fvsysid shortcourse 2 maneuvers

AIAA Short Course: Flight Vehicle System Identification in Time Domain, Aug. 2006 Maneuvers/1Dr. Ravindra Jategaonkar

Maneuvers for Flight Vehicle System Identification

ParameterAdjustments

Model Response

ResponseError

ActualResponseInput

Maneuver

ModelValidation

ComplementaryFlight Data

Identification Phase

Validation Phase

OptimizedInput

Flight Vehicle

IdentificationCriteria

EstimationAlgorithm /Optimization

MathematicalModel /

Simulation

Data Collection& Compatibility

easurementsM

ethodsM

odelsM

A Priori Values,lower/upperbounds

Model Structure

-+


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Fundamentals of Data Gathering

The process of performing experiments and recording system inputs and outputs

“If it is not in the data, it cannot be modeled”

Basics:

1) Define the scope of flight testing

2) Define the suitable sequence of flight maneuvers to be performed at each test point

3) Choose an adequate form of the inputs to excite the aircraft

motion in some optimum sense “Optimal Input Design”


1) Flight Testing for performance evaluation

2) Flight Testing for system identification

- FT for aircraft certification – first category

- FT for aerodynamic database development – 2nd category

- A large number required in each category- Some are common to both; some are characteristically different

- Proof-of-Match maneuvers - Required to demonstrate fidelity of aerodynamic databases- JAA / FAA - About 100 – 120 test cases (ATG: Acceptance Test Guide).

Classification


1) Acceleration - deceleration

2) Pushover – Pullup (Roller coater)- Primarily to determine lift and drag characteristic

3) Windup turn- To determine the gradient of ‘Stick force per g’ (design criterion)

4) Climb / sawtooth climb,- To determine the best climb rate

5) Bank-to-bank roll- To determine the maximum roll capability

6) Steady sideslip, - To determine the gradient of the rudder deflection

7) Landing and takeoffs

Flight Testing for Performance Evaluation


• Short period maneuver

• Phugoid maneuver

• Pushover-pullup (Roller coaster)

• Level turn

• Thrust variation

• Bank-to-bank roll

• Dutch roll maneuver

• Steady heading steady sideslip

+ windup turn,

+ acceleration-deceleration,

+ flybys, landings

Flight Testing for System Identification


Two Approaches:

a) Based on the estimation error criterion

b) Based on spectral behavior of the model

Approach a): Rigorous, involved, and theoretical

Approach b): Engineering approach, easy to practice

Guiding Principle:... the optimum input in a given case is that which best excites thefrequency range of interest, and hence the harmonic control of theinput should be examined before the test ...(Milliken, 1951)

Optimal Input Design


Optimal Input design (2)

Excitation level (input amplitude):- Sufficient excitation to result in individual

components greater than the measurement accuracy +-σ

- Resulting response negligible (within measurement accuracy), but the individual components large enough

- Large amplitudes may result in nonlinear response behavior.

Dynamic motion:- Excite different natural frequencies (modes of motion)- Preferably one control input at a time

(While moving the stick, many pilots are trained to maintainAoA through throttle variation)

Independent control inputs:- Manual or computerized inputs


Input Design by Estimation Error AnalysisStatistical properties of the parameter estimates:

bias and covariance matrix of the estimates.

The maximum likelihood (ML) estimation is based on maximization of p(z|θ), the conditional density function of the measurements z for a given parameter vector θ.

Fisher Information matrix: ,

Indicator of information content

The ML estimation: Bias free and efficient in a statistical sense

Inverse of F provides a good approximation to error covariance matrix P

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

Θ∂Θ∂Θ∂

−=ji

ijzLE )|(2

F ))|(ln()|( Θ=Θ zpzL


Input Design by Estimation Error Analysis (2)Estimation error covariance matrix:

Thus, the parameter error covariance matrix, P, depends upon:

1) response sensitivity.

2) number of data points N, (depends on the length of the record),

3) weighting matrix R, (depends on the measurement noise)

Response y and response gradient obtained from a-priori model

At this stage the parameters Θ are known and kept fixed.

Information content in F mainly determined by response gradients.

