- 

NASA Technical Paper 1459 

Kenneth W. Iliff, Richard E. Maine, 

and T. D. Montgomery 

APRIL 1979 

https://ntrs.nasa.gov/search.jsp?R=19790013942 2018-05-14T12:43:39+00:00Z
TECH LIBRARY KAFB, NM 

0334834 

NASA Technical Paper 1459 

Important Factors in the 

Maximum Likelihood Analysis 

of Flight Test Maneuvers 

Kenneth W. Iliff, Richard E. Maine, 

and T. D. Montgomery 

Dryden Flight Reseurch Cetzter 

Edwards, California 

National Aeronautics 

and Space Administration 

Scientific and Technical 

Information Office 

1979 
CONTENTS 

Page 

INTRODUCTION . .1 

SYMBOLS . .2 

THE MAXIMUM LIKELIHOOD ESTIMATOR . .4 

OUTLINE OF FLIGHT DATA ANALYSIS EFFORT . .6 

Preflight Procedures .7 

Basic .7 

Data Preprocessing . .7 

Data . .8 

Summary Plots . .8 

INSTRUMENTATION . .8 

Signals Required . .8 

Data . .9 

APPLICATIONS AND SPECIAL PROBLEMS . . 14 

Turbulence . . i6 

Structural Modes . 18 

Linear Dependence roblems . . 20 

Time-Varying Systems . . 23 

Nonlinearities . . 24 

Cross  . . 26 

Estimation of Pitching Moment Due to Vertical Acceleration . . 30 

CONCLUDING ARKS . . 32 

REFERENCES . . 38 

APPENDIX-DESCRIPTION OF MMLE 3 PROGRAM . . 34 

iii 

.... .. . .. . ...."... -.- .... , . , , ,-,,... _.""" _.I rn .. 11. I I 
IMPORTANT FACTORS IN THE MAXIMUM LIKELIHOOD 

ANALYSIS OF FLIGHT TEST MANEUVERS 

Kenneth W . Iliff, Richard E. Maine and T . D . Montgomery 

Dryden Flight Research Center 

INTRODUCTION 

One of the primary goals of an aircraft flight test program is to estimate flightdetermined aircraft characteristics y such as the aircraft's performance y structural, 

and stability and control coefficients. As these estimates become available during 

the flight test program they can be used to expand the flight test envelope update 

the simulators, and improve the aircraft propulsion and control systems. After 

the analysis of the flight test data, the estimated coefficients can be compared with 

calculated and wind-tunnel predictions and the results of this comparison can be 

used to update prediction methods for the improvement of future aircraft design. 

Some of the flight-determined coefficients can be obtained through an analysis 

that assumes static conditions (ref. 1). However the required flight test time can 

sometimes be reduced by using dynamic maneuvers. In addition, many coefficients 

of interest cannot be determined from static tests. The analysis of dynamic maneuvers has required the application of new and more complex analysis techniques. 

One of the techniques that is widely used to analyze dynamic maneuvers is maximum 

likelihood estimation (refs. 2 to 6). 

This paper discusses the application of a maximum likelihood estimator to 

dynamic flight data. The maximum likelihood estimator is described the procedure 

for flight data analysis is outlined instrumentation requirements and problems are 

discussed and examples of applications and special problems are presented. The 

discussion emphasizes the estimation of stability and control derivatives because 

most of the estimation experience to date has been with these derivatives. 
SYMBOLS 

A,ByC,D 

a 

n 

a 

X 

a 

Y 

b 

cA 

cD 

DI 

cL 

C 

cnl 

cN 

cn 

C 

f( 1 

GG" 

system matrices 

normal acceleration g 

longitudinal acceleration, g 

lateral acceleration, g 

reference span, m 

coefficient of axial force 

coefficient of drag 

coefficient of induced drag 

coefficient of lift 

coefficient of rolling moment 

coefficient of pitching moment 

coefficient of normal force 

coefficient of yawing moment 

coefficient of side force 

reference chord, m 

general function 

power spectral density of measurement noise 

acceleration due to gravity, m/sec 

general function 

inertias kg-m 2 

2 

2 
J cost functional 

Kcr7 Kp flow amplification factors 

L normalized rolling moment rad/sec 

M normalized pitching moment rad/sec 

2 

2 

m mass, N 

N normalized yawing moment rad/sec 2 

n state noise vector 

P , roll rate deg/sec or rad/sec 

4 pitch rate deg/sec or rad/sec 

R 

r 

S 

T 

t 

u 

V 

X 

dynamic pressure kN/m 

covariance of weighted residual measurement error 

yaw rate deg/sec or rad/sec 

reference area, m 

maneuver duration sec 

time sec 

input vector 

velocity m/ sec 

state vector 

distances of instruments forward of center of gravity m 

2 

2 

Y normalized lateral force rad/sec 

Y observation vector 

z normalized normal force, rad/sec 

Z measured observation vector 

z 7 za 7 zp distances of instruments below center of gravity, m 

Y 

3 
Kalman-filtered estimate of the observation vector 

a 

B 

8 

P 

6 

6e 

6r 

angle of attack, deg 

angle of attack induced by vertical velocity component 

of turbulence, deg 

angle of sideslip, deg 

general control 

aileron deflection, deg 

elevator deflection, deg 

rudder deflection, deg 

rl 

e 

5 

Subscripts: * 

m measured 

measurement noise vector 

pitch angle, deg 

vector of unknowns 

bank angle, deg 

p, q, r, a, 6, derivative with respect to indicated quantity 

p7 6, fja7 

trim immed value 

0 bias 

Superscript: 

* matrix transpose 

THE MAXIMUM LIKELIHOOD ESTIMATOR 

The problem considered is: Given a set of flight time histories of an aircraft's 

response and input variables, find the values of some unknown parameters in the 

system equations that result in the best representation of the actual aircraft 

4 
response. An intuitive mathematical approach to the problem is to minimize the 

difference between the flight response and the response computed from the system 

equations. This difference could be defined for each response variable as the 

integral of the error squared. The differences could then be multiplied by weighting 

factors proportional to the relative confidence in each signal and summed to obtain 

the total weighted response error. This approach defines an integral-squarederror criterion. 

A mathematically more precise probabilistic formulation can be made. For each 

possible estimate of the unknown parameters, a probability distribution for the 

noise-contaminated aircraft response can be defined based on the dynamic model. 

From each of these distributions, one can evaluate the relative likelihood that the 

noise-contaminated response will equal the response actually measured. Estimates 

are then chosen such that this likelihood is maximized. These estimates are 

referred to as the maximum likelihood estimates. 

The general model for the dynamic systems is 

The noise is assumed to be zero mean, white, Gaussian, and stationary. The 

maximum likelihood estimates are obtained by maximizing the likelihood functional 

or, equivalently, by minimizing the following negative log likelihood functional 

(ref. 7): 

where i is the Kalman-filtered estimate of y , and R is the covariance matrix of the 

weighted observation estimation error. This algorithm, in contrast to the extended 

Kalman filter method (ref. 81, uses the Kalman filter only to estimate the states 

and observations, not to estimate the unknown coefficients. When there is no state 

noise, R is the null matrix and i is obtained by simply integrating the system 

equations; no linearity asssumptions are required. If state noise is present, the 

system equations must be linear in order to rigorously define the likelihood 

functional as in equation (4). For nonlinear systems with state noise, an estimator 

can be defined by replacing the Kalman filter for 2 with an extended Kalman filter, 

but such an estimate will not be maximum likelihood. 

