
The estimated values were compared with values calculated from the steady and quasi-steady measurements using the methods of referen c e 13. As can be seen from figure 5, the values of _ n _ calculated from quasl-steady measurements showed similar trends and magnitudes as the results of the ma x imum likelihood estimation.
Cmq +
The values of Cmq + C r _ for the full power case tended to be more negative as CL increased (fig. 5). Otherwise there was very little variation with CL. Both parameter estimation methods showed the similar trends and magnitudes. The values obtained using the regression method tended to be m ore negative.
A definite power effect was observed, especially at the largest C L.
This was expected since the dynamic pressure ratio at the tail was greater at full power for the largest CL than at full power for the lower CL'S.
Cm6e The values of Cm5 e determined from perturbation data be c ame more negative as C L increased for both the trim and full power cases and for both extraction methods (fig. 5). This increased elevator effectiveness a t higher CL values is due to a higher dynamic pressure ratio at the tail for tri m med and full power than for the lower CL values. The results from the idle power tests showed that Cm6 e remained approximately constant with increasing C L (fig. 5), which would be expected since the propeller slipstream is minimal at idle power. The effect of power is similar to that noted for Cmq + Cn_ , the effect of power being greater at the larger C L.
The values determined using the q uasi-steady measurements and max i mum likelihood estimation showed similar trends with C L. For the trim power case the Cm8 e determined from the quasi-steady measurements showed a much larger variation with increasing CL than the Cm5 e determined from the ma x imum likelihood estimates.
1 2 Lateral The lateral derivatives estimated by the maximum likelihood method ate shown on figure 7. A linear fit to each set of these derivatives is also • shown on figure 7. Values from this linear fit to the derivatives determined by maximum likelihood are the preferred values to be used to mathematically represent the subject aircraft within the flight regimes covered by the tests.
" Other values are presented for comparison with the values determined using maximum likelihood. The lateral derivatives estimated by the linear regression method are shown on figure 8. The individual parameters for each run as estimated by each method are given as tables VIiand VIII. The run numbers are shown on the tables to enable the reader to compare the results of applying each method to the same data. For comparison the linear fit to each of the maximum likelihood derivatives was also shown on the plot of the corresponding derivatives as determined using the linear regression method (see fig. 8). The derivative values determined by both methods in general showed similar trends and in most cases had similar magnitudes. As another comparison of the maximum likelihood and equation error parameter estimates, the characteristics of the dutch roll, roll, and spiral modes were computed based on the two sets of estimates. These computed characteristics are shown in table IX. The charac- teristics estimated are similar for parameters obtained from both techniques.
The most obvious discrepancy is in the description of the spiral mode, but these differences are not considered important since the data runs were not long enough to accurately describe the spiral mode.
Where possible the trends of the parameter values determined using maximum likelihood were also compared with trends obtained from references I0 and 16. Also, the estimated parameter values will be compared with values taken from reference 17. These are given in table X. The derivative values shown are for a high-winged, single-engine, general aviation aircraft, but only the values from reference i? are for the specific configuration of this report.
The Cramer-Rao bounds of the lateral derivatives estimated were examined and CyB, C_, C_p, C_6a, CnB , Cnr , and Cn6 r were found to have bounds that were less than 2 percent of the extracted values (table XI). An examination of figures 7 and 8 reveals that the derivatives listed above had the least scatter, as would be expected since the Cramer-Rao bounds were small. In most cases, the derivatives Cyr, Cy6r , C_r , and Cnp had Cramer-Rao bounds of less than 5 percent of the estimated values, the derivatives C_6r, and On6 a had a Cramer-Rao bounds of less than I0 percent of the estimated values, and Cyp had a Cramer-Rao bound which varied considerably from run to run. The deriva- tives with Cramer-Rao bounds of 5 percent or less of the estimated value were considered well determined. Also, the fit to the flight data was considered good since the mean squared fit error for each state was less than I percent of the full scale range of the instrument used to measure that state. A typical fit to the lateral motions is shown in figure 9.
No apparent effect of control input was seen in the values obtained using the maximum likelihood estimation method. However, when the linear regression estimation method was used, several of the parameters showed some effect of input form (see fig. I0). With the exception of Cy_ and C_r none of the parameters that were considered well determined showed an effect of input. The values estimated for Cy B from the data generated when input B was used to perturb the air c raft were c onsistently lessnegative thanwhen the data generated by the other inputs were used. Some of the parameters that were not as well determined also had apparent input effects for some CL'S , but the scatter in the values estimated by the linear regression method made a definite c onclusion difficult.
The reasons for input effects occurring when the linear regression method was used and for their absence when using the maximum likelihood method was used is unclear. One clue may be in the fact that the cost fun c tions are defined differently for the two methods. In the linear regression method, the unknown parameters are estimated for each state equation independently of the other state equations. So, when using linear regression, since the individual inputs excite each of the states in a different manner, several parameter values could be estimated to describe the same state. For the maximum likeli- hood method all the states to be fitted are estimates simultaneously and the unknown parameter values are determined to give a best overall fit to all the states simultaneously.
The parameters Cysr , C_, C_p, C_sa, CnB, and Cnp showed good agreement between the estimation methods (figs. 7 and 8). These parameters had trends with CL similar to those seen in references 16 and 17 where comparisons could be made. With the exception of C%_ and Cn_ , the estimated values agreed with the values shown in table XI from other references. The lateral parameters which had differences between the values determined by maximum likelihood and the values determined by the other methods will now be discussed individually.
CY B Cy_ was negative and became less negative as CL increased. The maximum likelihood results showed no effect of control input and the scatter was small.
The results using linear regression were less negative than the maximum likeli- hood results and the scatter was greater. The general trend for the Cy B values extracted by the linear regression method was the same as for the maximum likelihood method, but the linear regression showed some effect of input for input B.
C yp Cyp was not determined well by either estimation method as was Indloated • by the scatter of the estimated values. When using the linear regression program, particularly large run to run variatlons were noted for the runs with input B.
&
C_ r
The v a lues es t ima t edfor C_r were positive and increasedwith increasing CL, This trend was similar t o t he t rends shown in reference 1 7 , bu t t he ma gni- t udes were lower than expected. The values determinedby t he linear regression showed t he same t rend as the maximum likelihoodbut t he values es t ima t edfor t he parame t erswere larger. For both es t im at ionmethods t he v a lues de t er m ined using the data for input A were generally lower a t the larger CL'S than t hose determinedusing the d at a from the o t her inpu t s.
C_6r The values estimated for C_6 r were positive and increased with increasing CL. Both the maximum likelihood and linear regression estimation methods generally resulted in similar trends and magnitudes for the estimated parameter.
The exception was the values estimated from the data obtained from input A using the regression method. These values tended to show a decrease in C_6 r with increasing CL.
The general trends seen in the estimated values of C_6 r are opposite those indicated in reference 16 for a stralght-wlnged aircraft. Also, the values estimated were smaller than the value determined from reference 17 and shown in table X. However, since C_6 r is not a strong parameter and is not well determined, the apparent discrepancy did not significantly affect the fit to the data or the other parameter values estimated.
Cn r The Cnr values which were estimated by both methods were negative and in general the trend of the derivative values was more negative as CL . increased. While both extraction methods gave similar overall trends and scatter for the estimated derivatives some effect of input was seen in the regression results for input D which showed a reverse trend with CL and for input A which tended to have values less negative than for the other inputs.
4 The scatter in the data hid any possible input effec t in t he maximum likelihood results. When compared to the values given in table X, the magnitudes of C . nr estimated seemed reasonable and reference 16 implies that the trend with C L is reasonable.
Cnsa The values determined for Cn8 a were generally negative, indicating an apparent proverse yaw with aileron deflection for the sign convention used in this report. Experience with the subject aircraft has demonstrated that this aircraft actually has adverse yaw with ailerons. This apparent c ontradiction can be explained in part when it is realized that the inputs required to determine Cn8 a also produce considerable rolling motion. This rolling motion induces a yawing moment described by the parameter Cnp. Thus, the same aileron inputs which cause a yawing moment through Cnsa, also produce a rolling motion which causes a yawing moment through Cnp. Therefore, the effects described by these two parameters is difficult to separate, as is indicated by the high correlation (.95) between this pair of parameters. This implies that both parameters are required to properly describe the adverse yawing motion observed for the subject aircraft, and not Cn8 a or Cnp alone.
