Design and evaluation of dynamic flight test manoeuvres

PROEFSCHRIFT

TER VERKRIJGING VAN DE GRAAD VAN DOCTOR IN DE TECHNISCHE WETENSCHAPPEN AAN DE TECHNISCHE UNIVERSITEIT DELFT, OP GEZAG VAN DE RECTOR MAGNIFI­ CUS, PROF.DR. J.M. DIRKEN, IN HET OPENBAAR TE VERDEDIGEN TEN OVERSTAAN VAN HET COLLEGE VAN DEKANEN OP DINSDAG 28 OKTOBER 1986 TE 14.00 UUR

DOOR

JAN ALBERT MULDER,

GEBOREN TE 'S-GRAVENHAGE ,

VLIEGTUIGBOUWKUNDIG INGENIEUR.

TR diss

1505

prof.dr.ir. O.H. Gerlach en

I

In de door Iliff en Maine gegeven verklaring voor het verschil tussen de theoretische en de werkelijke spreiding van uit dynamische vlieg-proefmanoeuvres berekende waarden van stabiliteits- en besturings­ afgeleiden is ten onrechte geen rekening gehouden met de beperkte nauwkeurigheid van het gebruikte aërodynamisch model.

K.W. Iliff, R.E. Maine, 'Observations on maximum likelihood estimation of aerodynamic characteristics from flight data', Journal of Guidance and Control, Vol. 2, No. 3, 1979.

II

In de onderlinge vergelijking van de prestaties, maar ook bij de se-lektie van een geschikte lengte van dynamische vliegproefmanoeuvres moet worden uitgegaan van een gemiddelde informatiematrix per

meet-tijdstip en een zodanige schaling van de ingangssignalen dat het totale gemiddelde vermogen steeds gelijk is.

J.A. Mulder, 'Aircraft control input optimization for aerodynamic derivative estimation in dynamic manoeuvres', 7th IFAC Symposium on

'Identification and System Parameter Estimation', York, UK, July 1985.

Ill

De in het proefschrift beschreven methode voor de optimalisatie van ingangssignalen voor systeem-parameterschatting is ook toepasbaar op het probleem van de optimalisatie van de begintoestand van autonome systemen.

Hoofdstuk IV van dit proefschrift.

-1-u(t) voor systeem-parameterschatting worden geoptimaliseerd met be­ trekking tot een gekozen kriterium J. Behalve de vorm van het geopti­ maliseerde ingangssignaal met energie E is bij implementaties in de praktijk eveneens van belang een berekening van het verloop van de tweede-orde partiële afgeleide:

o2J o u ( t )2

Hoofdstuk IV van dit proefschrift.

Het Kalman filter voor de recursieve schatting van de toestand van lineaire dynamische systemen kan ook worden toegepast voor de schat­ ting van de parameters in niet-lineaire statische systemen.

VI

In de vlucht kunnen de bruto stuwkracht en het massatransport van straalmotoren worden bepaald met in de straalpijp geïnstalleerde sensoren voor de totale druk en de totale temperatuur. Daarbij moet gebruik gemaakt worden van calibratie-faktoren voor de bruto stuw­ kracht en het massatransport die door de motorfabrikant uit metingen op een statische proefbank worden bepaald. Het is echter ook mogelijk simultaan met de vliegtuigpolaire de calibratie-faktoren af te leiden uit nauwkeurige reconstructies van de vliegtuigbeweging tijdens kwasi-stationair versnelde of vertraagde meetvluchten op verschillende nominale vlieghoogten.~

J.A. Mulder, J.M. van Sliedrecht, 'Estimation of drag and thrust of jet propelled aircraft by non-steady flight test manoeuvres', AGARD Conference Proceedings No. 223 on 'Flight Test Techniques', Porz Wahn, Germany, October 1976.

-2-De mathematische modellering van de pharmacokinetiek van geneesmidde­ len, gevolgd door schatting van de modelparameters aan de hand van concentratiemetingen van het oorspronkelijke geneesmiddel en eventuele metabolieten is niet alleen van potentieel klinische betekenis maar kan ook worden gezien als een uitbreiding van de tot nu toe beschik­

bare analytische technieken.

F.W. Schultz, K. Nooter, P. Sonneveld, J.A. Mulder, 'Modelling of in vivo distribution dynamics of Daunomycin, an anti cancer drug', 7th IFAC Symposium on 'Identification and System Parameter Estimation', York, UK, July 1985.

VIII

Bij de toepassing van mathematische pharmocokinetlsche modellen in een klinische context moet rekening gehouden worden met inter-individuele verschillen in modelparameterwaarden. Metingen van het werkelijke con­ centratieverloop in klinisch relatief gemakkelijk toegankelijke lichaamskompartimenten kunnen echter worden benut om tenminste een gedeelte van de a priori parameterwaarden te verbeteren.

P. Sonneveld, J..A. Mulder, 'Development and identification of a multi-compartment model for the distribution of Adriamycin in the rat' , Journal of Pharmacokinetics and Biopharmaceutics, Vol. 9, No. 5, 1981.

IX

Een meer frekwente toepassing van mathematische modellen in de medi­ sche wetenschap kan resulteren in een belangrijke vermindering van het aantal benodigde proefdieren.

-3-keersleidingsorganisaties en vliegers zich getroosten om te komen tot een steeds verder gaande reduktie van de door verkeersvliegtuigen ver­ oorzaakte geluidsoverlast, is de ogenschijnlijk gelaten publieke akseptatie van het opzettelijk geproduceerde, zeer hinderlijke en zelfs niet zelden de pijngrens overschrijdende bromfietslawaal op z'n zachtst gezegd verwonderlijk.

XI

De invoering van alternatieve technieken - zoals digitale data­ transmissie en beeldschermen - voor de tot nu toe gebruikelijke ver­ bale kommunikatie tussen verkeersvliegtuigen en verkeersleidingscentra op gemeenschappelijke en vaak zeer druk bezette radiofrekwenties zou de veiligheid in de verkeersluchtvaart bevorderen.

XII

De ingevoerde twee-fasenstruktuur in het wetenschappelijk onderwijs zal resulteren in een kwantitatieve en kwalitatieve teruggang van het universitaire wetenschappelijke onderzoek.