( ) ( )1

1

11−

=

−−

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

⎥⎦

⎤⎢⎣

⎡Θ∂

∂⎥⎦

⎤⎢⎣

⎡Θ∂

∂≈≈ ∑

N

k

kT

k tyR

tyP F

Θ∂∂ /y


Input Design by Estimation Error Analysis (3)Proper tuning of input shape will result in optimal excitation of the

modes of a system defined by the a-priori model

leads to maximization of information matrix

In other words, minimizes the error covariance matrix P.

such inputs lead to parameter estimates having loweststatistical errors

Different measures, all based on Fisher information matrix:

1)

Which is same as maximizing determinant of F :D-optimal: overall measure,

reduces redundancyleads to better identifiability of individual parameters

{ } )()det(min)(

tuP opttu

⇒


Input Design by Estimation Error Analysis (4)

The other criteria are the sum or product of the diagonal elements of the matrix P:

2)

3)

They minimize the standard deviations(square root of the diagonal elements of P).

Criterion 2) is called A-optimal: A for averageIt is not scale invariant

Define input space to be searched, and perform optimization.

{ } )()(min)(

tuPtr opttu

⇒

)(min)(

tuP opti

iitu

⇒⎭⎬⎫

⎩⎨⎧∏


� � � ��

�

�

� � �

��

� � � � � � � � � �

� � � � ��

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� � � � � � � � �

Input Design by Estimation Error Analysis (5)

- Combine integer multiples of basic freuqnecy- Summation of sine functions

Not suitable to be flown manually.

Needs on board computer implementation


Two step procedure:

- determine the range of frequencies needed for accurate estimation

- design multistep input to cover these frequencies

Range of frequencies:

- Synthesize contributions due to each parameter

- make use of Bode diagram to determine the frequencies which must be included in the input signal

Example:

ee

e

e

qu

u

qu

MZX

q

u

MMMUZUZ

gXXX

q

u

δ

θ

α

θ

α

δ

δ

δ

α

α

α

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

+

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡ −

=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

00100001// 00

&&

&

&

Design of Multistep Input Signals (1)


⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

=

010002.04-3.49-0010.867-0.0022-

9.80665-09.430.0091-

A T0] 5.09- 0.11- [0 ==B

Now, for each equation, the frequency response magnitudesof the various terms in that equation are plotted as a functionof the input signal frequency.

Consider, pitching moment equation:compute the frequency response magnitudes for each of the terms

uM αM qM eM δ as well as for the total pitch acceleration q&

with respect to the elevator input eδ



In other words, it amounts to computing the magnitudes:

)(~)(~

,)(~

)(~,

)(~)(~

,)(~

)(~,

)(~)(~

ωδ

ωδ

ωδ

ω

ωδωα

ωδω

ωδω δα

e

ee

e

q

ee

u

e

MqMMuMq&

where ~ denotes the Fourier transform.

The individual components for and are computed

from the output equations:

q& α q eδ

eT DquCy δθα += ][

by defining the observation matrices C and D as:

]004.249.30[ −−=qC & ]09.5[−=qD &1) For :

2) For α:

q&

]0049.30[ −=αC ]0[=αD

3) For q:

4) For :eδ]004.200[ −=qC ]0[=qD

]0000[=eCδ ]09.5[−=eDδ



Mqq/δe

Mαα/δe

Mδe

q/δe.

Optimum frequency ranges for determination of

MδeMα, MqMα

Mag

nitu

de

Frequency0.01 0.1 1 10 100

40

20

0

-20

-40

-60

-80rad/s

db

Bode magnitude plot of the pitching moment equation terms:

At any given frequency, a large magnitude of any particular term compared to the other contributions suggests a dominant influence of that derivative, which indicates good information content necessary for estimation of the parameter.

Conversely, the derivative cannot be accurately estimated, if the contribution is small.

Rule of thumb: a derivative is considered identifiable when its termhas a magnitude of at least 10% of the largest term’s magnitude



0.01 0.1 1 10 100rad/s

Mδe

Mq

Short periodPhugoid

XαZα, Zδe

Mα, Mq, Mδe

Xu, XαZu, Zα

Frequency

Mα

Xu

Xα

Zα

Zδe

Zu

Regions of Identifiability




The two regions surround the natural frequencies ofPhugoid and Short period motion

Logical conclusion: System excited at its natural modes exhibit dominant dynamic motion.

Basis for designing multistep input:1) Optimum range of frequencies covers a range below and above the natural frequencies

2) Eigen-frequencies based on a-priori model are subjected to uncertainties

3) Eigen frequency changes with flight condition.

Bandwidth of 1:10 desirable.


