On-Line Mu Method for Robust Flutter Prediction in Expanding a Safe Flight Envelope for an Aircraft Model Under Flight Test

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US006216063BI D c-oq -o01

12) United States Patent

(lo) Patent No.: US 6,216,063 B1

Lind et al.

145) Date of Patent: Apr. 10, 2001

4,343,447 8/1982 Reed ................................. 244/137.4 (54) ON-LINE ltt METHOD FOR ROBUST 4,475,385 10/1984 Farmer ................................... 73/147 FLUTTER PREDICTION IN EXPANDING A 5,615,119 3/1997 Vos ........................................... 701/4 SAFE FLIGHT ENVELOPE FOR AN 10/1998 Vos ........................................... 701/4 5,819,188 AIRCRAFT MODEL UNDER FLIGHT TEST (75) Inventors: Richard C. Lind; Martin J. Brenner, Prima_ 3, Evaminer--Jacques H. Louis-Jacques both of Lancaster, CA (USt (74) Attorne); Agent. or Firm--John H. Kusmiss The United States of America as (73) Assignee: (57) ABSTRACT represented by the Administrator of the National Aeronautics and Space A structured singular value (_t) analysis method of comput- Administration, Washington, DC (US) ing flutter margins has robust stability of a linear aeroelastic model with uncertainty operators (A). Flight data is used to (*) Notice: Subject to any disclaimer, the term of this update the uncertainty operators to accurately account for patent is extended or adjusted under 35 errors in the computed model and the observed range of U.S.C. 154(b) by 0 days.

aircraft dynamics of the aircraft under test caused by time- varying aircraft parameters, nonlinearities, and flight (21) Appl. No.: 09/074,024 anomalies, such as test nonrepeatability. This I_-based (22) Filed: May 6, 1998 approach computes predict flutter margins that are worst- case with respect to the modeling uncertainty for use in (51) Int. Cl. 7 ................................ G05D 3/_; GO6G 7/76 determining when the aircraft is approaching a flutter con- (52) U.S. CI ...................................... 701/3; 701/4; 701/10; dition and defining an expanded safe flight envelope for the 701/14 aircraft that is accepted with more confidence than tradi- (58) Field of Search .................................... 701/3, 4, 5, 7, tional methods that do not update the analysis algorithm 701/8, 9, 10, 11, 13, 14, 15, 16, 120, 121, with flight data by introducing _t as a flutter margin param- 122; 244/194, 118.1, 137.4; 73/147, 178 T eter that presents several advantages over tracking damping trends as a measure of a tendency to instability from References Cited (56) available flight data.

U,S. PATENT DOCUMENTS 4,116,056 1 Claim, 11 Drawing Sheets 9/1978 Bulychev ............................... 73/147 REVISE A/C MODEL ]_

{

al I Generate Computer Model I ^ ICompute flutter margins using # and optionallyl az also compute flutter margins using known trad tional method

i

optionally computed using known traditional I a3 Check # flutter margins against flutter margi-_ method J OK, PROCEED .___ NOT OK Take A/C to flight condition F and measure fligh-_ date including dynamic pressure j bl Compute flutter points at flight condition F I b2 using singular value pt

t

Determine dynamic pressure difference between ] b3 that at present flight condition F and at predicted flutter pont Fp

+

If dynamic pressure difference is large take aircraft to 1 flight condition Fi=F+A F and repeat steps bl-b4 but if I small declare F_ a point on expanded _ght envelope and repeat step,_ bl-b4 until expanded flight envelope b4 has beendefined

U.S. Patent

Sheet 1 of 11

Apr. 10, 2001 US 6,216,063 BI

(_ System model

Flight data

System

p-k

_t

Identification

Nominal

Robust Nominal

estimate

margin

margin

FIG. 1

A

W

P

FIG. 2

Y < U

a

P

k

FIG. 3

U.S. Patent Apr. 10, 2001 Sheet 2 of 11 US 6,216,063 B1

W

Z

FIG. 4

g

A

FIG. 5a

P u

<

Y

U

Y p

FIG. 5b

A

U

U.S. Patent Apr. 10, 2001 Sheet 3 of 11 US 6,216,063 B1

A

W

P

FIG. 6a

A

W

P

FIG. 6b

A

W

d

FIG. 6c

U.S. Patent Apr. 10, 2001 Sheet 4 of 11 US 6,216,063 B1

A

W

Z

P

FIG. 7

W

Z

P

FIG. 8

I

I

FIG. 9

U.S. Patent Apr. 10, 2001 Sheet 5 of 11 US 6,216,063 B1

0oo o lo

0 AOKAc

i" I

zC

0 ,AAQO -

-6c_ 0 0 -

_0 0 Z_BQ_

FIG. 11

ZAQ p I_..

zcl t' I<

ZBQ ]_

WBQ

FIG. 12

I< I wPl

Zpl

Qo(s) _ W13 2

yQ .._---

I-_--- WQ

U.S. Patent Apr. 10, 2001 Sheet 6 of 11 US 6,216,063 B1

FIG. 13

Y_-- Po

U

E. Plant P

Nominal plant P0

• ._. ...... Maximum leo (1 + WA)I

Magnitude

10-1

10-2

10-3

10-4

100 101 102

Frequency, rad/sec

FIG. 14

A

FIG. 15

f

y -_--( U

Po ]_

U.S. Patent Apr. 10, 2001 Sheet 7 of 11 US 6,216,063 B1

.#" _.

10 0

- ,\ ..'/\'<, ,

-- I\ , / \,_

- , \, I Y,...

10-1 -- " _ \ "'-,...

= Y

Magnitude

10-2

= -- Plant P ' 'Y

\\

10-3

_ --- Nominal plant PO . \', =--- ...... Maximum IPo (1-+ WA)I "_,,,

-- at each frequency \

10-4 I IIIIIIII I II111111 I III11111 I IIIIIII

10-1

100 101 102 103

Frequency, rad/sec

FIG. 16

A AN

FIG. 17

Po No

U.S. Patent Apr. 10, 2001 Sheet 8 of 11 US 6,216,063 B1

-- Nonlinear system N

- - - Nominal linear system No

- •..... Maximum and minimum

_ bounds on set N = No + AN .....-""

Y

FIG. 18

-10

-20

•""1 I I I I I I 1 I

-8 -6 -4 -2 0 2 4 6 8

U

-- Nonlinear system N •.

--- Nominal linear system No

2O

....... Maximum and minimum _.. .

bounds on set N = No + AN ..", ,... ,-"

Y

FIG. 19

-10

-20

-30 '" ] ] I I I I I I I

-10 -8 -6 -4 -2 0 2 4 6 8

U

U.S. Patent Apr. 10, 2001 Sheet 9 of 11 US 6,216,063 B1

3O

Nonlinear system N ."

Nominal linear system No .-""

2O

- ..... Maximum and minimum

bounds on set

Y 0

-10

-20

I I I I I 1 I I I

-3O

-10 -6 -4 -2 0 2 4 6 8 -8 10

U

FIG. 20

W

z[4 !

U

Y_ I

FIG. 21

U.S. Patent Apr. 10, 2001 Sheet 10 of 11 US 6,216,063 B1

([ System model_

Flight data _

Finite element

Model validation

and aerodynamics

A

P

Robust margin

FIG. 22

U.S. Patent Apr. 10, 2001 Sheet 11 of 11 US 6,216,063 B1

REVISE A/C MODEL_

al [ Generate Computer Model 1

rCompute flutter margins using IXand optionally 1

a2 I also compute flutter margins using known l

, traditional method 1

Check ix flutter margins against flutter margins

a3

optionally computed using known traditional

method

OK, PROCEED

NOT OK

I Take A/C to flight condition F and measure flight

bl

date including dynamic pressure.

' {

Compute flutter points at flight condition F

b2

using singular value IX

Determine dynamic pressure difference between

b3

that at present flight condition F and at

predicted flutter point Fp

If dynamic pressure difference is large take aircraft to

flight condition Fi=F+AF and repeat steps bl-b4, but if

small declare Fp a point on expanded flight envelope

b4

and repeat steps bl-b4 until expanded flight envelope

has beendefined

I FIG. 23

US 6,216,063 B 1

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3, May-June 1993, pp. 395-402; and K. E. Kadrnka, "Mul- ON-LINE it METHOD FOR ROBUST FLUTTER PREDICTION IN EXPANDING A timode Instability Prediction Method," A1AA Structure, SAFE FLIGHT ENVELOPE FOR AN Structural Dynamics, and Materials Conference (Orlando, AIRCRAFT MODEL UNDER FLIGHT TEST Fla.), AIAA-85-0737, April 19185, Volume 2, pp. 453--442.)

These flutter margin testing techniques have been used for ORIGIN OF INVENTION wind tunnel and flight test programs. (R. M. Bennett, "'Applications of Zimmerman Flutter-Margin Criterion to a The invention disclosed herein was made by employees of Wind-Tunnel Model," NASA-TM-84545, November 1982 the United States Government and may be manufactured and and H. Katz et al., "F-15 Flight Flutter Test Program,'" used by or for the Government for governmental purposes Fluner Testing Techniques, NASA-SP-415, October 1975, without the payment of any royalties thereon or therefor, l0 pp. 413-431.) However, the method is of limited applica- FIELD OF THE INVENTION bility for general flight flutter testing because the assump- tions of few modes coupling and the requirement to observe This invention relates to flight flutter testing, which is the those modes may be too restrictive.

process of expanding the envelope that determines a range 15 Stability parameters were also introduced in determining of flight conditions within which an aircraft is safe from flutter margins that consider an autorepessive moving aver- aeroelastic instabilities. This testing must be done for all age process to describe the aeroelastic dynamics. One new and reconfigured aircraft, parameter was based on determinants from a stability cri- BACKGROUND OF THE INVENTION teflon for discrete-time systems that are excited by random turbulence. (Y. Matsuzaki and Y. Ando, "Estimation of Traditional methods of flight flutter testing analyze sys- Flutter Boundary from Random Responses Due to Turbu- tem parameters, such as damping levels, that vary with flight lence at Subcritical Speeds," Joumul of Aircraft, Vol. 18, condition to monitor aircraft stability. (M. W. Kehoe, "A No, 10, October 1981, pp. 862-868 and Y. Matsuzaki and Y.

Historical Overview of Flight Flutter Testing," NASA-TM- Ando, "Divergence Bounda_'. Prediction from Random 4720, October 1995.) A real-time method to estimate the 25 Responses; NAS's Method." Jtm roul of Aircraft, Vol. 21, damping levels was developed based on a recursive No. 6, June 1984, pp. 435-436._ A similar parameter was prediction-error method. (R. Walker and N. Gupta, "Real- developed by extending the deternlinant method to consider Time FLutter Analysis," NASA-CR-170412, March 1984.)

short data segments with assumptions of local stati0narity.

This method was extended to improve the estimates by (Y. Matsuzaki and Y. Ando, "'Flutter and Divergence Bound- considering an extended Kalman filter in the formulation. 30 ary Prediction from Nonstationary Random Responses at (R. Roy and R. Walker, "Real-Time Flutter Identification," Increasing Speeds," AIAA Strm'tun,s. Structural Dynamics, NASA-CR-3933, October 1985.) On-line methods using and Materials Cot!fere.ce lOrlando, Fla.), AIAA-85-0691, both time-domain and frequency domain characteristics of April 1985, Vol. 2, pp. 313-32(1._ Another extension to this turbulence response data have also been formulated to method derived a similar stability parameter but relaxed the estimate dampings. (C. L. Ruhlin et at., "Evaluation of Four 35 requirements for stationarincss. I H. Torii and Y. Matsuzaki, Subcritical Response for On-Line Prediction of Flutter "Flutter Boundary, Prediction Ba-_ed on Nonstationary Data Onset in Wind Tunnel Tests," Journal of Aircraft, Vol. 20, Measurement," Journal _f Air_'ratt. Vol. 34, No. 3, May- No. 10, October 1983, pp. 835-840.) These methods moni- -June 1997, pp. 427--432. I The,.e techniques of determining tored stability at test points, but they were of limited flutter margins can be applied to complex systems and usefulness for predicting the onset of aeroelastic flutter 40 require only turbulence fi)r excitation: however, the flutter because damping may be highly nonlinear as flight condi- boundary is computed by extrapolating a nonlinear function tions vary, so damping trends may indicate stability despite and may be misleading.

proximity to an explosive flutter condition. An alternative In view of the foregoing, it is clear that in the past the eigenspace method was formulated based on orthogonality actual flight envelope developed for aircraft operation is between eigenvectors, but this method uses a parameter that, 45 essentially determined only b_ flight testing. The edges of similar to damping, indicates stability and may vary non- the envelope are points where either the aircraft cannot fly linearly with flight condition. (D. Afolabi et al., "Flutter any faster because of engine limitations or, with a 15% Prediction Using an Eigenvector Orientation Approach," margin for error, where the damping trends indicate a flutter A1AA Journal, Vol. 36, No. 1, January 1998, pp. 69-74.)

instability may be near. After flight testing, the envelope thus The concept of predicting the onset of flutter by analyzing 50 empirically determined is used for regular operations. It flight data at subcritical airspeeds has been introduced in would be desirable to use both the aircraft model computa- conjunction with a method for formulating a flutter margin tions and the test flight data in determining flutter margins in envelope. (N. H. Zimmerman and J. T. Weissenburger, order to provide a more expanded and robust flutter margin "Prediction of Flutter Onset Speed Based on Flight Testing envelope.

at Subcritical Speeds," Journal of Aircraft, Vol. i, No. 4, 55 July-August 1964, pp. 190-202.) This method considered STATEMENT OF THE INVENTION the interaction of two modes in the flutter mechanism to An object of this invention is to improve flight test formulate a stability parameter that varied quadratically with efficiency along with maintaining a high level of safety in dynamic pressure. This technique has been extended to 60 the process of producing a robust flutter margin envelope for consider several modes interacting as the flutter mechanism an aircraft model.

in order to demonstrate a prediction method for higher-order flutter. (S. J. Price and B. H. K. Lee, "Development and A further object is to provide an on-line method of Analysis of Flight Flutter Prediction Methods," AIAA producing a robust flutter margin envelope that is essentially Dynamics Specialists Conference (Dallas, Tex.) AIAA-92- model based so that it not only has the desired predictive 2101, April 1992, pp. 188-200; S. J. Price and B. H. K. Lee, 65 nature of a traditional method but also utilizes flight test data "Evaluation and Extension of the Flutter-Margin Method for to obtain the desired accuracy of predictive estimates. This Flight Flutter Prediction," Journal of Aircraft, Vol. 30, No. combines the strengths of both the traditional p-k method

US 6,216,063 B1

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and the new method of on-line estimation of the damping (6) If trends at any flight test condition are not OK, then method. This new method, referred to hereinafter as a unique declare that condition F as a flutter margin, a point on p method of flight testing for flutter margins is tor on-line the edge of the aircraft's safe flight condition envelope.

flight test prediction based upon both analysis data of the Note that the information from the computation step ( I ), Part ,aircraft model and flight test data in order that analysis data 5 I, is not used during the flight test, Part II. Conversely, the be updated during the testing procedure for continual cor- information from Part II is not used during Part I. This is rection of the aircraft model data.

essentially why the computational Part I is not mentioned when discussing flight testing Part II. Instead, only the What is required is a robust stability approach to formu- lating a flutter margin envelope that considers a state-space computation of margins in Part I are used as a first flutter model of the aircraft. This method is based on a formal lo m.argin estimate. If the flutter margin computation is good, mathematical concept of using a structured singular value, _, _tis assumed there are no obvious problems with the aircraft, but even if so, a robust flight envelope has still not been that guarantees a level of modeling errors to which the aircraft is robustly stable as described in a technical paper by determined because the computational results can never be completely trusted. This is so because the model can only be the present inventors R. C. Lind and M. J. Brenner, "Robust Flutter Margin Analysis that Incorporates Flight Data," an approximation to the real aircraft and there are often J5 important oddities about the real aircraft resulting in errors NASA-TP, March 1998, and presented orally on Sep. 9, that are not anticipated. The estimated flutter margins that 1997, the presentation of which has been documented by a technical memorandum, NASA/TM-97-206220, titled "A are then adopted as the boundaries of the safe operating envelope are only determined by flight testing in Part II Presentation on robust Flutter Margin Analysis and a Flutterometer," both of which by this reference are incor- without any method of correcting for such unanticipated 20 errors.

porated herein.