5 

5 

5 

Figure 1 illustrates the maximum likelihood estimation concept. The measured 

response of the aircraft is compared with the estimated response, and the difference 

between these responses is called the response error. The Newton-Balakrishnan 

(ref. 7) computational algorithm (formerly referred to as the modified NewtonRaphson algorithm) is used to find the coefficient values that maximize the 

5 
Turbulence Noise 

Control Measured 

input response 

aircraft 

1 jF;'"d Mathematical model response 

Iof aircraft (state estimator) I - 

Response 1 

Balakrishnan likelihood 

w-5 Maximum likelihood 1 aircraft parameters I 

I 

t 

Figure 1. Maximum likelihood 

estimation concept. 

likelihood functional. Each iteration of this algorithm provides new estimates of the 

unknown coefficients on the basis of the response error. These new estimates of 

the coefficients are then used to update the mathematical model of the aircraft, 

providing a new estimated response and, therefore, a new response error. The 

updating of the mathematical model continues iteratively until a convergence 

criterion is satisfied. The estimates resulting from this procedure are the maximum 

likelihood estimates. 

The maximum likelihood estimator also provides a measure of the reliability of 

each estimate based on the information content of each dynamic maneuver. This 

measure of the reliability, analogous to the standard deviation, is called the 

Cramer-Rao bound (ref. 3) or the uncertainty level. The Cramer-Rao bound as 

computed by current programs should usually be taken as a measure of relative 

accuracy rather than absolute accuracy. Only recently (ref. 9) has insight been 

gained into the computation and interpretation of the absolute magnitude of the 

bound. When carefully used, the Cramer-Rao bound has proven to be a useful tool 

for assessing the validity of the estimates. 

OUTLINE OF FLIGHT DATA ANALYSIS EFFORT 

A multitude of problems can occur in the analysis of flight data, especially if the 

data processing is hasty and attempts to untangle problems and interpret results are 

6 
left until the end (ref. 3). However, with careful attention to detail at the appropriate times, the analysis can proceed smoothly and quickly enough to meet the 

requirements of the most demanding flight test schedules. 

This section outlines a desirable procedure for the analysis of flight data. The 

outline does not include such special situations as the concurrent use and updating 

of a flight simulator. The procedure may change during the flight test program as 

the data system stabilizes and the expansion of the flight envelope begins. 

Preflight Procedures 

Before flight, the maneuvers and flight conditions are chosen, the details of the 

instrumentation systems are specified, and the basic analytical model is chosen. 

In planning a flight test program, one important point should be realized: Because 

of the nature of parameter identification, no single maneuver, no matter how 

carefully analyzed, can provide a definitive description of an aircraft or even of an 

aircraft at a given flight condition. This is true even when automatic tests for 

validity, such as the uncertainty levels, are available, because all such tests make 

certain a priori assumptions about the model. Thus, there is no substitute for 

obtaining several maneuvers at a single flight condition or a series of maneuvers 

that show a consistent trend as the flight condition changes. The purpose of the 

maximum likelihood estimation method is to prevent the flight time and the effort 

required to analyze these maneuvers from being prohibitive. The test pilot should 

be informed of the requirements for each maneuver and the reasons for these 

requirements. Keeping the pilot informed can improve the quality of the maneuvers, 

particularly if unexpected difficulties require innovations in flight. Getting the 

pilot's opinion of the feasibility of a given maneuver can also prevent wasted 

flight time. 

Basic Data Inspection 

During or after a flight, the raw data should be inspected for obvious data 

acquisition system problems. In addition, the analyst should make a quick check 

for violations of basic modeling assumptions such as mode coupling, varying flight 

conditions, or large bank angle excursions. If such problems are recognized in 

time, the maneuver can be repeated during the same flight. 

Data Preprocessing 

The flight data measured by the aircraft instrumentation must be recorded. 

The data should be digitized and converted to engineering units, then processed to 

remove data spikes and, if necessary, filtered to remove structural effects. Any 

effects of time skews in the recording system should be corrected at this point. 

Other preprocessing tasks, such as the computation of air data parameters from 

pressure data (ref. lo), should also be performed. 

7 
Data Analysis 

At this point, a program for the analysis of dynamic maneuvers, such as MMLE 

(ref. 11) or MMLE 3 (see appendix), is run using the aircraft model selected 

previously. All the cases should be examined for problems, and fit errors and other 

abnormalities should be studied. Particular attention should be paid to whether 

any of the abnormalities observed may have been caused by violations of modeling 

assumptions. Suspected problems should be checked using external sources if 

possible (for instance, if the flight condition seems to be misidentified). Cases where 

rectifiable problems are identified should be corrected and rerun. This step should 

be repeated as necessary depending on the urgency of the analysis, the economics 

of analyst and computer time, and the extent of the problems encountered. 

Summary Plots 

To examine the results, the derivatives and associated uncertainty levels should 

be plotted. Unexpected results should be studied to see if they have been caused 

by the misidentification of the flight condition, the improper specification of instrument location, or errors in the model or the data. If any significant errors are 

indicated, the data analysis should be redone; otherwise the preliminary analysis 

is finished. 

The emphasis of the procedure described in the previous sections is on the 

continual reevaluation of the modeling assumptions. When the analysis is finished, 

analysts often question only the data acquisition system or the maximum likelihood 

estimation method as obvious sources of difficulty, but they should also reexamine 

the modeling assumptions. 

INSTRUMENTATION 

This section discusses the data requirements for performing maximum likelihood 

estimation. A list of the signals required is presented, as well as a discussion of 

data quality considerations. 

Signals Required 

The signals required for maximum likelihood estimation fall into two partially 

overlapping classes. The first class consists of signals for which only an average 

value is required for each maneuver. The second class consists of signals for which 

the complete time history must be available. Some signals can fall in either class, 

depending on the particular maneuver. 

Signals for which only an average value is required include those that define the 

vehicle configuration, flight condition, and mass characteristics. These signals 

need not be recorded on the data tape; pilot lap notes or similar hand-kept records 

8 
may be adequate in some cases. However, for large numbers of maneuvers, it may 

prove convenient to record these signals on the data tape in order to automate the 

bookkeeping. 

The vehicle configuration includes the positions of all flaps, canards, landing 

gear, wings (sweep angle), engine controls, and other items that affect the 

aerodynamic characteristics of the vehicle and are held fixed during a maneuver. 

The flight condition is defined by velocity, altitude, Mach number, dynamic 

pressure, angle of attack, and other quantities used to nondimensionalize the 

derivatives or plot the results. Some flight test programs may only require the 

estimation of dimensional derivatives, in which case some of the nondimensionalizing 

parameters may not be necessary. The mass characteristics that should be known 

are the weight, center of gravity, and inertias. These mass characteristics are 

usually determined from tables based on vehicle configuration, fuel weight, and 

cargo loading. An accurate determination of the mass characteristics is essential, 

because errors in the mass characteristics result in proportional errors in the 

nondimensional derivatives. If feasible, the weight and inertias of the vehicle 

actually tested should be measured (ref. 12) and compared with calculated values. 

A frequently overlooked quantity is the vertical center of gravity position, the 

importance of which is discussed later in this report. 

The signals for which time histories are most necessary are the positions of 

the controls that move during a maneuver and the response variables to be matched. 

For stability and control analyses where cross coupling is important (see Cross 

Coupling), the longitudinal response variables are needed to match the lateral 

responses, and the lateral response variables are needed to match the longitudinal 

responses. The longitudinal response variables are a, q , 8, an, ax, and 4; the 

lateral directional response variables are p , p , r , cp , a 0, and i-. Measurements 

of 6, 0, and ? are often not available; however, differentiated values of p , q , and 

r should not be substituted as they add no new information and may even be 

detrimental. Not all the response variables must be measured for each maneuver, 

but the more that are available, the more reliable will be the derivative estimates. 