28 okober 1986 J.A. Mulder

-4-The accuracy of aerodynamic model parameters estimated from measure­ ments of dynamic flight test manoeuvres depends among other things on the form of the control input signals. Several types of longitudinal and lateral control input signals were evaluated in flight using an automatic control system for precise implementation. Sample covariance matrices could be determined by repeating each type of control input signal a number of times.

The performance of different types of control input signals with respect to sample standard deviations of parameter estimation errors could subsequently be compared to corresponding theoretical predic­ tions. Significant differences were shown to exist between the performance of different types of control input signals.

A technique is described for the optimization of control input signals with respect to certain norms of the information matrix. The technique applies to linear as well as nonlinear systems and is based on the representation of control input signals in terms of a finite number of orthonormal functions. It is shown that in case of linear systems the computational effort required for the optimization can be significant­ ly reduced by computing and storing a set of so-called elementary information matrices. The parameter-state estimation problem of dynamic flight tests is discussed in detail, in the linear and in the nonlinear case. It is shown that under certain conditions the parameter-state estimation problem can be solved in two consecutive steps pertaining to the reconstruction of the state and the estimation of the aerodynamic model parameters respectively.

CONTENTS

0 NOTATION, ABBREVIATIONS AND REFERENCE FRAMES 1

0.1 Frequently used notation 1

0.2 Abbreviations 10 0.3 Reference frames 11

1 INTRODUCTION AND RATIONALE OF THE PRESENT WORK 15

2 MATHEMATICAL MODELS 23 2.1 Kinematic models 24

2.1.1 Nonlinear kinematic models 24 2.1.2 Linearized kinematic models 26

2.2 Aerodynamic models 30 2.2.1 Nonlinear aerodynamic models 30

2.2.2 Linearized aerodynamic models 36

2.3 Observation models 39 2.3.1 Nonlinear observation models 39

2.3.2 Linearized observation models 41 2.4 Models of instrumentation measurement errors 43

2.5 Conclusions 44

3 FLIGHT PATH RECONSTRUCTION AND AERODYNAMIC MODEL IDENTIFICATION 45 3.1 Linear flight path reconstruction and aerodynamic model 47

identification

3.1.1 Linear reconstructibility 48 3.1.2 Linear flight path reconstruction 50

3.1.3 Parameter identiflability 60 3.1.4 Linear aerodynamic model identification 63

3.1.5 Relation between one- and two-step methods for 72 analysis of dynamic flight test measurements

3.2 Nonlinear flight path reconstruction 83 3.2.1 Nonlinear models for flight path reconstruction 83

3.2.2 Solution of the nonlinear flight path reconstruction 86 problem

3.3 Nonlinear aerodynamic model identification 88 3.3.1 Principles of regression analysis 89 3.3.2 Characteristics of simplified models 92 3.3.3 Model development via residual analysis 94

3.4 Conclusions 100

4 OPTIMAL INPUTS FOR AIRCRAFT PARAMETER ESTIMATION 101 4.1 Optimization of multidimensional input signals for 104

nonlinear and linear system parameter estimation

4.1.1 Representation of multidimensional input signals 104 4.1.2 Input signal optimization for nonlinear system 112

parameter estimation

4.1.3 Input signal optimization for linear system 120 parameter estimation

4.1.4 Application of the method of Newton and Raphson 128 4.2 Effect of decomposition of system parameter-state 130

estimation problems

4.3 Conclusions 140

5 FLIGHT TEST PROGRAM 141 5.1 Flight test measurement system 143

5.2 Electro-hydraulic control system 155 5.3 Measurement of moments and products of inertia 157

5.4 Flight tests 162

6 RESULTS OF EXPERIMENTS 165 6.1 Flight test data analysis 165

6.1.1 Flight path reconstruction 165 6.1.2 Aerodynamic model identification 181

6.2 Longitudinal and lateral input signals 190 6.2.1 Design of DUT input signals 190 6.2.2 DOUBLET, 3211, MEHRA and SCHULZ input signals 210

6.3 Evaluation of longitudinal and lateral input signal 215 performance

6.3.1 Sample statistics of the estimated parameters 215

6.3.2 Comparison of input signal performance 216

7 GENERAL DISCUSSION 235

APPENDICES

A Maximum likelihood estimation theory and its application to 247 the problem of estimating parameters and state of dynamic

systems

A.l General properties of maximum likelihood estimates 247

A.2 Continuous time nonlinear systems 249 A.3 Continuous time linear systems 257 A.4 Discrete time linear systems with unknown stochastic inputs 260

B Calculation of reconstructibility matrices Q. for the 267 observations y of the longitudinal and lateral linear

flight path reconstruction problem

B.l Reconstructibility matrices of the longitudinal flight 267 path reconstruction problem

B.2 Reconstructibility matrices of the lateral flight 270 path reconstruction problem

B.3 Reconstruetible subspaces 273

C Newton-Raphson method for nonlinear function minimization 275

D First and second order partial derivatives of criteria for 279 input signals with respect to spherical coordinates

D.l First and second order partial derivatives of J with 281 respect to cartesian coordinates

D.l.1 Partial derivatives of J = tr H 282 D.l.2 Partial derivatives of J = In det M 282 D.l.3 Partial derivatives of J = tr M 284 D.2 First and second order partial derivatives of cartesian 285

coordinates with respect to spherical coordinates

D.3 Partial derivatives of simplified criteria 289

E Estimated aerodynamic derivatives of 47 longitudinal and 291 47 lateral dynamic flight test manoeuvres

Samenvatting in het Nederlands (Summary in Dutch) 303

0 NOTATION, ABBREVIATIONS AND REFERENCE FRAMES

0.1 Frequently used notation

Latin symbols

a parameter vector A constant matrix

A , A , A specific aerodynamic force along X, Y and Z axis respect x y z

ively b wing span

c mean aerodynamic chord

C Cramér-Rao lower bound, constant matrix

L

C. , coefficient of aerodynamic rolling moment ipV2Sb C. c o n s t a n t p a r t of C. o C C

P

Cl Ö C*

P °fv

d C* I j, r b 9 2V ÖC,

p

*£

ÖCA a

r

ipV2Sc

M

— , coefficient of aerodynamic pitching moment

constant part of C m

ac

m Apt

a

— L

± P V2 ÖC m 5 ^ V

ac

m

aa

ac

m 3a2

ac

m aó e N

> coefficient of aerodynamic yawing moment ipv2sb constant part of C

n

ac

n

a

sï

2V

ac

n ft r b Ö 2V

ac

n

ap

ÖC n dC n 96 a dC n dó r

sidewash correction factor in (3-vane calibration formula upwash correction factor in a-vane calibration formula

» coefficient of aerodynamic longitudinal force ipV2S constant part of C„ d CX Apt

ipv

2 a c

x

an*

da 5 a2 öó Y

, coefficient of aerodynamical lateral force

ipv

2

s

ÖCY

P

C.