CONCLUDING REMARKS The maximum likelihood and a linear regression parameter extraction program were used to determine the longitudinal stability and con t rol parameters from data taken using a high-wlnged, general aviation aircraf t .
The parameter values obtained using the maximum likelihood method were considered the most reliable representation of the subject aircraft, while the values determined by other methods are presented for comparison.
The longitudinal parameters estimated using maximum likelihood showed the trends expected with variations of power and C L and were considered to give a reasonable mathematical representation of the aircraft. For a majority of the parameters the same trends were apparent using the linear regression method. A definite effect of power setting was observed in the derivatives CZSe, Cm_ , Cmq + % , and Cmse; with some power effects observed in Cz_ at the largest CL. The values estimated for Cz_ , Czq , Cz6e, Cm_ , and Cmq + Cm_ were noticeably different when different parameter estimation methods were used. The parameter Cm8 e showed a lesser variation with estimation method. Also, the derivatives Cx_, CZse, Qm_, Cmq + % , and Cm8 e varied with CL?
The parameters Cz_, _, and C m 6e were compared with values deter m ined using steady and quasi-steadytest data. The t rends with CL were the same for these derivativesregardlessof the method of determination , but the - • magnitudeswere different. The greatestdifferenceswere observed in Cz_ where the power effects predictedby the steady and quasl-steadytes t s were differentfrom those predic t edby perturb at ion t ests• b The lateralpar a metersestimatedby both t he maximum likelihoodand equa t ion error methods generallyshowed t he same trends with CL. For G_8, C_p, C_6a, CnB , Cnp , Cnr , and Cn6 r t he values es t imated using the two techniques agreed very well. The trends of the parameters estimated were as expected and the magnitudes determined were reasonable. As with the longi- tudinal results, the parameters estimated using maximum likelihood were considered the parameters to be used in any mathematical representation of the • subject aircraft• The parameters Cnp and Cn8 a were correlated indicating that a description of the adverse yaw due to aileron input of this aircraft required a combination of these parameters• The estimated parameter values obtained using the maximum likelihood extraction method resulted in fit errors less than the uncertainty in the measurements for both the longitudinal and lateral data. The es t imated longi- tudinal parame t ers all had Cramer-Rao lower bounds that were less than 2 percent of their value, as did the lateral parameters Cys, C_, C_p, C_6a, • C np Cn8 , Cnr , and Cn6 r The lateral parameters Cyr, Cy6r , C£r , and had Cramer-Rao lower bounds that were less than 5 per c ent of the estimated parameter value, while C_6 a and Cn6 a had bounds less than I0 per c ent of the value.
Cyp had a bound of appro x imately 25 percent of the e x tra c ted val u e.
The agreement of the two estimation methods, the reasonable trends and values of the extra c ted derivatives , the good fit to the data, and the low values of the Cramer-Rao lower bounds gave confiden c e that for the flight data e x am i ned, the mathemat ic al model estimated using the maximum likelihood method was a reasonable representation of the subje c t air c raft.
REFERENCES I. Wolowicz, Chester H.: Considerations in the Determination of Stabill t y and Control Derivativesand Dynamic Characteristics From Flight Data. AGARD Rep. 549-Part I, 1966.
2. Rampy, John M.; and Berry, Donald T.: Determination of S t abili t y Derlva- tives From Flight Test Data by Means of High Speed RepetitiV e Operation Analog Matching. FTC,TDR-64-8, Air Force Flight Test Center, May 1964.
3. Howard, J.: The Determin a tion of Lateral Stabillty and Control Derivatives From Flight Data. Canadian Aeronautics and Space Journal, vol. 13, March 1967, pp. 127-134.
4. Shinbrot, Marvin: On the Analysis of Linear and Nonlinear Dynamleal Systems From Translent-Resp0nse Data. NACA TN 32881 19 5 4. ......
5. Grove, Randall D.; Bowles, Roland L.; and Mayhew, Stanley C.: A Procedure for Estimating Stability and Control Parameters From Flight Test Data by Using Maximum Likelihood Methods Employing a Real-Time Digital System.
NASA TN D-6735, 1972.
6. Suit, William T.: Aerodynamic Parameters of the Navion Airplane Extracted From Flight Data_ ,NASA TN D-6643, March 1972.
7. Cannaday, Robert L.; and Suit, William T.: Effects of Control Inputs on the Estimation of Stability and Control Parameters of a Light Airplane.
NASA Technical Paper 1043, December 1977.
8. Klein, Vladislav: Determination of Stability and Control Parameters of a Light Airplane From Flight Data Using TwoEstimation Methods. NASA TP 1506, 1979.
9. Fink, Marvin P.; Freeman, Delma C., Jr.; Greer, H. Douglas: Full-Scale Wind-Tunnel Investigatio n of the Static Longitudinal and Lateral Characteristics of a Light Single-Engine Airplane. NASA TN D-5700, 1970.
I0. Greer, H. Douglas; Shivers, James P.; and Fink, Marvin P.: Wind-Tunnel Investigation of Static Longitudinal and Latera! Characteristics of a Full-Scale Mockup of a Light Single-Englne High-Wing Airplane. NASA TN D-7149, 1973.
ii. Gerlach, O. H.: Determination of Performance, Stability and Control Characteristics From Measurement in Non-Steady Maneuvers. AGARD Specialists Meeting on Stability and Control, September 20=23, 1966, pp. 499-523.
12. Langdon, S. D.: Fi x ed-Wing Stability and Control Theory and Flight Test Techniques. USNTPS-FTM-No. 103, AD703681, August i, 1969.
13. Klein, V.: Determination of Longitudinal Aerodynamic Derivatives From Steady-State Measurement of an Aircraft. Proceedings of the AIAA 4th Atmospheric Flight Mechanics Conference, Hollywood, Florida, August 8-I0, 1977.
14. lliff, Kenneth W. ; and Maine, Richard E.:_ Practical Aspects of Using a Maximum Likelihood Estimation Method to Extract Stability and Control Derivatives From Flight Data. NASA TN D-8209, 1976.
15. Etkin, Bernard: Dynamics of Flight. John Wiley & Sons, Inc., c. 1959.
16. Roskam, Jan: Flight Dynamics of Rigid and Elastlc Airplane. Roskam Aviator and Engineering Corp., c. 1972.
17. Smetana, Frederick 0.; Summey, Delbert C.; and Johnson, W. Donald: Riding and Handling Qualities of Light Aircraft--A Review and Analysis. NASA CR-1975, 1972.
APPENDIX EQUATIONS OF MOTION The equations used in this program are perturbation equations f rom trimmed level flight and are written relative to the set of body axes shown in " figure i.
The equations used to describe the longitudinal motions were
I v2s [ c c_(_ _t )] (At)
=-qw+rv- g sine+y _-7-[x,0 +
V2S _C = -pv + qu + g cos _ cos _0+ _ P q 2V
1 -_-_z, o. c{=(= -=t) . Czi
+ CZ (6e _ 6e ,t) _ (A 2 ) 6e
(Iz- Ix) Ixz 2 v Z s_- + Cm(_ at )
= pr Iy + _--- (r - p2) + P _--_ _m,o + Cm. 2"V + Cm 2V Cm6e(6e e,t q = q cos •- r sin _ (A4)
1 . (AS)
aX = g(U + qw - rv + g sin 8) I . (A6) az = g(W+ pv - qu - g cos 0 cos _) V 2+v2+w
=_u z (AT)
-i w (A 8) o' = tan u • 3 (A9) u -_ (used in maximum likelihoodextra c tion) (AI0) GZ6e - _t Cm6e 2 0 The values of the lateral states v, p, r, and _0 used in the longitudinal equations were the flight-measured quantities.