Y’ 

In special situations, time histories are required for some of the normally 

constant signals. The most common example of this is the necessity for time 

histories of velocity and dynamic pressure if those signals change enough during 

a maneuver to have significant effects on the estimates. In general, any signal that 

changes enough during a maneuver to affect the analysis should be recorded as a 

function of time. 

Data Quality 

Two basic factors in data quality are the instrument resolutions and the sampling rates. Experience indicates that in general neither of these factors alone is 

critical (ref. 3). However, the additive effects of these and other factors can be 

significant. Fairly low resolution can be tolerated for any noncontrol signal if 

measurement noise is low. This is particularly true if many signals are used for 

time history matches and some of the signals have good resolution (ref. 9). 

A resolution of 1/10 of the maximum signal amplitude for a maneuver is about the 

minimum acceptable under ideal conditions. Resolutions on the order of 1/ 100 of 

9 
the maximum signal amplitude for a maneuver are preferred. If the data clearly 

define the aircraft response and no other data problems exist, the estimator can be 

expected to perform well. 

Much the same conclusion can be drawn in regard to the sampling rate. For 

typical aircraft at typical flight conditions, sampling rates of10 samples per second 

are often adequate for the analysis (refs. 13 and 14); however, problems can arise 

for aircraft with fast responses or for data resulting from rapid control inputs. 

Sampling rate requirements are also related to the duration of the maneuvers 

(ref. la). As discussed later under Structural Modes, the sampling rate is often 

dictated by considerations other than the estimation procedure. Sampling rates as 

high as 200 samples per second are sometimes needed to filter out structural 

vibrations. 

The maximum likelihood estimator is very sensitive to small time or phase shifts 

between parameters (ref. 15). Most data sampling techniques result in small time 

skews between parameters. For example, if data are sampled at 10 samples 

per second, the sample of one parameter may be separated by up to 0.1 second from 

the sample of another parameter in the same time frame of data. The maximum likelihood estimation algorithm assumes that all measurements in one time frame are 

sampled simultaneously, so any time shift causes errors in the estimated coefficients. 

The time shift becomes particularly important when the control input is sampled at a 

significantly different time than one or more of the response measurements within 

the sample interval. If time skews are unavoidable, the effect can be compensated 

for by time shifting the appropriate signals before the analysis is begun. 

A phase shift due to instrumentation filters can cause a similar problem. All 

filter rolloff frequencies should be kept much higher than the aerodynamic 

frequencies of interest. If this is not possible, all the measurements should be 

filtered with the same filter, or phase shift corrections should be applied to the raw 

data for all the filtered measurements. Another possible source of time errors is 

the lag in the response of pressure instrumentation, particularly when long tubes 

are involved. 

The effects of time and phase shifts in the flight data on the stability and control 

derivatives are documented in reference 15. An example from reference 15 of the 

effect on L of a time shift in p P , or 6a is shown in figure 2, The yaw rate, r, 

and lateral acceleration a , were also used in the analysis but they were not time 

shifted. The zero-shift value is assumed to be the correct value, and a positive 

shift indicates that all the other signals lead the shifted variable. As shown in the 

figure, shifts in p or 6 have significant effects on the estimated value of L 

A positive time shift of 0.1 second for 6 results in a 50-percent error in L 

A negative time shift of 0.1 second in p also results in a 50-percent error. Time 

shifts larger than 0.1 second have been observed in flight data. Reference 15 

shows similar results for most of the stability and control derivatives of five 

aircraft, although the magnitude and direction of the effect of the shift on the 

derivatives are not necessarily the same as shown in figure 2. 

P 

Y 

a P' 

a P' 

10 
o pshift 

o pshift 

A ba shift 

L 

p’ 

radlsec /rad 2 

-80 

-120 -401 -.48 

-.32 -.I6 0 .I6 .32 .48 

Time shift, sec 

Figure 2. Estimated L as a function of 

time shift. 

P 

Instrument positions and angular orientations are important factors in analyzing 

flight data. There are usually no particular requirements on where the instruments 

must be, but it is important to know precisely where they are and how they are 

alined in order to account for the effects of their displacement from the center of 

gravity or their misalinement from the body axes. 

Knowing the positions of the accelerometers and the angle-of-attack and angleof-sideslip vanes is particularly important. If these instruments are offset from the 

center of gravity, corrections can be made in the analysis. If the data are not 

corrected for vane location, the fit of the data will be poor, particularly when 

angular rates are high. If the data are not corrected for accelerometer position, 

some of the estimated derivatives (Cy and CL , in particular) will be affected. 

P a 

If the correction for accelerometer position has not been made, it usually 

becomes evident when the measured and estimated data are compared. In figure 3 (a), for example, the fit of the flight and estimated data for the 3/8-scale F-15 

remotely piloted research vehicle (RPRV , ref. 16) for roll rate, p , is excellent, but 

there are some discrepancies in the fit of the data for a particularly where fi is 

large. This is the type of mismatch that occurs if the accelerometer location is 

different from that assumed in the model. If a more precise determination of the location of the lateral accelerometer is made and included in the estimation process, a 

better fit results. The comparison that resulted when the assumed vertical location 

of the lateral accelerometer on the RPRV was changed by 15 centimeters (which 

corresponds to 40 centimeters on an F-15 aircraft) is shown in figure 3 (b) . The fit 

for a is much better and the fit for p is slightly improved. 

Y’ 

Y 

It might be thought that such a small inconsistency would have an insignificant 

effect on the estimates of the derivatives. Figure 4(a) shows the coefficient Cy 

estimated with the accelerometer position assumed in figure 3 (a); figure 4(b) 

P 

11 
p, deglsec - 

Flight 

Estimated 

" 

-100 I I 

r 

-200 I I I I I I 

0 1 2 3 4 5 6 7 8 9 

Time, sec 

(a) Incorrect accelerometer location. 

loo r - Flight "_ Estimated 

I ! I ! I 

I I I I 

r 

-200 I 1 I I ~ -1 I I I I 

0 1 2 3 4 5 6 7 8 9 

Time, sec 

(b) Correct accelerometer location, 

Figure 3. Effect of assumed lateral accelerometer location 

on fit of flight and estimated data. 

12 

" 
I Uncertainty level 

(a) Incorrect accelerometer location. 

Accelerometer 

location 

-0- Correct 

"- Incorrect 

1 Uncertainty level 

o! 

-2 I I 1 I 1 

0 4 8 12 16 20 24 28 

(b) Correct accelerometer location. 

Figure 4. Values of Cy found when lateral accelerometer 

location is specified correctly and incorrectly. 

P 

shows the coefficient estimated when the assumed accelerometer position was 

changed by 15 centimeters, as in figure 3 (b) . The values of Cy in figure 4 (b) are 

approximately 50-percent greater in magnitude than those in figure 4(a). 

Obviously, the stability and control derivative estimates are sensitive to 

instrument location. 

P 

Aircraft carry a wide range of instrumentation to sense angle of attack and 

angle of sideslip, including boom-mounted vanes, "cheek" or "chin" vanes, 

differential pressure ports, and inertial navigation systems. Care should be exercised in choosing the type of instrumentation for derivative estimation purposes. 

"Cheek" and "chin" vanes are subject to local flow effects and require extensive 

13 
calibration , pressure ports are noted for measurement lags, and inertial navigation 

systems do not account for wind shears or turbulence. Boom-mounted vanes have 

been found to be the most satisfactory method of measuring these parameters. 

However, local flow effects must still be evaluated and vane dynamics may be a 

problem. In addition , installation of a rigid boom may involve significant modification to the vehicle. 

