NASA Technical Memorandum 101722

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Estimating Short-Period Dynamics

Using an Extended Kalman Filter

Jeffrey E. Bauer and Dominick Andrisani

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June 1990

(NASA- ]'M-101 72Z)

DY|WAMICS U31NC AN

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ESTIMATING SHORT-PERIOD

_XT_NDED KALMAN FILT_

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NASA

National Aeronautics and

Space Administration
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NASA Technical Memorandum 101722

Estimating Short-Period Dynamics

Using an Extended Kalman Filter

Jeffrey E. Bauer

Ames Research Center, Dryden Flight Research Facility, Edwards, Califomia

Dominick Andrisani

Department of Aeronautics and Astronautics, Purdue University, West Lafayette, Indiana

1990

National Aeronautics and

Space Administration

Ames Research Center

Dryden Flight Research Facility

Edwards, California 93523-0273
!
ESTIMATING SHORT-PERIOD DYNAMICS USING AN EXTENDED

KALMAN FILTER

Jeffrey E. Bauer*

NASA Ames Research Center

Dryden Flight Research Facility

Edwards, California

and

Dominick Andrisani**

Department of Aeronautics and Astronautics

Purdue Unversity

West Lafayette, Indiana

Abstract

In this paper, an extended Kalman filter (EKF)

is used to estimate the parameters of a low-order

model from aircraft transient response data. The loworder model is a state space model derived from the

short-period approximation of the longitudinal aircraft dynamics. The model corresponds to the pitch

rate to stick force transfer function currently used

in flying qualities analysis. Because of the model

chosen, handling qualities information is also obtained. The parameters are estimated from flight

data as well as from a six-degree-of-freedom, nonlinear simulation of the aircraft. These two estimates are then compared and the discrepancies noted.

The low-order model is able to satisfactorily match

both flight data and simulation data from a highorder computer simulation. The parameters obtained

from the EKF analysis of flight data are compared

to those obtained using frequency response analysis of the flight data. Time delays and damping ratios are compared and are in agreement. This technique demonstrates the potential to determine, in near

real time, the extent of differences between computer

models and the actual aircraft. Precise knowledge

of these differences can help to determine the flying

qualities of a test aircraft and lead to more efficient

envelope expansion.

"AerospaceEngineer.Member AIAA.

"*Professor.MemberAIAA.

Copyright ©1990 by the American Institute of Aeronautics and

As_'onautics, Inc. No copyright is assertedin the United States

under Title 17,U.S. Code. The U.S.Government has a royalty-free

license to exercise all fights underthe copyright claimed herein

for Governmental purposes. All other rights are reserved by the

copyright owner.

d

EKF

F(x, t)

f(x, t)

g

H

h

I

K

Lv

I.,,_

Ls

LOES

M,

Mt_

P

Q

q

R

Nomenclature

distance from angle-of-attack vane to aircraft

center of gravity

extended Kalman filter

linearization of f about x at time t

nonlinear function dependent on

variable x at time t

gravitational constant

linearized output matrix, defined in

equation (22)

nonlinear output matrix

identity matrix

Kalman gain matrix

derivative of & with respect to q

derivative of & with respect to o_

derivative of 6t with respect to/5

low-order equivalent system

dimensional variation of pitching

moment with pitch rate

dimensional variation of pitching

moment with angle of attack

dimensional variation of pitching

moment with surfa_ deflection

normal acceleration

Riccati error covariance

model noise variance (process noise)

pitch rate

measurement noise variance
V

X

Z

Ot

6

6p

P

T

rd

/,o

w_

Subscripts

C

I

k

0

Superscript

T

^

velocity

measurement noise

state vector

estimate of x taking advantage

measured data at k

estimate of x before measured

data at k is available

output vector

angle of attack

pseudo control surface deflection

pilot pitch stick signal

correlation coefficient

standard deviation

first-order time lag coefficient

pure time delay

process noise

short-period natural frequency

corrected

flight measured

at discrete time k

measured parameter

initial

matrix transpose operator

estimate

Introduction

Many near-real-time techniques have been used in

the envelope expansion of recent research aircrafflike

the X-29. I In the area of flight controls, two real-time

techniques were developed. One is a frequency response technique 2 where the aircraft's open-loop frequency response is determined from measured flight

parameters_-_s frequency response is then compared

with a p_cted f-r_-uency response based on the current aircraft simulation. The second technique is a time

history overplot comparison, 3 conducted in real time.

Overplots are obtained of the aircraft response with

time histories generated from a previously determined

linear model of the aircraft, at that flight condition,

driven by the pilot's input signal as it is telemetered

to the ground.

To provide more detailed information, an additional

real-time technique, the subject of this paper, has been

investigated. This technique uses an extended Kalman

filter (EKF) to estimate a low-order equivalent system (LOES) model of the aircraft from flight-measured

time histories. References 4-9 detail other work in

this area. For other work in parameter estimation see

Refs. 10and 11.