Since thrust changes are not explicitly modeled in the equations of motlon_ are not necessarily pure CX_ and but may contain small • C_ and C_ __ CZ_ contributions due to changes in thrust. Therefore , C_ and C_ , as determined in this study, are given by _CX ! _ - CX _-- + CT cos e _CZ ' - + CT sin e Cz _ Since, in this study, thrust was held constant and the angle-of-attack changes ' and were no more than 7° peak to peak, the contributions of thrust to CX ' were considered minimal.
CZ The equations used to compute the lateral motions were I V2S_ C + = -ru + pw + g cos 0 sin q0+ _ 0 m _Y,o Cy_ + Cyp _ + CYst(6 r - 6r,t) _ (All) - _X r+ qr+ pq+ p +
xz vrc
_Ix ) _ Ix _,o c _B rb + z v + (AI2) C_p C_r _ + C_6r(6 r - 5r,t) + C_6a(5 a - 5a, t r- I Z P q_ IZ Pq "t_jqr +_ P'_Z-Z tln,o + CnB _ • Cn p Cn r r + 2 _ V+ _ + C nsr(6 r t ) + C n 6 a (8 a 5a,t
rb - 5 - 0 ( AI 3 )
= p + (q sin _ + r cos _ ) tan e ( AI 4 )
• ay = _(v + ru - pw - g cos 0 sin q0) (AI 5 ) 2 1
v =_u 2 + v2 + w2
-iv 8 = sin V The values of longi t udinals t ates u, w, q, and 8 used in the la t eral equa- tions were the fligh t -measured quan t lties. The equationswere used to compu t e the airplane s t a t e responses. The compu t edrespo,seswere t hen comparedwi t h t he recorded responsesfr o m t he fligh t t es t s and the differenceswere used t o update the Parame t ers(stabili t yand con t rolderiv at ives)to improve the fi t .
The longitudinal measured and c o mpu t ed responses,or s t a t es,used in the algori t hm for t his st u dy were u, w, q, 8, ax, and aZ. The lateral sta t es used were v, p, r, _, and ay. Discussionof the iden t ification algorithm is given in reference 7.
2 2 TABLE I.- GEOMETRIC C HARACTERISTICS • Mass kg 837 93 Inertia: IX kg-mp2 1395 Iy, kg-m 2 .......................... l k 80.
I 7. , kg-m 2 ........................... 2 563.
IXZ, kg-m ................ ." .......... . 1 2 3.
Fuselage length, m ........................ 8. 2 Wing: 2 Area, m ........................... 16.2 Aspect ratio ......................... 7.47 Span, m ............................ ii.0 Mean geometric chord, m .................... 1.49 Vertical tail: Area, m2............................ 1.0h Aspect ratio ......................... 3.96 Span, m ............................ 2.03 m 2 " 68 Rudder area, ...................
Horizontal2Tail: Area, m ........................... 3.35 Aspect ratio ......................... 3.44 Span, m ............................ 3.45 Tail length, m ........................ 4.36 a, 8, V Boom location relative to c.g.: D x, m ............................ 1.h7
Z,m ............................. • -5.43
z,m ............................ - .77
TABLE II.- INSTRUMENT R A NGES Instrument Range .
Airspeed, m / sec 0 to 63.0 m Angle of attack, deg -8.0 to 3 9 .0 Angle of sideslip, deg _ +__23.0 Altitude, m 0 to Normal acceleration, g units -.5 to 4,0 Longitudinal acceleration, g units +_l.0 Lateral acceleration, g units +l.0 Elevator position, deg + 2 5.0 to - 2 9.0 Aileron position, deg +16.0 to -20.0 Rudder position, d6g- +19.0 Throttle position Total Throttle travel Pitch rate, deg / sec +__30.0 Roll rate, deg / sec +_30.0 Yaw rate, deg / sec +_30.0 Pitch attitude, deg +__30.0 Roll attitude, deg +--60.0 TABLE III.- FLIGHT TEST CONDITIONS L01GITUDINAL Trimmed . _Airspeed Power _ 31.6 m / sec h0.7 m / see 54.2 m / see Setting e IDLE 1 run 2 runs i r u n LEVEL FLIGHT 2 runs 2 runs ° 2 runs FULL 2 runs 2 runs same ru n s as Level Flight LATERAL Trimmed _Airspeed Power _ 31.6 m / see 31.6 m / see 31.6 m / see 31.6 m / see Setting LEVEL FLIGHT Input A Input B Input C Input D 3 Runs 2 Runs 3 Runs i Run _Trimmed _irspeed 40.7 m / see 40.7 m / see 40.7 m / see 40.7 m / see Power Setting LEVEL FLIGHT Input A Input B Input C Input D i Run 3 Runs 2 Runs i Run Trimmed _irspeed Power _ 54.2 m / see 54.2 m / see 54.2 m / see 54.2 m / see Setting LEVEL FLIGHT Input A Input B Input C Input D 2 Runs 3 Runs 3 Runs i Run TABLE IV.- LONGITUDINAL PARA_TER VALUE MAXIMUM LIKELIHOODESTIMATIONMETHOD (CRAMER-RA0 BOUNDS IN PARENTHESIS)
AIRSPEED 31.6m / sec
Power Setting IDLE TRIM TRIM FULL FULL Parameter C 1.68 1.40 1.41 1.44 1.81 x (.0620) (.0293) (.01831 (.0223) (.0247) C -1.076 -.951 -.980 -i.00 -.972 z
o (.0024) (.0017) (.0030 (.0026) (.0026)
Cz -4.28 -4.74 -4.92 -4.76 - 4.95
(.i01) (.0622) (.05771(.0841) (.0 6 70)
c -9.77 -9.78 -10.29-13.09 -12.62
z
q (. 9 6 9 ) (.754) (.714)(i , i03)(. 9 05)
C -.355 -.425 -.466 -.512 i-.496 cZ6e -1.22 -1.13 -1.06 -.862 -.947 m (.0206) (.0110) (.0095) (.0132) (.0105) C . - 4 . 0" - 4 . 0" -4.0* - 4 .0" -4.0* m C e -11.82 -14.55 -16.71 -19.67 -18.41 m q (.298) (.188) (.214) (.272) (.218) C -1.04 i-1.25 -1.37 -1.50 -1.45 m6e (.0113) (.0071) (.0086 (.0102) (.0083) *Parameter Fixed LINEAR REGRESSION ESTIMATION METHOD AIRSPEED 31.6 m / sec-- Power Setting IDLE TRIM TRIM FULL "FULL Parameter C 1.58 1.46 1.50 1.60 1.56 X C a -.954 -.866 -.888 -.89 -.87 z o C -4.23 -4.74 -.477 -4.88 -4.85 z C _ -16.0 _13.71 -12.50 -18.98. -16.49 z q C -.43 -.39 -.34 -.56 -.44 Z qe C -.963 -.927 -.916 -.744 -.815 m C + C -17.36 -19.32 -19.20 -22.73 -21.50 m m q C -1.02 -1.21 -.121 -1.39 -1.34 m6e TABLE IV CONTINUED M A XIMUM LIKELIHOOD ESTIMATION METHOD (CRAMER-RA0 BOUNDS IN PARENTHESIS) AIRSPEED 40.7 m / sec.- • Power Setting IDLE IDLE TRIM TRIM FULL FULL , : Para m eter
C 1.22 1.01 .836 .867 .937 .884
x c (.0442) (.0306) (_01 6 91 (.0208) (.0174 (.0189) C -.641 -.665 -.645 -.612 -.639 -.614 z
o (.0026) (.0039) (.0025 (.0022) (.0025 (.0029)
C -4.64 -4.56 -5.26 -5.07 -4.92 -4.98 Z (.0838) (.0947) (.0472 (.0520) (.0636) (.0722) C -io.17 -io.76 -8.95 -9.09 -ii.28 -io.51 z q (1.091) (1.149) (.585) (.642) (.855) (.929) C -.364 -.36 9 -.422 -.4!3 -.440 -.440 cZ_e -.877 -.879 -.853 -.886 -.855 -.825 m (.0114) (.0155) (.0086_ (.0078) (.0098 (.0118) C -4.0* -4.0* -4.0* -4.0" -4.0* -4.0* m.