V1 V2 V3

V4 V5

V6

V7


Multistep input signal of arbitrary shape can be synthesized by a suitable combination of pulse inputs

⎥⎥⎦

⎤

⎢⎢⎣

⎡Ω+

Ω

Ω−Δ=Ε ∑ ∑ ∑

=

−

=+

−

=

N

i

N

jji

jN

iii VVjVt

1

1

1 1

22

2 cos2cos12)(ω

The power (or energy) spectrum:

tΔ=Ω ω tNT Δ=

tΔ

tΔ

iV

duration of each pulse

amplitudes

normalized frequency Total duration


0 4 8 12frequency

rad/s

Ener

gy

0

0.1

0.2

0.3

t

u

Δt

Δt=1.2 s

Δt=0.8 sΔt=0.4 s


Energy spectrum of pulse inputs:



Energy spectrum of doublet inputs:

Ener

gy

0.2

0.4

0.6

0.8

1.0

0

1.2

0 4 8 12frequency

rad/s

t

u

Δt

Δt=1.2 s

Δt=0.6 s

Δt=0.8 s

0 2 4 50

0.2

0.4

0.6

0.8

1.0

Normalized frequency, ωΔtE/Δ

t2

t

u

Δt

1 3

(ωΔt)mid

Δt= 1.2 s 0.8 s 0.6 s

ω1 ω2

1 : 3

noscillatioofperiodtnn

DBLT ⋅≈≈≈Δ7.2

17.223.2ωπ

ω



Energy spectrum of 3211, doublet and pulse inputs

noscillatioofperiodtnn

⋅≈≈≈Δ41

26.1

3211 ωπ

ω

noscillatioofperiodtn

⋅≈≈Δ311.2

3211 ω

ω2ω1

Normalized frequency, ωΔt0 1 32 4 5

Ener

gy sp

ectru

mStep

Doublet

3211 Signal

Modified3211 Signal

t

-1.1

0 3 5 7

1.

-1.1.

-1.1.

-1.1.

-1.

1.1

-1.2

0.8

Δt

(ωΔt)mid

(ωΔt)2/3

Bandwidth

1 : 101 : 3



Multistep inputs: - Time step for 3211 input is somewhat smaller than that for

the doublet- 3211 provides good excitation of short period mode

- 1123 input has same spectrum 3211 input

- Phugoid is excited by long duration pulse

- Dutch roll is lightly damped; Doublet input provides adequate excitation


Multistep inputs: - Onboard generated computer inputs

- Step changes within single ΔtMay lead to excitation of structural modesLarge accelerations (at off CG locations, e.g. cockpit)

- Pilot applied inputs- Simple counting procedure (21,22,22; 21,22; 21; 21)- Audio/optical cueing (series of beeps, track displayed signal)- Extremely sharp changes are automatically filtered out

Exact time step and shape is not that critical, Use rule of thumbfor Δt

Practical Aspects of Input Signals (1)

10150 kts200230

260

0 30

Speed period Doublet(kts) (s) 2Δt (s)150 4.7 3.6200 3.7 3.9230 3.3 3.9260 2.8 3.1

2010time

s-6

6deg

0

0

deg/s

-10

r

δr


Elev

ato

r Small Input

3 2 1 1 3 2 1 1

Ban

k A

ng

le

20 s 20 s

Elev

ato

r

Elevator Pulse for PhugoidExcitation10s 1 Oscillation Period

Aile

ron

/Sp

oile

r

Aileron/SpoilerBank to Bankmaneuver

Ailerons only

30° Bank Angle10° Bank Angle

Level Turn with60° Bank and

Elevator Doublets

3-2-1-1 ElevatorMultistep Input

Rudder Doubletfor Dutch RollExcitation

Δ t 1s

Ru

dd

er

Small Input

1 1 1 1

Thru

st

10s 10s

20% ThrustVariation

Δ t 1s

Ru

dd

er

15 s

Rudder pulsesFor steady sideslips

Elev

ato

r Elevator pulland push forRoller-coaster

Practical Aspects of Input Signals (2)

Typical manual inputs


Elevator 3-2-1-1

Short Period

Elevator pulse

Phugoid

Bank angle

Level Turn Maneuver

Aileron/Spoiler

Bank to BankManeuver

Rudder Doublet

Dutch Roll

Thrust Doublet

FL 300

FL 260

FL 160

FL 80

FL 20

100 150 200 250 300 True Airspeed (Kts)

0.2 0.3 0.4 0.5 Mach No.