While instrumentation and data acquisition can pose certain problems for the 

analyst, these problems can usually be overcome by care and planning. The 

analyst need only know what the instruments actually measure and to what effects 

the estimator is sensitive. 

APPLICATIONS AND SPECIAL PROBLEMS 

For 12 years, the NASA Dryden Flight Research Center (DFRC) has used the 

maximum likelihood technique to estimate stability and control derivatives from 

flight data. During this time, a considerable amount of experience has been gained 

in both the theoretical and practical aspects of using this method. Stability and 

control derivatives have been obtained from over 3500 maneuvers from 32 aircraft, 

including many unusual configurations. Among these aircraft are a hypersonic 

rocket-powered research vehicle (the X-15); a series of lifting bodies (the M2-F2 , HL-10, M2-F3, X-24A, and X-24B); two large commercial airliners (the CV-990 

and B-747); several general aviation aircraft (including the Piper Comanche and 

the Beech 99); a high-altitude reconnaissance aircraft (the YF-12); two large 

supersonic bombers (the B-70 and B-1); several remotely piloted aircraft , including 

an oblique wing vehicle; and a large number of fighter aircraft (the F-8 , F-111, 

F-15, YF-16, and YF-17). The flight conditions have included Mach numbers up 

to 5 , altitudes up to 30 kilometers, angles of attack from -2OO to 53O , and normal 

accelerations up to 4 g's. Virtually all derivative extraction at the Dryden Flight 

Research Center is now done with maximum likelihood estimators. 

Most of the data acquired have been successfully analyzed using a simple linear 

model. For a linear model, equations (1) and (2) can be written in the 

following form: 

The simple linear two-degree-of-freedom model of the longitudinal dynamics of an 

aircraft is given by the following equations from reference 17. 

k(t) = 

14 

10 cos cp 0 

a (t) [e"::: + 
I 

’0 O1 

J(5) of equation (4) is then minimized over the time interval T by adjusting the five 

stability and control derivatives and the three bias terms in the vector 5 to minimize 

the difference between the measured response and the computed response, z (t) and 

z5 (t> . 

The simple linear lateral-directional model is given by the equations I 

LP LP Lr 

PP Nr 

0 

NN 0 

01 (cos ‘P) (tan 8) 0 L 

L J 

1 

0 11 + 

0 0 

0 0 

0 0 :I 0 

(sin a) L L Lr N N Nr Yg Yg Y L Lg L N N6 N (b Q 

PP PP ar O ‘a r O ‘a r Ybias] 

The MMLE program (ref. 11) was designed to handle these simple longitudinal 

and lateral-directional models. While these models are representative of most 

aircraft at many flight conditions, occasionally unusual aircraft configurations, 

15 
unusual analysis requirements, or unusual flight conditions create special problems 

for the analyst. Several of these problems are discussed below. Some of these 

problems require the use of a more versatile program such as MMLE 3 (see 

appendix). 

Turbulence 

Most flight data are analyzed with algorithms that do not model the effect of 

turbulence. These algorithms perform poorly when significant turbulence is present. Figure 5 is a comparison of longitudinal flight data obtained in atmospheric 

- Flight . . . . . . . Estimated 

IIIIII 

4r 

..... 

e, deg 0 

-4 I 

2r I 

a, deg 4 - 

0 I 

Figure 5. Comparison of flight data 

obtained in turbulence with computed 

data from estimator that does not 

account for turbulence. 

16 
turbulence with the computed data from a maximum likelihood estimator that does 

not account for turbulence. The estimator uses the linear model described by 

equations (7) to (9). The fit is obviously unacceptable, and poor derivative 

estimates resulted. The maximum likelihood estimator can use a model that includes 

turbulence effects (ref. 18) , but this results in increased computational complexity. 

The method then estimates the turbulence time history, in addition to the unknown 

coefficients. The flight data of figure 5 were analyzed with a maximum likelihood 

estimator that accounts for turbulence, and the resulting fit is shown in figure 6. 

The fit is excellent, and reasonable derivative estimates were obtained (ref. 18). 

lo r - Flight . . . . . . . . Estimated 

4r 

-4 c I I I I I I 

0 2 4 6 8101214 

Time, sec 

Figure 6. Comparison of flight data 

obtained in turbulence with computed data 

from estimator that accounts for turbulence. 

Thus, the capability for obtaining derivative estimates in the presence of 

turbulence does exist. However, several cautions must be stated. First, the 

computation time required is generally about 30 percent greater than with the 

estimator that does not account for turbulence. Second, it is conceivable that 

17 
the aerodynamic derivatives are different in turbulence than in smooth air. 

Although such differences have not been documented in flighty care must be 

exercised in using derivatives obtained in turbulence for application in smooth air. 

Of course if handling qualities in turbulence are being studied the converse problem 

exists. Third questions arise about the accuracy of the turbulence model used and 

the sensitivity of the algorithm to errors in modeling the turbulence. Because of 

these three considerations every attempt should be made to obtain most data in 

smooth air. However for data that are difficult or expensive to repeat or for 

experiments explicitly related to turbulence the algorithm that accounts for 

turbulence is available. 

Structural Modes 

The identification of structural coefficients from flight data is of great importance for modern aircraft. Research is currently being conducted in this area by 

many aircraft organizations, and certain successes have already been reported 

(ref. 19). In addition structural modes are of interest when performing a stability 

and control analysis. Several techniques are available for treating data corrupted 

by structural vibration to improve the results of the stability and control analysis. 

Three of these techniques-the removal of the structural effects pseudostatic 

structural modeling and dynamic structural modeling-are discussed in this section. 

The most common approach to structural vibration problems is to remove the 

effects of the structural vibration. This approach assumes that the structural 

vibration is in relatively high-frequency modes uncoupled from the rigid-body 

oscillation. The power spectra of the signals are obtained and digital notch or 

low-pass filters are designed (ref. 20) to remove the structural effects without 

degrading the rigid-body data. Care must be exercised to avoid introducing data 

time skews by filtering. Such time skews could present more problems than the 

original structural vibration (ref. 15). The advisable approach is to filter all the 

signals with the same digital filter. The necessity for such digital filtering must be 

considered when the data system sampling rates are chosen: Sampling rates of 100 

to 200 samples per second are often required for input to the filter, even though 

the filtered data are used at only 10 to 50 samples per second (ref. 21). Although 

the increased sampling rate requirement can be eliminated by using active analog 

filters before the data are sampled this method creates other problems. The 

unfiltered data cannot be recovered and the filter characteristics cannot be changed 

after the flight. Also the addition of analog filters requires hardware changes in 

the data system. Passive analog filters should be avoided if possible, because the 

filter break frequency and thus the time lag, changes with the impedence loading. 

Unless the lags are very small, each passive filter in the system must be checked 

to determine its actual lag. 

High-frequency structural vibration in the measurements is often caused by 

mounting the instrument package on a flexible member. Therefore, the effects 

of such vibrations on the data can often be alleviated by mounting the instrument 

package at a location less subject to vibration. Obviously such a solution is not 

always practical. 

18 
The second approach to accounting for structural effects is pseudostatic 

structural modeling. This approach is appropriate for aircraft that deform under 

load, causing the vehicle shape to be significantly different for different flight 

conditions. For such aircraft, the aerodynamic coefficients are functions of dynamic 

pressure or load factor. This pseudostatic deformation effect requires no modification of the analysis technique. It does, however, require the investigation of a 

wide range of flight conditions to determine the trends of the coefficients as a 

function of flight condition. 