A Kalman filter is an optimal state estimator of

a linear system. The estimate is adjusted based

on each measurement of the system. An extended

Kalman filter extends linear Kalman filter theory to

nonlinear systems.

The LOES model is based on the longitudinal shortperiod approximation in the MIL-STD-1797 for flying qualities.12 This technique is able to extract flying qualities parameters in near real time, which may

quantify discrepancies between the aircraft and the aircraft simulation.

_Before the flight, the aircraft's six-degree-offreedom, nonlinear simulation is analyzed at a specified flight condition, and a LOES is estimated. During

flight test, parameters of the low-order model are estimated and these flight-data based estimates are compared with the simulation based estimates in real time.

The results of the flight estimation are intended to help

diagnose discrepancies between the aircraft response

and the predicted response. This comparison Could be

done before the aircraft reaches its next test point, giving the flight controller time to decide whether the discrepancy is serious enough to curtail further testing.

The aircraft used in developing this technique was

the X-29 (Fig. 1). The X-29 airframe has a negative

static margin of approximately 35 percent. This instability requires a high degree of control augmentation. The control system is entirely fly-by-wire, consisting of a triply redundant primary channel operating

at 80 Hz. The longitudinal Control laws use pitch rate

and normal acceleration feedbacks for primary stability and control. Of particular importance to the EKF

algorithm is the airplane's three surface pitch control,

using canards, symmetric flaps, _strake flaps. Data

from the airplane is telemetered to the control room at

40 samples/see (40 Hz).

The estimation was made of the closed-loop system

rather than the open-loop system. Because of the high

degree of control augmentation on the X-29, the researchers thought that estimating the closed-loop system offered the best chance of success. The control

2
system was designed to make the closed-loop aircraft

behave much like the low-order model. Detailed postflight analysis would provide greater insight into any

discrepancy by separating the effects of the control

system from the aerodynamics.

If a major discrepancy between the simulation and

actual airplane is present, this algorithm may be able to

determine what the discrepancy is more quickly than

currently available technology can.

Not only general information about the aircraft performance, compared to its predicted performance, is

available, but handling qualities data is obtained as

well. First, an equivalent time delay is available. This

value can be monitored to identify potential stability

problems. The time delay computed in near real time

would be used to identify large discrepancies from the

anticipated values.

In addition, plots of normal acceleration divided by

angle of attack (r_/c_) could be obtained for both the

simulation and real-time estimations. This would allow researchers to determine if the flight values are

moving towards an unacceptable region as defined by

Ref. 12.

With the information provided by this technique in

near real time, more intelligent decisions about the ongoing flight test could be made, possibly saving time

and money. The results obtained by this technique

could also increase the efficiency and safety of the

flight test program.

The potential for reducing postflight data processing is also great. This technique could determine the

area responsible for any discrepancies, and postflight

analysis could then identify the cause.

Techniques and Procedures

Modeling

Only the transient longitudinal dynamics of the aircraft in 1-9 level flight are studied. The basic model is

the longitudinal short-period approximation

d¢ = Laot + Lqq + L66 (1)

(t = Maot + Mgq + M_6 (2)

where La, Lq, and L6 are the derivatives of d with

respect to c_, q, and 6, and Ma, Mq, and M_ are the

derivatives of (1 with respect to or, q, and 6. This approximation is equivalent to the short-period approximation for the pitch rate to stick force transfer function

given in Ref. 12, allowing handling qualities informa3

tion to be obtained. Aircraft angle of attack, pitch rate,

and normal acceleration make up the systems observation equations.

Because of the aircraft's open-loop instability, the

closed-loop dynamics were estimated rather than the

open-loop dynamics. If the open-loop equations were

used, then any small measurement errors in the surface

positions or the numerical differentiation would cause

the model to diverge from the measurements.

The aircraft uses three surfaces for pitch control.

This configuration made defining a stick force to aircraft response transfer function difficult. Since there

was no single pitch generating surface, the effects of

the three surfaces were collected into a single pseudo

control surface. A third state equation is added to the

model to represent the pseudo control

= -I 6(t) + -1 p(t - 7-d) (3)

7- ,7.

where _ is the pseudo control surface deflection and

/Spis the pilot pitch stick signal. It was reasoned that

this equation should have the appearance of an actuator model. The coefficient of the model is estimated

by the filter. Equation (3) contains a first-order lag of

the pilot's stick signal. The first-order lag accounts for

only part of the system time delay. The rest is modeled

through a pure time delay. 13

To determine the pure time delay, several a priori

estimates are made, and several filters are run in parallel. To determine which of these filters is operating

with the proper amount of pure time delay, the likelihood function is used.14 The proper amount of pure

time delay is the sum of each filter's time delay multiplied by its likelihood. However, in all cases examined

one delay always resulted in a likelihood much higher

than the others. For this reason, and to save computational complexity, the time delay associated with the

largest likelihood is used as the correct pure time delay.

Figure 2 shows that the pure time delay is implemented

outside of the EKF.

A detailed investigation of identifiability was not

conducted. However, the fact that one delay resulted in

a likelihood far larger than the others suggested identifiabilty is not a problem.