Alt

itu

de

80 K

CAS

100

KCAS

120

KCAS

160 K

CAS

195 K

CAS

277 KCAS

230 K

CAS

140 K

CAS

Practical Aspects of Flight Testing

Typical flight test program for system identification


Practical Aspects of Flight Testing (2)

Important Aspects:

Flight testing is costly; optimize the test procedure and maneuver sequence

Define the specific goals of flight testing- System identification

- dynamic maneuvers: small, medium and large amplitude- linear or nonlinear model identification- global model or point-models at selected trim conditions- configuration changes- flight (trim) conditions: angle of attack, sideslip angle, Mach, altitude, ...- special effects- repetition of dynamic maneuvers- influence of atmospheric conditions

- Model verification

On-site preliminary verification of recorded data to insure adequacy for off-line analysis


p

r

φ

δr

δaL

δaR

deg/s

deg/s

deg

deg

deg

20

0

-2020

0

-2015

-10

0

0

0

15

-10

1.5

-1.00 80time sec

Aerodynamic database:- Lateral-directional motion Identified from bank-to-bankand Dutch roll maneuvers.

- Multi run analysis

Example:- validation of Dutch roll- Rudder doublets- three maneuvers with differentinput amplitudes

- Beta variations up to +- 10°

Observation:The model response matches with flight measurement of p, r, β. Match for bank angle does not show any particular discrepancies.

Dynamic maneuvers



Combined motion:in yaw (due to directional stability) and roll (due to dihedral effect)

Beta-Sweep:- slowly scan β from 0° to βmax to 0° to βmax to 0°

- 30 to 50 seconds

Steady state sideslip:- 0° to +4°, hold for ~15 sec;

increase to 8°, hold for 15 sec- repeat for –ve sideslip angles

In both cases try to keep wings level.

Observation:Some discernible discrepancies in the match for Bank angle.

sideslip maneuvers

p

r

φ

δr

δaL

δaR

deg

5

0

-5

10

0

-1010

-10

0

0

0

10

-10

4

-425 50 75 100

time

sec



p

r

φ

δr

δaL

δaR

deg/s

deg/s

deg

deg

deg

5

0

-5

10

0

-1010

-10

0

0

0

10

-10

4

-425 50 75 100

time

sec

Flat Yaw Maneuver?- Banked turn: classical, looks elegant

- Yaw-only turn, bank 0°

sideslip maneuvers

Asymmetric configuration:Beta sweeps and steady sideslip maneuvers provide additional information – directional stability, lateral-directional control, coupling effects.

Database update:-Nonlinearity in aileron effectiveness for small deflections.

-Pitching moment nonlinear function of sideslip angle.



Downwash Lag Effects - Pitch Damping DerivativesSpecial Flight Test Techniques

ATTAS VFW-614

Bank Angle

-12

12deg/s

-75

75deg

φ

0 20 40 stime

qα-dot

Large bank angle maneuver with elevator input

-12

12deg/s

q

0 20 40stime

Pitch rateα-dot

α-dot

conventional elevator input maneuver

Modeling: Two Complementary Approaches

Two point aerodynamic model

- wing and tail are modeled separately

- physically more realistic

- automatically accounts for q and α-dot effects- nonlinear model with time delay

- necessarily requires NL estimation program

Conventional approach - Taylor series

- simplification based on linearization- linear parameter estimation program adequate- generally total pitch damping C mmq+C α

)sinsincoscos(cosVgqLCVm

qS αθ+αφθ++−=α&

independent component of α−dotthrough gravity term

- Cmqand Cmαseparation require special tests



Correlated Inputs (1)

X-31A: Pilot Input and Separate Surface Excitation α σδTEδcan

deg

deg

deg

deg

20

35

0

2

-15

-4520

-20

time s200 10 15 255

PID command

pitch command

angle of attack

canard

sym. trailing edge flaps

TV deflection in pitch

2.5

-2.5

deg20

-20

deg

pilot inputpitch doublet

canard 3211

elevator 3211

separate surface excitation (SSE)


20 40 60 80Angle of Attack, deg

0.6

0.4

0.2

0

-0.220 40 60 80

Angle of Attack, deg

Pilot Input Separate Surface Excitation

Single maneuverData partitioningWindtunnel predicted

EstimatedWindtunnel

Cmδcan

X-31A: Canard Effectiveness

α σδTEδcan

Correlated Inputs (2)


Flight Test Instrumentation and Measurement (1)A typical set of measurements for aerodynamic model extraction