°Y

S C \ _r Ap CZ 2V ÖCY Y a rb Ö 2V ÖCY

s

p

w

ÖCY

p

ȣ

ÖCY CY_ "öó" o a a Y. 96" o r 2

C„ , coefficient of aerodynamic vertical force

Z , '. ipV2S C„ constant part of C_ a c

z

t ö ipV2 Ö C

z

v

^ z

CZ 9a a

as

_Ö6

e

parameter in a-vane calibration formula parameter in p-vane calibration formula matrix of elementary input signals model residual

model residual; elementary input signal mathematical expectation operator; energy vector function

right handed rectangular reference frame; linear system matrix

acceleration due to gravity linear system input matrix geometric altitude

linear system observation matrix identity matrix

moment of inertia of propeller and rotating engine compo­ nents

moment of inertia about X, Y and Z axis respectively corresponding products of inertia

matrix in linear system observation model; performance index

integer

Kalman filter gain matrix

one stage prediction gain matrix

b2 m

— » m c

b ' m

integer

likelihood function; aerodynamic rolling moment about X axis; orthogonal matrix

aircraft mass; integer

information matrix; aerodynamic pitching moment about Y axis

characteristic function corresponding to multlvariate gausslan probability density function of the zero mean random variable v

integer

aerodynamic yawing moment about Z axis; total number of samples

origin of reference frame

rate of rotation about X axis; probability density function static pressure of air

total pressure of air in propeller slipstream

engine power; covariance matrix of state estimation errors rate of rotation about X axis; integer

impact pressure of air

rate of rotation about Z axis; integer

R* partial correlation coefficient

R n dimensional Euclidean space

s integer

s vector in M (s)

v

-S wing area; sensitivity matrix

t time

T observation time interval; state transformation matrix;

temperature

u speed along X axis

u vector of input signals

U state transformation matrix

v speed along Y axis; measurement error

v vector of measurement errors

V true airspeed; covariance matrix

w speed along Z axis

w vector process noise; vector of measurement errors

W aircraft weight; constant atmospheric wind

x coordinate along X axis

x system state vector; row vector of independent variables in

regression analysis

x„ x coordinate of a-vane in F

D

x„ x coordinate of (3-vane in F

B

X aerodynamic longitudinal force along X axis; matrix of

independent variables in regression analysis

y coordinate along Y axis

2 observation vector

y y coordinate of a-vane in F

0

Y aerodynamic lateral force along Y axis; vector of dependent

variables in regression analysis

coordinate along Z axis z coordinate of p-vane in FR

aerodynamic vertical force along Z axis

angle of attack vane angle of attack

side slip angle; cartesian coordinates vane side slip angle

flight path angle Kronecker delta

aileron deflection angle 6 = 6 - 6

a a a.

r x.

left aileron deflection angle, aileron down is positive right aileron deflection angle, aileron down is positive elevator deflection angle, elevator down is positive rudder deflection angle, left rudder is positive increment

aerodynamic model error pitch angle

transducer bias error correction discrete system input matrix

m pSb

m pSc

Kalman filter innovation air density

standard deviation roll angle

spherical coordinate

t r a n s i t i o n matrix

yaw angle

angular frequency in rad/s

recoristructlble variable or Identifiable parameter estimated quantity

small deviation from nominal value matrix transpose

matrix inverse

matrix pseudo inverse

derivative with respect to time mean value

nominal value

body fixed reference frame F_ datum reference frame FD

engine

earth fixed vertical reference frame F. measured quantity

partial derivative of vector x_ with respect to scalar 9., 5x.

vector with elements -r^— , i-l(l)n, in which n denotes dimension of x

partial derivative of scalar x. with respect to vector 6, ox.

vector with elements -rzr- . j=l(l)m» in which m denotes

Ö Ö

J

dimension of 6

partial derivative of vector x with respect to vector 6, n x m matrix which can be partioned as:

ox 5x ox r_JL_ _ I _ ... -JL.1

L

öe, se. ae

J

*

\ l m

in which n and m denote the dimensions of x and 6 respec­ tively reference frame F„ reference frame F„ reference frame Fv center of gravity column

Cramêr-Rao lower bound

determinant of square matrix A distance measuring equipment Delft University of Technology logarithm to base e

left hand side maximum likelihood right hand side V, true air speed

cov a covariance matrix of a

A = 0 all elements of the matrix A are equal to zero a = 0 all elements of the vector a are equal to zero

0.3 Reference frames

Datum reference frame F

The location of characteristic points relative to the aircraft - as for instance the center of gravity - is expressed in terms of coordinates in a body fixed, rectangular and left-handed reference frame which is named here the datum reference frame, see Fig. 0-1. The X_^ axis Is in the plane of symmetry of the aircraft. The Y_ axis is perpendicular to this plane of symmetry and points to port. The direction of the Z-. axis is upwards in normal flight. For the particular aircraft used in the present flight tests, the origin 0Q coincides

with the projection on the plane of symmetry of a reference point on the star­ board wing leading edge at 1.4 m distance from the plane of symmetry. The direction of the Xp axis is chosen parallel to a reference wing chord connecting the leading edge and the trailing edge at the same distance from the plane of symmetry.

The reference frames Ffi, Fg, F_, F„ and F_, below are rectangular and

right-handed.

Body-fixed reference frame FB

See Fig. 0-1. The origin 0fi of the body-fixed reference frame is placed in the

aircraft's center of gravity. The Xg and Z„ axes are in the aircraft's plane of symmetry, the YR axis points to starboard, the ZB axis points downwards In

normal flight. In the present work the direction of the Xg is chosen to be anti-parallel to X-, axis.