C -13.00 -13.51 -14.94 -14.75 -15.92 -15.87 m q (.248) (.319) (.245) (.171) (.228) (.296) C -1.07 -1.08 -1.24 ]-1.21 -1.2 9 -1.2 9 m6e (.0090) (.0117) (.0103_ (.0062) (.0087) (.0121) *Parameter Fixed LINEAR REGRESSION ESTIMATION METHOD
AIRSPEED 40.7m / sec
Powe r Setting IDLE TRIM TRIM FULL FULL Paramete r C .98 .85 .99 .94 .90 X C -.586 -.573 -.56 -.55 -.57 z • C -4.72 -4.77 -4.72 -4.74 -.485 z C -16.8 -14 . 18 -16.27 -17 . ll -15.39 . Z q
C -. 36 -. 37 -. 42 -. 46 -. 33
Z6e
C -. 66 -. 64 -. 67 -. 63 -. 64
m
C a + C -17.76 -19.o -19.5 -19.76 -20.14
mq m_ C -i . o4 -i.i6 -i . 17 I-I . 21 I-I.23 m6e i TABLE IV CONCLUDED MAXIMUM LIKELIHOODESTIMATIONMETHOD (CRAMER-RAO BOUNDS IN P A RENTHESIS) AIRSPEED 54.2 m / sec Power Setting IDLE FULL = FULL =
TRIM TRIM
Parameter C .652 .52 .56 " x a (.0244) (.0151) (.0221)
C -.339 -.37 -.35
Zo (.0024) (.0027) (.0025)
C -5.02 -5.18 -5.27
ze (.0612)i (.0672) (.0553)
c -8.81 -8.65 -8.48
Zq (.705) (.968) (.755) c -.335 -.405 -.398 Z_e C -.722 -.68 -.676 ms (.0130) (.0101) (.0101) C -4.0* -4.0* -4.0* m.
e C -12.30 -15.34 -15.35 mq (.301) (.337) (.318) C -.98 -1.19 -1.17 m_e (.0112) (.0134) (.0127) *Parameter Fixed LINEAR REGRESSION ESTIMATION METHOD AIRSPEED = 54.2 m / sec Power Setting IDLE FULL= FULL= TRIM TRIM Parameter C .62 .54 .54 x e
C -.357 -.339 -.336
z o
C -. 4. 4 9 -4.64 -4.62
z e
C -19.54 -18.87 -1 9 .56
z q
¢ -.38 -.45 -.46 "
Z6e C -.50 -.482 -.485 m e Cmq + Cm & 17.89 -19.86 -19.44 c -.985 1.13 -i.ii m_e | . • • • TABLE V.- THE PERCENT THE CRAMER-RAO BOUND IS OF THE ESTI_iTED LONGITUDINAL PARAMETER Power Cx Cz Cz Cz Cm Cm Cm6e Setting Airspeed _ o _ q _ q TRIM 31.6 m / sec 3.690 0. 2 23 2.360 9.918 1.689 2 .521 1.087 TRIM 31.6 m / sec 2.093 0.179 1.31 2 7.710 0.932 1. 2 9 2 0.568 TRIM 40.7 m / sec 1.298 0.306 1.173 6.939 D.896 1. 2 81 0.628 TRIM 40.7 m / sec 1.549 0. 2 60 1.767 8.4 2 6 1.531 1.383 0.680 TRIM 54. 2 m / sec 1.365 0. 2 67 1.354 7.171 1.109 1.184 0.641 TRIM 54. 2 m / sec 3.623 0.406 1.806 10.7 2 8 1.300 1.908 0.841 FULL 31.6 m / sec 3.030 0.586 2.077 10.678 1.7 6 3 2.347 1.083 FULL 31.6 m / sec 2.022 0.388 0.8 9 7 6.536 1.008 1.640 0.831 IDLE 31.6 m / sec 2.399 0.359 1.026 7.063 0.880 1.159 0.512 FULL 40.7 m / sec 1.857 0.391 1.293 7.580 1.i46 1.432 0.674 FULL 40.7 m / sec 2 .138 0.47 2 1.450 8.839 1.430 1.865 0.938 IDLE 40.7 m / sec 3 . 742 0.708 1. 2 19 8.002 1.801 2.447 l.lh3 IDLE 40'7 m / sec 2 .903 0.730 1. 2 97 ll.191 1.485 2 .197 1.126 IDLE 54.2 m / sec 3.946 0.714 1.049 8.903 1.494 2 .072 1.085 a o TABLE VI.- PERIODS AND TIMES TO DAMP TO 1 / 2 AMPLITUDE 31.6 m / sec 31.6 m / sec 40.7 m / sec 40.7 m / sec 54.2 m / sec 54.2 m / sec 31.6 m / sec Flight T r im Trim Tri m T r im Full Full Full condition power power power power power power power Extraction method M.L. Regression M.L. Regression M.L. Regression M.L.
Period 1.80 sec 1.82 sec 1.58 sec 1.64 sec 1.26 sec 1.33 sec 1.94 sec Time to damp to 1 / 2 amplitude .2 9 5 sec .268 sec .235 sec .217 sec .17h sec .160 sec .262 sec r 31.6 m / sec 31.6 m / sec 31.6 m / sec 54.2 m / sec 5h.2 m / sec Flight condition Full powe r Idle powe r Idle powe r Idle power Idle powe r Extraction method Regression M.L. Regression M.L. Regression Period 1.90 sec 1.94 sec 1.91 sec 1.29 sec 1.36 sec Time to damp to 1 / 2 amplitude .241 sec .372 sec .271 sec .197 sec .173 sec | • • • • TABLE VII.- LATERAL PARAMETER VALUES DETERMINED USING MAXIMUM LIKELIHOOD ESTIMATION (CRAMER-RAO BOUNDS IN PARENTHESIS) Case 20 21 22 23 24 25 26 27 28 29 30 31 Input A A A A A A B B B B B B CL 1.09 .975 1.102 .633 .388 .363 1.03 l.lO .67 .576 .562 .342 C -.546 -.550 -.542 -.589 -.595 -.563 -.563 -.536 -.580 -.58 -.57 -.59 Yfl (.0036) (.0045) (.0030) (.0049) (.0042) (.0056) (.0060) (.0034) (.0052) (.0086) (.00 3 7) C .058 .060 .027 -.089 -.041 -.'081 .i07 .140 .070 .053 .042 .054 Yp (.0152) (.0208) (.0155) (.0119) (.0105) (.0172) (.0148) (.0108) (.0143) (.0288) (.0107) C .150 .148 .160 .147 .130 .096 .152 .185 .166 .127 .145 .133 Y_r (.004 6 ) (.0071) (.003 9 ) (.007 6 ) (.0052) (.004 6 ) (.0045) (.0054) (.0047) (.00 6 2) i(.0042) (.oo055) (.00054) (.000 6 4) (.oo13) (.ooo 9 o](.ooo 6 7) (ooo71) (.00047) ( . 00o 6 7) ( . ooo8 9 (.ooo55) 0_8 -.057 -.0 59 -.058 -.070 -.071 -.073 -.0 6 1 -.0 6 2 -.074 -.0 6 7 -.0 6 1 -.078 C£ -.405 -.400 -.438 -.430 -.404 -.44 -.44 -.439 -.484 -.440 -.391 -.477 p (.0041) (.003 6 ) (.0052) (.0058(.004 6 )(.0041) (.003 9 ) (.0033) (.0046) (.004 9 ) (.0041)
C£ .0787 .i00 .128 .0 9 0 .047 .07 9 .132 .137 .108 .i01 .125 .078
r (.0021) (.0023) (.0023) (.0045)(.0029)(.0022) (.0025) (.0020) (.0022) (.0027) (.0019)
C£6 r .00176 .0051 .0082 .0101 .0O30 .0055 .0078 .012 .0077 .0O58 .0110 .0079
(.00041) (.00045)(.00045) (.0013) (.000 6 01(.00041) (.0039)(.00049) (.00041) (.00044)(.00043)
0£ 8 a -.0742 -.081 -.08 9 -.0 96 -.0 9 5 -.098 -.lOl -.i00 -.114 -.i04 -.087 -.114
(.00067) (.00065) (.00095) (.0013) (.00099 (.00084) (.00o77) (.o0069) (.00086) (.00085[(.00088)
C .038 .038 .035 .039 .044 .042 .033 .038 .04 6 .042 .031 .051