The stability and control derivative estimates are often quite insensitive to 

pseudostatic structural deformations. Figure 7 shows flight-determined estimates 

0 1.0 

0 1.5 

A 2.0 

+ 3.0 

x 1.0 

I Uncertainty level 

.126 CI Fairing is of lg flight data 

~ 

0 2 4 6 8 10 12 14 16 

a, deg 

Figure 7. CN for elevated g flight. 

a 

of C for an F-11lA airplane with a wing sweep angle of 58O and load factors of 

1 to 3 g's (ref. 22). Although the F-11l.A airplane is large and relatively flexible for 

a fighter airplane, no effects of the structural deformation are apparent in the 

data shown. 

Na 

The third method of accounting for structural effects is to dynamically model 

the structural modes and their interactions with the aerodynamics. For large flexible 

aircraft, where the structural modes have low frequencies and couple significantly 

with the rigid-body motion, this may be the only applicable analysis technique. 

This is a currently active research area, and, although many investigations are 

addressing the structural identification problem, no completely satisfactory 

practical application has been demonstrated. Therefore, no definitive list is 

19 
available of the instrumentation, maneuvers, and ground test data required to 

obtain accurate coefficient estimates in the presence of dynamic aerostructural 

interaction. 

Linear Dependence Problems 

Many of the difficulties in estimating stability and control coefficients from 

flight data are linear dependence problems (ref. 3): Two or more of the variables 

to be estimated are linearly dependent and cannot be separately estimated from the 

data available. There are two basic causes of these problems. 

The first cause of linear dependence problems is an overly complex model. 

There is a temptation for the analyst to make the model as detailed and accurate as 

possible, including every term that might conceivably arise in any situation. The 

increased computational costs are accepted in exchange for the presumably increased 

accuracy and greater generality. If the additional terms are completely known, as in 

the case of kinematic cross-coupling terms (see Cross Coupling section), there are 

no particular problems with this approach. However, the additional terms often 

include unknown coefficients, which must be estimated along with the basic stability and control derivatives. If too many coefficients are unknown, there will not 

be enough information in the data to obtain accurate estimates. Not only will the 

estimates of the added coefficients be poor, but the estimates of the basic stability 

and control derivatives will also be degraded. In other words, rather than 

increased accuracy, a complicated model can actually result in poorer estimates. 

In some cases, the algorithm will not converge and no estimates can be obtained. 

Because of these difficulties, a program should not be designed to handle all 

situations, then left in the hands of an inexperienced technician. Rather, the 

program should be designed for a basic set of equations, with options or modifications that apply to specific circumstances. The analyst must have the experience, 

or be guided by someone with the experience, to recognize the important factors in 

each situation and invoke the appropriate option or modifications. 

The second cause of linear dependence problems is inadequate control input. 

If the control input does not adequately excite all the dynamic modes of the model, it 

may not be possible to estimate all the coefficients. Problems with inadequate 

control inputs are common when the airplane is flown with the stability augmentation system (SAS) on. Without additional information, it is impossible to distinguish 

between the basic airframe damping and the damping induced by the control 

feedback. To obtain derivatives from SAS-on maneuvers, each control surface 

must have an independent control input in addition to the SAS feedback. 

A similar problem occurs when two or more control surfaces move nearly in 

phase. A common example occurs when deflection of the lateral stick moves both 

the differential tail and the wing ailerons. In this case, only an overall equivalent 

derivative that combines the differential tail and aileron effects can be estimated. 

To separate the individual control effects, each control surface must have an 

independent input. 

20 
The Cramer-Rao bounds (uncertainty levels) are useful for evaluating whether 

linear dependence problems exist (ref. 3). These bounds give an estimate of the 

uncertainty in each coefficient. Figure 8 shows estimates of Cn for a PA-30 

P 

per rad 

.80 

.40 

0 

-.40 

-.80 

I$ A A 

Flap 

deflection 

0 Zero 

0 One-half 

A Full 

0 

-1.20 I I I I 

-2 0 2 4 6 8 10 

a, deg 

Figure 8. Variation of Cn with angle of 

attack without uncertainty levels. 

P 

airplane at three flap settings. There is a significant amount of scatter, and little 

information about C can be gleaned from this figure. In figure 9 the same data 

are shown with uncertainty levels. The data points with small Cramer-Rao bounds 

can be faired readily when the points with large Cramer-Rao bounds are ignored. 

n 

P 

In an attempt to determine the reason for the poor Cramer-Rao bounds associated 

with some of the data in figures 8 and 9 the data from rudder and aileron maneuvers 

were plotted separately (fig. 10). The data from the aileron maneuvers form a welldefined line with little scatter and good Cramer-Rao bounds but the rudder 

maneuvers result in a large amount of scatter and poor Cramer-Rao bounds. It is 

obvious that the rudder pulse does not excite the airplane motion adequately for 

identification purposes. To best identify all the stability and control derivatives 

of the PA-30 airplane maneuvers with both aileron and rudder inputs should be 

used. As a substitute, an aileron maneuver can be analyzed together with a rudder 

maneuver to obtain a single set of estimates based on both sets of data. Figure 11 

shows the same data as in figure 10 reanalyzed with the multiple maneuver 

approach (refs. 3 and 9). The scatter and poor Cramer-Rao bounds have disappeared. The fairing from the aileron maneuvers (fig. 10) is shown to be a good 

representation of the response to both aileron and rudder pulses. 

21 

I 
Flap 

deflection 

0 Zero 

o One-half 

.80 

.40 

0 

'n I 

P 

per rad -.40 

-.80 

-1.20 

-2 0 2 4 6 8 10 

a, deg 

Figure 9. Variation of Cn with angle of 

attack with uncertainty levels. 

P 

.80 Rudder maneuvers Flap deflection 

.40 Zero 

0 One-half 

A Full 

'n 

per rad 

0 1 Uncertainty level 

P "_ Fairing 

-.40 

-.80 

-1.20 1. - L" 

Aileron maneuvers 

'n 9 

PO 

per rad ""-"-" ~" "" "" ~ - __ 

Figure 10. Variation of Cn with angle of 

attack for rudder and aiZeron maneuvers. 

P 

22 
Flap 

deflection 

Zero 

0 One-half 

A Full 

1 Uncertainty level 

- - - Fairing from aileron 

maneuvers (fig. IO) 

-40 r I 

Figure 11. Variation of Cn with angle 

of attack for multiple maneuver analysis. 

P 

These PA-30 data provide a good example of a linear dependence problem and 

the use of the Cramer-Rao bound to help deduce the reasons for the problem and 

devise a solution. The particular problem illustrated here, the inadequacy of the 

rudder inputs, is symptomatic of many aircraft. In general, both rudder and aileron 

inputs should be included in each lateral-directional maneuver to insure adequate 

excitation. 

Time-Varying Systems 

One of the most common simplifications used by the data analyst is the assumption that the system being studied does not vary significantly during the brief time 

span of a maneuver. This simplification translates into especially simple models. 

In equations (7) to (12), this means that the dimensional derivatives, the biases, 

and the terms composed of V, 8, and cp are all assumed constant during the 

maneuver. However, under many circumstances the system varies significantly 

during the maneuver. In some of these cases, the analysis technique can be 

extended to cover the time variation of the system. 

If dynamic pressure or velocity changes significantly during a maneuver, the 

dimensional derivatives will change correspondingly. Fortunately, the dependence 

of the dimensional derivatives on dynamic pressure and velocity is well known. 

The nondimensional derivatives do not usually vary significantly with velocity and 

dynamic pressure. The approach taken, therefore, is to identify the nondimensional derivatives. The dimensional derivatives in the equations of motion can then 

be written as the nondimensional derivatives multiplied by the dimensionalization 

ratios, which are computed as known functions of time using the measured velocity 

23 
and dynamic pressure. By writing the equations in this manner, the known time 

variation of the system can be accounted for. 