Stability reference frame Fg

The stability reference frame is a special body-fixed reference frame used in the study of small deviations from a given nominal flight condition. The difference between F_ and F~ is the orientation of the Xg-axis. In the case of a symmetrical nominal flight condition, the Xg axis is chosen parallel to true

axis is chosen parallel to the projection of V on the aircraft's plane of symmetry.

Earth-fixed vertical reference frame F

The origin of the earth-fixed vertical reference frame can be placed in principle at an arbitrary position. Often it is convenient to locate the origin at the position of the aircraft's center of gravity at the start of a flight test manoeuvre. The Z„ axis points downwards parallel to the local direction of gravitation. The X„ axis points to the North, the Y„ axis points to the East.

Vehicle-carried vertical reference frame F„

See Fig. 0-2. The origin of the vehicle-carried vertical" reference frame is attached to the aircraft's center of gravity. Except for this difference, Fy is

identical to the earth fixed, vertical reference frame F„.

Vehicle-carried vertical reference frame F„

See Fig. 0-2. The reference frame F_ was found to be convenient in the analysis of the linearized flight path reconstruction problem. The origin is attached to the aircraft's center of gravity. The Z„ axis points downwards parallel to the local direction of gravitation. The X_, axis coincides with the projection of the Xg axis at the start of a flight test manoeuvre on the local horizontal plane.

Fig. 0-2: The body-fixed reference frame F„ and the vehicle-carried vertical reference frames F„ and F_, at the start of a flight test manoeuvre.

1 INTRODUCTION AND RATIONALE OF THE PRESENT WORK

The primary goals of most flight test programs of civil aircraft are the cer­ tification for air worthiness and the measurement of performance characteristics for the calculation of tables and graphs in the aircraft operating manuals. An objective of increasing importance is the measurement of stability and control characteristics to Improve the validity of the mathematical models used for flight simulation and for the design of automatic control systems.

Flight test programs are very expensive and stretch over long periods. Among other things, this Is due to the total amount of flight test time required for the measurement of aircraft performance, stability and control characteristics. Consequently, the development of techniques leading to more efficient measure­ ment of these characteristics is of great practical interest.

The past two decades have been a period of considerable progress In this res­ pect. In the first place, it has become possible to measure aircraft performance in quasi stationary flight test manoeuvres. This resulted in large reductions of required flight test time as compared to classical performance measurements in strictly stationary rectilinear flight conditions; see Refs. 3, 4, 14, 42 and 49. Furthermore, advances in relevant analytical and numerical methods have resulted in algorithms which allow estimation of stability and control derivatives from measurements in dynamic flight test manoeuvres on an almost routine basis as reported for instance in Ref. 20.

Stability and control derivatives are the parameters in a linear 'aerodynamic model' of the aircraft. Linear aerodynamic models consist of homogeneous polynomials of the first degree in the state and control input variables of the linearized equations of motion. Such polynomials are widely used as linear approximations of aerodynamic forces and moments acting on the aircraft In nonstationary flight conditions; Ref. 10. In general the domain in which linear models are valid is restricted to 'small* deviations from a nominal flight condition, which is stationary with respect to the aerodynamic forces and moments acting on the aircraft. The interest in techniques for the estimation of stability and control derivatives from flight test measurements can be explained from their essential role in mathematical models for flight simulation and for the design of aircraft (automatic) control systems.

Ref. 13 describes a flight test technique to estimate the parameters in non­ linear aerodynamic models. Nonlinear aerodynamic models consist of polynomials in state and control input variables of nonlinear equations of motion. These polynomials, which may be nonhomogeneous and of any degree, are used to represent aerodynamic forces and moments in stationary as well as in nonstation-ary flight conditions. The technique of Ref. 13 applies also to the estimation of stability and control derivatives, as homogeneous polynomials of the first degree form a subset of the more general set of non homogeneous polynomials of any degree. In general the domain of nonlinear models covers much larger deviations from a given nominal flight condition, as compared to linear models. The advantage of using nonlinear models is, that dynamic flight test manoeuvres are - in principle - much less constrained with respect to the amplitudes of for instance angle of attack and airspeed excursions. Furthermore, it is - again in principle - possible to analyze measurements in those flight regimes in which inherently nonlinear aerodynamic phenomena occur. A typical example is the high angle of attack regime, where hysteresis effects may be present in the process of airflow separation and reattachraent.

The different aspects and the rationale of the present work are described in the following.

Analysis of measurements in nonstationary flight conditions. Analysis of dynamic flight test data, in the sense of estimating stability and control derivatives from measurements of the dynamic response of the aircraft to control input signals, can be formulated in the theoretical frame work of maximum likelihood estimation theory; see for instance Ref. 46. This requires the stability and control derivatives to be interpreted as unknown parameters in a dynamical system model of a given form. It is assumed that the response of the system to precisely known input signals has been observed, by measuring the outputs of the system at discrete instants in time. The measurements are assumed to be corrupted by additive, mutually independent and normally distributed random errors. It is shown in Appendix A, that the likelihood function of these measurements depends on the parameters, as well as on the initial condition of the system. Maximization of the likelihood function with respect to the para­ meters and the initial condition, constitutes a nonlinear optimization problem. The optimum values are called the maximum likelihood estimates of the system parameters and initial condition.

Perhaps the contemporary method most frequently used for estimating stability and control derivatives from measurements in dynamic flight test manoeuvres is based on the maximum likelihood method as described above; see also Refs. 17 and 20.

It is a considerable disadvantage of this method, that its application is limited in practice to the case of linear system models. In principle, the maximum likelihood method may also be applied to the case of nonlinear system models. This, however, will result in a considerable increase of the required computing time. An even more serious problem is due to the fact, that the form of the nonlinear system model may not be known beforehand. This means that different nonlinear system models must be evaluated in terms of parameter estimation accuracy and model fit to the measurements. In general, any model change will require a software modification. This means that the development of adequate nonlinear system models is a cumbersome process.

In the present work it is shown that, if certain conditions concerning accuracy and type of the variables measured in flight are met, the original maximum likelihood estimation problem can be decomposed into two separate estimation problems which are much easier to solve than the original estimation problem. These two steps are indicated below as step 1 and step 2.

In the case of linear equations of motion, step 1 corresponds to a linear state reconstruction problem. After this reconstruction of the state, step 2 can be formulated as a linear-in-the-parameters estimation problem.