Moreover, there is a dramatic risk associated with that A realistic representation of the modeling errors can be flight testing procedure because of the unreliability of look- formulated by describing differences between predicted ing at damping trends. Damping does not smoothly change flight and measured flight data. In one application of this as an instability condition is approached: rather, the damping invention, the method has been successfully used to corn- __5 may often undergo sudden changes. Thus, the damping pule flutter margins for an F/A-18 Hornet aircraft modified into a Systems Research Aircraft (SRA) at NASA Dryden trends may seem good at flight condition E but there is no Flight Research Center. The robust flutter margins were guarantee that the damping will not sharply change as the determined form aeroelastic flight data to demonstrate the flight condition is changed to F+A F and aeroelastic flutter of potential errors that may exist in the flutter margins corn- the aircraft may occur. This lack of ability to predict an puled by a traditional p-k analysis. (R. Lind and M. Brenner, 30 instability is the reason for the risk in flight flutter testing in "Robust Flutter Margins of an F/A-18 Aircraft from this tradition procedure. This results in a greater time and Aeroelastic Flight Data," Journal of guidance. Control and cost in flight testing because the envelope of safe operating Dynamics, Vol. 20, No. 3, May-Jane 1997, pp. 597-604.)

conditions must be expanded more cautiously using very small increments of AF.

In accordance with the present invention, on-line estima- tion of flutter margins are computed during a flight test in 35 In contrast, the present invention comprises a method contrast to the prior-art flight flutter testing techniques used with a singular value, _t, to predict fight margins from referred to above which actually involve a two-step process flight data at successively higher dynamic pressure condi- that first requires a computational analysis of the aircraft tions of flight at which an onset of a flutter condition will model to estimate off-line flutter margins and then does a occur. Thus, for given dynamic pressures, such as a given flight test of the margins. Although the computational part is altitude and successively greater airspeed., the point at often not discussed in conjunction with the actual flight which aeroelastic flutter will occur is predicted, and the testing part, it is in fact a necessary part.

process is repeated for the full range of dynamic pressures The basic prior-art procedure that is fairly universal with at which the aircraft is capable of safely operating, such as industry and military organizations around the world con- altitudes and airspeeds. Ideally, each flutter margin predicted sists of two parts: 45 at a given altitude will be the same. To expand a full operating envelope, the procedure is repeated at all altitudes Part I: Pre-flight estimate of flutter margins at which the aircraft is capable of operating. In the ixmethod (1) Generate a computer model that is a 'best-guess" of the of this invention, the flight data is not used to estimate model aircraft model.

parameters or to identify a transfer function; rather, the flight (2) Compute flutter margins for the aircraft model using 50 data is only used to update the uncertainty description for the a well known p-k method.

theoretical model of the aircraft being tested. This approach (3) If margins are too small, redesign the aircraft and avoids several difficulties in trying to estimate a high-order modify the computer model accordingly. Then repeat model from flight data that has low signal-to-noise ratio but steps 1 and 2.

still accounts for time-varying dynamics by updating the (4) If margins are OK, flight test the aircraft.

55 uncertainty description to describe changing errors between Part II: Flight test to determine envelope the aircraft and the nominal model.

(1) Take aircraft to test point at flight test condition E a The stability parameter, ix, that is central to this new flutter dynamic pressure defined by, for example, altitude and margin method is essentially linear with changes in flight airspeed.

condition. Consequently, instabilities can be accurately pre- (2) Telemeter flight data to ground control room, unless 6o dicted. In addition, this method can easily consider realis- the following step 3 of this Part It can be carried out by tically high-order models and does not make assumptions computers already mounted in the aircraft.

about the number or type of modes that interact as the flutter (3) Estimate dynamics from flight data such as damping.

mechanism. The process used in a flight flutter test is as follows: (4) Evaluate levels and trends of dynamics.

(5) If trends are OK, take aircraft to a flight condition 65 Part I: Pre-flight estimate of flutter margins closer to last estimated aeroelastic flutter condition (1) Generate a computer model that is "best-guess' of the aircraft model.

(F,.=F+A F) and repeat steps 2, 3, and 4.

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during the flight tests, but the flutter margins are updated (2) Computer flutter margins for aircraft model using the well known p-k method (optional). correspondingly. This updating is essentially related to the manner in which flight and model data are combined in (3) Compute robust flutter margins for aircraft model generating an operator A that describes changes in the using the singular value _t method.

model. This is so because the structured singular value _t is (4) If the flight envelope determined by computed flight a function of the plant or system P (i.e., aircraft) and is margins is too small under either computation method.

defined by the following: redesign aircraft and repeat steps 1.2 and 3.

(5) If the flight envelope thus computed is OK, flight test aircraft.

_IP) = minff'(A):A _ A, det { 1 - PA) = 0 Note that step 3 is what distinguishes over the prior art.

Part II: Flight test to determine actual safe aircraft flight envelope where: _t(P)=O if no A exists such that det(1-P&)=0, and A=a (1) Take aircraft to a safe test point at flight test condition structured uncertainty operator (modeling errors and F, a dynamic pressure defined by, for example, altitude perturbations) which is allowed to lie within a norm- and airspeed.

bounded set such that (2) Telemeter flight data to ground control room, unless the following step 3 of this Part II can be carried out by .x={,x:lAIl__1}, and computers already mounted on the aircraft.

(3) Compute a predicted flutter point Fp at a higher 20 is a set of perturbations such that the size of the perturbations dynamic pressure using a flutterometer algorithm based [IAII_ is =<1. This means that for robust stability of the plant, on both flight data and singular value _.'

no perturbation A greater than size Ikxll_ < 1 can destabilize

the plant. Thus, the increase of _ is the smallest destabilizing (4) If the difference between the present flight test con- perturbation, i.e., an exact measure of stability.

dition F i and the predicted flutter point Fp is large, then 25 An algorithm is provided for updating the aircraft model expand the next flight text condition to F=Fi+A_- and using the uncertainty operator A during flight testing under repeat steps 1, 2 and 3.

different conditions, and a procedure that uses the structured (51 If the difference is small, then declare the last pre- singular value _ to determine if the aircraft system P with its dicted flutter margin Fp as a point on the edge of the set of uncertainty matrix operators A is not invalidated. If so, envelope.

30 the set is adjusted until it is not for robust stability of the The algorithm is repeated for all altitudes at which the system, This model validation is then used to generate aircraft is capable of operating until a completed and robust reasonable norm bounds for a sufficient set of uncertainty envelope is developed.

operators scaled such that _t is always less than unity.

The main advantage to using this _ algorithm over tradi- Two separate algorithms, local and global, are provided tional methods is that with each iteration at a new flight 35 for using flight data sets to update uncertainty operators condition F, a new predicted flutter point is determined, associated with the aircraft plant model P. In the local whereas in the prior-art damping algorithms that is not the algorithm, flight data at identical flight conditions is used.

case. Moreover, the envelope can be expanded more confi- This is done by independently computing uncertainty dently and in greater increments of AF because there is not descriptions, i.e., sets of operators A for models at different so much concern about sudden changes in damping and 40 flight conditions, resulting in smaller uncertainty operators stability. The basic flowchart for the flutterometer algorithm required for subsonic plants and a less conservative worst- is as follows: case flutter margin. In contra_t, the global algorithm uses the 1. Read flight dada D from test point at flight condition F.

entire set of flight data at all flight conditions to generate a 2. Generate computer model P of aircraft at E single uncertainty description, i.e., set of operators A for all 3. Generate A to relate P and D.

45 normal aircraft models. A disadvantage is possibly more 4. Compute analysis of (P,A) using the _t method.

conservative flutter margins, but an advantage is that the 5. Compute prediction of flutter margin from p., a struc- uncertainty description used for computing a flutter margin tured singular value from step 2. (estimate of distance to a flutter point) is truly a worst case.

A key point is that the flutterometer algorithm works by Although this statement of the invention speaks of devel- 50 oping the flight envelope for new or reconfigured aircraft by using both the computed model data and the flight data. This is important because using only one set of data is insuffi- testing at elevations with progressively increasing airspeed (to progressively increase dynamic pressure) in search for cient. The model is only an approximation, so it cannot be trusted completely, and the flight data only indicates the robust flutter margins (distance to a flight envelope on a current state of the aircraft so it cannot be used alone to graph of altitude versus airspeed), it should be understood 55 that it is possible to hold true airspeed constant for each predict the flutter margins. Using both provides robust flutter margin prediction which is most important because it flutter margin prediction while progressively increasing dynamic pressure by changing altitude or Mach number enables the safe flight condition envelope to be reliably established without the aircraft actually reaching or even which is a function of air density that depends on altitude closely approaching a flutter point. Reliable prediction and air temperature. Consequently, while holding a lower throughout the entire process of developing the flight enve- 60 true airspeed constant, the computed flutter margin may lope is due to the fact that the flutterometer algorithm uses predict a point below sea level. That is a valid prediction, but flight data to continuously update the structured singular the predicted point is outside the flight envelope to be value /a. promulgated as the universe of safe operating conditions with a conventional 15% margin for error at lower altitudes.

Another key point is that time-varying changes in the 65 While the _ method of computing robust flutter margins aircraft dynamics that give rise to modulating errors are taken into account. For example, the weight of a fighter has been developed for promulgating an envelope of safe operating conditions, it should be noted that in aircraft plane undergoes significant changes as fuel is consumed

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FIG. 20 illustrates system responses for hysteresis having greater computer data processing capability the IX method can be used with a display as a cockpit flutterometer example.

to provide the pilot with virtually real-time information FIG. 21 illustrates a linear fractional transformation sys- about changes in flutter margin in terms of altitude, airspeed tem for robust stability analysis and model validation with and dynamic pressure to indicate how far the aircraft can 5 forcing and measurement signals.

drop before reaching a predicted flutter condition as well as FIG. 22 information flowchart to generate plant and damping trend to indicate when the flutter condition is near.

uncertainty operators from a system model and flight data with the IXmethod.

BRIEF DESCRIPTION OF THE DRAWINGS FIG. 23 is a flow chart for the over-all IXmethod for robust FIG. 1 illustrates diagramatically an information flow- flutter prediction and safe flight envelope expansion of an aircraft model under test.

chart for traditional and IX methods to compute flutter margins.

DETAILED DESCRIPTION OF THE FIG. 2 illustrates a block diagram for the small gain INVENTION theorem, t 5 FIG. 3 illustrates a block diagram with uncertainty for the 1. INTRODUCTION example system.

Aeroelastic flutter is a potentially destructive instability FIG. 4 illustrates a block diagram for robust stability resulting from an interaction between aerodynamic, inertial, analysis of the example system using the small gain theo- and structural forces. The stabifity properties of the aeroelas- rem. 20 tic dynamics must be investigated to determine a flight FIG. 5a illustrates a linear fractional transformation F,(E envelope that is clear of flutter instabilities for new aircraft A), and FIG. 5b illustrates a linear fractional transformation designs or new configurations of current aircraft. Analytical FI(EA).

predictions of the onset of flutter must be accurate to reduce FIG. 6a illustrates a family of plants P=P(I+AW) with dangers and costs associated with experimental estimation.

input multiplicative uncertainty, FIG. 6b illustrates a family Critical flutter conditions are the points closest to the of plants P=-P(I+AW)P with output multiplicative uncer- flight envelope at which flutter instabilities occur. This tainty and FIG. 6c illustrates a family of plants P=-P+AW concept of closeness is formally defined here as a flutter with additive uncertainty.

pressure that considers the critical dynamic pressure for a FIG. 7 illustrates a linear fractional transformation system 3o constant Mach value. Obviously, different flutter measures for robust stability analysis using _t.

such as a flutter speed can be defined because a unique equivalent airspeed is associated with each dynamic pres- FIG. $ illustrates a linear fractional transformation system sure for a given Mach number; however, the following for nominal stability analysis in the Ix framework with definition 1.O. 1 for a flutter pressure will be used to parameterization around perturbation in dynamic pressure.

describe the critical flutter flight conditions.

FIG. 9 illustrates a linear fractional transformation system 35 Definition 1.O.1 for robust stability analysis in the IXframework with param- eterization around perturbation in dynamic pressure and A flutter pressure in the smallest value of dynamic pres- structured uncertainty. sure for which an aircraft at a particular Mach number experiences a flutter instability.

FIG. 10 illustrates a linear fractional transformation sys- tem for robust stability analysis in the _ framework with 40 The flutter pressure is used to compute a stability margin, parameterization around perturbation in dynamic pressure or flutter margin, that indicates the distance between the and uncertainty in structural stiffness and damping matrices. flutter pressure and a reference point. A common flutter margin, F, considers the difference in dynamic pressure FIG. 11 illustrates a linear fractional transformation sys- between the flutter pressure and a point on the edge of the tem for robust stability analysis in the IXframework with flight envelope. Another common flutter margin, _, consid- parameterization around perturbuation in dynamic pressure ers the percentage difference between equivalent airspeeds and uncertainty in A o and B o matrices of the state-space at the flutter condition and a point within the flight envelope.

Q(s) model.

Definition 1.0.2 FIG. 12 illustrates a linear fractional transformation sys- Aflutter margin relates a measure of distance between the tem describing Pad_ approximation to represent unsteady 50 aerodynamic force matrix in the IXframework with uncer- flight condition associated with the flutter pressure and a tainty in lag terms. reference point.

The traditional p-k method has been extensively used to FIG. 13 illustrates a family of plants P=Po(I+AW) with compute flutter margins for a variety of military and com- input multiplicative uncertainty.

mercial aircraft. (H. J. Hassig, "An Appropriate True Damp- FIG. 14 illustrates transfer functions for example system 55 ing Solution of the Flutter Equation by Determinant with multiplicative uncertainty.

Iteration," AIAA Journal of Aircraft, Vol. 8, No. 11, Novem- FIG. 15 illustrates a family of plants P=Po+WA with ber 1971, pp. 885-889.) This iterative method uses an additive uncertainty.

analytical dynamic model coupled with harmonic motion FIG. 16 illustrates transfer functions for example system 60 solutions for the unsteady aerodynamic forces. The p-k with additive uncertainty.

method predicts flutter margins entirely from a theoretical FIG. 17 illustrates linear fractional transformation system model that may not accurately describe the true dynamics of with nominal models and associates uncertainty operators.

the airplane. The resulting flutter margins do not account for FIG. 18 illustrates system responses for hardening spring possible variations between the model and the aircraft.

example. 65 The community studying aeroelasticity has identified the development of improved methods for characterizing flutter FIG. 19 illustrates system responses for softening spring margins as a vital research area. Flight testing for envelope example.