1) Control surface deflections -- essential, input to the aero model

2) Linear accelerations -- three axes; provide very good information

3) Angular rates -- three axes; provide very good information

4) Attitude angles -- primarily useful in the data compatibility check

Also, for Aero-model ID using output error method

5) Air data -- very useful in the parameter estimation

6) Static pressure,

7) Engine parameters

8) Pilot forces and inceptor positions (Hinge-moment database)

9) Angular accelerations -- Recommended

10) Signal-to-noise ratio -- 10:1- noise around system frequencies critical


θ2.5

-2.5

0θ

35

-35

0

q-do

t

9

-9

0q

0 2 4 6 8 10times

qq&

θS θ

θ(f)

0

0.12

0.08

0.04

0

12

8

4

0

1.2

8

4S qq(

f)

0 0.4 0.8 1.2 1.6 2Hz

q q&

S qq(

f)

frequency

degdeg/sdeg/s2

Flight Test Instrumentation and Measurement (2)

Three levels of signals: acceleration, rate, attitude angle


Current Trends

Current Trends Computerized Control Inputs

NASA LaRC Input Signal

DLR Improved 3211 Input Signal

Manual Multi step Inputs

Computerized Control Inputs

Design techniques robust toinaccurate a priori model

Optimal Input Design byDynamic Programming- nonlinear models- actuator dynamics- feedback control

Separate Surface Excitation- unstable aircraft

1050

-5

Time

0.8 1.1

-1.1-1.2

1.

-1.

0Δt 3Δt 5Δt 7Δt

Hypersonic flight vehicles- multi axis orthogonal phaseoptimized sweeps


References (1)Jategaonkar, R. V., Flight Vehicle System Identification: A Time Domain Methodology, Volume 216, AIAA Progress in Astronauticsand Aeronautics Series; Published by AIAA Reston, VA, Aug. 2006, ISBN: 1-56347-836-6http://www.aiaa.org/content.cfm?pageid=360&id=1447

Gates, R. J., Bowers, A. H., and Howard, R. M., “A Comparison of Flight Input Techniques for Parameter Estimation of Highly Augmented Aircraft”, AIAA Atmospheric Flight Mechanics, Conference, San Diego,CA, Aug. 11-13, 1996, Paper No. AIAA 96-3363.

Gupta, N. K. and Hall W. E. Jr., “Input Design for Identification of Aircraft Stability and Control Derivatives”,NASA CR-2493, Feb. 1975.

Hamel, P. G. and Jategaonkar, R. V., “Evolution of Flight Vehicle System Identification”, Journal of Aircraft, Vol. 33, No. 1, Jan.-Feb. 1996, pp. 9-28.

Koehler, R. and Wilhelm, K., “Auslegung von Eingangssignalen für die Kennwertermittlung”, DFVLR-IB 154-77/40, Dec. 1977.

Morelli, E. A. and Klein, V., “Optimal Input Design for Aircraft Parameter Estimation using Dynamic Programming”, AIAA Atmospheric Flight Mechanics Conference, Portland, OR, Aug. 20-22, 1990, Paper No. AIAA 90-2801.

Morelli, E. A., “Flight Test Validation of Optimal Input Design and Comparison to Conventional Inputs”, AIAA Atmospheric Flight Mechanics Conference, New Orleans, LA, Aug. 11-13, 1997, Paper No. AIAA 97-3711.

Mulder, J. A., Sridhar, J. K., and Breeman, J. H., “Identification of Dynamic Systems: Applications to Aircraft. Part 2: Nonlinear Analysis and Manoeuvre Design”, AGRAD AG-300, Vol. 3, Pt. 2, May 1994.

Plaetschke, E. and Schulz, G., “Practical Input Signal Design”, AGARD LS-104, Nov. 1979, Paper No. 3.


Stepner, D. E. and Mehra, R. K., “Maximum Likelihood Identification and Optimal Input Design forIdentifying Aircraft Stability and Control Derivatives”, NASA CR- 2200, March 1973.

Stepner, D. E. and Mehra, R. K., “Maximum Likelihood Identification and Optimal Input Design forIdentifying Aircraft Stability and Control Derivatives”, NASA CR- 2200, March 1973.

Weiss, S., Friehmelt, H., Plaetschke, E., and Rohlf, D., “X-31A System Identification using Single Surface Excitation at High Angles of Attack”, Journal of Aircraft, Vol. 33, No. 3, May-June 1996, pp. 485-490.

References (2)


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fvsysid shortcourse 2 maneuvers

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