Since the aircraft cannot obtain perfect trim in flight,

estimated bias terms are added to the state equations.

In the presence of trim errors these terms balance the

state equations and accurately determine the trim state.

Without the biases, the equations would integrate a

small error over the course of the maneuver.
The equationfor measured angle of attack needs to

be corrected for the fact that angle of aUack is measured at the noseboom of the aircraft and not the ccnter of gravity. This can be modeled to the first order by

correcting for pitch rate

d

_m=_--_q

(4)

where am is the measured angle of attack and d is the

distance from the angle-of-attack vane to the aircraft

center of gravity.

The equivalent system model can now be defined

and is given by equations (5-10)

& = L,,,a + Lqq + Ltdi + Lo (5)

?1= Maa + Mqq + MrS + Mo (6)

where x(0)

= -16 + -l p + (7/ ance Po.

7" T

d

am = a- _-q (8)

qm = q (9)

r_,, = -V[Loa+ (Lq- l)q

g

+L68 + Lo] + n.,o (10)

where the subscript 0 indicates initial state.

Estimation

The estimation problem consists of estimating La,

Lq, L_, M,., Mq, Mr, L Lo Mo, 8o, r_ o of the Ion- .7.j r

gitudinal short-period approximation model as a function of time for the closed-loop aircraft response, in

near real time. To do this using an EKFr these coefficients must be made states of the filter. Therefore the

state vector (x) becomes

x = [a,q,,5, L_,,Mo,,Ls, Mt,Lq,

Mq, l, Lo,Mo,8o,r_o] T (11)

T

The extra state equations required are L,, = O, M,, =

0r =0, M6----Or/,q--0, Mq=0, =0, L0 - 0,

._/'o = 0r _0 = 0, and/,_ = 0. The short-period state

equations (5-7), augmented by those previously listed

and output equations (8-10), form the state and measurement equations for the EKF.

The EKF equations used are for a continuousdiscrete system and are listed in the following

equations. 15

The continuous system model is

_(t) = f(x(t), t) + to(t) (12)

where to is a white noise process with spectral density

Q(t) where Q is the model noise variance.

The discrete measurement model is

z_=h_(x(tk))+uk ; k=lr2,3 .... (13)

where the ,_ 's are independent Gaussian noise vectors

with mean zero and covariance Rk. It is also assumed

that to and u are independent.

The initial conditions are

x(O) (14)

is Gaussianr with mean _o and variThe state estimate propagation between samples is

_(t) = f(_(t),t) (15)

The error covariance propagation between sampies is

15(t) = F(_(t), t) P(t)

+ P(t)FT(i(t),t) + Q(t) (16)

where P(t) is the Ricatti error covariancer

F(_(t), t) is defined in equation (20).

The state estimate update is

and

_k(+) = Xk(--) + Kk[Zk -- hk(]:k(--))] (17)

where (+) is the estimate using information at time

k, (-) is the estimate before data from time k is

available, and K is the Kalrnan gain matrix defined in

equation (19).

The error covariance update is

4

Pk(+) = [!- KkHk(xk(-))]Pk(-) (18)

where H_(xk(-)) is defined in equation (21).

The Kalman gain matrix is

Kk = P_(--)HT(_:k(--))

[ gj,(:_( ,))pk(_)g w _:

(]:k(-)) + Rk] -1 (19)
Definitions

F(_(t), t) =

nt(_(-)) =

Of(x(t), t) [ (20)

az(t) I,(0=_(0

tghk(x(tt))ax(tk) _:(th)=i_(-) (21)

Equation (12) for the system model is a continuous nonlinear equation describing the state propagation with time, between samples. An approximation

that F is constant between samples has been made.

In this application, the state is propagated through the

nonlinear equations using a fourth-order Runge-Kutta

technique. The measurement model (equation (13))

is discrete. In equation (20), F is the linearized state

matrix. This matrix must be recomputed for each update of the state vector since the elements of F are also

states of the EKF and will change with each update of

the filter.

The first step in estimating the flight data is obtaining an estimate from the simulation. There are two

primary objectives in estimating the low-order model

from nonlinear simulation data. First, determine the

coefficients of the low-order model (filter states 4-11)

as accurately as possible. These will be the initial state

estimates for the estimation from the flight data. In

addition, the simulation estimates will be compared to

the estimates obtained from the flight data to check for

major discrepancies.

The second objective is to determine the amount of

process noise to use in estimating the data from flight.

For the X-29A, earlier flight tests have shown that the

nonlinear simulation accurately predicts the response

of the airplane. As a result, the process noise weighting matrix is selected by adjusting the weighting matrix until both the EKF and LOES indicated that good

results had been 0bt,3ined. Overplots of the LOES system with the original data are used to evaluate the estimates qualitatively. Innovations, time histories of the

estimates, correlations, and state variances from the

EKF are used in the evaluation. A tradeoff is then

made between the EKF and LOES results to obtain the

final estimates and weightings. These weightings are

then used to estimate the parameters during flight. In

the cases examined, the process noise weightings were

kept quite low. 16

To determine values for the measurement noise

(R), sample time histories are obtained, From these,

bounds are drawn defining a noise envelope for that

signal. These bounds are then selected as 4-2 cr (two

standard deviations) bounds about the mean. The

value for Ri,i in this case is o'_e This is possible because the process noise was kept low.