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9 10

transfer functions obtained by flight data with transfer expansion incurs dramatic time and cost because stability margins are not computed with a high level of confidence functions predicted by the analytical model. The norm bound on A is chosen based on these observed errors. A using traditional methods. The flutter dynamics often exhibit model validation condition is used to ensure the A is an explosive behavior that results in a sudden change in stability for a small change in flight conditions. Thus, small 5 sufficient to account for multiple data sets without being errors in predicted margins could have grave consequences excessively conservative. With respect to A. a worst-case for aircraft and crews operating near the flutter conditions.

flutter boundary is computed that directly accounts for flight Several approaches exist for characterizing accurate flut- data.

ter margins using flight data generated by the aircraft. These This method illustrated diagrammatically in FIG. 1 is data describe the true dynamics and can be used to generate to inherently different from traditional algorithms based on p-k realistic models and compute confident flutter margins.

methods or parameter identification and robustness Parameter estimation algorithms have been developed to approaches. The Ix method uses information from both an directly identify an aeroelastic model from the flight data.

analytical system model and flight data; traditional The accuracy of the resulting model can deteriorate as the approaches use only one of these sources, namely the system complexity and number of degrees of freedom of the system 15 model in the p-k method. Methods that use only an analyti- increase and signal-to-noise ratios decrease from optimal cal model can be inaccurate, and methods that use only the wind-tunnel conditions to realistic flight levels. Modal fil- flight data can fail if the data are of poor quality. The Ix tering has been introduced in association with parameter method uses the flight data to improve the analytical model estimation algorithms m simplify analysis by decoupling the system into a set of first-order responses. This type of by adding uncertainty operators. Poor quality flight data will filtering does not guarantee robustness and may not perform 20 merely increase the difficult) of obtaining a reasonable well for systems with many closely-spaced modal natura/ uncertainty description resulting in a small A. The robust frequencies that cross and shift as flight conditions change.

margins will be similar to the nominal margins in this case, Other approaches toward computing confident flutter mar- which makes intuitive sense because any information gins evaluate the robustness of a stability margin with obtained from the data should only enhance the plant model respect to changes in the model as an indication of the 25 and improve the accuracy of the "flutter margin.

confidence in that margin. A flutter margin robust to pertur- The concept of computing robustness in flutter margins bations to the model is a confident margin because model has been recognized for its importance and has recently been inaccuracies do not affect that margin. An algorithm has termed a state-of-the-art rescm-ch area in aeroelasticity.

been developed to compute the most critical flutter margin Informal measures of robustness are not necessarily useful with respect to first-order perturbations in a model. This 30 because the informal measurc_ provide no guarantee for the method considers only parametric perturbations and can be system stability. The Ix method is based on operator theory computationally expensive. A robust control framework has and provides a well-defined concept of robustness that has a been adopted using a feedback structure to relate the struc- clear set of guarantees as to the stability properties of the tural model and the aerodynamic model. This approach uses system.

highly conservative robustness conditions with respect to an uncertainty structure that may not be physically meaningful.

ROBUST STABILITY A similar approach is adopted allowing unmodeled dynam- A common definition of a signal is a Lebsegue measurable ics and high-order parametric perturbations based on series function that maps the space of real numbers R into R _. A expansion. Statistical approaches are also considered to formulate a flutter probability measure. These approaches 40 space of such signals is denoted S.

will converge to a robustness indicating using Monte Carlo Definition 2.1. I simulations, but the computation time can be prohibitive for The space of signals that are Lebesgue measurable func- complex systems. The robustness measures for these per- tions is S.

turbation and statistical approaches are suspect because no global guarantees can be made as to perturbations not 45 s= _f:R_R '_1 ( l J explicitly considered by the minimization algorithms or the Monte Carlo simulations.

Analog measurements xtt_ of physical systems are real An approach to computing flutter margins that guarantees vector functions of the real parameter t describing time and a level of robustness and directly accounts for flight data is thus are valid members of the space of signals, x(t)_S.

presented herein. (R. Lind and M. Brenner, "Robust Flutter 50 Values of the time parameter, which are often arbitrarily Margins of an F/A-18 Aircraft from Aeroelastic Flight numbered as a distance from some reference point, actually Data," A1AA Journal of Guidance. Control and Dynamics, extend to positive and negati ve infinity. Stability for physical Vol. 20, No. 3, May-June 1997, pp. 597-604.) An aeroelas- systems must ensure stability for all values of time. A tic model is formulated in a formal robust stability frame- time-domain 2-norm is defined as a measure of size (or work that uses a set of norm-bounded operators, A, to 55 energy) for time-domain singlas x_t/eS that considers all describe modeling errors and uncertainty. A multivariable time.

robust stability measure known as the structured singular Definition 2.1.2 va/ue, _, computes flutter pressure that are robust to the The 2-norm measures the energy of a signal x(t)eS.

amount of modeling errors as determined by A. (G. J. Bales et al., Ix-Analysis and Synthesis Toolbox, Musyn Inc. and The MathWorks Inc., Minneapolis, Minn. and Natick, Mass., IIX,r)II: = (___lx(t)l z d ,} 1;" (2) 1995.) A robust flutter margin problem is posed by ques- tioning what is the largest increase in dynamic pressure for which the plant is stable despite possible modeling errors One characteristic of a stable system is that only finite- described by A.

65 energy output signals are generated in response to finite- Flight data are easily incorporated into the analysis pro- energy input signals. Signals with finite energy are known as cedure. The modeling errors are determined by comparing "square integrable" because the integral of the square of the

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signal is finite. The Lebsegue space of square integrable notation by rarely distinguishing between time-domain and frequency domain signals except where the context does not signals is defined as ff2(--_ooo). This space is also referred to make it clear. The notations for the 2-norm of domain and as the infinite-horizon Lebesgue 2-space to denote that the frequency domain are also not distinguished because the norm uses an integral over infinite time.

notations are equivalent, as demonstrated by Parseval's Definition 2.1.3 identity.

The space _2(-oo,_} consists of square integrable time- An important subspace of ff 2 is the Hardy space, _t_,.This domain signals.

space contains the complex variable tions that are analytic in the open right-half of the complex plane and have finite 2(_.°*)=l _ t):x_, p,r( t_l].,<o,} 13) 2-norrn.

Signals associated with physical systems are only known Definition 2.1.8: The Hardy space, 9-Q,i2, consists of the for values of time greater than the time at which measure- following functions.

ments are started. Stability analysis and norm computations 9_.--{f(s):f(sl_kf: and f{s_is analytic in Re(sl>0} (8t using these signals cannot use properties of the signal before the starting time because no information is known. The t5 System Plant traditional method of characterizing these signals is to A system P is defined as an operator mapping the space of assume the signal is identically zero for all times before the input signals Si, , to the space of output signals So,,,. This starting time. The time value at which measurements are definition implies that for any w¢S,, and z=Pw, then Z¢So, ,, started can be chosen without loss of generality and is usually chosen to be t=0. The space ff2(0,oo) is defined as a 20 P:S_,---)s .... (% subset of the infinite-horizon Lebesque 2-space to empha- Linear, time-invariant systems defined by state-space equa- size such signals. They are identically zero for all t<0.

tions are considered.

Definition 2.1.4 The space ff2[0,oo)ct4--oo, oo) consists of square inte- 25 H0) grable time-domain signals that are identically zero for all x(t) D/, II u(tJ I t<0.

£:[0. oo1= .t(tl: -' The signal x(t)¢R"' is the state vector, u(t) CR ''' is the input I £_ [[x(t)[] dt ..... It) = 0 for all t < 0} (47 3o vector, and y(t) cR "° is the output vector. The state update matrix is A;, eR"_'% Bp eR ''_m determines how the input affects the states; C e eR ...... computes the outputs as a linear A similar space i_(--_.O) is defined for signals x(t) that combination of states; and Dp eR .... ' is the direct are assumed to begin at t=--oo and are identically zero for all feedthrough from inputs to outputs. The operator S={Ap, times t=0. Thus, the integral to compute the energy for 35 Be, Ce, Dp} denotes the time-domain state system.

elements of this space considers t=--oo until t>0.

Linear time-invariant state-space system are commonly Frequency-domain signals are often considered in stabil- represented by transfer-function operators. These function, ity analysis but do not fall into the set of signals, S. These P(s), are complex-valued matrices of the complex Laplace signals .f(jt0) are complex-valued functions of the imaginary transform variable, s. Such a transfer-function matrix exists unit j=v--r, and the real frequency variable co is expressed in 40 if and only if the state-space system is linear and time- rad/sec. The set, Sj, o is defined for frequency-domain sig- invariant.

nals.

Definition 2. 1.5 P( sP=Dp+C,,_ sI_ -Ap)-t Bt, ( 11 ) The space of frequency domain signals is Sjo,.

Stability must be considered over the infinite-horizon time lengths so that the operators used map i2 (.,,_ oo) into _2 sj,o={f(jtm:jR-+C" and f*O(ol=.f'(-jo:,)} (51 (_oo,oo) Properties of the Fourier transform relating i_ (oo,_) A frequency-domain 2-norm is formulated to compute a and if2 simply a state-space system, S: t, (-oo,,,o)--+i 2 measure of energy.

(.oox,oo), is linear and time-invariant if and only if the Definition 2.1.6 50 associated transfer-function matrix P is such that y=Pu _i2.

The 2-norm measures the energy of the signal fClo_)eSj,,,.

This condition leads to consideration of the gain for these signals.

(12) A frequency-domain Lebesgne space,if,, is defined for finite-energy signals.

This ration of 2-norms will be finite if the system is stable.

Definition 2.1.7 Properties of the 2-norm are used to derive a condition on The space ft,_ consists of frequency-domain signals with 6o the system transfer-function operator, P. This condition is finite energy. referred to as an induced norm because it results from consideration of signal norms associated with the operator.

_ _ {f(jo)):J'%o,l_/2<_} {7_ The t2 induced norm is defined as the _ oo-norm.

The spaces i2(--o*,oo) and ff 2 are isomorphic Hilbert Definition 2.2.1: Define the _-(_-norm for transfer-function spaces under the appropriate inner produce through the 65 operators.

Fourier transform, which means the spaces have equivalent algebraic properties. TN relationship is used to simplify

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The second condition in theorem 2.3.1 is associated with A space of operators with finite 9{-nortn is denoted a.s ff_.

the first condition guaranteeing a well-posed and stable Definition 2.2.2: The space ff_ consists of system with system. This uniqueness condition can be understood by finite ,'_-norm.

consideration of the solution y for the loop equations shown in FIG. 2.

y=l_u+W I=(I-PAV_ Pu (17_ Transfer functions of linear time-invariant systems are stable if and only if z=Pw and w ¢_2 implies z¢3-( 2. This impli- The inverse term, (I-PA) -_ has a magnitude of infinity if cation results from the Laplace isomorphism between _2 [0, the norm of PA is allowed to he unity. Such a condition would allow the norm of signal y to be infinite despite a oo) and 9£ 2 space. These transfer functions are shown to be norm-bounded u input signal. Restricting I]PA]]_<I ensures analytic in the open right-half complex plane with finite the inverse term exists and a unique finite-nornl y is gener- 9£_-norm. Define the space _ to contain these operators.

ated in response to a finite-norm u. The issue of well- Definition 2.2.3: The space 3£_ consists of transfer func- posedness requires this condition to hold at s=_ and is tions of stable, linear, time-invariant system with finite automatically considered by the _f_-norm.

_-norm.

Robust Stability _]_z={P: P is analytic in Re(s0>0 and I[Pll_'<_} (15) The small gain theorem can be directly used to analyze A subspace _t _f_ is often defined for rational elements.

2o robust stability, of a plant model with respect to a set of Definition 2.2.4: The space, _9-f_, ,-- 9£_ consists of ratio- perturbations. These perturbations are used to describe nal elements of 9£_.

uncertainty in the analytical plant model caused by errors _H7 ={p: P¢9_ and P is rational} (16) and unmodeled dynamics. Usually, the exact value of the modeling error is not known, but a norm-bounded, real Transfer-function operators of linear, time-invariant state- 25 scalar, o_>0, can be placed on the size of that error. Define the space systems are rational functions of the Laplace trans- A of norm-bounded operators describing these perturbations form variable, s. These transfer functions P e_3-(_ if and that affect the plant P through a feedback relationship.

only if P is stable such that no poles lie in the closed A={A:IJAJi__-< o_} (18_ right-half plane. The space _= which may appear to be mathematical abstraction, is thus shown to have a physical 30 The small gain theorem allows consideration of the entire interpretation. _t _K_ is merely the operator theory repre- set of possible modeling uncertainties as described by all sentation of stable, rational, transfer functions.

AeA. The 9(_-norm of the loop gain cannot be explicitly Small Gain Theorem computed for these systems because an infinite number of Stability of a linear time-invariant system is determined 35 loop gains PA generated by the A exists. The triangle by location of all poles in the left-half plane. Robust stability inequality of norms can be used to generate a sufficient in the _ and V frameworks is determined by considering condition for robust stability of P.

an interconnection of stable operators. The basis for deter- mining stability of these interconnections of operators is the "small gain theorem. " IIPAI[- --< I[PIId]_L (19 4o The small gain theorem states that a closed-loop feedback A condition for robust stability of the close&loop system system of stable operators is internally stable if the loop gain can be stated.

of those operators is stable and bounded by unity. Several formulations of the small gain theorem are derived for Lemma 2.4.1: The plant P is robustly stable to the set of various signals and systems. Theorem 2.3.1 presents the uncertainty perturbations, & that enter the system as in FIG.

formulation used for this application. 45 2 with ][A[[_<c_ for all A_A if Theorem 2.3.1 (Small Gain Theorem): Given the feedback IIl_l,<'/_ 1201 interconnection structure of FIG. 2 for stable transfer- function operators P, A: ___---_ff2 with P, A_I-C_; if the Lemma 2.4.1 shows a sufficient, but not necessary, con- _K_-norm of the loop gain is bounded by unity such that 50 dition fo robust stability. The structured singular value, _t, is I]PA[G<I, then: introduced in the next chapter as a less conservative measure 1. the closed-loop system is well-posed and internally of robust stability that is sufficient and necessary.

stable.

An excellent illustrative example has previously been 2. a unique y, w ¢ff_ is associated with each u eff_. presented to demonstrate the issue of robust stability. This This small gain theorem is overly reslrictive in the sense 55 example uses classical arguments to compute a robust sta- bility condition for a simple system that is seen to be of requiring P, A¢ _ 3/_. A more general small gain theorem identical to the robust stability condition generated using the is formulated for operators not restricted to lie in the small gain theorem and lemma 2.4.1. A similar example is subspace P, Ae_t_; theorem 2.3.1 is a special case of this given below for the feedback interconnection in FIG. 3.

general theorem. The extended operator space in the general The single-input and single-output elements in the nomi- small gain theorem allows consideration of robustness for nal system model of FIG. 3 are p, which represents the plant systems composed of nonlinear and time-varying operators.

dynamics; a, which represents actuator dynamics; and k, The requirement of considering stable, rational, transfer- which represents a feedback controller. Each of the nominal function operators is explicitly stated in the theorem to system elements are stable transfer functions contained in emphasize that the nominal aeroelastic system considered in 65 81_. A modeling error exists on the output of the actuator this paper is assumed to be stable and the flutter margin is that is represented by a muhiplicative uncertainty operator, associated with a destabilizing perturbation to that nominal g>¢_9£_, on the output of the element a.

system.

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The transfer function from w to z can be computed as Be, Cp, De) matrices of a state-space realization. The follows.

transfer function can be written as an upper-loop LFT involving S and the Laplace transform variable s where l/s z=-I-( l+akp)-lakp)w (21 ) over s is an integrator.

Internal stability of the closed-loop feedback system is P(s) = Dr, + Ce(sl- At,)-J Bp (28) equivalent to stability of the feedback system shown in FIG.

4a with the operator g-_(l+akp)-Iakp.

x=At,._+B,o,, _s=IAp Be]= l) y = Cpx + D_u Ce Do = F.(S, Because the operators 6 g _3t _ are stable, the Nyquist criterion determines the closed-loop system is stable if and only if the Nyquist plot of fig does not encircle the -1 point.

The LFT is a useful framework for analyzing complex This stability condition is equivalent to the following norm systems with many feedback and series interconnections of condition.

operators. Property 3.1.3 shows the main property of LFFs that will used herein. This property allows complex systems sup_g0o)&j_o_<I (22) t5 of several interconnected LFTs to be expressed as an equiva- This condition is an __-norm condition on the loop gain, lent single LFT. The operators of the new LFT are block- g& Thus, classical Nyquist arguments derive an _-t,-norm structured with blocks composed of the individual operators condition that is equivalent to the stability condition imme- of the LFTs from the original system.

diately formulated by applying the small gain theorem.