The problem of determining time-varying coefficients is then replaced by the 

problem of determining constant coefficients with known time-varying multipliers. 

This technique relies on noise-free measurements of velocity, dynamic pressure, 

and other changing flight conditions. The technique cannot be applied to transonic 

data where significant changes in Mach number occur, because the dimensionless 

derivatives are functions of Mach number. 

In both the case of large variations in altitude and the case of varying flight 

conditions, the time-varying nature of the system is reduced to knowr, effects. 

Thus, the analyst is not determining time-varying derivatives in the general sense. 

The computer costs for the analysis of these "time-varying" systems are high. 

As implemented in the MMLE 3 program, the "time-varying" option triples the 

computer time used. Although expensive, the "time-varying" option permits the 

analysis of the most commonly encountered time-varying systems in aircraft 

stability and control derivative determination while retaining the basic simplicity of 

the model used. 

Nonlinearities 

Most airplanes show great nonlinearities in total forces and moments when 

considered over large ranges of flow angles. These nonlinearities are neglected 

in the typical derivative estimation process because the derivatives are local 

linearizations of the total forces and moments. For example, the pitching moment 

coefficient, Cm , as a function of c1 is quite nonlinear over a large angle-of-attack 

range. If the change in angle of attack can be kept small for a given maneuver, 

the locally linearized derivative, C , can be estimated and plotted as a function 

of angle of attack. Figure 12 (from ref. 23) shows Cm as a function of a for an 

angle-of-attack range of -2OO to 50°. The estimates are consistent and show a clear 

trend that is in fair agreement with wind-tunnel estimates. By using the linear 

perturbation model for each maneuver, an excellent comparison can be made with 

the globally nonlinear wind-tunnel data. This simple and widely used technique 

avoids many of the problems of modeling nonlinear systems and is readily applied to 

aircraft where maneuvers are typically small perturbations about a point in a much 

larger envelope. 

ma 

a 

In some cases, the linear model of the aircraft is inadequate to determine the 

needed information, and the known system nonlinearity must be addressed directly 

by the maximum likelihood technique. An example of this type of problem is the 

need to estimate the drag polar of an aircraft. The simplified drag polar is 

=cD +cDIcL 

2 

D 0 

24 

. . . .. 
.02 

.01 

0 

cm ’ 

(I 

per deg -.01 

-.02 

I 

I 

-. 03 I 

-30 

I 

I 

1- 

I 

I 

I 

-1 

I 

T 

k; I 

-20 -10 

I 

I 

I 

YI 

Ik 9 

I 

1 P 

i 

I 

0 Flight 

1 Uncertainty level 

- Wind tunnel 

30 40 50 

Figure 12. Comparison of flight and wind-tunnel 

estimates of Cm over a large angle-of-attack range. 

a 

where the second term, CD1CL2, accounts for the induced drag, This term results 

in an essential nonlinearity in the problem-that is, although C can be written as a 

linear function of the angle of attack and elevator position, CD cannot. Therefore, 

the estimation of C from dynamic data involves equations that cannot be written in 

the linear form given by equations (5) and (6), and the more general functional 

form of equations (1) and (2) must be used. A complete description of this analysis 

is given in reference 24, and the results shown in figures 13 and 14 are from that 

reference. Figure 13 is a comparison of the measured and computed responses of 

an aircraft during a pushover-pullup maneuver where the computed response is 

generated by a nonlinear model incorporating a drag polar. The agreement is 

considered excellent. In figure 14, the drag polar obtained from this maneuver is 

compared with wind-tunnel estimates of the drag polar. 

L 

D 

The fundamental problem of nonlinear maximum likelihood estimation is that, in 

practice, the form of the nonlinear model is unknown. In cases where the form of 

the model is known, as in the drag polar case, meaningful estimates can be expected 

if the maneuvers excite the nonlinearity of the system. Very little useful experience 

is available to guide the analysis of nonlinear systems where a linearized model is 

inadequate and the form of the nonlinearity is not known. If ad hoc techniques are 

used in modeling unknown nonlinearities , great care must be taken or meaningless 

results may be obtained. 

25 
~ Flight "" Estimated 

5- 

q, 0. deglsec 

-5- ' 

. 2or 

2- 

a 

a 

.2 - 

0 12 3 4 5 6 7 8 9 10111213 

9 

0 12 3 4 5 6 7 8 9 10111213 

C .5 

Ltrim 

.4 

"- Maximum 

likelihood estimate 

- ""- Wind tunnel 

I I I "_I". 

Dtrim 

O.02 .03 .04 .05 .06 .07 .08 

C 

Time, sec 

Figure 14. Comparison of drag 

Figure 13. Comparison of flight data polars obtained from estimates and data estimated by using a nonlinear based on wind-tunnel and 

model for a pushover-pullup maneuver. flight data. 

Cross Coupling 

The simplified aircraft equations of motion (eqs. (7) to (12)) are separated into 

longitudinal and lateral-directional modes, and it is assumed that no cross coupling 

exists. When significant coupling does exist, there are two approaches that can be 

used. The most obvious approach is to use the full nonlinear six-degree-of-freedom 

equations of motion with coupling terms. Although simple in principle, this method 

is plagued with practical difficulties. These include problems of complexity, 

maneuver design, and computer time and core. The second approach is to use the 

measured lateral-directional data as inputs to the longitudinal equations and the 

measured longitudinal data as inputs to the lateral-directional equations (ref. 21). 

This approach requires that the measurements of all the state variables be available 

and have low noise levels. 

26 
Two types of cross coupling exist. The first type is kinematic cross coupling, 

which is cross coupling arising from the equations of motion. A typical example 

is gyroscopic forces. Kinematic cross coupling can be a problem even with 

symmetrical aircraft. One of the most common kinematic cross coupling problems is 

nose slice during longitudinal pulses in steady turns. To accoun? for nose slice, 

the kinematic cross coupling term r(sin cp) must be added to the 0 equation. 

The second type of cross coupling is aerodynamic cross coupling, which 

appears in the expansions of the force and moment coefficients. Predictions indicate 

that aerodynamic cross coupling is significant at high angles of attack, even for 

symmetric aircraft (ref. 25). For asymmetric aircraft, of course, aerodynamic 

cross coupling exists at all angles of attack. 

Figure 15 is a three-view drawing of an oblique wing RPRV flown at the Dryden 

Flight Research Center (ref. 21). The wing of this aircraft can be skewed up to 

e Reference center 

Figure 15. Three-view drawing of remotely 

piloted oblique wing aircraft. 

27 
45O. The oblique wing concept is of interest because of its potential for transonic 

drag reduction. When the wing is skewed, both aerodynamic and kinematic cross 

coupling are important. Figure 16 shows a comparison of flight and estimated 

-10 J 

.....,,. 

0 2 4 6 8 10 12 14 16 18 

Time, sec 

Figure 16. Comparison of flight-measured and 

estimated lateral-directional motions of oblique 

wing aircraft with 45O of wing skew where 

cross-coupling terms are omitted. 

28 
lateral-directional data obtained with the wing skewed to 45O. This unintentional 

maneuver was caused by interference in the radio control system. All the crosscoupling terms were ignored for this fit, which is totally unacceptable. Figure 17 

lo 

O 

1 

~ Estimated 

-10 J 

-40 I 

20 1 

10 - 

0- 

-10 J 

.ll,....."- 

0 2 4 6 8 10 12 14 16 18 

Time. sec 

Figure 17. Comparison of flight-measured and 

estimated lateral-directional motions of oblique 

wing aircraft with 45O of wing skew where 

cross-coupling terms are included. 