For_ and q, the initial state variance (P(0)) is the

value for measurement noise on o_and q. For the other

states a value for (r is selected which equals one-fourth

the estimated initial values of the states, thus Pi,i =

b.2 • = (1⁄4_:i,i) 2 . For state variables with initial values

close or equal to zero, bounds are established based on

engineering judgement.

With the identification algorithm initialized, the

next step is to determine the pure time delay. Nine

filters are run with fixed time delays ranging from 0 to

200 msec, in increments of 25 msec. The likelihood

function is used to determine which filter is operating

with the correct amount of time delay.

Several analysis techniques are used to monitor the

filter and to aid in interpreting the results. The most

important of these are the filter innovations. The innovations are the difference between the predicted output

of the equivalent system and the measured response.

The innovations are calculated by

innovations = zk - ht:(xt:(-)) (22)

Results and Discussion

Simulation

A simulation time history is generated using the

six-degree-of-freedom, nonlinear simulation and flight

data inputs at this condition. Some of the flight data

analyzed is presented in Fig. 3. The actual time interval of the identification was varied to illustrate various points.

The filter is initialized as described earlier. The measurement noise is determined from the flight data and

then divided by 100 (to represent _ the root mean

square error) since the simulation would produce a perfeet measurement signal. A low-order equivalent system is then estimated from this data. The results of this

estimation are presented in Table 1.

Large correlations between the state 6 and the coefficients M0 and 5o are found. Some correlation is

also indicated between the coefficients Mq and Mt.

This is expected because the control system has the appearance of a pitch rate command system. Although

this correlation is not desirable, it is not felt to be

a problem.
These results are obtained for very low levels of process noise. Large values of process noise often result

in misleading estimates. The values for final variance

are much smaller than the initial values, which shows

that the EKF has obtained a good estimate.

Plots of the LOES and the simulation time histories

are shown in Fig. 4. The LOES matches the simulation

data exceptionally well. The only discrepancy occurs

between 9 and 10 sec. This discrepancy is believed

to be the result of a nonlinearity around the zero stick

position, possibly caused by break out forces or hysteresis in the stick at that point.

Flight

For the flight data estimation, the final results of the

simulation estimation are used as the initial state estimates. The values for process noise are those determined from the simulation. The initial variance of

the state is that used in the simulation estimation. The

flight data is then processed to arrive at the final estimate, which is compared to the simulation final estimate. The output of the flight data estimation is presented in Table 2.

The same correlations among parameters that are

seen on the simulation, _5with M0 and _0, and M6 with

M_, occur here. As before, this result is not desirable,

but it is expected and is not thought to be a problem in

the estimation.

Plots of the LOES and the flight data are presented

in Fig. 5 and demonstrate excellent agreement with the

flight data. However, for reasons which are not clear,

the filter underestimates the final peak.

Figure 6 shows the innovations. The bounds of the

innovations are obtained by the equation

bound = 4-2[ HPH r + R]i_ (23)

The structure, especially in the pitch rate data, indicates correlation over time because a low-order model

was used to describe higher-order dynamics, resulting

in process noise. This correlation also suggests that

information is still available in the signal, or that the

noise is not Gaussian and white. Additional tuning of

the gains may reduce the correlation.

Figure 7 shows the time histories for the state estimates. In almost all cases, the states settle by approximately 9 sec. These results are encouraging. If the

time histories for the states representing the parameter estimates vary wildly at the end of the estimation,

then little confidence could be held in the estimation

for that state. Large variations would also be reflected

in a large final variance for that state.

Simulation and Flight Comparisons

Figure 8 is a comparison of the final state estimates,

with a +2 tr error bound (based on the final P), of the

simulation and flight estimations. The frequency and

damping ratio are computed from the state estimates;

their bounds were computed by a simple linearization.

This is only an estimate of the bounds and is thought

to be sufficient because of the low correlation of the

parameters. The results were expected to agree better

than they do.

The damping ratio is estimated to be higher from

flight than from the simulation. This increased damping ratio has also been observed by the pilots and in

postflight frequency response analysis. This increase

in the damping ratio is also observed in the near-realtime open-loop frequency response analysis as an increase in the phase margin.

To further verify the results of the algorithm, the

damping ratio and pure time delay estimates are compared to the results from the postftight frequency response analysis. The EKF estimated a time delay between 140 msec and 146 msec; the frequency response

analysis estimated a time delay of 120 msec. For the

damping ratio, the EKF estimated 0.84 to 0.92, the frequency response analysis estimated 1.05. The agreement is good, considering frequency and damping ratio

tend to tradeoff without affecting performance.

Thus, the differences between the flight data estimates and the simulation data estimates are caused

by a discrepancy between the simulation and airplane.