Property 3.1.3: Feedback and series interconnections of 2o LFTs can be formulated as a single LFf.

(23) closed-loop stability--->lig& (_< l This issue of stability for LFT systems is associated with the concept of a well-posed interconnection. Stability analy- The error in the actuator command is unknown and sis based on the small gain theorem given in theorem 2.3.1 possibly time-varying, so the operator 8 is used to allow can be used to guarantee the LFI" is stable and well-posed.

consideration of a range of errors. Assume the actuator is 25 Structured Uncertainty weighted such that the range of errors is described by the set of perturbations, I_1 <1. Lemma 2.4.1 is used to generate a The concept of uncertainty is formulated as a set of condition that ensures the system is robustly stable to all norm-bounded operators, A, associated with a nominal plant, actuators errors described by & P, through an LFI" feedback relationship. A family of plants, 30 _ arises through consideration of F,(P,A) for every AeA.

closed-loop stabili ty *---{]g [1_< 1 (24) The true plant model is assumed to lie within this family of plants.

3. STRUCTURED SINGULAR VALUE Modeling the uncertainty as a norm-bounded operator can Linear Fractional Tra,_fformations lead to overly conservative models. The uncertainty descrip- The linear fractional transformation (LF1F) is a common tion can be made more accurate by including frequency framework suitable for robust stability analysis using argu- 35 information. Formulating a model of a physical system that ments based on the small gain theorem. An LFT is an is accurate at low frequencies but less accurate for repre- interconnection of operators arranged in a feedback con- senting the system response at high frequencies is often figuration. These operators can be constant matrices, time- possible. A frequency-varying transfer function, W, is gen- domain state-space systems, or frequency-varying transfer erally associated with each uncertainty element to describe functions. Consider a linear operator PeC _°'+c_-_×_"÷_-" that is 40 magnitude and phase uncertainty as it varies with frequency.

partitioned into four elements.

Uncertainty can enter a system model in a linear fractional manner in several general ways. Two typical types of uncertainty are termed "multiplicative" and "additive" p=[e. p__,] (25) P21 P22 uncertainty. Multiplicative uncertainty can be either on the input or output of a system. Systems with these types of uncertainty are easily described in block diagram form. FIG. The LFT, F,(EA), is defined as the interconnect-ion matrix 6a shows the LFT for a plant with input multiplicative such that the upper loop of P is closed with the operator AeC+,_,.

uncertainty. FIG. 6b shows the plant with output multipli- Definition 3.1.1: Given PeC _°'÷°_-_×"'+j:_ and AeC l'_' define 50 ca(ire uncertainty. FIG. 6c shows additive uncertainty.

F,,tP.A) as the upper-loop LFT of P closed with A such that Uncertainty can also be associated with specific elements y=F,,(EA) u as in FIG. 5a.

of the system. These parametric uncertainties are usually associated with a system operator in a feedback relationship.

(26) F,,(P.A _P....+P21A(i-Pt iA)-_Pt- The number of input and output siguais of the system 55 operator is increased to account for the additional feedback A similar LFT is defined as Ft(P,A) to represent the signals associated with the uncertainty operator. This opera- interconnection matrix of the lower loop of P closed with an tion can be demonstrated by considering P generated by a operator A_C i:_-.

system with an unknown pole.

Definition 3.1.2: Given P_C _O_+°:-)×(h+i:) and A_C i-'×°:-, define F,,(P,A) as the lower-loop LFT of P closed with A such that y=E (P,A) u as in FIG. 5b.

1 [2, 129) _={(s+l)(s+x):X_ 31} (27) Fl(P,A_=Pu +Pt.A(I--P..A)-IP-I A norm-bounded, real, scalar, uncertainty parameter 5 can An example of an ,ereonnection that is common in be introduced to account for the possible variation in pole stability analysis is t; :epresentation of a time dependent value. The set of plants can be written in the LFT framework state-space system a._ _equency-varying transfer function.

using this uncertainty operator and definition 3.1.1.

Define S as the con._tant matrix whose entries are the (Ae,

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17 18

if [JPIJ_<I. This robustness condition can be overly conser- {30_ vative because it does not account for structure in the .r + 2.5 .,r + 2.5 uncertainty operator. The structured singular value, IX, is OR= 1 I " defined as an alternative measure of robustness.

-I -1 ] 5 Definition 3.3. !

(s'+ ll(v + 2.5I Is + l)(s+ 2.51 Given the complex transfer-function matrix Pe3tg-f_ and associated norm-bounded set of uncertainty operators A, define g.

I 132) _(P) = rain _(A}:del(/- pA} = O} AeA A complex system with several uncertainty operators can be expressed as an LFI" with a single uncertainty operator using property 3.1.3. This operator is structured as a block- Define g=O if no AeA exists such that det(l-PA)=0.

diagonal operator with each block associated with the indi- The structured singular value is an exact measure of vidual uncertainty operators. Two main types of uncertainty robust stability for systems with structured uncertainty. The blocks exist. A full-block uncertainty is a matrix with value of IX determines the allowable size of uncertainty unknown elements. This type of block is used to describe matrices for which the plant is robustly stable as demon- unstructured uncertainty in a group of signals.

strated in theorem 3.3.2.

Definition 3.2.1: A full-block uncertainty, AeC "x'', has 20 Theorem 3.3.2 unknown elements Az/for every ie(1,n) and je(l,m).

Given the system in FIG. 7, P is robustly stable with A repeated-scalar block introduces more structure into the respect to the A, which is norm-bounded by real scalar tx uncertainty description than a full block does. Only the such that ]_A[l__-<etfor all A_._Xif and only if IX(P)<I/ct.

diagonal elements of the matrix contain unknown elements: The model P is usually internally weighted such that the the remaining elements are zero. Furthermore, the diagonal range of modeling errors is described by the uncertainty set elements are identical. This type of uncertainty is used to ,5, which is norm-bounded by 1.

relate inpu-output signal pairs with the same uncertainty parameter.

I,AI}_ l for all Ae.5 (33) Definition 3.2.2: A repeated-scalar block uncertainty AEC ''<" has zero-valued elements except an unknown parameter 3O Theorem 3.3.3 presents the specific condition for robust along the diagonal such that A--_I,,, A scalar block is a stability that will be used I-or unity norm-bounded uncer- repeated-scalar block of dimension 1.

tainty sets.

The single structured uncertainty block used for robust Theorem 3.3.3 stability analysis is formally defined in terms of these Given the system in FIG. 7. P is robustly stable with blocks. Let integers m, n, and p define the number of real 35 respect to the A with liA[,.._- I for all AeA if and only if scalar, complex scalar, and complex full blocks respectively.

g(P)<l.

Define integers R_..... R,, such that the i 'h repeated-scalar A value of B<I implies no perturbation within A exists that block of real, parametric uncertainty is of dimension R_ by will destabilize the feedback x.xstem. This condition can also Rj. Define similar integers C¢..... C,, to denote the be interpreted as saying the true plant dynamics are stable, dimension of the complex repeated-scalar blocks. The struc- 40 assuming these dynamics lie _ tthin the range generated by tured uncertainty description A is assumed to be norm- the nominal model dynamics coupled with the set of mod- bounded and belonging to the following set.

eling errors.

Obviously, Ixis dependen! on the block structure of A. The ,._= IA= diag(6_l,_j .... 6Rl,v_ ,. 6_1cI.... .6,Clc,, Ab .... Ap): (31) robust stability properties computed by Ix will only be 45 accurate if a realistic uncertainty operator is chosen. The 6_ • k. 6', _ C. a_ _ C_il structured singular value ma) be arbitrarily greater when computed with respect to an unstructured uncertainty opera- tor as compared to a highly structured uncertainty, operator.

Real parametric uncertainty is allowed to enter the prob- Definition 3.3.1 demonstrates the _t condition of theorem lem as scalar or repeated-scalar blocks. Complex uncertainty 50 3.3.3 is equivalent to the small gain condition of lemma enters the problem as scalar, repeated-scalar, or full blocks.

2.4.1 when the uncertainty is unstructured.

Complex uncertainty parameters allow uncertainty in mag- Unfortunately, Ixis a difficult quantity to compute. Closed- nitude and phase to be modeled; uncertainty in physical form solutions exist to exactly compute IXfor only a small characteristics can be more accurately modeled with real number of block structures for A. Upper and lower bounds parameters. The robustness analysis will be less conserva- 55 are used to compute Ix for generalized uncertainty block tive by accounting for this structure to accurately describe structures.

the model uncertainty.

Structured Singular Value 4. ROBUST FLUTFER MARGINS FIG. 7 shows the general framework for robust stability analysis. The plant operator P(s) _N_ is a stable, rational, 60 Nominal Aeroelastic Model transfer-function matrix representing the aeroelastic dynam- Consider the generalized equation of motion for the ics. A norm-bounded A_I _)1_ is described such that A(s)_.A structural response of the aircraft.

describes the modeling errors in P through a feedback relationship.

M[l$]$':gti+O'l+Krl+OQ(s}q=O (34) The robustness of P with respect to the A can be deter- For a system with n modes, define MeR '_ as the mass mined using the small gain theorem as presented in lemma ma_.x, CeR '_ as the damping matrix, and KeR .... as the 2.4.1. This condition guarantees stability for any value AeA

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19 20

stiffness matrix. Define q_R as a scalar representing the and s-_. An approximation to these forms allows them to fall dynamic pressure, and Q(S)eC"'" as the matrix of unsteady within the framework of the method used. Including a aerodynamic forces. This equation is valid for a particular high-frequency pole in the nonproper term, such as replac- Mach number, with a different Q(s) describing the unsteady ing s with aerodynamic forces at a different Mach number.

Values of the aerodynamic force matrix at distinct fre- s+ 10000" quencies can be derived using finite-element models of the ,aircraft and panel methods for unsteady force calculations.

This research uses a computer program developed for NASA would not affect the low-frequency region of interest while known as STARS. This code solves the subsonic aerody- 10 ensuring stable and proper functions. With the namic equations using the doublet-lattice method. The approximation, the forms of Roger and Karpel can be shown supersonic forces are generated using a different approach to be subsets of this method.

known as the constant panel method.

Standard frequency-domain system identification algo- Formulating a linear time-invariant representation of the rithms can generate a system with an arbitrarily large aerodynamic forces to incorporate them into the robust 15 number of states. This state dimension does not greatly stability framework is desired. Pad6 approximations can be affect the computational cost of the robust stability analysis.

used to compute a rational function approximation to the Extending the robustness analysis to controller synthesis, transfer-function matrix.

however, places an emphasis on limiting the state dimen- sion. imiting the number of states in the identification s s (35) 20 process is not directly considered here, although standard Q, = Ao + sAi + sZA2 + s + [3---_A3 + s--_Aa model reduction techniques can be used on the state-space system to lower the state dimension.

Generating a state-space representation of the aeroelastic This form is often referred to as Roger's form. The equation presented here includes only two lag terms, 25 system, including the state-space form of the unsteady aerodynamic forces, is straightforward, Consider the force although more terms can be included. The poles of the lag vector, y, generated by the state vector, rI. Define xeR "_ as terms. I]p and ___, are restricted to be real and positive to the vector of aerodynamic states.

maintain system stability. The matrix elements of Roger's form can be computed using a least-squares algorithm to fit (38) the frequency-varying aerodynamic data.

l B°llxl

30 y = Q_s)q ¢= = Ce D o q The aerodynamic lag terms can be replaced in the for- mulation with a finite-dimensional state-space system rep- resented by a transfer-function matrix using Karpel's method. Using x, formulate the aeroelastic differential equation.

35 O=M[l$1$"gfi+Cr_+K_+#Q_srq=M[l$l$'g_i+C_+g_+4y=- Q_ s ,_a,o+sA j+s2A2CQ_sI-AQ) -I 8QS (36) M[l$]$gli+C_+grl+¢ _ CQ.r+D,_I) _MrI+Co+(K+_D_21q+qC_'39 ) Standard system identification algorithms, including A state-space system is formulated using the generalized curve-fitting or least-squares approaches, can be used to states, rland rI, and the unsteady aerodynamic states, x. The compute the elements in the state-space portion of the 40 state-update matrix is determined by the following three formulation. The A_ matrices are assumed to be known from differential equations.

the low-frequency aerodynamic force data or from experi- mental wind-tunnel data.

140) A matrix fraction approach is also formulated to represent -M-I{K + _Do) -M-IC -_M ICQ the aerodynamic forces as a linear time-invariant system.

This generalized form computes rational matrix polynomials BQ 0 AQ in a fractional form using a least-squares algorithm. Roger's form and Karpel's form can be shown to be subsets of the matrix fractional form.

Nominal Aeroelastic Model in the Structured The approach taken in this application is to fit the aero- Singular Value Framework dynamic force matrices to a single, finite-dimensional, state- The generalized equation of motion for the nominal space system. This form is most similar to Karpel's form, aeroelastic system can be expressed in a form suitable for except the additional A t matrices are not explicitly accounted for in the formulation.

using la analysis to compute a flutter margin. The flutter margin is dependent on the flight condition parameters that (37) 55 result in a flutter instability, and la is defined to be the = D o + CQ(sl - AQ )- I BQ AQ B o ] smallest perturbation among the A set that causes an insta- Q(s)= CQ DQ bility. The obvious approach to formulating flutter analysis in the _t framework is to introduce a perturbation to a flight Given the number of generalized states, n, and the number condition parameter and find the smallest perturbation that causes an instability.

of aerodynamic states, nQ, define AQeR "Q_'Q, BQ¢R "Q'_', 60 Essentially, the two subsystems in the nominal aeroelastic CQ¢R '_'Q, and DQ¢R '_" as the elements of the state-space and system approximating Q(s). model are composed of the structural dynamics, involving Fitting the aerodynamic data to a finite-dimensional state- mass, damping, and stiffness matrices; and the unsteady space system is equivalent to fitting each term in the matrix aerodynamics scaled by the dynamic pressure. The gener- to a real, rational, proper, u'ansfer function. This method 65 alized equation of motion demonstrates the dynamic pres- seems to contradict the methods of Roger and Karpel, which sure linearly affects the dynamics at a constant Mach con- form nonproper transfer functions caused by the terms in s dition. Perturbations to dynamic pressure can thus enter the

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The dimension of the uncertainty block is the dimension system through a feedback operator in a linear fractional of the signal z. The state-space equations for P_sl demon- manner that is perfectly suited to IXanalysis.

strate this dimension is the number of modes in the system, Consider an additive perturbation, _y_R, on the nominal n. The number of free variables in the p upper-bound dynamic pressure, qo.

optimization, and consequently the computational cost of IX, is a function of the uncertainty dimension. In this way, the number of aerodynamic states, n_, does not directly affect the cost of the flutter estimation. The only cost increase Separate terms in the system dynamics that involve _.

caused by these additional states is computing the frequency O=M[l$l$'g¢i+C_+_K+qO,,_q+qC,ax response of the state-space matrix, which is generally much Jo lower than the cost of computing ix.

=M'q+[Cq_K+_toDe_q+qoC,7,,l+Sa[OQq+C_.,x] Demonstrating the aeroelastic system formulated in the IX framework in FIG. 8 is straightforward. The dynamic pres- ---rl+lM -jCq+M" J(K+qoD Q)q+qo M- J C,ax] sure parameterization is equivalent to the nominal state- space system given in the previous section. Simply compute +ta[M -joeq+M-' Ce_. ] 15 the closed-loop transfer function with no uncertainty, _q=0, --q+lM-_ C_+M-' _K+qoO_,_rl+qoM-' C_,vl+tq= and the nominal system is recovered.