n8 (.00042) (.00039) (.00045) (.00034 (.00035)(.00033) (.00037) (.00020) (.00034) (.00052)(.00022) -.139 -.104 -.124 -.104 -.062 -.071 -.108 -.117 -.058 -.078 -.144 -.031 Cnp (.0029) (.0030) (.0040) (.0015) I(.0016)i(.0021) (.0020) (.0016) (.o020) (.0028) (.0018) Cnr -.121 -.117 -.109 -.i01 7.'098 i[.095 -.144 -.124 -.i00 -.113 -.132 -.105 (.0014) (.0015) (.0016) [.0013 ) <.0010) (.0012) (.0013) (.00078) (.0010) (.0019) (.00082) Cn_r -.0 6 3 -.0 6 1 -.0 6 2 -.056 -.054 -.051 -.0 66 -.061 -.057 -.059 -.060 -.057 (.00034) (.00037)(.00032) (.00037)(.00032)(.00035) (.0003 6 ) (.00028) (.00030) (.00043](.00024) C -.011 -.0082 -.012 -.012 -.0061 -.0068 -.0015 -.0084 -.0011 -.0029 I-.0134 .0019 n_a (.ooo59)i(.ooo58) (ooo78) (.00033](.00037)(.00048) (.00044) (.00038) (.000_2) (.00042 (.00041)
TABLE VII.- CONTINUED
Case 32 33 34 35 36 37 38 3 9 40 41 42 43 44
input B B C C C C C C C C C C C
!C L .320 .325 .326 .322 .326 .385 .376 .5 6 8 .662 .949 1.00 . 9 88 . 9 26 C -. 58 -.58 -.582 -.583 -. 581 -. 577 -. 577 -.602 -. 596 -. 592 -.59 -.574 -.53 YB (.0037) (.0036)(.0037) (.0035) (.0036) (.0033) (.oo3o)!(.o04o) (.o032) (.0054) (.0032) (.0o26) (.o044)
C .0057 .o41 -.122 -.092 -.116 -.025 -.Oll -.181 -.122 -.070 -.079 -.032 .210
Yp (.0148) (.0154) (.0102) (.0098) (.0100) (.0110) (.0113) (.0139) (.0119) (.0195) (.0132) (.0104) (.0141) C .104 .121 .091 .iii .095 .114 .121 .134 .125 .178 .155 .139 .240 Y6r (.0057) (.0067)(.0054) (.0054) (.0054) (.0051) (.0044) (.0055)(.0050)(.0064)(.0042)(.0039) (.0075) (.00088) (.00088) (.00066)(.00072)(.00055)(.00074)(.000911(.00065) (.00069) (.00066) (.00042) _[00041)[[00059
CZB -.087 -.081 -.074 -.074 -.072 -.081 -.082 -.06 9 -.0705 -.053 -.05 9 064 O64
(.0072) (.0072) (.0041) (.0044) (.0034) (.0048) (.0059) (.0045) (.0048) (.0048) (.0048) (.0036) (.0032) C£p -.557 -.505 -.461 -.457 -.456 -.49 -.485 -.448 -.442 -.377 -.416 -.445 -.45 .o47 .o59 .076 .080 .o71 .062 .078 .iii .093 .161 .124 .113 .128 CZr (.0029) (.0037)(.0021) (.0026) (.0018) (.0022) (.0025) (.0020)(.0026)(.0028)(.0016)(.0016) (.0026)
C£6 r -.0005 .0045 .0035 .0060 .0023 .0036 .00 9 4 .0083 .0087 .013 .0094 .0071 .0125
(.00068) (.00090) (.00049) (.00058) (.00042)(.00055)(.000571(.00049) (.00062) (.00065) (.00045) (.00042)(.00057) C£_a -.127 -.117 -.103 -.105 -.102 -.i14 -.115 -.102 -.103 -.094 -.099 -.103 -.i14 (.0015) (.0015)(.00084) (.00092)(.00069),(.0010) (.0013) (.00091) (.0010) (.0010) (.00077) (.00065)(.0007 9 C .051 .051 .044 .044 .045 .049 .046 .040 .0404 .034 .038 .029 .034 nB (.00030) (.00027) (.00022) (.00023) (.00019](.00023](.00027)(.00030) (.00029) (.00041) (.00022) (.00024)(.00026)
-.028 -.030 -.063 -.053 -.061 .037 .055 -.090 -.086 -.0 9 5 -.067 -.130 -.073
(.0020) (.0019)
Cnp (.0026) (.0025)(.0014) (.0014) (.0012) i.0016) _.0019) (.0023)(.0023)(.0034)(.0019)
Cnr -.i01 -.iii" -.096 -.093 -.096 -.i00 .i01 -.099 -.104 -.125 -.137 -.139 -.094
(.0010) (.0012) (.00069) (.00075) (.00063 (.00064 (.00064)(.00089) (.0010)(.0016)(.00083) (.00090)(.0011)
c -.055 -.057 -.052 -.051 -.054 -.054 -.053 -.056 -.056 -.062 -.064 -.062 -.056
n{r (.00028) (.00032) (.00029) (.00029)(.00026 (.00023 (.00021)(.00032) (.00031) (.00045) (.00026) (.00025)(.00027
c .oo14 .002 9 -.0054 -.0037 -.005 ,.0o13-.0035-.oo81 -.0075 -.00036 .006 9 -.0064 .oo15
nsa (.0005 9) (.00057) (.00030)(.00031) (.00026 (.00036 (.00043)(.00053) (.00051) (.00077) (.00045) (700044)(.00045 w • • • TABLE VII.- CONCLUDED Case 45 56 Input D D CL .559 .320
c - . 56 - . 59
Y_ (.oo46)( . oo32)
C .055 -.041
Yp (.oi43) (.oo84)
• .134 .131 Cy_r (.0074) (.0041) C£_ -.076 -.078 (.00095) (.00073) C£ -.47 -.47
p (.oo58)(.oo42)
.080 .071 CZr (.0038) (.0025) .0114 .0097 C£_r (.00062) (.00048)
C£6 a -.113 -.108
(.0012) (.00090) C .043 .047
n8 (.00033) (.00021)
C -.072 -.049 n p (.0024) (.0013) -.124 -.i18
Cnr ( . 0015) (.00079)
-.o65 -.o58
Cn_r (.00038) (.00021) C .00036 -.0015
n_a (.00052) (.00028)
o_ TABLE VIII.- LATERAL PARAMETER VALUES DETERMINED USING LINEAR REGRESSION ESTIMATION Case 20 [ 21 22 23 24 [ 25 I 26 I 27 I 28 I 29 30 [ 3BI 32 33 -Inpul A J A A A B B B
_.__1_._o _.___z_o i_._ i _._/_._ i_._ i_._o_ _._o I_._ _.__._
_ ° o_ /_ o_ _. o_ _° o_ l_. o_L_ ° o_o_ / _ ° oo_L _° o_l _° _° o_L _. _o_ _ ° _o_ _. _o_
° oo_ / . oo_ ° oo_ ° oo_ I° o_ I . oo_. L ° oo_ I ° oo_ I ° o_o ° o_ I ° oo_ ° oo_ ° oo_
_ ° o_/ _. o_ _° o_o_ _° o_ 1_° o_ i _° _o_/_ ° o_1 _° o_ /_° o_o _ ° o_o [ _° o_ _ .o_ _° o_
_°oo_/_.oo_ _,oo__.oo_o__,oo_1_.oo_/_.oo_l_,oo_
s • p • I i, D • TABLE VIII.- L A TERAL PAR A METERVALUES DETERMINEDUSING LINEAR REGRESSIONESTIMATION (CONCLUDED) Case 34 35 36 37 38 39 40 41 42 43 44 45 46 Input C C C C C C C C C C D D D CL .326 .322 .326 .385 .376 .568 .662 .949 1.00 .988 .926 .559 .320 C -.508 -.516 -.509 -.502 -.515 -.500 -.49 -.447 -.452 .464 -.537 -.56 -.552 YB C -.056 -.068 -.073 -.028 -.013 -.065 -.061 .085 .012 .081 .047 -.037 -.057
yp
C .066 .073 .069 .082 .087 .124 .118 .155 .159 .144 .14 .082 .060 Y_r C£8 -.076 -.075 -.076 -.076 -.077 -.071 -.071 -.057 -.058 -.064 -.060 -.073 -.083 C£ -.44 -.45 -.45 -.45 -.45 -.453 -.451 -.412 -.421 -.438 -.44 -.475 -.48 P