Some discrepancies between the simulation _ air--

craft performance are expected. This demonstrates the

EKF's ability to identify such discrepancies.

Robustness

The data was also analyzed in more detail to determine the robustness of the algorithm. In one instance,

the window over which the estimation was performed

was shifted 1.5 sec. This shift only has a slight effect

on the final state estimates.

In another check for robustness, the filter was reinitialized with the final state estimate from the previous run and the process was repeated in an iterative

fashion until convergence. Although the results differed slightly, they showed that the initial estimation

would be sufficient. This is where the Kalman filter
possesses an advantage over other identification techniques, which require multiple processing of the data

to obtain a final estimate. The results of the above two

investigations are presented in Table 3.

The most dramatic evidence of the robustness of the

algorithm, however, is its reaction to a data spike in the

stick input signal at approximately 15 sec (Fig. 3(a)).

This channel is the most sensitive to data spikes since

the model in the filter algorithm will be driven by this

input, producing outputs inconsistent with the aircraft

outputs. As a result large differences between measured and predicted values can occur altering the state

estimates unfavorably.

The flight data was estimated in the presence of this

spike with the process noise weighting matrix equal to

zero. The effects of this spike can be seen most clearly

in the plots for the filter state (Fig. 9), at approximately

10.6 sec (the difference in time is due to the window

over which the flight data is identified). The LOES

model and flight data (Fig. 10) still agree well especially considering that the process noise was zero. The

ability of the algorithm to survive a data spike of such

magnitude on the most critical channel indicates that

the algorithm will perform well during flight test.

Since the technique is in the developmental stage

all the analysis so far has been postflight analysis. The

goal of this technique is implementation during flight

test. While proving the concept, however, execution

speed was not considered. Once the technique was

proven sound and acceptable, computation time was

considered. By then, there was time for only a cursory investigation. The resulting execution time was

approximately 2.5 min (central processing unit (cpu)

time) to process 10 sec of flight data. Major sections

of the code may be written more efficiently, reducing

the execution time. When the program is moved to

the computer used during flight test, faster computation times may occur because of the increased capabilities of this system.

Concluding Remarks

A third-order model based on the short-period approximation is an acceptable model to describe the

pitch axis transient response of highly augmented and

very large order aircraft dynamics. This equivalent

system model can be estimated by using an extended

Kalman filter. Results suggest such an estimation

could be conducted in near real time during flight test.

The results of this technique have been verified

against the existing near-real-time open-loop frequency response analysis and postflight frequency domain handling qualities analysis.

The ability to determine equivalent time delay, undamped natural frequency, and damping ratio in near

real time provides the flight test organization with

quick and accurate information about the aircraft, contributing to safe, quick, and efficient flight testing.

References

I Hicks, John W., and Kevin L. Petersen, "RealTime Flight Test Analysis and Display Techniques for

the X-29A Aircraft," paper number 6, AGARD 73 ra

Flight Test Techniques Symposium, Flight Mechanics

Panel, Oct. 17-20, 1988.

2Bosworth, J.T., and J.C. West, "Real-Time OpenLoop Frequency Response Analysis of Flight Test

Data," AIAA 86-9738, Apr. 1986.

3Bauer, Jeffrey E., David B. Crawford, Joseph Gera,

and Dominick Andrisani, II, "Real-Time Comparison of X-29A Flight Data and Simulation Data,"

AIAA 87-0344, Jan. 1987. Also available in J. Aircraft, vol. 26, no. 2, Feb. 1989.

4 Pischoff, David E., "Longitudinal Equivalent Systems Analysis of Navy Tactical Aircraft," AIAA

81-1775, Aug. 1981.

5Burton R.A., B.T. Kneeland, U.H. Rabin, and

R.S. Hansen, "Flight Testing and Development of the

F/A- 18A Digital Flight Control System," Active Control Systems, RTP# AGARD-CP-384, 1985.

6Speyer, Jason L., and Edwin Z. Crues, "OnLine Aircraft State and Stability Derivative Estimation Using the Modified-Gain Extended Kalman Filter," J. Guidance, vol. 10, no. 3. Also available as

AIAA 85-1762.

7 Klein, Vladislav, and James R. Schiess, Compatibility Check of Measured Aircraft Responses Using

Kinematic Equations and Extended Kalman Filter,

NASA TN D-8514, 1977.

7
SKain,J.E.,C.M. Brown, Jr., J.G. Lee, S. Pallas,

and E.S. Sears, "Missile Aerodynamic Parameter and

Structure Identification from Flight Test Data," AIAA

78-1341, Aug. 1978.

9 Kain, J.E., "An Evaluation of Aeroballistic Range

Projectile Parameter Identification Procedures," AIAA

79-1687, Aug. 1979.

to Maine, Richard E., and Kenneth W. lliff, Identification of Dynamic Systems. Theory and Formulation,

NASA RP-1138, 1985.

llMaine, Richard E., and Kenneth W. Iliff, Application of Parameter Estimation to Aircraft Stability and Control. The Output-Error Approach, NASA

RP-1168, 1986.