Wind-tunnel and ground vibration testing can experimen- =q+lM- t Cfl+ M- I_ K+qoDe _tl+qoM- t CQXI+W (42 I tally determine aerodynamic stiffness and damping matrices that are more accurate than the matrices approximated by a The signals z and w are introduced into this formulation 20 finite-element model. These matrices can be incorporated to associate the perturbation in dynamic pressure to the into a nominal state-space model in the , framework.

nominal dynamics in a feedback manner. The signal z can be This procedure considers variations in dynamic pressure generated as an output of the plant because z is a linear for an aeroelastic model at constant Mach number. The combination of states.

unsteady aerodynamics are highly nonlinear with variation 25 in Mach number, and attempts to model Mach variations in z=M-_Dorl+M-I C_.,v _43) an LFT may produce highly conservative flutter margins.

The ;a method presented herein is considering flutter margins The signal w is related to z by the dynamic pressure in terms of dynamic pressure as measured along lines of perturbation.

constant Mach. Flutter is a function of the two variables, *_'=-tqz (44) 30 dynamic pressure and Mach, so computing the dynamic pressure causing flutter for a dense set of discrete Mach The state-space aeroelastic model for nominal stability values will generate an accurate portrait of the flutter mar- analysis in the IXframework is formulated using the state- gins.

update matrix. This matrix is determined by the dynamics at Robust Aeroelastic Model in the Structured Singular Value the nominal dynamic pressure, and the additional input and 35 Framework output signals that introduce perturbations to the dynamic A robust aeroelastic model in the ix framework can be pressure. That perturbation, _q, is not an explicit parameter generated by associating uncertainty operators, A, with the in the state-space model because _ only affects the plant nominal model and including the parameterization around a through a feedback relationship as deten-nined by the sig- perturbation in dynamic pressure. These uncertainty opera- nals z and w. Define the transfer function P(s) generated by 40 tors can resemble any of the forms presented in section 3.2, state-space matrices such that z=P(s)w.

including parametric uncertainty and additive and multipli- cative representations of dynamic uncertainty.

0 l 0 Choosing a reasonable uncertainty description is crucial -M-I(K + _oDo) - M-I C -_M-t CO -1 0 for determining a valid robust flutter margin. This choice can 0 x BQ 0 AQ 45 arise logically from consideration of weaknesses in the 0] I r/ (45) modeling process, previous experience with aeroelastic

o

M-I DQ 0 M-I C_2 analysis, and comparison with observed flig."-,- dynamics.

The following section 5 on UNCERTAINTY DESCRIP- FIG, 8 shows the feedback interconnection between the TIONS IN AEROELASTIC MODELS gives a noncompre- perturbation in dynamic pressure and the nominal plant 50 hensive examination of several obvious uncertainty descrip- model parameterized around that perturbation. This inter- tions that may be associated with an aeroelastic model.

The LFT is a valuable framework for formulating the connection is an LFT, and the small gain condition of lemma robust aeroelastic model so that the model is suitable for IX 2.4.1 or the IXcondition of theorem 3.3.3 can be directly applied to analyze stability with respect to a variation in the analysis. The various system blocks composed of the nomi- flight condition q. 55 nat state-space model with associated uncertainties and any Formulating the nominal aeroelastic dynamics in the ix additional subsystem blocks and their associated uncertain- framework immediately demonstrates the procedure used in ties can be expressed as a single model and uncertainty computing a flutter margin. Traditional flutter analysis algo- operator.

rithms such as the p-k method and the ix method as applied FIG. 9 shows the block structure used for Ixanalysis of the to FIG. 8 are searching for a value of dynamic pressure that 60 uncertain aeroelastic system. The structured singular value results in a flutter instability. The nominal flutter margin is computed with respect to a single, block-diagonal, struc- question may be posed, which is answered by these meth- tured operator that contains the perturbation to dynamic ods, pressure and the structured uncertainty operator along the Question 4.2. l diagonal. The perturbation to dynamic pressure is explicitly Nominal Flutter Margin 65 shown to distinguish _q from the modeling uncertainty and What is the largest perturbation to dynamic pressure for emphasize _ as a special operator used to describe a range which the nominal aeroelastic dynamics are stable? of flight conditions.

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The flutter margin computed for the uncertain system (FIG. 9) is a more accurate margin than one computed with (i = rtfin I_ : F.(P. 6q)is unstable} traditional methods such as p-k. These traditional methods address the nominal flutter problem in question 4.2.1; the robust flutter margin must consider the effect of the model- Then _t(P)=l/6 such that qa,,j'°"'-_o+6 is the nominal ing uncertainty. The robust flutter margin actually finds the flutter pressure and smallest perturbation to dynamic pressure for the entire set r',o,,=_i of plants formulated by the interconnection of the nominal represents the nominal flutter margin answering question dynamics and all elements A_A. Question 4.3.1 poses how to l0 4.2.1.

compute these mar_ns.

Proof Question 4.3.1 This result is immediately obvious using definition 3.3.1 for _twith respect to a scalar uncertainty parameter _. The Robust Flutter Margin Ix is the inverse of the smallest destabilizing perturbation, What is the largest perturbation to dynamic pressure for 15 and b is computed as the smallest value of _ that destabi- which the nominal aeroelastic dynamics are robustly stable lizes P; thus Ix(P)=I/& [] to the entire range of modeling errors as described by the Lemma 4.4.1 indicates a computational strategy to com- norm-bounded A?

pute a nominal flutter margin that does not require a search over a set of frequency points. The flutter pressure is found This question can be answered by computing Ix for the by increasing values of _ until an eigenvalue of the state system in FIG. 9.

2o matrix of F,(P,_) has a negative real part indicating F,(E8 Computing a Flutter Margin with the Structured Singular _) is unstable. Algorithm 4.4.2 demonstrates a bisection Value search implementation that efficiently computes upper and The nominal flutter problem posed by question 4.2.1 and lower bounds on the minimum destabilizing _, perturbation to within a desired level of accuracy.

the robust flutter problem posed by question 4.3.1 can be 25 Algorithm 4.4.2 (nominal flutter margin): solved as a Ix computation. The value of Ix is a sufficient Given plant P at nominal dynamic pressure, qo, affected direct indication of the flutter margin for the nominal sys- by perturbation _ as in FIG. 4.1: tem; additional information regarding the norm bound on the Define scalars _i_,p_,.._ t..... ,>0 as bounds on the smallest uncertainty is required to derive the robust flutter margin.

destabilizing _.

The nominal aeroelastic model is formulated for stability 30 Define scalar e>0 for accuracy.

analysis in the g framework in section 4.2 by introducing a perturbation, _, to dynamic pressure. A nominal flutter while (6,pro., - d/l_. > e) { margin is computed to answer question 4.2.1 by considering the smallest value of this perturbation that destabilizes the model. The nominal flutter pressure can be directly calcu- if FdP. 6zt)has an unstable pole), then 6.,_r,_. = 6q lated by computing _t with respect to the perturbation else 6to,,,, = operator _q.

The exact value of g can be analytically formulated to I compute a nominal flutter pressure because a closed-form 40 _'_tl_,<_ = qo + 6_em'_ solution for Ix with respect to a single, real, scalar operator is known. This solution is the spectral radius of the frequency-varying transfer-function matrix.

Robust flutter margins that address question 4.3.1 cannot u(,,,) = ma_p(t:'t/,,,) (46) 45 be computed using algorithm 4.4.2 because art additional search over the set of uncertainty operators A must be included. These margins must be computed using the aug- The spectral radius of P(jto) is a discontinuous function of mented plant P, which includes feedback signals relating the frequency, so computational algorithms based on searches perturbation to dynamic pressure and the uncertainty over a finite set of frequency points may not guarantee the 50 description as shown in FIG. 9. Define the block-structured correct computation of robustness values. A small amount of set of operators _, which considers a particular perturba- • q complex uncertainty can be added to the real uncertainty that tion _ and set of operators A describing model allows a continuous Ixfunction to be analyzed but introduces uncertainty-- unrealistic conservatism. An heuristic robustness indicator (47) can be substituted for _t that considers stability over a 55 frequency segment but is not considered here.

A relatively simple approach can be used to compute Ix with respect to a single real parameter by considering the A set of plant models F,(E A8 ), exists for each value of destabilizing value of the parameter. A search over the 60 _q that defines the nominal plant at dynanuc pressure q=qo+6 parameter space will result in computation of _t and the and variations to that plant caused by the set of uncertain- desired flutter margin. Lemma 4.4. l presents the principle of ties A. The robust flutter margin corresponds to the smallest this approach.

perturbation _q for which an unstable plant, F,(P, Aa.), exists Lemma 4.4.1 with S 8 F._8 . If every member of the set of plants I_,,(P, A_.)

Given the plant, E derived at nominal dynamic pressure, 65

is stabl_, _ien q:=_o+_q is not a flutter pressure and

formulated at _ is robustly stable to the uncertainty descrip- 90, with a perturbation to dynamic pressure, _, arranged in tion A.

the feedback relationship of FIG. 9 define 8.

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destabilizing perturbation to dynamic pressure is at least 6 The smallest destabilizing perturbation _ corresponding to a robust flutter margin can be computed by a IXcompu- ca=l, which corresponds to dynamic pressure q-=qo+Wq-_q - tation. The _t framework analyzes robustness with respect to q,+Wca. Thus, i5 is guaranteed to be robustly stable to the a single, structured operator, so the operator set used to uncertainty set A for any perturbation to dynamic pressure compute a robust flutter margin must contain the set of 5 less than WCa,to WCais a robust flutter margin.

uncertainty operators A along with a range of dynamic ---)(sutticient) pressure perturbations. The IXwill compute the robustness of Assume _t(P)>l. Define real, scaler ct<l such that u (P)= P with respect to this operator set to find the smallest _1. which implies, from theorem 3,3,2, that P is robustly destabilizing perturbation to dynamic pressure and the stable to all uncertainties T_8S with IlAli_<a<l. Thus, P is not smallest destabilizing uncertainty operator. Define the ,5 that 10 guaranteed to be stable over the entire range of modeling uncertainty defined by the unity norm-bounded set A, so this contains A6_ sets for a norm-bounded set of _ operators.

perturbation is not a valid robust flutter margin and does not answer question 4.3.1.

= . _ _ & II,all_ -< 1, II_qll -< I 01 } ,48, Assume HF<I. Define real, _>1 such that u(P)=rfl, which implies, from theorem 3.3.2, that P is robustly stable to all uncertainties AeA with llAl/_<ct. Thus, P is robustly stable to an uncertainty description larger than that defined by the Imposing the norm bound for &_operators as It_l__ -< 1 may seem overly restrictive because the units of _ are the same unity nonn-bounded set A, so this condition defines a valid as the units of q in the model. This condition implies the S flutter margin but is not the least conservative robust flutter considers the range of flight conditions q=q+l lbf/ft -_ for ._0 margin and does not answer question 4.3.1.

plants formulated by dynamic pressure in units of lbf/ft 2.

The only difference between models P and P results from Such a small rb_nge of flight conditions is not useful for the scaling WCawhich scales the feedback signals between P stability analysis unless qo is extremely close to the flutter and _. No external scaling matrix is allowed to affect the pressure. This limitation is avoided by introducing a weight- feedback signals between P and A because A is defined with ing function, W_, to the computation of q. 25 a unity norm bound. Computing g of the plant P with an additional scaling on the lower-loop signals would consider a scaled set of operators A that does not accurately represent qw---Oo= Wq,_q (49_ the uncertainty description. Therefore, P only scales the &q feedback signals.

A WCa>I allows a large range of flight conditions to be Theorem 4.4.3 can be modified to compute a nominal considered despite the unity norm-bound constraint on &q. 30 flutter margin by changing the identity matrix in the scaling This weighting is incorporated into the stability analysis by used to compute P to a zero matrix. This modification scaling the feedback signals between the _ operator and the eliminates the feedback interconnection between the model plant P to form the scaled plant. P.

and the uncertainty description A, so IXconsiders only the nominal dynamics and computes the smallest destabilizing perturbation to dynamic pressure and F,obF .... • _=_W_o oil (50) 35 Including the uncertainty, description A ensures the robust flutter margin will be no greater than the nominal flutter margin. The robust flutter margin considers the model used A robust flutter margin is computed by analyzing Ix(F) with respect to the A. The robust flutter pressure is deter- a0 to compute the nominal flutter margin that corresponds to the uncertainty operator A=0aA and the models that corre- mined by_ _iterating over scalings Wfi u_ntil the smallest spond to the remaining operators A_A. The conservatism in pressure q=qo=W_ is found for which the P is robustly stable to the set of uncertainties A. Theorem 4.4.3 formally dem- the robust flutter margin makes intuitive sense because the onstrates this concept.

nominal flutter margin is the worst-case stability boundary Theorem 4.4.3: Given the plant P derived at nominal 45 for a single model and the robust flutter margin is the dynamic pressure qo with a perturbation to dynamic pressure worst-case stability boundary for a family of models.

and set of uncertainty operators A norm-bounded by one arranged in the feedback relationship of FIG. 9, define the ¢: Fret, = Fno m ( 53 plant P with real diagonal matrix WCascaling the feedback signals relating &q and P.

The proof demonstrating the necessary and sulficient condition IX(P)=I also makes intuitive sense because the equality sign is needed to ensure the flutter margin is valid _=p[W_o 0It (51) without being overly conservative. If IX(P)<I, then no AeA causes an instability and the flutter margin is too conserva- -- rot, "X +W tive. If g(P_<l, then the system is not robust to all modeling Then qfl,,_ =qo cais the robust flutter pressure if and errors AeA and the flutter margin is not a valid robust flutter only ff g(P)=l. Also.

margin.

Theorem 4.4.3 demonstrates a robust flutter margin can be F_o_,=Wq (52) computed by determining a scaling matrix Wra for which _t( 15)=1. Algorithm 4.4.4 implements an iterative approach to represents the least conservative robust flutter margin compute a scaling matrix for which _t(15)¢(1+_) for some answering question 4.3.1.

desired level of accuracy e.

Proof Algorithm 4.4.4 (robust flutter margin): _---(necessary) Given plant P at nominal dynamic pressure qo affected by The condition _t(Pl=l implies that the smallest destabi- 65 unity norm-bounded _q and A as in FIG. 9: lizing perturbation to P is described by some AeA with I_1[_, so no destabilizing AeA exists and the smallest positive Define initial weighting WCa.

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27 28

Define scalar ¢>0 for accuracy. function derived in appendix A must be used. Algorithms 4.4.4 and 4.4.5 can be adapted by replacing the _ calculation with a _t upper-bound calculation to compute flutter margins.

A search over frequency points is required when using the 5 upper bound (appendix A), so the accuracy of the robust flutter margin requires a dense grid of frequencies associated with the natural frequencies of the worst-case dynamics to be considered.

whih, (/a(PI > I + x)OR (/ziP) < 1 -e){ A simple approach can be implemented if the natural 1o frequency of the unstable dynamics at the nominal flutter

Wq

pressure can be assumed to be similar to the natural fre- _(P) quency of the unstable dynamics at the robust flutter pres- sure. This assumption can be justified if the uncertainty does not change the critical flutter mode between the nominal and z5 robust flutter pressures, which is often true for systems that --rob -- q aum,_ = q0 + l,_q have relatively small levels of uncertainty and clear sepa- ration between critical and subcritical modal frequencies.

F_ = Wq Algorithm 4.4.6 presents this approach, which first com- putes the frequency of the nominal flutter mode and then The dynamic pressure qo defining the flight condition for 20 computes a robust flutter margin from the _ upper bound the nominal plant dynamics must be chosen carefully to evaluated near that frequency.

ensure the robust flutter margin computed with algorithm Algorithm 4.4.6 Irobust flutter margin with reduced fre- 4.4.4 is valid. The nature of _t and the upper bound is such quency grid): that all norm-bounded operators centered around the origin Given the system in FIG. 9: are assumed to be valid perturbations to the plant dynamics, 25 Compute frequency co associated with nominal flutter so positive and negative perturbations to dynamic pressure dynamics using algorithm 4.4.2.

are considered by the robust stability analysis. Thus, the Define dense frequency grid _ centered around m.

robust flutter margin computed by _(P)=I could correspond Compute robust flutter mating from V upper bound evalu- to either perturbation q_¢=l or _=-1.