c£ .o 6 9 .072 .070 .071 .073 .o9 6 .0 9 2 .145 .134 .i17 .134 .094 .074
r C£_ a -.106 -.106 -.106 -.108 -.109 -.108 -.108 -.102 -.104 -.105 -.iii -.115 -.'I14
C£_ r .0078 .0073!.0075 .0071 .0087 .0064 .0068 .0102 .0085 .0047.012 .0075 .0103
Cn_ .046 .046 .047 .046 .045 .042 .042 .034 .035 .029 .0302 .042 .047 C -.054 -.049 -.0503 -.'059 -.062 -.069 -.0709 -.094 -.086 -.126 -.i01 -.078 -.056 n p C -.i14 -.116 -.115 -.12 -.i18 -.i18 -.119 -.138 -.14 -.143 -.iii -.122 -.127 n r C -.0031 -.0021 -.0025 -.0033 -.0045 -.0034 -.0041 -.0021 _000046-.0068-.0045 -.0028 -.0029 n_a c -.o55 - . o56 -.o56 -.o57 -.o5 6 -.058 -.058 -.o 66 - . o 6 7 -.o 6 3 -.o 6 o -.o 6 5 -.o 6 1 n_r ( 2 1 w TABLE IX.- CHARACTERISTICS CALCULATED FROM ESTIMATED LATERAL PARAMETERS MODE CL ESTIMATIONMETHOD PERIOD(SEC) TIME TO HALF OR DOUBLE
AMPLITUDE(SEC)
Dutch Roll 1.0 MaximumLikelihood 4.263 1.495 1.0 Linear Reg r ession 4.370 1.600 .6 MaximumLikelihood. 3.402 1.497 .6 Linear Regression. 3.416 1.522 •35 Maximum Likelihood 2.598 1.337 •35 Linear Regression 2.611 1.382 Roll 1.0 Maximum Likelihood --- .162 1.0 Linear Regression .174 .6 Maximum Likeliho o d .122 . 6 Linear Regressi o n .12 6 •35 Maximum Likelihood .0 9 2 •35 Linear Regression .0 9 2 Spi r al l.O Maximum Likelihood 43.95* 1.0 Linear Regression 57.98* . 6 Maximum Likelihood 81.27 .6 Linear Regression 254.85 •35 Maximum Likelihood 41.70 •35 Linear Regression 46.00 *Time to double amplitude • t • TABLE X.- COMPARISONOF DERIVATIVEVALUES DETERMINEDBY THE MAXIMUM LIKELIHOOD METHOD WITH THOSE FROM SELECTED REFERENCES Values from least squares' linear Reference Reference Reference fit to the M.L. estimated para- I0 1 7 16 meters fo r th r ee CL values CL = i.i CL = .25 CL = .9 CL = .3 CL = .6 CL = 1.0
.Cy B - .54 - .22 -...303 - .59 - .57 - .55
C - .047 - .213 - .045 - .005 .04 Yp
C .143 .15 .Ii .14 .17
YSr
C£8 - .0785 - .08 2 - .1 22 - .078 - .070 - .058 C£ - .47 - .494 - .47 - .45 - .41 P
C£ .07 .07 . 09 5 .1 3
r
.o13 .0058 .oo7 .0o95
C£_r
C _6 a - .10 2 - .198 - .ii - .10 2 - . 0 9_
C .057 .035 .0701 .048 .042 .033 nB C - .077 - .096 - .05 - .078 - .115 n P C - .062 - .115 - .i0 - .ii - .125 n r C - .057 - .046 - .053 - 0.58 - .063 n_r
c .o14 - .003 - .004 - .oo5
n_ a
TABLE XI.- THE PERCENT THE CRAMER-RAO BOUND IS OF AN ESTIMATED LATERAL PARAMETER CASE NUMBER PARAMETER 20 21 22 24 25 26 27 28 2 9 C 0.659 0.818 0.554 0.824 0.746 0.995 1.12 0.586 0.897 YS
C 26.21 34.67 57.41 29.02 12.96 16.07 10.57 15.43 26.98
Yp
C 3.220 4.692 2.507 5.800 3.957 4.305 3.963 3.00 3.890 Yr C 3.067 4.798 2.438 5.846 5.417 3.026 2.432 3.25 3.701 YSr
C£8 0.965 0.915 1.103 1.831 1.233 1.098 1.145 0.635 1.00
C£ 1.012 0. 9 00 1.187 1.436 1.045 0.932 0.888 0.682 0.909 P
C£ 2.6 6 8 2.300 1.797 9.574 3.671 1.667 1.825 1.852 2.178
r C£_r 23.30 8.824 5.488 43.33 10.91 5.256 3.25 6.364 7.069
C£_ a 0.903 0.802 1.067 1.368 1.010 0.832 0.770 0.605 0.827
C 1.105 1.026 1.286 0.773 0.833 1.000 0.974 0.435 0.810
n8
C 2.086 2.885 3.226 2.419 2.254 1.944 1.709 2.759 2.564 n P C .1.157 1.282 1.468 1.327 1.053 0.833 1.048 0.780 0.885 n r C 0.540 0.607 0.516 0.685 0.627 O.530 O.59O 0.491 0.508 n6r Cm6a 5.364 7.073 6.500 5.410 5.441 32.000 5.238 34.55 14.48 I i ii i • • J • TABLE XI.- CONTINUED CASE NUMBER PARAMETER 30 31 32 33 34 35 36 37 38 C 1.590 0.508 0.638 0.621 0.636 0.600 0. 6 20 0.572 0.520 Y8 C 68.57 19.81 259.65 37.56 8.36 10.65 8.621 44.00 102.7
yp
C 5.599 2.808 4.838 4.452 3.69 3.50 3.589 3.573 2.911
Yr
C 4.276 3.158 5.481 5.537 5.93 4.775 5.579 4.474 3.636 Y_r C_B i._5 9 0.705 1.011 1.086 0.892 0.973 0.764 0.914 i.ii0 C_ 1.253 0.860 1.293 1.426 0.889 0.963 0.746 0.980 1.196 P C£ 2.160 2.436 6.170 6.271 2.763 3.250 2.535 3.548 3.205 r C£6r 4.000 5.443 136 20.0 14.00 9.667 18.261 15.28 6.064 C_8a 0.977 0.772 1.181 1.282 0.816 0.876 0.676 0.877 1.130
c 1.677 o.431 0.588 0.52 9 o.5oo 0.523 0.422 o]46 9 0.587
n8 C 1.944 5.806 9.286 8.333 2.222 2.642 1.967 4.324 3.455 n P C 1.439 0.781 0.990 i.o81 o.719 0.806 0.656 0.640 0.634 n r Cn8r " 0.717 0.421 0.509 0.561 0.538 0.549 0.481 0.426 0.396 C 3.134 21.58 42.14 19.66 5.556 8.378 5.200 27.69 12.29 nsa O TABLE XI - CONCLUDED CASE NUMBER PARAMETER 39 40 41 42 43 44 45 46 C 0.664 0.536 0.912 0.542 0.453 0.830 0.821 0.542 Y8 C 7.680 9.754 27.86 16.71 32.5 6.714 26.00 20.49
yp
C 3.584 3.972 4.337 3.673 3.164 5.41 9.609 3.733 Yr C 4.104 4.000 3.596 2.710 2.806 3.125 5.522 3.130 C£8 0.942 0.979 1.245 0.712 0.641 0.922 1.250 0.936 C£ 1.004 1.086 1.273 0.865 0.719 0.800 1.234 0.894 P C£ 1.802 2.796 1.739 1.290 1.416 2.031 4.750 3.521 r C£_r 5.904 7.126 5.000 4.787 5.915 4.560 5.439 4.949 C£_a 0.892 0.971 1.064 7.778 0.631 0.693 1.062 0.833 C 0.750 0.718 1.206 0.579 0.828 0.765 0.767 0_447
n8
C 2.556 2.674 3.579 2.836 1.538 2.603 3.333 2.653 n P C 0.899 0.962 1.280 0.606 0.647 1.170 1.210 0.670 n r Cn_r "0.571 0.554 0.726 0.406 0.403 0.482 0.585 0.362 Cn_a 6.543 6.800 213.89 6.522 6.875 30.000 144.4 18.67 • e • o u P.
a X
Y q w
V
Z
Figure I.- System of body axes and positive sense of angles, forces, and moments.