12Flying Qualities for Piloted Vehicles, MILSTD-1797 (USAF) 1987.

13Shafer, M.E, "Low-Order Equivalent Models

of Highly Augmented Aircraft Determined from

Flight Data," J. Guidance, Control, and Dynamics,

vol. 5, no. 5, Sept.--Oct. 1982, p. 504. Also available

as AIAA 80-1627-R.

_4Andrisani, D., Multipartitioned Estimation Algorithms, Ph.D. dissertation, State University of New

York at Buffalo, 1978.

15Gelb, Arthur, ed., Applied Optimal Estimation,

MIT Press, 1974.

16Bauer, J.E., Identification of Equivalent Short

Period Dynamics of the X-29A, Masters Thesis, Purdue University, Dec. 1988.
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Fig. 1 The X-29A airplane.

(_c

.... . ..... .-. .... m ..... - ..... ° ....... ..............m.... ........ --- ............... I

! !

', Deadband and G'p Flight deflections X-29A _ Zm ,,

, stick dynamics syscontrol :em plant ] "- ,"

' Aircraft i ,

I II

Zm

I I cow-o 0er Pure time . = equivalent

delay td " system Extended Kalman _ Zc

model Xp filter corrections v

Xc --I logic i

Extended Kalman filter

8c Pilot command Zc Corrected angle of attack, pitch

5 'p Stick signal to flight control system

Zp

,5p Stick signal used to drive extended

Kalman filter aircraft model

Xp

Zm Measured angle of attack, pitch

rate, and normal acceleration Xc

Fig. 2 ExtendedKalman filter algorithm.

rate, and normal acceleration

Predicted angle of attack, pitch

rate, and normal acceleration

Predicted state vector

Corrected state vector

900032

I4
0

Stick

Input, -1 --

In.

-2

-3

-4--

-5

0

I

2

_/

I

4

.....I I I 1

6 8 10 12

Time, sec

(a) Stick input.

I !

14 16

900033

5.2

deg

5.0

4.8

4.6

4.4

4.2

4.0

3.8

3.6

0

m

I

m

I I i I ! I

|

2 4 6 8 10 12

Time, sec

(b) Angle of attack.

Fig. 3 Flight data as a function oftime.

I I

14 16

900034

15
w

q,

deg/sec

3

2--

1--

0

-1

-2

"3

-4

0

! I .I 1

2 4 6 8

Time, sec

(c) Pitch rate.

I I I I

10 12 14 16

900035

1.8 n

1.6

1.4 II

(d) Normal acceleration.

! I I

12 14 16

900036

Fig. 3 Concluded.

16
1.0 --

.8 -- --- (x f, slmulatlon estimate

.:-

0_,

•,-

0 ............. AA

-_"o I I I I I I I

• 0 2 4 6 8 10 12 14

Time, sec

(a) Angle of attack.

5 -- -- qc' low-order equivalen ;ysl

4 --- qf' slmulatlon estlmate

deg/sec ql1- I

-2 _

-- (x c, low-order equivalent system estimate

I

16

900037

low-order equivalent system estimate

-3 I I I I I I I I

0 2 4 6 8 10 12 14 16

Time, sec

900038

(b) Pitch rate.

Fig. 4 Low-order equivalent system response compared with simulation data (perturbation quantities) as a function

of time.

17
nz ,

g

1.0

.8

.6

.4

.2

0

-.2

-.4

-.6

".8

-1.0

0

__ m nzc,

--- n z f, slmulatlon estimate

-

low-order equivalent system estimate

1

\

!

m

! !

2 4

! I I I I I

6 8 10 12 14 16

Time, sec 900039

(c) Normal acceleration.

Fig. 4 Concluded.

deg

1.0

.8

.6

.4

.2

-.2

-.4

(x c' low-order equivalent system estimate

--- (x f, flight data

m

t.,J!,.,..!_f..rj _L_,i,_!=tl

, q'l' S ,_'.i f:;1 ;J'l / z":q:_i 'l,

' ! II _ ! I i U _I t

i

2

7_

!

= I

ill_,

-.6 I I I

0 4 6 8 10 12 14

Time, sec 900040

(a) Angle of attack.

Fig. 5 Low-order equivalent system response compared with flight data (perturbation quantities) as a function

of lime.

18
q_

deg/sec

4

3

0

-1

-2

-3

0

-- qc, low-order equivalent system estimate

--- qf, flight data

n

I I I I I I I

2 4 6 8 10 12 14

Time, sec

900041

(b) Pitch rate.

nz ,

g

1.00 --

.75 --

.50

.25

0

-.25

-.50 --

-.75 --

-1.00

0

-- nz c' low-order equivalent system estimate

--- flight data nzf,

V

I I I I I I I

2 4 6 8 10 12 14

Time, sec 900042

(c) Normal acceleration.

Fig. 5 Concluded.