29 
shows the data for the same maneuver when both kinematic and aerodynamic crosscoupling terms are computed with measured longitudinal signals. The fit is now 

very good considering the unconventional nature of the maneuver. This example 

shows that cross-coupling effects can be accurately accounted for using this 

simple technique. 

Estimation of Pitching Moment Due to Vertical Acceleration 

The estimation of Cm from flight data is a problem that exemplifies many of the 

d 

considerations discussed previously. The derivative C cannot normally be 

estimated from flight data because Cm and Cm are linearly dependent. Analysts 

have had to be content with estimating C + Cm . At DFRC , maneuvers specif- m 

ically designed to remove the dependence of C and C (ref. 26) have recently , 

been developed. Figure 18 shows a comparison of flight and estimated data for one 

of these maneuvers, an aileron roll with a series of elevator pulses. The fit is 

md 

(5 4 

(5 q 

m 

4 ml? 

excellent, and reasonable estimates of C and Cm , as well as all the other 

m 

9 6 

longitudinal stability and control derivatives, were obtained. In figure 19, the 

estimates of C and Cm from 13 maneuvers are compared with simplified 

analytical predictions. Because the airplane undergoes a complete 360° roll in each 

of these maneuvers, the cross-coupling effects of the lateral-directional motion 

on the longitudinal analysis are extremely important. In fact, the removal of the 

linear dependence of Cm and Cm is primarily due to the cross-coupling effects. 

For the maneuver shown in figure 18, dynamic pressure varies from 3.5 

to 7.5 kN/m2, so a time-varying analysis is necessary. (The maximum permissible 

variation in dynamic pressure is about 10 percent for computer programs that do 

not account for time-varying effects .) The altitude changes are sufficiently large 

and rapid that pressure lags in the static pressure measurements are significant. 

The 0.4-second lag of the static pressure system results in errors of up to 

10 percent in the uncorrected dynamic pressure. The successful analysis of this 

maneuver is a good indication of the state of the art in the estimation of stability 

and control derivatives from dynamic flight-test data. 

m 

4 ci 

q d 

30 
- Flight 

.__.. Estimated 

-200:11 

lq"--"-----\ 

-100 

-10 L 

0 2 4 6 8 10 12 14 16 18 20 

Time, sec 

> -. 

Figure 18. Comparison of flight-measured and 

estimated data for aileron roll with elevator pulses. 

Cm and C estimated independently. 

4 m& 

31 
Source of derivative 

estimate 

Two-point hesitation rolls 

Smooth rolls 

Calculations 

per rad I 

-a 

Maneuvers 

Figure 19. Comparison of 

independent flight estimates of Cm 

Q and C with calculated values. 

m& 

CONCLUDING REMARKS 

During the past 12 years at the NASA Dryden Flight Research Center, the 

maximum likelihood estimator has been used to analyze over 3508 maneuvers from 

32 aircraft. Most of the analysis has involved the extraction of stability and control 

derivatives from dynamic flight maneuvers. In this report, procedures for obtaining 

high-quality estimates from dynamic maneuvers were discussed and the signals 

required for analysis were listed. The importance of well documented and accurate 

instrumentation was stressed, and several common instrumentation problems 

were presented. 

The application of the maximum likelihood estimator to some special problems 

the analyst encounters when working with unusual aircraft configurations analysis 

requirements, or flight conditions was also discussed. The analysis of data gathered 

in atmospheric turbulence was outlined, and three methods for handling the effects 

of flexibility on the estimation process were given. Sources of linear dependence 

problems were noted and potential solutions to these problems were considered. 

A class of problems with time-varying dimensional coefficients was shown to be 

solvable by proper formulation of the equations of motion. In addition, the maximum 

likelihood method was shown to be applicable to some nonlinear aircraft problems, 

with the determination of an aircraft drag polar given as an example. Kinematic 

and aerodynamic cross coupling between longitudinal and lateral-directional modes 

was discussed and illustrated using an example of the successful estimation of the 

32 
derivatives of an oblique wing aircraft. The use of a specially designed maneuver 

for separating the derivatives for rate of change of angle of attack (6) and pitch 

rate (4) was discussed. The analysis of this maneuver required several of the 

special techniques discussed previously and is an excellent example of the 

state of the art in estimating aircraft stability and control derivatives. 

Dryden Flight Research Center 

National Aeronautics and Space Administration 

Edwards, Calif. , November 28, 1978 

33 

I 
APPENDIX 

DESCRIPTION OF MMLE 3 PROGRAM 

The MMLE 3 program is an outgrowth of MMLE (ref. 11) that is currently in 

use at the Dryden Flight Research Center. The new program, which was developed 

to satisfy the need for more versatility, is designed to handle a general set of linear 

or bilinear dynamic equations of arbitrary order. Measurement noise and, 

optionally, process noise (such as turbulence) are included in the equations. 

Because of its generality, MMLE 3 sometimes requires a large amount of input 

to completely specify a given problem. In addition, some program features that 

are useful for some problems have little or no meaning for other problems. To 

satisfy the conflicting requirements of generality and ease of use, the MMLE 3 

program is divided into two levels. The basic level consists of a general maximum 

likelihood estimation program, applicable to any linear system. This program can 

be run by itself, using input data to completely describe the system to be analyzed 

and the program options to be used. 

At the second level, the basic program is used as the core of a program adapted 

to a particular situation. The adaptation is accomplished by a set of user routines, 

so called because the user can write or modify them for his own application. When 

the user routines for a particular application are implemented, the program input 

need not contain the detailed system specifications, but only those items that change 

from case to case. The concept of a modular set of user routines allows the basic 

program structure to be very general, while retaining the simplicity of input 

possible in programs designed for specific systems. The use of user routines 

to simplify the required input deck is most advantageous when many cases are 

being processed. In addition , the user routines can be coded to perform computations or operations unique to a specific application. Examples of this function 

include reading data from special formats, normalizing or correcting the estimates 

to a given set of conditions, and punching the results in a form suitable to auxiliary 

programs for purposes such as plotting estimates or updating a simulator. 

The MMLE 3 program includes a set of user routines tailored for the estimation 

of aircraft stability and control derivatives. With these routines, the MMLE 3 

program is capable of analyzing a wide variety of aircraft stability and control 

maneuvers. Among the specific capabilities is the inclusion of kinematic and 

aerodynamic cross-coupling terms. Large variations in velocity and dynamic 

pressure during a maneuver can be handled by the program, as long as the 

nondimensional derivatives remain constant. 

These standard user routines are based on the equations of motion presented 

in the following paragraphs. Various extensions to these equations are used 

in specific instances. The nonlinear terms in these equations are implemented 

by linearization about the measured values. 

34 
The basic longitudinal state equations are 

- 

& = - (cL + + q + 91 (cos e cos cp cos a + sin e sin a) mV V 

- tan p (p cos a + r sin a) 

I Q = qsccm + rp (I~ - lX) + (r - p 

22 

Y 

6=qcoscp-rsincp+6 0 

and the longitudinal observation equations are 

a m = Ka (a - >q) 

4, = 4 + 4, 

The expansions of the longitudinal force and moment coefficients are 

cN=c a+c 6+CN Na 0 

cm=cm a+c =+cm 

m 2V 6+C 

a 4 6 mO 

C =C a+CA6+CA 

A Aa 6 0 

35 

I 
C =C COS~+~ sina LN A 

where the 6 terms are summed over all controls. 

The basic lateral-directional state equations are 

- 

P=*(c mV Y +po)+Bcosesincp+psina-rcosa V 

@Ix - ?Ixz = csbCIl + qr(Iy - Iz) + pqIxz 

PIz - @Ixz - CsbC, + pq (Ix - Iy) - qrIxz 

@ =P 

and the lateral-directional observations are 

Pm = P 

r =r 

m 

cp,=cp 

rj, = rj + Po 

i-,=P+P 0 

The expansions of the lateral-directional force and moment coefficients are 

cy=cy p+c 6+C 

P y6 

rb 

P r 

36 
I 

rb 

n 2V P 0 

where the 6 terms are summed over all controls. 