!
i I0. 91 -i
E l Figure 2 .- Three-vlew d raw i ng of the S u b j e c t a i r c raft. A ll linear d i mens i ons i n meters.
_ er a l • in p ut A inp u t B . 4 _ .4 1 - .2 -- . 2 - _ 2 - - .2_ -- %4 _ 111 IIII _ IIII III1_1111 IIII IIII IIII IIII _ 4 _ 11111111 IIlI IIII IIII IIII IIIIIIIII IIII IIII 0 4 8 1 2 1 6 20 0 4 B 12 16 20 T i m e , sec T ime , sec .4 - . 4 .2 _ . 2 _ ° _ 2 - m . _llltlllIiIIllli_IIll fillI111 fillI{IIIIII _4_I11 JilllIIJII_ ll_ _l_ _l _l l_ _ III 0 4 8 12 1 6 20 0 4 8 12 16 20 T i me, sec T i me , se c input C input D .4 _ . 4 _ .2 _ _ .2 _ -- _€ o - - . . _ -. j - _ .
-.2- ._ -.2_ 4_I_ l_{ _l _l_l _l_ _l_ _ _l_ l_ _l_ ".4_l{ll=_ _ _ _ _ l_ _ l_ l_ 0 4 8 1 2 16 20 0 4 8 1 2 t S 20 • T i me, s e c T i me , s ec .4 - .4 • . 2 _ . 2 - _..
% 4 _lll IllS Illl Illl Illl Illl IIll Illl Illl Illl - .v ='_lll, ,,,ll=,, ,i,, i,,, ,=,, IllS Sill Illl IISl 0 4 8 12 1 6 _ 0 4 8 t2 ,16 20 T i me , s ec Time, s ec F i g ure 3 . , Lon g itud l na l an dl ateral control inputs.
m / se c Flight Estimation Parameters Airspeed Power data method estim_ed 31.6 m / sec l_e ___(f_l power climbs and calc_ation Static longitudin_ 40.7 m / sec _i i lidle power descents for of Ref. 13 ! Stead _a_tical
• i
5h.2 m / sec i 1 2 c.g.'s J
il
Longitudinaldata 40.731"6 m / secm / Sec _ Trim (slowacceleration-deceleration at _--_calculation Static longitudinal 54.2 m / sec trim power for 2 i i of Ref. 13
c.g.'s)
|
31.6 m / sec l_e (linear regression) Complete longitudinal 40.7 m / sec Trim _ Pert_bation par_eters, 54.2 m / sec _l " (doublets) Cr_er-Rao bo_d, M_imam ___ Parameter correlation
morro
likelihood
i
Lateral _ Perturbation (Linear regression) Complete longitudinal -- --_ Equation error (rudder doublets 40.7 m / sec Trim ' _ Cra m er-Rao bound, followed by aileron data 31.6 m / sec _ _ parameters, 54. 2 m / sec ,._ d°ublets) _ Maximum ._ Parameter cOrrelatiOn likelihood Figure 4.- Flight data processing scheme.
0 M axlike l ih o o d Regression A S t ea d y and qu as i st e ad y t es t • Maximum likelihood results Flagged symbols idl e power Trim Idlean dfuj i
. Cx a
0 • ; I "0 ,, 0 0 - 1 0 - - 1 0 0 0 - 2. 0 - 2. 0!- 0 0 Cmq + Cm6 -2 0 _ _ _ C m q + Cm_ -20 O -4 0 -40 - • 0 0 m 6 e -L 0 - e -2.0 I I I -2.0 • t I 1 .4 . 8 L2 0 .4 .8 L2 CL CL Figu r e 5 .- E s ti m at e d lo ng itudi n a l pa r a m et e r s v e rsus CL fort h reeesti ma ti on me tho d s .
........... Flight data Fittodata 177- _ 5 3 2 0 _ .1 5 7 - 4 7 1 0 - .137 -, -- - , - -. ..,...... - -- 41"_ -' _ 0 " --. -- .- --- ,- -- - 117 35 =" -I 0 - i 9 7-_tliJil11 tilt fill tiff fill IIII IIII ilii!1111"-29 - 201111 Illl IIIlillfl IIII Iffl Ilil IIII IIII IIlll " O 2 4 6 8 lO 0 2 4 6 8 10 Time , s ec Time , sec 40 - 1 2 . 4 _ "
o- -oI'_ = o _
_ . _ _ _- j× -
-20 - -6 ". 2 .40_ ttl Ittt Vltllltll till tttl Vttllltil tilt tlt_ _ 1 2 - . 4 _lll li l t Il t i ! t lllJ l t il li lt ti l l t vi illt tl till 2 4 6 8 1 0 0 2 4 6 8 1 0 Time , se c Time . se c • 50 _ 1. 5 0 _ .2 5 ' n / _l _ .7 5 - t n i x , , -- 0 - - _ O_
/ \
= _ / ' ._ _
- " . 2 5 - . 7 5 - ' % ...50._ fff4fff l i fff f flll lllll'If l t i l l f f|t fl l f; f lll _ 1 . r., N' ttfl., . . , , , l'_ { A l ti l l ffff ffl l llll ' llll l lll, l |ll 0 2 4 6 8 I0 0 2 4 6 8 10 T i me , sec T i me, sec •50 -. 25 -,50 IIII flllJllll IIII fill Illl IIIf 0 2 4 6 8 I 0 Time, sec . 2 i • . _ll l I I llJIIII till IIII IIII lilt till till IIII 0 2 4 6 8 10 T i me , sec Figure 6 . - Ty p ical f it s to re sp on s e s toanelevator in p ut.
M aximum l ik elihood extraction I¥| MAIIIIUIII III _ . _ IIIIUU I J _A Lrd L , I.IO N C Y l _ _ in fi g ure3 L Input s as s hown -L0 O A [] B L0- 0 ¢ n D • Max. lik elihood best f i t li n e Cy p 0 _ _ _ 0 _ v _ ,q 6 a -. 1 - -L 0 - CZ .4 6r _ -. 2 - .I C_ -. 0 4 - 0 - . 08 - C n p - . I - ' 0 EJ .8 - - . 2 .2-- O - C l 1 - Cn 6 r 0 r . 0 5 - __ ", 10 - • 02 - , . 0 2 - 6 r . 0 1- a 0 A
c, O n 8
. 0 _ I 0 I - .0 2 I ! I 0 .4 . 8 1. 2 0 .4 .8 L 2 i CL CL F i gu r e 7.- Es t i mated la tera l par a me t e rs d e te r m i ned us in g maxi mu m l i k el i h o od ver s u sC L. , ; , _i! ' Z.L Li nearregresslon extraction C YI ) __ _ '_Z ]{_ in f i g ure 3 0 _ Inputs a _; sh o wn - 1 .0 O A Z] B
1 . o - 0 c
[] _. . . D
[] • M a x. l ik eli h o o d be s t f it l ine o C yp 0 _ 0 m - I .0 - C / s a -. l - ._ C y 5 r _ -.2 - 0 . 1
o
Cnp I C_ .0 4- _C) " 0 --,2 - C _ -.4,- _
Cn r
" . 8 - - .2 , 2 - 0- [] - -. 05 - C[ r . 1 [] C n5 0 -, 1 0 - •02 - . 0 2 i " I C{ A 5r .01' - C 0 I I C) I - . 0 2 0 l I .I 0 . 4 . 8 L2 . 4 . 8 1 . 2 CL C L F i g u re 8. - Est i mat e d lateral param e ter s dete r mined using l i n e arregre ss ion versu s C L . I_ " ......... Flig ht data •. . - -Fittod a t a • _! 14 6- '_ ,4 4
226 _- , 3 8 B _
25 Q ,,, --- - .. 1 0 6 _ " -_ _ "" " '-
/ r \
0 \ - .. - = 3 2 = "
/ = . _
= : - . 25 J 8 6_ 2 6 . . 50-_iil Itll IIII IIII IIIIIIIII IIII IIII I11111111 6 6_ 111 IIII I IIII IIII III1 ! 1111 IIII III1 IIII I l l 20 0 4 8 12 1 6 20 0 4 8 I2 I6 20 Ti m e , s ec Time, s ec •50 _ 20 _ < ., . 25_ 10 -- - .....