19
-.15

-.20 I I I I

2

iI I

0 4 6 8 10 12 14

Time, sec

900043

(a) Angle of attack.

qc- qf'

deg/sec

.4

qc- qf

Bounds

.3

-.2

I I I I I

0 2 4 6 8 10 12

Time, sec

(b) Pitch rate.

Fig. 6 Innovations for flight data idcntification.

14

900044

2O
nz c- nzf,

g

.100

.075

.050

A

A nzc-nzf

Bounds

.025 *,

0

-.025 A _ A

-.050

-.075

-.lOOr I I I I I I I

0 2 4 6 8 10 12 14

Time, sec 900045

(c) Normal acceleration.

Fig. 6 Concluded.

.025

.020

.015

.010

(X,

deg .005

0

-.005

Fig. 7

I I I I I I I

2 4 6 8 10 12 14

Time, sec 900046

(a) Angle of attack.

Estimated state time histories for flight data identification.

21
.10 D

q_

deg/sec

.08

.06

.04

.02

0

-.02

-.04

-.06

0

I I I 1

2 4 6 8

Time, sec

I I I

(b) Pitch ratc.

10 12 14

900047

deg

2.0

1.5

1.0

.5

"°5

-1.0 --

-1.5 --

I I I I I

2 4 6 8 10

Time, sec

(c) Psuedo control surface deflcction.

Fig. 7 Continued.

I

12 14

=

900048

22
-1.4 F

-1.5

-1.6

-1.7

L(X -1.8

-1.9

-2.0

-2.1

-2.2 I

0 2

I I I I I

4 6 8 10 12

Time, sec

(d) L,,.

14

900049

-7

-8

-9

-10

Mc_ -11

-12

-13

-14

-15

0 2 4 6 8

Time, sec

(e) M..

I I I

10 12 14

900050

Fig. 7 Continued.

23
.015

.010

.005

0

L8 -.005

-.010

-.015

-.020 I'-

-.025

0

I I I I I

2 4 6 8 10

Time, sec

(f) L_.

I I

12 14

900051

-.10 f

-.15

I I I I I

2 4 6 8 10

Time, sec

(g) M_.

Fig. 7 Continued.

I I

12 14

900052

24
1.15 1.20

1.10

1.05

Lq 1.00

.95

.90

.85

I I

.80

0

I I I I I

2 4 6 8 10 12 14

Time, sec 900053

(h) Lq.

4

0

-2

M q -4

-6

-8

-10 m

-12

0

I I I I I I I

2 4 6 8 10 12 14

Time, sec 900054

(i) M a.

Fig. 7 Continued.

25
16

15

14

13

12

11

10

0 2

I I I I I I

4 6 8 10 12 14

Time, sec 900055

"r"

L0

.0020

.0015

.0010

.0005

0

-.0005

-.0010

-.0015

-.0020

0

I I I I I I I

2 4 6 8 10 12 14

Time, sec 900056

(k) Lo.

Fig. 7 Continued.

26
.040 --

.035

.030

.025

M 0.020

.015

.010

.005

0

i I ! I I I I I

2 4 6 8 10 12 14

Time, sec 900057

0) Mo.

1.0

,8

.6

.4

50 .2

0

-.2

-.4

-.6 I I I I I I

0 2 4 6 8 10 12

Time, sec

(m) *o.

Fig. 7 Concluded.

14

900058

27
Lc¢

L_

-1.9 -

-2.0 -

-2.1 -

-2.2

rn

I I

0

-1

-2

-3

2x 10-4

1

I

1.05 -

Lq 1.00

.95

D

M(_

M 5

M

q

-.40

-2 -

-3 - []

"4 --

"5 -

-6

-7 -

"8 I

I

I

4.6 --

4.7 -

4.6 -

Frequency 4.5 -

4.4 -

4.3 -

4.2

Simulation

I

Flight

Damping

ratio

1.0 --

°9 --

•6 --

.7 --

°6 --

.5

[]

I

Slmulatlon

Fig. 8 Simulation results compared with flight data.

I

Flight

900059

28
-1.0 --

-1.2 --

-1.4 --

-1.6

L_-1.8

-2.0

-2.2

-2.4

-26 1 I I I ......1 ! 1 I I J

0 1 2 3 4 5 6 7 8 9 10 11 12

Time, sec 900060

(a) L_.

M(x -15

-35

0 1 2 3 4 5 6 7 8

Time, sec

(b) M_.

i I I I

9 10 11 12

Fig. 9 Estimated state time histories for input with data spike.

900061

29
.025 --

.020

.015

.010

L5.005

0

-.005

-.010

-.0150

! I I 1 1 ! I I I I I !

1 2 3 4 5 6 7 8 9 10 11 12

Time, sec

900062

(c) /.,8-

m

-.05 m

-.10

-.15 m

M 8 -.20 --

-.25 --_I_ -.30

-.35 '_

-.40 !

0 1

I I

2 3

L.---_

I,, I I I

4 5 6 7

Time, sec

(d) M6.

Fig. 9 Continued.