In the case of nonlinear equations of motion, step 1 becomes a nonlinear state reconstruction problem. For the case of strictly longitudinal flight test manoeuvres, this problem is known as the 'flight path reconstruction problem'; see for instance Refs. 22 and 40. In the course of the present work, flight path reconstruction was extended to encompass the more general case of both longi­ tudinal and lateral flight test manoeuvres. After nonlinear reconstruction of the aircraft's state, rather surprisingly, step 2 can be shown to be again a linear-in-the-parameters estimation problem. This is of great practical importance, as it allows the systematic and stepwlse development of adequate nonlinear models of the aerodynamic forces and moments during the flight test manoeuvre; see Ref. 2.

Optimal input signals. Parameter estimation errors are bounded from below by the so called Cramér-Rao lower bound; see Ref. 46. The significance of the Cramêr-Rao lower bound in estimation theory is due to its universality, i.e. its independence of the particular estimation procedure being used. A given estimation procedure is called efficient if the variances of the resulting estimation errors are equal to this lower bound.

In the case of a multidimensional set of parameters, the Cramér-Rao lower bound takes the form of a covariance matrix which is equal to the inverse of the so-called information matrix; see Ref. 11. The elements of the information matrix depend not only on the number and accuracy of the observation measurements, but also on the initial condition of the system, on the energy and form of the system input signals.

The Cramér-Rao lower bound is not merely of theoretical interest. For example, the covariance matrix of maximum likelihood estimation errors can be shown to approach asymptotically (i.e. for the number of measurements tending to infinity) the Cramér-Rao lower bound; Refs. 10 and 46. Therefore, It appears useful, to design Input signals such that they lead to the lowest possible value of some norm of the Cramér-Rao lower bound. Here, such input signals are called optimal input signals.

In the context of dynamic flight test manoeuvres, the possibility to design optimal input signals was first noted in Ref. 13. Ref. 46 was first In proposing to base the design of optimal input signals on the information matrix by maximizing the trace of this matrix. Although attractive from the computational point of view, it was not difficult to see that implementation of the resulting input signals would not guarantee small parameter estimation errors as expressed, for example, by the diagonal elements of the inverse of the informa­ tion matrix, i.e. the Cramér-Rao lower bound. However, the idea to use the Information matrix in input signal optimizations stimulated the development of new algorithms for the optimization of input signals based on different norms of the Cramér-Rao lower bound. A review is given in Ref. 16.

It is possible to divide these algorithms for the optimization of input signals - as proposed so far - in frequency domain and time domain techniques. In these techniques it is generally assumed that all stochastic processes involved, as for instance the measurement errors, are mutually independent and have a gaussian distribution.

Frequency domain techniques generate optimal input signals characterized by line-spectra; see Ref. 35. Although fairly efficient with respect to computation time, the problem remains to transform these spectra into input signals in the time domain, as required for actual implementation. Moreover, since the result­ ing input signals must be of finite length, they will be suboptimal to some unknown extent depending on the selected length of the input signals.

Time domain techniques appear to be cumbersome from the computational point of view; see Ref. 36. Another disadvantage is that the resulting input signals do not readily allow a physical interpretation. The user of these techniques might want to suppress a range of frequencies in order to avoid excitation of certain aircraft structural modes. However, present time domain techniques lack the flexibility to be able to account for such input signal design considerations.

A new technique for the calculation of optimal input signals was developed in the course of the present work. This technique is based on the description of input signals by linear combinations of elements of finite sets of orthonormal functions. Each of these descriptions contains a finite number of weighting factors as parameters. The optimization of input signals can thus be formulated in terms of a parameter optimization problem. Parameter optimization problems can be solved in a variety of ways. In the present work two different optimization methods were applied i.e. a direct search method, as proposed by Powell Ref. 53, and the method of Newton-Raphson Ref. 11. It is shown that in particular the latter method results in a very efficient algorithm for the op­ timization of input signals. The new technique combines the following advantages of the frequency domain and time domain techniques:

- computational efficiency,

- the resulting input signals are in the time domain,

- the frequency contents of the resulting input signals can be 'influenced' by varying the set of orthonormal functions mentioned above.

With respect to applicability of the resulting optimal input signals in dynamic flight test manoeuvres, it should be noted here that such Input signals may be difficult for the pilot to implement manually. This will be true in particular for multi-dimensional input signals. Examples of such multi-dimensional input signals are combined elevator control and throttle input signals for longitudi­ nal manoeuvres, and combined aileron and rudder control input signals for lateral manoeuvres.

It may be expected, however, that future transport type aircraft will be equipped with digital automatic flight control systems from the first flight on. Then, prior to the flight tests, optimal control input signals for a variety of flight conditions and aircraft configurations may be calculated and stored In digital data sets. These data sets can subsequently be used in flight for the execution of dynamic flight test manoeuvres via the automatic flight control system.

Flight tests. In principle, through experiments it has to be proved that control input signals which are optimal with respect to the Cramér-Rao lower bound, will also be optimal in practice, i.e. with respect to an experimentally determined covariance matrix. The need for this proof is based on the fact, that the theoretical assumptions underlying the optimization of input signals are not always fully satisfied in real flight. As mentioned above it is assumed that the observation measurement errors are mutually independent and have a gaussian distribution. Furthermore, input signal optimizations are based on models which only approximate the characteristics of the real aircraft under test.

Therefore it was thought to be essential to evaluate optimal input signals and so - perhaps even more important - to demonstrate the validity of the underlying theory by actual flight tests. A procedure to accomplish this is described below.

Parameters estimated by the maximum likelihood method are in fact stochastic variables. They are known to be asymptotically (for the number of measurements tending to infinity) efficient and unbiased; e.g. see Ref. 11. In practice, this means that for a sufficiently large number of measurements, the covariance matrix and the mean values of these parameters should be approximately equal to the Cramér-Rao lower bound and the true values of the parameters respectively. Measurements taken in one dynamic flight test manoeuvre can be used to calculate one set of stability and control derivatives. In terms of maximum likelihood theory, this can be seen to correspond to one sample of a multi-dimensional stochastic variable. In order to determine the covariance matrix and mean value of such a variable, it must be possible to obtain a set of such samples. These samples should, necessarily, be of the same ensemble. This implies that the nominal flight condition as well as the control Input signals should be identical for all members of the population. It is quite impracticable to achieve this by manual implementation of control input signals. It was for this reason that in the present work, use was made of an automatic control system,

which was designed to allow precise repetitions of control input signals of given form (and energy).