The actual dynamic pressure at the flutter instability is ated at _ using algorithm 4.4.5.

occurs at a negative dynamic pressure that may be unreal- 3o A large and dense frequency grid increases the confidence istic for classical flutter analysis. The value of the nominal that computed robustness measure is actually an upper dynamic pressure qo can slide along the real axis to a large bound for _. Algorithm 4.4.6 must be used with caution value without loss of generality in the _ analysis because this because the assumptions behind its use may not be satisfied.

parameter linearly affects the nominal dynamics. A simple Computing a robust flutter margin with a dense frequency approach to ensure that the robust flutter pressure is a 35 grid for a particular aircraft at several Mach numbers and positive dynamic pressure is demonstrated in algorithm then comparing the frequencies of the flutter dynamics to 4.4.5, which iterates over increasing values of qo until the those of the nominal flutter dynamics is recommended. If sc_aling associated with the robust flutter margin satisfies W these frequencies are similar, then algorithm 4.4,6 can be _<qo- considered for further analysis at different Mach numbers.

Algorithm 4.4.5 (robust flutter margin with qo itemtiont: 40 4.5 Properties of the Structured Singular Value as a Flutter Given parameters as in algorithm 4.4.4: Margin Given initial value of nominal dynamic pressure 90: The flutter computation method described herein uses _ as the worst-case flutter parameter. The structured singular value is a much more informative flutter margin than tradi- valid_margin = FALSE 45 tional parameters such as pole location and modal damping, while(validmargin == FALSE) { so _t presents several advantages as a flutter parameter.

The conservatism introduced by considering the worst- compute plant P at nominal d>71amw pressure qo case uncertainty perturbation can be interpreted as a measure compute ?]_,,, and associated W# from algorithm 4.4.4 of sensitivity. Robust _t values that are significantly different 50 than the nominal flutter margins indicate the plant is highly (f ( Wq > _oL then _lo = l.lW_ I sensitive to modeling errors and changes in flight condition.

else valid_margin = TRUE A small perturbation to the system can drastically alter the flutter stability properties. Conversely, similarity between the robust and nominal flutter margins indicates the aircraft 55 is not highly sensitive to small perturbations.

Robustness analysis determines not only the norm of the The exclusion of low and negative dynamic pressures for smallest destabilizing perturbation but also the direction.

stability analysis may not be desirable for all applications This information relates exact perturbations for which the related to aeroelasticity. An example of such an application system is particularly sensitive. Thus, p can indicate the is the analysis of aeroservoelastic dynamics for a high- 6o worst-cast flutter mechanism, which may naturally extend to angle-of-attack aircraft that concerns instabilities at low indicate active and passive control strategies for flutter dynamic pressures. The procedure for these types of analysis suppression.

chooses a low value of qo and finds the scaling W_ corre- Damping is only truly informative at the point of insta- sponding to q6_--1, which computes the low dynamic pres- bility because stable damping at a given condition does not sure instability. Algorithm 4.4.5 can be modified for these 65 necessarily indicate an increase in dynamic pressure will be applications.

a stable flight condition. The structured singular value The flutter margin computation must allow for an arbi- computes the smallest destabilizing perturbation, which trary structure of operators A. so an upper bound such as the

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29 3O

indicates the nearest flight conditions that will cause a flutter tire. The actual choice of Ax and A. is problem-dependent and can vary with different aircraft and different finite- instability. In this respect. IX is a stability predictor and damping is merely a stability indicator. element modeling procedures.

These characteristics of IX make the worst-case flutter The unqertainty operators are described in this section as 5 Ax-AceR ..... with real elements because the operators algorithm especially valuable for flight test programs.

Aeroelastic flight data can be measured at a stable flight describe perturbations to the generalized stiffness and damp- condition and used to evaluate uncertainty operators. Unlike ing matrices that are usually real. These operators are often damping estimation, the _ method does not require the additionally constrained to be diagonal operators with n independent parameters because the generalized stiffness aircraft to approach instability for accurate prediction. The g can be computed to update the stability margins with respect l0 and damping are often computed as real diagonal matrices.

to the new uncertainty levels. The worst-case stability mar- The real and diagonal nature of these uncertainties is not required for IXanalysis, so fuU-block complex uncertainties gin then indicates what flight conditions can be safely considered.

can be used if they better describe the nature of the modeling errors.

Safe and efficient expansion of the flight envelope can be performed using an on-line implementation of the worst- l.s Also, the weighting functions,WwWceR "_' are presented as constant and real matrices because the functions are case stability estimation algorithm. Computing Ix does not introduce an excessive computational burden because each associated with constant and real stiffness and damping F/A-18 flutter margin presented herein was derived in less matrices. These constraints on the nominal plant through a than 2 rain using standard off-the-shelf hardware and soft- feedback relationship. The feedback operation w is a linear ware packages. The predictive nature of Ix and the compu- 20 combination of the states of the plant. The weightings can be tational efficiency allow a flutterometer tool to be developed relaxed if the nature of the uncertainty is best described by that tracks the flutter margin during a flight test. complex and frequency varying weighting functions, Substitute the uncertain K and C into the equation of 5. UNCERTAINTY DESCRIPTIONS IN motion, including the state-space representation of the AEROELASTIC MODELS 25 unsteady aerodynamic forces Q(s) presented in section 4.1.

Parametric Uncertainty in Slructural Models Introduce a perturbation, _q, to dynamic pressure, and sepa- Parametric uncertainty denotes operators that describe rate the nominal dynamics from the unknown terms, errors and unmodeled variations in specific elements and coefficients in dynamic system equations. Recall the gener- alized aeroelastic equation of motion for state vector qeR'-.

30 O=M[l$l$'gti+Cr_( l(+qD Ol'q+qC_ =Mtl+Cotl+Wo-Sct_+_Ko+14k,.Sh.+(q,,+_q_Dt,)Wl4"qo+6a)C_"( M[l$l$Ui+Ol+g_2+#Q_ s_rl---o (54 ---,]+IM- ' Co_+M-' ( Ko+eloD_,n]+qoM-' CQr I Robust flutter margins computed with the _t method are 35 +6_[M-'Dq'q+M'-'Cc'x]+A"c[M-'Wr<'ql+AcIM-'C°We_] strongly affected by the choice of uncertainty descriptions associated with these dynamics, so this uncertainty must be ---rl+fM-ICo rl+M-' _Ko+_oDe_rl+qoM -j C_t 1+6_%+._ ^.zh4-_cZc a reasonable indicator of potential modeling errors. Para- --rl+lM-ICo'd+M-'(Ko+(loDo_rl+qoM-*Ce_:]+w_+w,,.+wc (57) metric uncertainty can be directly associated with the struc- tural matrices to indicate specific errors in the finite-element 40 The signals z_ and wfi are inlxoduced in section 4.2 to model.

relate the perturbation in dynamic pressure to the nominal Define an operator, Ax_R "'_' that describes additive uncer- plant through a feedback relationship. The feedback opera- tainty of a nominal stiffness matrix K o. Associate a weight- tion wfi--_qz_ is used where z_ is a linear combination of the ing matrix, W,_R""' with this uncertainty such that a states of the plant.

stability analysis should consider a range of stiffness matri- ces described by all A k with [_c_[_ < 1.

zct=M- _ DorI+M- t CQa (58 ) K=Ko+ WxA K (55) Additional signals are introduced to the plant formulation, where z K and w h. are associated with the uncertainty in the Parametric uncertainty can also be associated with struc- 5o stiffness matrix, and z c and w c are associated with the tural elements in a multiplicative relationship. Define an uncertainty in the damping matrix. The outputs of the plant operator, Ace.R ''°' that describes multiplicative uncertainty z K and z c are formulated as linear combinations of the states.

of the nominal damping matrix C o. A weighting Wc_R _'' is again associated with the uncertainty such that the antici- z _--M- _ Wt,.'q pated range of damping matrices for robust stability analysis 55 is described by all AC with I[Act[,-_ < I.

z_=M -_ WcCoq (59_ The feedback mechanism to describe the modeling uncer- C=Co(I+WcAcJ ¢561 tainty uses a relationship between these output signals and The choice of additive uncertainty for A t. and multiplica- 60 the w^. and w c input signals.

rive uncertainty for A c does not reflect any generalized assumptions regarding the proper way to model errors in Wa=.5,vZx stiffness and damping; rather, each type is included to demonstrate the different mathematical derivations. Addi- Wc=.5eZ c (60) live and multiplicative operators are common types of uncertainty models, so demonstrating how these types of The state-space plant malxix can be formulated using these uncertainty are associated with a structural model is instruc- additional input and output signals.

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31 32

0 1 0 -1 -1 -M L(I_) +_I,,D_) -M-tC -'_M-tCo q BO 0 AQ M-I D_: 0 M-) C_

00°11 0 0 O w o

/il

M iB' h 0 0 0 0 0 Ilw_, 0 M- I Co Wc 0 0 0 0 I l wc FIG. 10 shows how the uncertainty operators and pertur- which type of uncertainty is most suitable to describe errors bation to dynamic pressure affect this plant formulation in a in A e and B e is problem-dependent.

feedback manner Also, defining the uncertainty operators as real and The formulation does not directly allow uncertainty in 15 weightings that are real and constant is not a requirement.

mass to be described by a feedback operator• The LFq" and This section presents the problem formulation with these frameworks require uncertainty operators to affect the definitions because associating these types of uncertainties nominal dynamics in a linear manner, and this requirement and weightings with the constant real A e and BQ matrices precludes introducing mass uncertainty. The inverse of the makes intuitive sense. Certainly St can also compute robust mass matrix scales most terms in the state matrix of P(s), 20 flutter margins with respect to complex frequency-varying including terms involving _. Associating a mass uncertainty weighted uncertainties.

The nominal aeroelastic model in section 4.2 defined the operator A M with the mass matrix scaling q would introduce vector x as the states associated with the state-space model terms of Ate5-4, which is a nonlinear funcdon of uncertainty operators and cannot be directly considered by the _tmethod.

Q(s). The matrices A e and B o directly affect the aeroelastic Parametric Uncertainty in Aerodynamic Models 25 dynamics only through the state derivative equation for x The unsteady aerodynamic forces Q(s) _ C "_' can be and do not appear in the other state derivative equations. The aeroelastic plant can be formulated in the B framework by represented as a state-space model with n o states.

substituting the uncertain values ofA e and B e into this state _$)=DQ+Ctd(sI-A q) IBf2 162) derivative equation without considering changes in the 3o derivative equations for-the remaining states rI and 11 Parametric uncertainty can be associated with the matrix elements of this state-space representation to describe errors.

.;_=AQ.x+Be_ These errors can result from several sources in the modeling procedure, including computational fluid dynamic algo- rithms that determine the frequency-varying forces and the system identification algorithms that represent the compu- -'=Aoox+Beoq+AaQWacr+ABvWsoBQorl tational values as a state-space system.

rlQ_'l --Ae_r,+Boo rl+A_oz,_e +Ast:,zao Define an operator Aa e R Q to describe uncertainty in the state matrix of Q(s_. This operator directly affects a =A_x+BooTI+W,4Q+WB 0 nominal Ae0 and describes errors and variations in the poles of the state-space representation of the unsteady aerody- Several signals are introduced to this equation, where zae namic forces. Include a weighting function W a _ R ''_" such and w A are associated with the uncertainty in A_ and zs • . Q that the range of state mamces to be constdered by robust and w% is associated with the uncertainty in BQ. The sign's z a and zn are output from the plant matrix as linear stability analysis is described by all A% with [IA_tl[__- < 1.

0 . . fl combmatron of the states.

Ac=Aoo+W_c._ae (63) zAe=W,4ex Define also an operator Asee R "q_" to describe muitipli- cative uncertainty in a nominal B o matrix of Q(s) A Q_ zee=WBeBeo_ (66) weighting function Ws _ R "_" is associated with this uncer- . Q . • .SO The feedback mechanism to describe the modeling uncer- tamty such that the range of possible B e matrices _s tainty uses a relationship between these output signals and described by all with I[AA¢It_= < 1.

the w K and w c input signals.

BQ=BQo(I+WneAse) (64) The choice of additive and multiplicative operatorsis w_e=aaez_ ¢ made for reasons similar to those presented above in the first subsection of this section 5. One of each type of uncertainty w eQ=Asozs¢ (67) is included to demonstrate the derivation procedures of how each uncertainty is associated with the nominal aeroelastic The state-space plant matrix can be formulated using dynamics in a feedback relationship. The actual choice of these additional input and output signals.

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0 1 o 0 0 0 r/ (68) -M-t(Ka + _oDo -M'tCo -_M-ICQ -1 0 0 q 0 1 I ._ BQ_ 0 A qo 0 0 0 w_ M-I DQ 0 M-r Co 0 0 0 WA9 0 0 t_,:AQ 0 0 6' wa_j =-1_,,r_-, -13, dk,-wt_, (71} FIG. 11 shows how the uncertainty operator and pertur- bation to dynamic pressure affect this plant formulation in a Perform a similar derivation for the state equation of x> feedback manner.

Associating a AB uncertainty operator with the B,, matrix q .,_2=- ft-_,_-13 :uQ may not seem immediately useful because considering errors in the poles determined by the AQ matrix is often =-(l_zo+l_]t,.%,n zml_:_+W_:-x_: me intuited. This An uncertainty can be essential to accurately =-_, _-:fhoUe--% l W_.,.<,+w_?,_,) describe the modeling errors because errors in terms com- mon to both A o and BQ may exist. Such a situation arises for =-_2_<.-f_2oue-%:z& certain modeling representations of aerodynamic lags. Con- sider a simplified Roger's form Q(sj matrix that uses two =--[_20XZ--_Z0I/Q--W[$ 2 (7 1 } Pad6 approximations to represent lag terms.

The signals zl_ ' and w13' are introduced as plant output and {69) 25 input signals to relate the uncertainty Az, in a feedback manner. The signals z_, and w_, are similarly introduced to Qts_=--+-- = 0 -_2 - : s+/h s+&

s °l q

relate the uncertainty An: in 9 feedback manner. The state- 1 1 space matrix can be formulated to describe the nominal Qo(s) with these additional input and output signals.

The poles of this system are determined entirely by the A o matrix, so uncertainty in the poles can be entirely described (73) -13_ ° 0 -1 o -fl,o

;i

by a Aae operator associated with the AQ matrix. A similar o -&, o -t -I_,__ uncertaanty A n should also be associated with the B,_ matrix _ = in this case because the [_,, and [_2 terms appear in both Ale

II

35 Z_ 0 V_, 0 0 W_..

and B o, Allowing variation in A o but not in B o will 1 1 0 0 2 introduce unwanted zeros to the system, so the proper way ]1 uQ YO to model pole uncertainty for this formulation is to include both Aa_ and Aee operators.

Dynamic Uncertainty The lbad6 approximation appears often in aeroelastic 40 Dynamic uncertainty operators are often associated with models, so demonstrating the LFT formulation of Q(s), aeroelastic models to account for modeling errors that are which includes the uncertainties in poles _ and _2 in a not efficiently described by parametric uncertainty. Unmod- feedback relationship, may be useful. The uncertainties in eled dynamics and inaccurate mode shapes are examples of Q(s) can be developed distinctly from the structural uncer- modeling errors that can be described with less conservatism tainties because LFT operations allow the structural and 45 by dynamic uncertainties than with parametric uncertainties.

aerodynamic models to be combined into a single plant These dynamic uncertainties are typically complex in order model with a structured uncertainty description.

to represent errors in both magnitude and phase of signals.

Define real scalar operators As, and An, to describe Consider a system P having two modes with natural uncertainty associated with nominal values of the poles [_zo frequencies at 4 rad/sec and 30 rad/sec.

and 132o.Real scalar weightings WI_j and Wls: normalize the 5o uncertainty such that the range of poles to be considered by (74) robust stability analysis is described by all As, with / 16 _ 900 / P= ks 2 +O.'-4s + 16]k s _- +0.6+900/ IIAB,I[_< 1 and all AI_: with I_1_..11_ < 1.