I I i ! I

8 9 10 11 12

900063

30
1.3

1.2

1.1

1.0

Lq .9

.8

.7

.6

.5

0

I I I I I I I I ! I I I

1 2 3 4 5 6 7 8 9 10 11 12

Time, sec 900064

(e) Lq.

m

0

-1

M q -2

-3

-41

-5

-6

0

m

:1

I

1

I I I I I I I I I I I

2 3 4 5 6 7 8 9 10 11 12

Time, sec

900065

(_ M,.

Fig. 9 Continued.

31
12

11

10

9

1

T8

6

4

0 1 2 3 4 5 6 7

Time, sec

I I I 1 I

8 9 10 11 12

900066

.003--

.002 --

.001 --

0

L 0 -.001

-.002 -

-.003 --

-.004 --

-.005

0

! I I I I I I I I I I I

1 2 3 4 5 6 7 8 9 10 11 12

Time, sec

900067

(11) Lo.

Fig. 9 Continued.

32
°6E

.04

.02

0

M 0 -.02

-.04

-.O6

-.08

-.lO I I { I

0 1 2 3 4 5 6 7 8 9 10 11 12

Time, sec

900068

(i) Mo.

1.0 m

.5

0

-1.5

-2.0

-2.5

-3.0

0

I i........... I I I I I i I I 1 I

1 2 3 4 5 6 7 8 9 10 11 12

Time, sec

(j) _o.

Fig. 9 Concluded.

900069

33
deg

.8

.6

.4

.2

-.2

-.4

-.6

-.8

0

w

I ., i I

--- (x f, flight data

I L [I l_o_'

'

O_c, low-order equivalent system estimate

1 I I

1 2 3

'I

I I

I '

I I ! I I I I I I

4 5 6 7 8 9 10 11 12

Time, sec

900070

(a) Angle of attack.

3

2

qc, low-order equivalent system estimate

qf, flight data

q' 0

deg/sec

-4

0 1 2 3

II

1 • I I I I I I

4 5 6 7 8 9 10 11 12

Time, sec

900071

(b) Pitch rate.

Fig. lO Low-order equivalent system response compared with flight data (perturbation quantities) for input with

data spike as a function of time.

34
N/ A Nalcnd/_o11_ _d

IIm_ ,_gc_lmton

I. Report No.

NASA TM-101722

4. TitJeand Subtitle

Report Documentation Page

2. Government Accession No. 3. Recipient's Catalog No.

5. Report Date

Estimating Short-Period Dynamics Using an Extended Kalman Filter

7. Author(s)

Jeffrey E. Bauer and Dominick Andrisani

9. Performing Organization Name and Address

NASA Ames Research Center

Dryden Flight Research Facility

PO Box 273, Edwards,, CA 93235-0273

12. Sponsoring Agency Name and Address

National Aeronautics and Space Administration

Washington, DC 20456-3191

J_e1990

6. Performing Organizalion Code

8. Performing Organization Report No.

H-1625

10. Work Unit No.

RTOP 533-02-51

1I. Contract or Grant No.

13. Type of Report and Period Covered

Technical Memorandum

14. Sponsoring Agency Code

15. Supplementary Notes

Prepared as AIAA 90-1277 for presentation at the AIAA/SFI'E/SETP 5th Biannual Flight Test Conference,

Ontario, CA, May21-24, 1990.

16. Abstract

In this paper, an extended Kalman filter (EKF) is used to estimate the parameters of alow-order model from

aircraft transient response data. The low-order model is a state space model derived from the short-period

approximation of the longitudinal aircraft dynamics. The model corresponds to the pitch rate to stick force

transfer function currently used in flying qualities analysis. Because of the model chosen, handling

qualities information is also obtained. The parameters are estimated from flight data as well as from a

six-degree-of-freedom, nonlinear simulation of the aircraft. These two estimates are then compared and

the discrepancies noted. The low-order model is able to satisfactorily match both flight data and simulation

data from a high-order computer simulation. The parameters obtained from the EKF analysis of flight data

are compared to those obtained using frequency response analysis of the flight data. Time delays and

damping ratios are compared and are in agreement. This technique demonstrates the potential to

determine, in near real time, the extent of differences between computer models and the actual aircraft.

Precise knowledge of these differences can help to determine the flying qualities of a test aircraft and lead

to more efficient envelope expansion.

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

Kalman filter, Low-order equivalent model, Near-realtime, Parameter estimation, X-29A airplane

18. Disuibution Statement

Unclassified-Unlimited

Subject Category - 05

19. Security Classif. (of this report)

Unclassified

NASA FORM 1626 OCTS8

20. Security Classif. (of this page)

Unclassified

21. NO. of Pages

38

For sale by the National Technical Information Service, Spnngfie/d, Hrginia 22161

22. Price

AO3
nz c' low-order equivalent system estimate

--- nz f, flight data

nz,

Ib/in. 2

2 3

! I I 1 I ! I I I

4 5 6 7 8 9 10 11 12

Time, sec

(c) Normal acceleration.

Fig. 10 Concluded.

900072

35