'- " - . 25 -lO - -.50_t li lliitliililt till_III tilttililltll ilii .20_II} I ll}fill il_i,=llI lIIllli_,i_tl l_li'lil_ 0 4 8 12 1 6 20 0 4 8 12 1 6 20 Time , s ec Time , s ec 40! _ _ 12 . 4 _
,o f \ / ' , ,, - - o / -
-6 _ ' > " --.2
- 2 0_ _ _ trill IIII IIII IIII qo_lll 1111_1111 IIII IIII III1,1111 IIII IIII 111_.12 -o!_lll,llll IIII 1111 IIII 1111 4 8 12 16 2 0 0 4 8 .... 12 ze 26 v Time , sec T i m e, sec •5 0 O _ " -. 2 5 i . . _1111 1111 IIII I111 IIII Illl III II1! IIII III1 0 4 8 12 1 6 20 T i m e , s e c l .4_ . 4 _ m . 2 - _ _ . 2 0 ' , = -- I 0 - ,,. m " L -" -.2 - " " l_2Z I ..- = ' II -. 4 _l till lilt lilt Illllllll lilt IIIl lilt IIII ,. .4_ lll Illllllll IIII IIII IIII IIII Jill,Ill| IIII 0 4 8 1 2 16 2 0 0 4 8 1 2 16 20 T i m e , sec Time , se c Fi gure 9 . - Typical fits tores p onses toarudder - ai l eron in p ut .
0 Maximum likelihood A Linear regression •3< C L<.4 • 0 - .2- 0 - C Y i)- . 5 - _ © e 6 C l 'r " 1" 0 ¢ nsr - 1 . 0 - . 1 0 - , . 4 O. 5!- .02 - .O l -
o 8
-0 . 5 - 0 0 - . 0 1
A B C D
1.0 O- CY r .5 0 C l .s a -. 1 _ _ O_ O - _2 .2- .10- 0 0 0 Cysr .1 0 0 __ A Cn_.05 - _ _ _ O 0 A 0- O- O-
©
A 0 C _ _0 5 Cnp_0 5 _ _ _ _ o _IG _ I - k - 0 _ • - . 8 -2 A B C D A B C D F i gure 10 . - E ff e _ o fc on t rol in p ut f orm onestimated p arameter values f orthelateral in p uts sh o wn in f igure 3.
0 Maximum likelihood A L inear regression . 5 < CL <.7 O - .2 O- -,5- O O O Q 2 O n o r Cyi B A A C l . r . 1 A _ 8 A C - 0 5 _ _ a -I . 0 - - " , I0 • 5 - .02'- .005 h, O
,_ 0
Cyp 0 - _ _ , _ 2 C_r . 01- O (_O AO C _ 0 A &" n6 a-. 005 A v A. _ _ --.0 1 0- --, 5 - - -. 0 15 A B C D L0- 0- CY r .5-0 0 8 0 C l' 6 a "l _ _ _ m . 2 I . 2 - .1 0- O A j_ O Cnl).05- CY6r .1- A _ _I _) @ U 0 - 0 0'- 0 C,]B I .., 0 5 - Cnp -, 1 O O -.I0 --.21- " 0- 0- -.8 -_ A B C D A B C D Fi gu r eI 0.- C o n tin u ed , O Maximum likelihood A L inear regression .9<CL< L 102.
0- . 2 - 0 -- ,, _ , , cz o _i _ _ : . .
ir -1 . 0 - - .10 - -- O j • 5 - .0 2 -- .00 . 5 -- /x O A O 0 -- /x
Cyp 0- C[sr 01- Cn6a-.005- A _
0 " o _ _ o
A - . 010 - • - . 5 - O - _ " . 0 1 5 - A B C D 1 .0 - 0 - 0 0 0 0 CYr..5- O C_a -.I-
oo _ __
O-- % 2 -- .3 . . I 0- O .2
o 6
- 0- 0- 0 - O Cil3- . 0p _ _ z_ Cnp-. - _ _ A
" o _ _ 60
- 10 - -. - L O_ 0 _ •
c_ Cn-1 - A 2
'
-.8 - . 2 A B C D A B C D F i gu re 10 . - C o n c lude d.
1 Re p ort No 2 Government Access i on No. 3. Recipient's Catalog No , NASA TM- 8 016 3 4 Ti tle and Subtitle 5 . Re p ort,Date COMPARISON OF STABILITY AND C ONTROL PARAMETERS FOR A September 19 7 9 LIGHT, S INGLE-ENGINE, HIGH-WINGED AIR C RAFT US ING 6 Performing O rganiz a tion Cod e DIFFERENT FLIGHT TEST AND PARAmeTER ESTIMATION TECHNIQUE_ 7 Aut h or(s) 8 Performing O r ga n i zatio n Re l_ ort No William T. Suit and Robert L. Cannaday 1 0. W o rk Unit" No .
9. Per fo rm in g O r gan iz a ti o n Na m e a nd A ddres s 5 0 5- 0 6 - 6 3 -02 NASA Langley Resear c h Center 1 1 Cont ra ct or Gr ant No Hampt o n, VA 23 665 13 Typeof Re po rta n d P e r i o d Co ve r ed 12 . S po nso r in g Ag en cy Name and A d dr es s Techni cal Mem o randum National Aer o naut ic s and Spa c e A d m i nistrat io n Washington, DC 20 546 14 A rmy P roj e ct No .
15 S u ppl e m e n ta r y N o tes . = . ..
16 . Abs t r act L o ngitudinal and lateralstability and co ntr o l parameters were estimated fr o m flight data for a high-wing, general aviat i on, airplane using fl i ght data obta i ned at various flight c ond i t i ons within t h e normal range of the air c raft. These parameters were est i mated us i ng an output error te c hn i que (ma xi mum l ik el i hood ) and equation error technique (linear regress i on ) . Longitud i nal stati c parameters were also estimated from c limbing , des c ending, and quas i -steady-state fl i ght data. For the lateral ex ci tat i ons, four input forms were used i nvolv i ng some c omb i nat i on of rudder and a i lerons. The resulting longitudinal and lateral parameter estimates were used to c ompute the per io ds and t i me-to-damp to one-half ampl i tude of the various air c raft modes of motion to determin e the sensitiv i ty of these mot i ons to variations in t h e parameter estimates.
• _ . K ey w or ds ( S ug g_ ted b y A uthor(s) ) 18. Di st r i b ution S tat em ent Parameter estimation Stability and c ontrol parameters Unclass i fied - Unl i mited Ma x imum likelihood Subject Category 08 19 S ecu ri ty Cl ass if . (o f t h is re por t ] 2 0 Se cur i t y C la ss if ( o f t h i s p a ge) 2 1. N o of P ages 2 2 Price " Un c lass ifi ed Un c lass i f i ed 5 2 $5 .25 • Forsaleby theNationalTechnical Information Service , Springfield , Virginia 22161