37 
REFERENCES 

1. Klein, V . : Determination of Longitudinal Aerodynamic Derivatives From 

Steady-State Measurement of an Aircraft. AIAA Paper 77-1123, Aug. 1977. 

2. Iliff, Kenneth W . ; and Taylor, Lawrence W . , Jr . : Determination of Stability 

Derivatives From Flight Data Using a Newton-Raphson Minimization Technique. 

NASA TN D-6579, 1972. 

3. Iliff, 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. 

4. Hamel, P . G . : Status of Methods for Aircraft State and Parameter Identification. 

AGARD CP-187, 1976, pp . 8-1 to 8-16. 

5. Eykhoff, Pieter: System Identification. John Wiley & Sons, Inc. , c. 1974. 

6. Goodwin, Graham C . ; and Payne , Robert L . : Dynamic System Identification: 

Experiment Design and Data Analysis. Academic Press, 1977. 

7. Balakrishnan , A. V. : Stochastic Differential Systems I. Filtering and ControlA Function Space Approach. Lecture Notes in Economics and Mathematical 

Systems, 84, M. Beckmann, G. GOOS, and H. P. Kunzi, eds., SpringerVerlag (Berlin), 1973. 

8. Powell, J . David; and Tyler, James S . , Jr .: Application of the Kalman Filter 

and Smoothing to VTOL Parameter and State Estimation. 1970 Joint Automatic 

Control Conference of the American Automatic Control Council, Paper 18-G, 

American Soc . Mech. Eng . , June 1970, pp . 449-450. 

9. Iliff, Kenneth W.; and Maine, Richard E .: Further Observations on Maximum 

Likelihood Estimates of Stability and Control Characteristics Obtained From 

Flight Data. AIAA Paper 77-1133, Aug. 1977. 

10. Herrington, Russell M . ; Shoemacher , Paul E. ; Bartlett , Eugene P . ; and 

Dunlap, Everett W .: Flight Test Engineering Handbook. AF Tech. Rept. 

No. 6273, Air Force Flight Test Center, May 1951 (rev. June 1964). 

11. Maine, Richard E. ; and Iliff, Kenneth W , : A FORTRAN Program for Determining 

Aircraft Stability and Control Derivatives From Flight Data. NASA TN D-7831, 

1975. 

12. Wolowicz , Chester H . ; and Yancey , Roxanah B . : Experimental Determination 

of Airplane Mass and Inertial Characteristics. NASA TR R-433, 1974. 

38 
13. Suit, William T . : Aerodynamic Parameters of the Navion Airplane Extracted 

From Flight Data. NASA TN D-6643, 1972. 

14. Brenner , Martin J . ; Iliff, Kenneth W. ; and Whitman, Robert K. : Effect 

of Sampling Rate and Record Length on the Determination of Stability 

and Control Derivatives. NASA TM-72858, 1978. 

15. Steers, Sandra Thornberry; and Iliff, Kenneth W . : Effects of Time-Shifted 

Data on Flight-Determined Stability and Control Derivatives. 

NASA TN D-7830, 1975. 

16. Holleman , Euclid C . : Summary of Flight Tests To Determine the Spin and 

Controllability Characteristics of a Remotely Piloted, Large-Scale (3/8) 

Fighter Airplane Model. NASA TN D-8052 1976. 

17. Wingrove R . C .: Estimation of Longitudinal Aircraft Characteristics Using 

Parameter Identification Techniques. Ninth Annual Symposium ProceedingsWhy Flight Test? SOC . Flight Test Eng . Oct. 1978, pp . 9-1 to 9-24. 

18. Iliff, Kenneth W . : Estimation of Characteristics and Stochastic Control of an 

Aircraft Flying in Atmospheric Turbulence. Proceedings of AIAA 3rd 

Atmospheric Flight Mechanics Conference c. 1976 pp . 26-38. 

19. Rynaski, Edmund G. ; Andrisani, Dominick, 11; and Weingarten, Norman C.: 

Identification of the Stability Parameters of an Aeroelastic Airplane. 

AIAA Paper 78-1328 Aug. 1978. 

20. Edwards, John W . ; and Deets Dwain A. : Development of a Remote Digital 

Augmentation System and Application to a Remotely Piloted Research 

Vehicle. NASA TN D-7941, 1975. 

2 1. Maine Richard E. : Aerodynamic Derivatives for an Oblique Wing Aircraft 

Estimated From Flight Data by Using a Maximum Likelihood Technique. 

NASA TP-1336, 1978. 

22. Iliff Kenneth W. ; Maine, Richard E. ; and Steers, Sandra Thornberry: 

Flight-Determined Stability and Control Coefficients of the F-11lA Airplane. 

NASA TM-72851 1978. 

23. Iliff, Kenneth W . ; Maine, Richard E. ; and Shafer , Mary F . : Subsonic 

Stability and Control Derivatives for an Unpowered , Remotely Piloted 

3/8-Scale F-15 Airplane Model Obtained From Flight Test. 

NASA TN D-8136 1976. 

24. Iliff, Kenneth W.: Maximum Likelihood Estimates of Lift and Drag Characteristics Obtained From Dynamic Aircraft Maneuvers. Proceedings of AIAA 3rd 

Atmospheric Flight Mechanics Conference, c ,1976, pp . 137-150. 

39 
II I I 

25. Orlik-Ruckemann, K. J .: Aerodynamic Coupling between Lateral and ' 

Longitudinal Degrees of Freedom. AIAA J., vol. 15, no. 12, Dee. 1977, 

pp . 1792-1799. 

26. Maine, Richard E. ; and Iliff, Kenneth W . : Maximum Likelihood Estimation of 

Translational Acceleration Derivatives From Flight Data. AIM 

Paper 78-1342, Aug. 1978. 

40 
1. Report No. I 2. Government Accession No. 

NAS'A TP- 1459 - "" ~ 

4. Title and Subtitle 

IMPORTANT FACTORS IN THE MAXIMUM LIKELIHOOD 

ANALYSIS OF FLIGHT TEST MANEUVERS 

~ 

7. Author(s) 

Kenneth W . Iliff, Richard E. Maine, and T . D . Montgomery 

9. Worming Organization Name and Address 

NASA Dryden Flight Research Center 

P. 0. Box 273 

Edwards, California 93523 

I 12. Sponsoring Agency Name and Address National Aeronautics and Space Administration 

Washington, D .C . 20546 

~~ 

Supplementary Notes 

act 

3. Recipient's Catalog No. 

- 

5. Report Date 

April 1979 

6. Performing Organization Code 

H-1076 

8. Performing Organization Report No. 

10. Work Unit No. 

505-06-64 

11. Contract or Grant No. 

13. Type of Report and Period Covered 

Technical PaDer 

14. Sponsoring Agency Code 

This paper discusses the application of a maximum likelihood 

estimator to dynamic flight test data. The information presented 

is based on the experience in the past 12 years at the NASA Dryden 

Flight Research Center of estimating stability and control derivatives 

from over 3500 maneuvers from 32 aircraft. The overall approach 

to the analysis of dynamic flight test data is outlined. General 

requirements for data and instrumentation are discussed and 

several examples of the types of problems that may be encountered 

are presented. 

17. Key Words (Suggested by Author(s)) 

Stability and control derivatives 

Dynamic flight data analysis 

Maximum likelihood estimation 

18. Distribution Statement 

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Unclassified Unclassified 

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