Is,=lS,o+%_, Define a nominal model Po, which will be used for 55 stability analysis of the system but does not model the high-frequency mode of P.

Define states x_ and x, of Q(s), and consider an input signal u 0 that generates the output signal Yo by the rela- p=(;, 16 + 0.4s + 16) (75) tionship ye--Qu e. Substitute the uncertain 13_into the state equation of x t.

The large difference in natural frequency between the high-and low-frequency modes of P precludes parametric uncertainty associated with the low-frequency mode of Po 65 from being a reasonable description of the modeling errors in Po. The magnitude of any parametric uncertainty associ- ated with the low-frequency mode would need to be _f_,oh-_bue-Al%zl_

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_P=lP.+WzX:I/_i__ -< 1 } 181) extremely large to account for the unmodeled high- frequency dynamics, so the stability analysis would be FIG. 15 shows the block diagram for robust stability relatively meaningless because the large uncertainty would analysis of (P.

imply the plant is not accurate at an)' frequencies.

FIG. 16 shows the magnitude of the transfer function A multiplicative uncertainty operator A • C, can be used 5 from input to output for P and Po, and the maximum to describe the high-frequency modeling error without intro- magnitude of IPo+WA{ at each frequency. The additive ducing the excessive conservatism resulting from a para- uncertainty bounds the modeling error at each frequency, metric uncertainty description. Associate a complex, scalar, including the frequencies near the zero of the nominal plant, weighting function W(s_ • C with this uncertainty to reflect because the output of P is bounded above by the maximum the frequencyvarying levels of modeling errors. _0 magnitude of the members of the set 9.

_ + 0.5 (76) These multiplicative and additive uncertainties are par- W: 100-- s + 500 ticularly important when comparing an analytical transfer function with experimental transfer functions derived from 15 flight data. Analytical models are often computed for a low The set of plants _P used for robust stability analysis is range of frequencies because the high frequencies add formulated to account for the range of dynamics as complexity to the model but do not always affect the stability described by the norm-bounded multiplicative uncertainty margins of the aircraft. The experimental data may indicate A.

a high frequency mode that is not included in the analytical 20 model, so a frequency-weighted dynamic multiplicative _={PdI+W_:PlA[I__ l I t77) uncertainty can be associated with the model.

The issue of mode shape uncertainty is often encountered FIG. 13 shows the block diagram for robust stability when comparing low-frequency-predicted dynamical analysis of 9 and FIG. 14 shows the magnitude of the responses with flight data because sensor measurements are transfer function from input to output for P and Po, and the 25 directly affected by the mode shapes. Both multiplicative maximum of [Po(I+W)] which is an upper bound for the and additive uncertainties may be required to accurately output of P at each frequency. The multiplicative uncertainty model mode shape errors and account for inaccurate is able to bound the modeling error at the frequency mode response levels (which may be higher than predicted at some without introducing excessive conservatism from large frequencies but lower than predicted at others) and inaccu- uncertainty associated with the low frequency mode.

30 rate frequencies associated with poles and zeros.

Dynamic additive operators may also be required in the Uncertainty Associated with Nonlinearities uncertainty description to account for errors that are not The _t framework described uses linear operators to rep- efficiently described by either multiplicative and parametric resent dynamical models and associated uncertainties but uncertainties. Modeling errors associated with a zero of the does not admit nonlinear operators. The It framework is system dynamics are an example of an error that is best 35 useful for analyzing aircraft stability despite the constraints described by additive uncertainty. Multiplicative operators of linearity because physical systems, which are always are not useful in this case because Potjto)=0 at the frequency nonlinear, can often be approximated by linear models to a to associated with the zero of nominal model and, high degree of accuracy. A classic example of this situation correspondingly, every member of the set of plants Po(Jto} notes the linearized dynamics are often an acceptable rep- (I+WA)=0 at this frequency. Additive uncertainty allows the 40 resentation of an aircraft operating near trim flight system output for some member of the set of plants to be conditions, so linear models work well in practice for control nonzero even at frequencies of the zeros of the nominal synthesis and stability analysis.

plant, Nonlinear dynamics cannot always be accurately Consider a plant P with several poles and zeros.

described by a linearized dynamics model, so the stability (78) 45 analysis should consider the effects of these nonlinearities.

16(s 2 + 0.48s + 64_ 900 The _t framework can associate linear uncertainty operators

e-- _t_/_s: + oT+ 9o0)

with linear models to describe the errors resulting from some types of unmodeled nonlinear dyn.amics. The uncertainty does not actually represent the nonlinearity; rather, the Assume the nominal plant Po of this plant is similar to P uncertainty allows the linear system responses to vary with and has the correct number of poles and zeros, but the 5o sufficient magnitude to bound a range of nonlinear system coefficients of the system equations are incorrect.

responses.

Separating the nonlinear dynamics that cannot be linear- ized from the nonlinear dynamics that can be accurately P0 = 36_0s._ +0.4s+ 16)l,s 2 + 0.-'_'+ 900) represented by linear models is useful. Separate uncertainty descriptions can be formulated for each dynamical block, and the resulting operators can be combined using the LFT Define an additive uncertainty operator A _ C that is framework to formulate a single, linear, plant model with a normalized by a complex, scalar function W to reflect the structured uncertainty description. FIG. 17 shows an frequency-varying levels of the modeling errors.

60 example LFT system representation for a nominal plant Po _so) and associated uncertainty A, and a nominal linear model N O s+50 W =O.l-- and associated uncertainty AN representing a system element s+5 with nonlinear dynamics.

The system shown in FIG. 17 is commonly used to The set of plants 5P used for robust stability analysis is 65 describe the coupling between the aeroelastic dynamics and formulated to account for the range of dynamics as actuators affecting an aircraft through control surfaces.

described by the norm-bounded additive uncertainty A. Actuators can display many types of nonlinear behaviors

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and should be considered in the aeroelastic stability analysis, because pilot and autopilot commands that maintain trim No=2 (88) during flight ensure the control surfaces are continuously moving. The errors in linear models resulting from unmod- Associate an additive weighting operator A,, with N O such eled actuator dynamics such as nonlinear stiffness param- that stability analysis considers the set of plants N.

eters or hysteresis functions can sometimes be described by a linear uncertainty operator.

Consider the response y of a nonlinear system N that N=._',,+-_x: II±AI_<- - t 18,_ models a system that has a nonlinear stiffness corresponding to a hardening spring that is valid for the bounded input Io FIG. 20 shows that the maximum and minimum magni- signal u c [-10, 10]. Such a system can represent an element tudes of the responses of the set N are able to bound the of an actuator model or a nonlinear structural model.

response of the nonlinear system N with the hysteresis for the range u _ [-10, 10]. Again, the bounds resulting from a )=Nt_2tt+O.O2u2+O.OO82t¢ 3 _82_ linear uncertainty description are overly conservative 15 throughout this operating range, _ut they achieve the desired Define a linear nominal model N o such that y=N o u goal of describing errors caused by the unmodeled hysteresis approximates the response of N.

nonlinearity.

Explicitly constraining the operating region of the input signal u e [-10, 10] can be important to developing reason- No=2 (831 20 able tmcertainties to describe errors resulting from unmod- eled nonlinearities. The errors in the linear model can grow Associate an additive weighting operator A N with N o such excessively large when considering a large range of inputs, that stability analysis considers the set of plants N.

so the uncertainty magnitude would need to also grow N= {NoA/¢:]LAN_i_ < | } (84_ excessively large. The conservatism resulting from such a 25 large uncertainty description may be unacceptable and FIG. 18 shows that the maximum and minimum magni- require the input range to be constrained to more reasonable tudes of the responses of the set N are able to bound the limits.

response of the nonlinear system N for the range u e [ 10, 10].

The uncertainty description, even for a constrained oper- These bounds are overly conservative throughout this oper- ating region, will usually be overly conservative when ating range, but they achieve the desired goal of describing 30 describing modeling errors for certain parts of the operating errors in the linear system response resulting from the region. The uncertainty is able to bound the errors in FIGS.

unmodeled nonlinearity.

19 and 20, but the maximum and minimum bound are A similar procedure can be used to describe error caused clearly not optimal. Some amount of conservatism is by an unmodeled nonlinear softening spring. Consider the expected when describing errors resulting from unmodeled response y of a system represented by N that is valid for an 35 nonlinearities because a linear model, whether a single plant input signal u el-10, 10].

or a set of plants, will usually not be an accurate represen- tation of a nonlinear system that cannot be linearized.

.',=N=2u+O.OO5uZ"4).OO5u 3 (85) Also, several types of unmodeled nonlinearities exist that are frequently encountered in aircraft systems but are not Define a linear nominal model N O such that y=N o u 40 easily described by linear uncertainty operators associated approximates the response of N.

with the linear models. Examples of these types of nonlin- earities include free play, dead-band responses, friction, and No=2 (86_ rate limiting of actuators.

The robust stability margins computed from g with Associate an additive weighting operator AN with No such 45 respect to the linear uncertainty operators describing unmod- that stability analysis considers the set of plants N.

eled nonlinear dynamics will always be somewhat suspect, because this approach cannot consider stability properties _87) unique to nonlinear systems such as bifurcation points and limit-cycle behaviors. This approach limits the usefulness of HG. 19 shows that the maximum and minimum magni- 50 the _t method to systems for which the nonlinearities have tudes of the responses of the set N are able to bound the small effects on the response and do not introduce nonlinear instabilities to the critical flutter mechanism.

response of the nonlinear system N for the range u e [-10, 10]. These bounds are also overly conservative throughout Uncertainty Associated With Hight Data this operating range, but they achieve the desired goal of A theoretical model with an associated uncertainty describing errors in the linear system response resulting 55 description can be an accurate representation of the aeroelas- from the umnodeled nonlinearity.

tic dynamics of an aircraft, but responses from that model Another nonlinearity that commonly affects aeroelastic may not identically match flight data. Additional uncertain- ties can be associated with the model to describe errors that systems is hysteresis. The response of a hysteretic system depends on the trend of the input signal such that an are observed between the predicted responses and the mea- increasing input signal generates a certain response, but a 6o sured responses from a commanded excitation to the aircraft.

decreasing input signal generates a different response. Such These uncertainties do not necessarily indicate errors in the model; rather, these uncertainties indicate errors in the hysteresis dynamics are difficult to express as a simple mathematical formula, so for illustrative purposes, assume N process used to generate aeroelastic responses and measure is a nonlinear hysteresis function whose response y depends flight data.

on the trend of the input signal. Define a linear nominal 65 One source of error is an incorrect assumption of excita- model N O whose response y=N o u approximates the tion force used to generate the predicted and measured response of the hysteretic N. responses. The measured excitation force associated with

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the flight data may not correctly account for poor hardware beyond the true modeling errors and can be overly optimistic performance and nonuniform spectral distribution of the if the uncertainty does not sufficiently account for the true modeling errors.

force. Also, inexactly phased excitation between multiple force mechanisms can excite modes that are not anticipated Model validation algorithms can be used to indicate if an by a theoretical analysis. A frequency-varying dynamic -_ uncertainty description is reasonable with respect to flight uncertainty can be associated with the force input of the data. These algorithms consider whether or not a set of data analytical model to describe errors in the excitation.

measurements could have been generated by a proposed The phenomenon of nonrepeatibility can cause discrep- model that includes the nominal dynamics and associated ancies between predicted and measured responses from uncertainty operators and noise and disturbance signals.

multiple occurrences of excitation signals. Nonrepeatibility 1o Uncertainty operators associated with the nominal plant affects flight data by introducing slight variations in model specifically in the LPT framework can be considered responses, even for data recorded at identical flight condi- by validation algorithms. Frequency-domain algorithms are tions with identical excitation signals. This unexplained generated that consider the model validation problem in the behavior may result from some unmodeled nonlinear context of control design. Tune-domain approaches are also dynamic or inexact excitation that is not correctly measured.

15 developed that can be solved with convex optimization External disturbances such as wind gusts or turbulence can algorithms for certain uncertainty structures.

introduce an unmodeled dynamic that inconsistently affects A model validation procedure has been formulated that the aircraft responses. A frequency varying dynamic uncer- uses att condition to determine if an uncertainty model is tainty can be associated with the model to describe nonre- invalidated. This procedure uses frequency-domain transfer- 2o peatible data variations.

function data to determine if some perturbation AeA the Another source of error between predicted an measured nominal plant could produce the measurements. Consider responses is an incorrect assumption of flight condition.

the block diagram for robust stability analysis of systems Flight data sets are sometimes generated at test points that with measurement y and forcing u signals shown in FIG. 21.

attempt to maintain a constant flight condition to match the The model validation question, as applied to uncertainty data sets predicted from a model describing the aeroelastic models, is given in question 6.1.1.

dynamics at that same flight condition. Slight variations in Question 6.1.1 (uncertainty validation): Is there some flight conditions while the experimental response is mea- frequency-varying AeA for FIG. 21 such that F,(EA) could sured may cause some discrepancy between the predicted generate the set of observed data y and u?

response and the flight data. Parametric uncertainty associ- ated with the unsteady aerodynamic model can be used to 3o Define P(s) ec _*+"'_'_m' as a stable transfer-function sys- account for these errors because flight condition variations tem matrix such that P _ _, 5-f. Partition this matrix into four only affect the aerodynamic model and not the structural elements such that Pll(S) EC ...... , P22(S) EC .... i and P2j(s) are model.

of appropriate size. Pa2(s) is the nominal-plant transfer The model may accurately represent the mode shapes of function in the absence of uncertainty.

the aircraft but have a poor representation of the sensor locations. The responses measured by sensors are inherently (901 dependent on sensor location, with respect to mode shapes, to determine the magnitude and phase of the signal. Addi- tional errors in magnitude and phase are introduced when considering transfer-function estimates generated by signals 40 Robust stability analysis of this system is determined that violate assumptions used in computational algorithms.

using the transfer function P], which comains the feedback A frequency-varying dynamic uncertainty can be associated relationship between the plant dynamics and the uncertainty with the output of the plant model to describe errors in both operator. The robust stability condition given by theorem mode shape and sensor location predictions.

3.3.3 requires _PH)<I.

The choice of signal-processing algorithms can also intro- 45 The model validation condition uses the_ elements of P duce errors between predicted an measured responses. The and the finite-energy frequency-domain signals y, u _ Fourier transform, which is the traditional tool for signal where the measurements are scalar functions y, u _ C.

processing, assumes several characteristics of the data that Formulate the following two matrices.

may be violated with transient-response aeroelastic data.

Filtering and wavelet-based algorithms may be used to reduce errors introduced by signal processing and reduce ]St .-,=P_zu conservatism in the resulting stability margins. Parameter uncertainty associated with the modal parameters of the 15_.=.=Pz2u-y (91 ) linear model may be used to describe some errors in the 55 The following theorem formulates the model validation test natural frequencies and dampings observed using flight data using _t.

analyzed by incorrect algorithms. Dynamic uncertainty may Theorem 6.1.2: Given measurements y generated by also be required to describe errors introduced by leakage and inputs ix, then the system P with associated uncertainty set, aliasing effects.

A is not invalidated if MODEL VALIDATION OF UNCERTAINTY Model Validation Using the Structured Singular Value _PH-P,.-P2._ -'PzO>l (92) Robust stability analysis considers the stability of a sys- tem subject to a set of modeling errors and perturbations This condition may seem counterintuitive in that the represented by a norm-bounded A. A logical question that desired condition for validation is _t greater than one, while arises is how to reasonably determine this set. This issue is 65 the robust stability condition seeks a value less than one.

important for computing robust flutter margins because I_ This condition can be explained by considering the follow- can be overly conservative if the uncertainty is excessive ing relationship of y=F,,(P, A) u as shown in FIG. 6. 1.