In sofar as the applicability of the Cramér-Rao lower bound in input signal optimizations is concerned, it must be shown that different types of input signals, ordered according to some norm of the Cramêr-Rao lower bound, would be ordered identically according to the same norm of an experimentally obtained covariance matrix. This would guarantee that input signals which are optimal with respect to a norm of the Cramêr-Rao lower bound, would also be optimal in practice.

The practical implication is, that flight tests must be made with a set of different types of input signals. A certain number of times, each member of this set has to be repeated precisely. Then it is possible to calculate for each type of input signal the corresponding covariance matrix of the stability and control derivatives and to evaluate the Cramér-Rao lower bound. In addition, it is also possible to calculate the mean values of the stability and control derivatives for each type of input signal. It is of practical and theoretical interest to determine to what extent these mean values are independent of the type of input signal as predicted by maximum likelihood theory.

In the course of the present work, a flight test program was carried out with a DeHavilland DHC-2 'Beaver' experimental aircraft, equiped with an automatic control system as mentioned above. This system allowed precise implementation in real time of quite arbitrary elevator, aileron and rudder control input signals. Also a high accuracy flight test measurement system was installed in the air­ craft. The flight test program was prepared and executed in close cooperation by the following Institutes or organisations, in alphabetical order:

- Delft university of Technology (DUT), Department of Aerospace Engineering, Delft, the Netherlands,

- Deutsche Forschungs- und Versuchsanstallt für luft- und Raumfahrt (DFVLR), Braunschweig, Federal Republic of Germany,

- National Aerospace Laboratory (NLR), Amsterdam, the Netherlands. The flight tests were performed in 1977 and 1978.

The present work is organized as follows. Chapter 2 starts with a description of the kinematic, aerodynamic, and measurement error models as used further on. The

analysis of dynamic flight test data is discussed in detail in Chapter 3. Chapter 4 gives a detailed description of the technique for the optimization of control input signals as developed in the present work. The flight test program, flight test measurement system, automatic control system and the method used for measuring the aircraft moments and products of inertia are described in Chapter 5. Chapter 6 begins with a description of five different types of longitudinal and lateral control input signals as implemented in flight, one of these being the result of the optimization technique of Chapter 4. Next, the results of the flight test program are described* Conclusions from thé preceding Chapters are summarized and commented upon in Chapter 7.

2 MATHEMATICAL MODELS

The optimization of control input signals, flight path reconstruction and aero­ dynamic parameter estimation require explicit formulation of several mathematical models. These models will be briefly discussed in the following.

Kinematic models. Kinematic models of aircraft rigid body motion with respect to earth consist of first order ordinary differential equations in which specific aerodynamic forces (quantities sensed by ideal accelerometers in the center of gravity) and body rotation rates appear as forcing functions; e.g. Ref. 43. Mathematical models for flight path reconstruction consist of a kinematic model, an observation model and a model of the measurement errors. High accuracy flight path reconstruction dictates accurate and, consequently, usually nonlinear models. Linearized models, however, may be good enough for qualitative evaluations of the flight path reconstruction problem. For example in Chapter 3 an analysis is made of the reconstructibility properties of a linearized mathematical model and different hypothetical measurement configurations. Nonlinear and linearized kinematic models are presented in Section 2.1.1 and 2.1.2 respectively.

Aerodynamic models. If the atmosphere is in uniform motion with respect to earth and the effects of elastic deformations of the airframe are neglected, the components of total aerodynamic force and total aerodynamic moment depend on the present and all past values of control surface deflections, engine power settings and aircraft motion with respect to the surrounding air mass. This leads to aerodynamic models consisting of integrals of 'indicial functions'; Ref. 61.

A more practical alternative is to expand each of the above mentioned aero­ dynamic variables in the form of a truncated Taylor series backwards in time. This results in aerodynamic models in the form of (usually nonlinear) algebraic functions of present time values of the above mentioned variables and their time derivatives. The validity of this latter type of aerodynamic models is experi­ mentally verified in Chapter 6. A priorily postulated models of the longitudinal and lateral aerodynamic forces and moments are listed in Section 2.2.1. Their linearized counterparts, which are needed in Chapter 6 for the calculation of optimal input signals, are derived in Section 2.2.2.

Observation models. Observation models relate on-board measured variables as specific aerodynamic forces, body rotation rates, airspeed and altitude

vari-ations, to the components of the aircraft state vector. Nonlinear as well as linearized observation models are developed in Section 2,3.1 and 2.3.2 respectively. In Section 2.4 a model is presented for flight test measurement errors.

2.1 Kinematic models

2.1.1 Nonlinear kinematic models

For the case of the atmosphere moving uniformly with respect to a flat earth and if the effect of airframe elastic deformations is negligible, the equations of motion for an aircraft having a geometrical (i.e. not necessarily an aero­ dynamical) plane of symmetry can be written as e.g. in Ref. 10, in terms of a set of relations between the components of the total aerodynamic force (i.e. including the aerodynamic effects of engines) X, Y and Z and the components of

• • •

the kinematic acceleration u, v and w along the axes X^, YR and Z of the body

fixed reference frame F :

X = m (u + qw - rv) + mg sin 9 ,

Y = m (v + ru - pw) - mg cos 9 sin cp , (2.1-1) Z = m (w + pv - qu) - mg cos 9 cos cp ,

in which u, v and w denote the components of true airspeed (TAS) along - and p, q and r denote rates of rotation about the axes of FR, 9 and cp denote pitch

angle and roll angle respectively, m denotes aircraft mass and g acceleration due to gravity; and a second set of relations between the total aerodynamic moments (i.e. including the aerodynamic effects of engines) L, M and N and the

• • •

rotational accelerations p, q and r about the axes of F_:

L = Z

J ~

(I

y - V «

r

"

X

zx

(

'

+

P

q)

•

M - Iyq ~ (Iz - Ix) rp - Iz x (r2 - p2) + I ^ r , (2.1-2)

N = I r - ( I - I ) p q - I (p - qr) - I u q .

z x y ^ zx y H e eM

in which I , I and I denote moments of inertia and I the only (due to

x y z zx symmetry) non-zero product of intertia in FR. In (2.1-2) the gyroscopic effects

of rotating propellers or compressors and turbines of jet engines with spin axes parallel to XD have been taken into account by terms with to .