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-continued I O=[P.._u-y]+P,_IA(I-PnA}-J[PI2u]=P._2+PzlA(I-PllA)-J"fit2 (93) Pii = aP_l (equivalent to _ = aA) Define the plant P={PI i, Pl2, P2J, P__2}. A value of g(P)< 1 implies this system is robustly stable to all perturbations AeA. This robust stability of g (i5)<1 also implies that the A is the required norm-bounded uncertainty loop gain F.(15_ A) is not singular for any value AeA. Thus u (P)<I would contradict the relationship shown in the The norm bound on the uncertainty set A is not actually above equations that requires F. (15, A) to be singular to 10 .

increased with this method as denoted by the parentheses satisfy the input-to-output relationship of the data. In this respect, the model validation test is actually an inverted around the statement A=(_. Scaling the uncertainty and robust stability test.

retaining these scalings throughout the procedure would be The phrase "model validation" may be misleading. No difficult. The algorithm is simplified by always considering analytical model can ever be truly validated by considering 15 the uncertainty is scaled such that g<l is always the desired a finite set of experimental data. A model may not be result. This scaling of the uncertainty is actually accom- invalidated by the data, but no guarantee exists that a plished by the statement P_ _--o.P_ t which scales the feedback different data set could not be generated from the physical signals between the plant and uncertainty operators. The system that invalidates the model. The finite sets of mea- norm bound for the uncertainty that is needed to ensure the surement data cannot record the response of the system to an uncertainty levels are not invalidated by the data is scaled infinite number of input signals subject to an infinite number into the plant by the parameter _. Robust flutter margins can of initial conditions. Theorem 6.1.2 reflects this fact by be computed directly from the new scaled plant using explicity stating the condition only determines if the model algorithm 4.4.6 with a desired g<l condition.

is invalidated by the data. The system is assumed to be Algorithm 6.2.1 is a straightforward implementation of nearly linear, and the data is assumed to sufficiently repre- the model validation test. The g values are only compared sent the behavior of the dynamics, so theorem 6.1.2 repre- with 1 to determine if the model is invalidated. More sents a model validation test.

sophisticated algorithms could use the value of y to deter- Validating Norm Bounds for Uncertainty mine the factor c_ to scale the uncertainty. Also, the values The value of g computed using theorem 6.1.2 can be of g across frequency could be exploited. Frequencies with interpreted as a measure of how reasonable the uncertainty low y value indicate areas where the uncertainty set is least description is. A value of g=2 implies the uncertainty can be scaled by 2 before the model is invalidated. Alternatively, a conservative: frequencies with high Ix are overly conserva- value of IX=0.5 implies twice as much uncertainty is required tive. The scaling 0_ could vary with frequency to reflect this information.

for the model to not be invalidated. This interpretation of g, as relating to the size of allowable perturbations, emphasizes The situation may arise when the initial value chosen for the relationship of the model validation condition to a robust the norm bound of the uncertainty set may be overly stability condition.

conservative. In this situation, the model validation condi- This model validation is used in practice to generate tion will pass for each data set during the first processing of reasonable norm bounds for an uncertainty description. The the outer loop. The initial norm bound can simply be following algorithm can be used to determine a sufficient 40 decreased by some level consistent with the lowest Ix value level of uncertainty required such that the model is not computed during the validation checks, and algorithm 6.2.1 invalidated by multiple data sets. A small scalar (x>l is can be run again to compute a less conservative uncertainty chosen to scale the uncertainty set and increase the amount description that does not invalidate the model.

of allowable errors if the size of A is not sufficient.

Theorem 6.1.2 is only valid for scalar data signals gen- Algorithm 6.2.1 (Model Validation): erated by systems with a single input and single output.

Given frequency-domain data sets {Yl, Yz..... y,} and Multiple data signals can be considered by applying theorem {u,, u_..... u.}: 6.1.2 to each combination of singleinput and --output sig- Given frequency-domain transfer-function P with ele- nals. Algorithm 6.2.1 can still be used by simply including ments P_I, PJ2, P21, P2--: outer loops to cycle over the number of input and measure- Given uncertainty set A with initial norm bound: ment signals.

Given update scalar _x>l: The model validation algorithm using theorem 6.1.2 is a departure from traditional methods of analyzing flight data valid = FALSE to assess accuracy of an analytical model. The most widely used algorithms for analyzing flight data estimate natural while_ valid = FALSE){ frequencies and modal damping. The model validation pro- valid = TRUE cedure using g considers the response dynamics at each frequency without explicitly comparing modal properties.

for/= l:nl This procedure enhances the ability of the g method to 191z = Pizul analyze flutter stability without requiring damping esti- mates.

]5".2 = Pzz u_ - Yl Another interpretation of algorithm 6.2.1 uses the uncer-

if.(e. - r,:_,) < ]l

tainty A to bound the magnitude and phase of the possible transfer functions generated by the family of plants F,(P,A).

valid = FALSE The s_uctured singular value ensures the experimental data transfer function lies within these analytical bounds at every frequency.

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PROCEDURE FOR THE _t METHOD the plant but has the equivalent effect of scaling the norm bound of the uncertainty description. The algorithm loops Model Updating over the model validation procedure until an uncertainty Generating a model by analyzing flight data is essential description is determined that is not invalidated by the flight for computing a confident stability analysis. A nominal 5 data. The final step is to compute a robust flutter margin model generated purely from analytical equations of the from the scaled plant by using a _t<l condition as in predicted aircraft dynamics may not accurately describe the algorithm 4.4.6.

true aircraft. A model must be generated that accounts for the Algorithm 7.l.l (robust flutter margins with model flight data to ensure the predicted dynamics represent the updating): true dynamics.

1o Given nominal plant P: The most direct method of generating a model from the Given uncertainty set A associated with P: flight data is to identify a system model entirely from the Given input excitation data u: data measurements. Many system identification algorithms exist that have become standard tools for systems and Given output response measurement data y: control engineers. Direct application of these methods to 15 Define W_-<I as the scaling update for A aeroelastic systems rarely produces an accurate model that accounts for the dynamics of the aircraft. Aeroelastic while (F_(P, AI is invalidated by u_ y using algorithm62.11t response data is typically of poor quality relative to ground 4= WA vibration test data because of the low signal-to-noise ratio and unobserved dynamics in the response measurements that } may drastically lower the effectiveness of system identifi- cation algorithms.

_[lu rob "¢: An alternative method is to use the nominal aireratt t,_r l_ flutter margin computed from algorithm 4.4.6 flutter dynamical model as an initial estimate to model the true 25 Several advantages exist to using this method as com- aircraft. The flight data are then used to update the elements pared to traditional model updating methods. The typically of this model. Several methods have been devised to update poor quality of flight data, in association with aircraft an analytical structural model using experimental data.

dynamics consisting of many modes, makes updating a Model updating can be performed on the full stress model or nominal model difficult. Traditional methods of norm-based a subset computed with Guyan reduction. Generally, con- 30 update algorithms often generate a nonunique set of model sidering the full model is preferable because the reduction updates that have no way to determine which update has the may distribute local errors throughout the entire model if an most logical physical interpretation. The method of updating orthogonality condition is violated.

the uncertainty operators based on a worst-case magnitude There are two basic methods for updating the full struc- avoids this problem.

tural model using comparisons between experimental and 35 Also, this method can work with flight data of varying predicted data. One method updates the mass and stiffness quality. The updated uncertainty descriptio, _, J highly accu- matrices of the finite element model. This method suffers rate if the data show a high signal-to-noise r.ltio and much from lack of physical interpretation of the matrix updates of the dynamics are observed by the sensor;, if the data do ,and possible numerical conditioning. Another method not have these desired characteristics, however, the method updates specific parameters in the model. This method is 40 can still compute a flutter margin. An uncertainty description accurate for small systems but may require an excessive may be difficult to compute if the data do not indicate the computational cost for large systems.

aircraft dynamics well, so the model validation procedure Aeroelastic models have the additional freedom of updat- will not require a large magnitude for the uncertainty opera- ing the aerodynamic and the structural elements. Another tors. The robust model in this situation will closely resemble method of updating the linear model in a modern control 45 the nominal model. In this way, the Ix method will always framework has been developed. (K. D. Gondoly, Application generate a more accurate flutter margin, and at worst, the of Advanced Robustness Analysis to E.rperimental Flutter, robust IXflutter margin will be equivalent to the nominal Masters of Science in Aeronautics and Astronautics Thesis, flutter margin.

Massachusetts Institute of Technology, June 1995.) This Approaches to Use Flight Data method may be overly conservative for describing 50 A flight test generally consists of maneuvers at several test nonlinearities, and the corresponding stability margins are points that may be at identical or different flight conditions.

only accurate for flight conditions near the instability. A The entire flight test program will use many flight tests to parametric identification algorithm has been developed that measure response data at test points throughout the flight uses flight data to update specific terms in the aerodynamic envelope. The model-updating method that generates uncer- model through a nonlinear optimization. This method suffers 55 tainty operators can use the entire set of flight data from the with flight data because of unobserved dynamics and the low different test points.

signal-to-noise ratio in the measurements.

Several approaches are formulated to use multiple flight The approach taken in this invention is to update only the data sets to update the uncertainty description associated uncertainty operators of the robust aeroelastic model, using with a nominal plant model. The uncertainty description the flight data, and leave the nominal dynamics model 60 may be different for each approach, and the resulting flutter unchanged. The model validation condition of theorem 6.1.2 margin will be different for each approach because of the is used with the nominal plant P to generate a reasonable dependence of IXon the uncertainty set. Two approaches uncertainty description A to associate with P. FIG. 22 shows discussed here are denoted as local and global.

the flow of information through the _ method.

A local approach uses flight data from test points at Algorithm 6.2. I presents a procedure to implement the Ix 65 identical flight conditions. These data are used to generate an method in a manner corresponding to FIG. 22. The model uncertainty description for the nominal model at the par- update procedure based on algorithm 6.2.1 actually scales ticular flight condition associated with the data. The mag-

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nitude of the uncertainty operators is chosen such that a The following algorithm 7.2.2 outlines the global robust model at the single flight condition is not invalidated approach to use flight data and compute flutter margins.

by any of the flight data sets measured at the same flight Algorithm 7.2,2 (global approach): condition with no consideration of data from other flight Define scalar nf as the number of flight conditions.

conditions. 5 Define scalar n i as the number of data sets at flight The local approach shows the benefit of independently condition i.

computing uncertainty descriptions for models at different Define vectors {u, t, uf ..... u/"} as input excitations at flight conditions. This approach allows each uncertainty flight condition i.

description to be more accurate because, for example, the Define vectors {yil y2 ..... y;"_} as response measure- flight data may indicate much smaller uncertainty operators J0 ments at flight condition i.

are required for subsonic plant models even though large Define matrices {Pt, P ...... P,,,} as plant models at flight uncertainty operators are required for transonic plant mod- condition i. - eis. The resulting worst-case flutter margins will be less Choose initial A_ conservative because less uncertainty is required for the model. 15 The following algorithm 7.2.1 outlines the local approach to use flight data and compute flutter margins, for j = 1 : nr Algorithm 7.2.1 (local approach): for i = t : n_ { Define scalar nf as the number of flight conditions. Validate F. (P., A) using yai and tP i increase size of A if necessary to validate Define scalar n, as the number of data sets at flight 20 } condition i. } fori=l :nr{ Define vectors {ut t, u_-_ ..... uF _} as input excitations at compute flutter margin _Si_._from u(F.tPi. A)) flight condition i. } Define vectors {y,?, y,r_..... yi ''_} as response measure- ments at flight condition i. 25 Hybrid approaches have also been formulated that mix the Define matrices {P,, P__ ..... PJ as plant models at flight local and global approaches. One straightforward hybrid condition i. appro_ich is to generate an uncertainty description using all data from a small range of flight conditions. This approach 30 can be useful for separately considering sets of plant models for i = 1 : nf { that are generated using different techniques. For example, Choose initial _ the model-generating package used for this paper computes for j = 1 : n_ { all subsonic plant models with a doublet-lattice algorithm Validate F. (P_. _) using yi and uJi and the supersonic models are generated with constant-panel increase size of A i if necessary, to validate I 35 algorithms. A hybrid approach could be used to reflect this compute flutter margin 8_.._from ud:_(P_, AiD knowledge and consider groups of subsonic, supersonic, and } transonic plants independently.

The approaches outlined here are certainly not exhaustive.

A global approach uses the entire set of flight data from A weighted-norm approach can be formulated that uses test points throughout the flight envelope to generate a single 40 flight data from the entire envelope but depends heavily on uncertainty description for all nominal aircraft models. The a particular subset of that data. Other approaches could magnitudes of the uncertainty operators are chosen such that concentrate on particular dynamics through modal filtering all nominal models with the associated uncertainty descrip- techniques to generate separate uncertainty descriptions for tion are not invalidated by any of the flight data sets. individual modes.

Several advantages and disadvantages exist to using the 45 Although particular embodiments of the invention have global approach instead of the local approach. One disad- been described and illustrated herein, it is recognized that vantage is a possible large increase in conservatism of the modifications may readily occur to those skilled in the art.

flutter margin because the uncertainty description is not For example, although matrix algebra has been used minimized at each flight condition. A single particularly throughout to describe the invention in order to facilitate its inaccurately plant model will require large uncertainty 5o implementation by computer programming since the matri- operators that may be highly conservative for plant models ces define for the progranuner how the elements are to be at flight conditions that are better representations of the true processed, the invention is not limited to the implementation dynamics, disclosed in this manner. Any other form of mathematical One advantage to this approach, however, is that the expression and its implementation by computer program- uncertainty description is truly worst-case with respect to the 55 ming will be equivalent. Consequently, it is intended that the entire flight envelope, The worst-case errors from the worst- claims be interpreted to cover such modifications and case fight condition are used to generate the uncertainty equivalents thereof.

description for all conditions. Also, this approach is not very What is claimed is: sensitive to poorly measured flight data. A poorly modeled 1. An on-line method for robust flutter prediction in modal response may only appear in certain data sets. The 6o expanding a safe flight envelope for an aircraft to define an local approach would not include uncertainty for these envelope of safe flight conditions, comprising two parts, a dynamics at conditions that did not clearly observe this first part comprising the steps of modal response, so the resulting flutter margin would not (all generating a computer model of said aircraft, account for the true level of modeling errors. The flutter (a2) computing flutter margins of said aircraft from said margin generated with the global approach may be more 65 computer model using a singular value _t to define a first conservative than the local approach, but this approach safe flight envelope and optionally using a well known introduces a corresponding higher margin of safety, traditional method for defining a second safe flight

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envelope used in double checking the safety of said first (b41 if said dynamic pressure difference is large, take said safe flight envelope defined by its computed flutter aircraft from present flight test condition to a new flight margins, test condition F,=F+A F, where AF are the changes to (a3j if computed flutter margins thus double checked are parameters of flight condition F comprising altitude to found to define an envelope of safe flight conditions, or 5 take said aircraft to a new flight condition Fp from flight an optional method of defining a safe flight envelope is condition E to incrementally expand testing said first not used. proceed to a second of said two parts, safe flight envelope and repeat steps b 1, b2, b3 and b4; said second part comprising in-flight steps of.

but if said dynamic pressure difference is small, declare (b I ) taking said aircraft to a safest point within said first to the last predicted flutter margin Up as a point on said safe flight envelope at a flight condition, E and m,'J- aircraft's expanded safe flight envelope, and repeat suring flight data at said present condition F. includ,::g steps bl-b4 until sufficient ffigti: conditions within said dynamic pressure defined by parameters comprising first safe flight envelope have been tested to expand altitude and air speed; margins thereof; (b2) compute a predicted flutter point, Fp, at a higher dynamic pressure than at said condition F using an whereby a robust expanded flight envelope is determined for algorithm based on said flight data and said singular said aircraft using said singular value g and said in-flight value _t; steps.

(b3) determine dynamic pressure difference between dynamic pressure of said aircraft at said present flight condition F and at said predicted flutter point Fp;