If m is known, the total aerodynamic forces X, Y amd Z can be measured directly in dynamic flight conditions according to:

X - A m , x Y = A m , y Z = A m , z (2.1-3)

in which A , A and A denote the 'specific aerodynamic forces' , i.e. the total x y z

external non-field forces per unit of mass. These quantities are sensed by ideal accelerometers along the axes of F . Substitution of (2.1-3) into (2.1-1) and devision of all terms by m results in:

- qw + rv , + g cos 6 sin cp - ru + pw ,

w

A - g sin 9 x A y A + g cos 9 cos cp - pv + qu . z (2.1-4)

Because aircraft mass m has disappeared, (2.1-4) may now be interpreted as a set of kinematic relations. The orientation of Ffi with respect to the earth-fixed

vertical reference frame F„ is governed by a set of kinematic relations for the Euler angles 4», 9 and cp:

<I> = q sin cp/ cos 9 + r cos cp/cos 9 ,

9 = q cos cp - r sin cp , cp = p + q sin cp tan 9 + r cos cp tan 9 .

(2.1-5)

The geographic position coordinates of the origin of F„ (i.e. the center of gravity) with respect to F satisfy the following relations:

hi

u V w + W *E W yE W ZE _ (2.1-6)

EB cos 9 cos 4» cos 9 s i n <\> - s i n 9 s i n cp s i n 6 c o s <\> - cos cp s i n c|> s i n cp s i n 0 s i n <|> + c o s cp cos 4» s i n cp c o s 9 cos cp s i n 9 cos 4* + s i n cp s i n cj> cos cp s i n 9 s i n <\> - s i n cp cos 4> cos cp cos 9 (2.1-7)

and W , W and W denote the components of the constant atmospheric wind

XE yE ZE

along the axes of F„(*).

The set of kinematical relations (2.1-4), (2.1-5) and (2.1-6) can be written in the form of a continuous nonlinear system, e.g. Ref. 54:

x = f (x, u) ,

with state vector:

x = col [u, v, w, <\>, 6, cp, xp, yv, zK] ,

E' JE '

"E-(2.1-8)

and input vector:

u = col [Ax, Ay, Az, p, q, r]

2.1.2 Linearized kinematic models

The nonlinear equations of motion (2.1-1) and (2.1-2) in the body fixed reference frame F_ may readily be transferred into an equivalent set of equa­ tions in the stability reference frame Fg. Next, these equations are linearized for small deviations from a nominal flight condition of steady, rectilinear flight with side slip angle equal to zero. It may readily be ascertained that in the nominal flight condition the components of TAS along- and the rates of rotation about the axes of F_ have the following values:

(*) In the case that curvature and rotation of the earth are to be taken into account it may be more convenient to express geographic position in terms of longitude and latitude and to decompose the local atmospheric wind along the axes of the vehicle carried vertical reference frame F or F„; see Ref. 43.

u = V , v = 0 , w = 0 ,

°s °s °s

p = O , q = 9 , r = O,

°s °s °s

while the nominal pith angle is equal to the nominal flight path angle:

9 -

Y

,

°S °

the subscript o referring to the nominal flight condition.

The linearized versions of the equations of motion (2.1-1) may now be written

as:

X

s

- m u

s

+ mg cos

y

Q

9

g

,

Y

g

= m (v

s

+ V

Q

r

g

) - mg cos Y

Q

<P

g

, (2.1-9)

Z = m (w - V q_) + mg sin

y 9 ,

L„ « I p„ - I r_ - I co sin a q„ ,

S x_

r

S zx_ S e e o ^S »

5

S

= X

y

c

^S

+

Ve

8 i n a

o PS

+

V e

C

°

8

"o

7

s » (2.1-10)

S

S ~

\ *S - ^xg P

S

- V e

C 0 S

"o *S *

in which the superscript indicates small deviations from the steady, recti­

linear nominal flight condition mentioned above. In the nominal flight

condition, the side slip angle is defined to be zero. However, due to for

instance the rotation in the propeller slipstream as in the present case or due

to asymmetrical engine trust in the case of multi engine aircraft, the nominal

aerodynamic flow condition may well be asymmetrical. This means that, in order

to maintain rectilinear flight, a small roll angle must be established. In

connection with the reconstructability analysis of the absolute rather than of

the relative roll angle in Section 3.1, cp is, in contravention of what Is

stated above, defined to be a small deviation from zero rather than from cp .

The aerodynamic force increments X

g

, Y and Z

g

may be expressed in terms of

corresponding increments of accelerometer readings according to:

*S

= m

\ '

Y_ = m A ,

S y

s

Z = m A .

S z„

(2.1-11)

The linearized forms of the kinematical relations for the Euler angles of F_

are:

*e = _{S COS Y '} O

e

c

- q«

(2.1-12)

9

S

- p

s

+ tan e

Q

r

s

.

Now it is convenient to express geographical position in terms of coordinates

x™, y

T

, z™ along the axes of the vertical reference frame F,_. Eq. (2.1-6) is

then written as:

*T

TS

- 1 u

s

v

s

w

s

+

"

w

w yT w

L

Z T

.

(2.1-13)

in which the transformation matrix L_,

g

is written as, cf. (2.1-7)

hi

cos9

s

cos(|;

s cosG^sin cjL - sin6„ s incp- s i n 9g cos<L - coscps8in(jIs sincpg sinögSinipg + coscp cos(J*s sincpgcosög coscpgSin6gCOS(|>s + sincpgSin^g co8(Pq sin9_sin«pq - sincpgcosög coscpgCOsGg

L i n e a r i z a t i o n of ( 2 . 1 - 1 3 ) r e s u l t s i n :

xT = cos Y- u„ + s i n Y„ WQ - v„ s i n Y„ 6 + W ,

1 O O O O O O i > X - ,

yT - vs + VQ cos YQ bs + Wy , ( 2 . 1 - 1 4 )

ts* r*j rw» *N/

z„ = - s i n Y u0 + cos Y w_ - V cos Y ©o + W

E o S o S o o S z„

Because of the definition of the nominal flight condition given above it follows

that:

v

s

- v

s

, w

s

= w

s

,

P

S

= P

S '

q

S

=

V

r

S

= r

S

*S

=

V

With:

W- V n* r<*

_{s ^ ~ s}

a -

Y~

and P -

y-o y-o

Eqs. (2.1-9), (2.1-12) and (2.1-14) may then be written as the following sets of

linear first order differential equations for the longitudinal variables:

u

s

= - g cos Y

Q

6 + A

x

,

- g sin Y 9 + A

~

_a

_{_ +}

2

S , ~

_{q §}

_,

O

e = q

g

, (2.1-15)

^r

= c o s Y

o -"s

+ V

o

8 i n Y

o " "

V

o

s i n Y

o *

+

\

»

•

z: = - s i n Y u„ + V cos Y a - V cos Y e + w ,

and l a t e r a l v a r i a b l e s : cos Yo <ps + A ^ - r , V '•S » o rS cos Y ' o

(2.1-16)

fs* «%/ <p = ps + t a n YQ rs , y - V B + V cos yn <\> + W r o o o y„ 2.2 Aerodynamic models

2.2.1 Nonlinear aerodynamic models

In this Section models are developed for C.., C„ and Cy, the dimensionless compo­

nents of the total aerodynamic force and for C , C„ and C , the dimensionless rax n

components of the total aerodynamic moment.

For a given aircraft configuration the aerodynamic force coefficients C„, C„ and C„ and aerodynamic moment coefficients C , C- and C depend on the following variables and their first and higher order time derivatives: angle of attack a, side slip angle P, airspeed V, body rotation rates p, q and r, control surface angles 6 , 6 and 6 and engine power settings. In the present case of low

ci c r .

-speed flight the effect of compressibility can be neglected. Also, scale effects are not taken into account since, at the relatively high Reynolds number of ~ 5 * 10 in flight, the effect of Reynolds number variations as occurring in flight is considered to be negligible.

It is common practice, cf. Ref. 10, to disregard the influence of all time derivatives except for a and p.

If the propeller is represented as an ideal pulling disc, it is possible to derive the following relation:

A pt P

= a + b (2.2-1)

in which Ap denotes the increase of total air pressure in the propeller slip­

stream and P denotes engine power; cf. Ref. 13. In the case of piston engined

aircraft, variations of air speed V and engine power settings (engine speed and

manifold pressure) affect the aerodynamic force and moment coefficients only

indirectly through changes of Ap /ipV

2

. Consequently, air speed and engine power

settings can be replaced by one single variable Ap /ipV

2

in the list of

variables mentioned above.

Next it is assumed that the aerodynamic force and moment coefficients are

analytic functions which may be expanded in a Taylor series.

If the effects of the lateral variables 0, p, r, P, 6 and 6 on the

longitudi-nal coefficients C

x

, C

z

and C and vice versa, the effects of the longitudinal

variables Ap^/ipV

2

, a, q, a and 6 on the lateral coefficients C

v

, C„ and C

t e i x. n

are neglected (*),then first order models for the longitudinal and lateral aero­

dynamic force and moment coefficients can be written in terms of dimensionless

variables as:

-x -x ~x. ^

2 + c

x °

+ c

z

a&

'

-?

t

ipV

2

a q _ _

e C

" ' ^

+

_{o Ap ipV}

\ 7^2

+_z

_{a q a 6}

°X "

+ C

Z V

+ C

X '

^

+

\

6

e>

CZ "CZ + CZA ~Z2+CZ a + C Z ^ + C Z . - T + C Z A V <2-2"2> o Apk ipVz a q a 6 t e Ap, "o mApt ipV2 ,ua mq ' mct ' m6

C - C + C — -

+ C a + C - P - + C •££■ + C 6 ,

m m

„

m

A„ i«w2 m~ m. V m» V m

c

e'

(*) In cases where an aerodynamical plane of symmetry exists (coinciding with

the geometrical plane of symmetry) it follows that these 'cross coupling'

effects cannot exist in first order aerodynamic models.

a n d : Pb C

Y =

C

Y

+_{o p p r P 6}

V *

+

^ ^

+

^ ^

+

S V

+

° I .

6

a

+

V

6

r •

a 6r ■Cl = Ci + 'Cl P + CI — + °l ^ + Cl - ^ + CA 6a + S 6r > ( 2*2 _ 3 ) C = C + C p + C ^ 7 + C ö T f + C | - + C 6 + C 6 n n nQ ^ n 2V n 2V n» V nA a n , r o B p r p 6 O r r a r

It is important to note here that (2.2-2) and (2.2-3) result in nonlinear relations for the dimensional aerodynamic forces and moments. For example, substitution of (2.2-1) into the model for Cx in (2.2-1) results in:

c

x " (

c

x

+ a c

x

A

)

+ b c

x

A

• T5v3'

+

°x

a + c

x

Y-

+

o Apfc Apt a q

+ C

X.f

+

\

6

e •

a o e

which, when substituted in the expression for the dimensional aerodynamic force X:

X = (^ ipV2S

results, for the case that engine power P is kept constant, in the following nonlinear model for the dimensional force X:

X - Xv 2 V2 + X _x V"1 + Xa v 2 aV2 + X qV + Xj Sv + X& y2 6 V« .

V H e

Similar expressions may be derived for Y arid Z and the dimensional aerodynamic moments L, M and N. These expressions are linearized in the following Section.

Prior to the flight test program described in Chapter 5, static wind tunnel measurements were made on a 1:11 scale model of the test aircraft in the low subsonic wind tunnel of Delft University; see Ref. 44. A systematic evaluation was made of the dependence of the force and moment coefficients on a, {3,

-0.20 L +0.20 r 0.20 -(3=0C 5e= 5p=0°

p = 0°

öp = 0°

p=o°

5e=0°

P<

c ♦0.40 T -0.20 x -4 0

* - p (

e öe = 0° a=8° A pt - 1.0 0.5 0 r i — - -0.8-

-1.6-i

c

z

> 1 öe = 0° a = 8° ^ 0

t

A pt ►0.20-r -H 1

-0.20-1

4 0 + 4

^ - p ( ° ) - ^

5e = 0° ^ 1.0

t

Apt

ipv

2 +8 a = 8°

Fig. 2 - l ( a ) : Longitudinal aerodynamic force and moment coefficients of the DHC-2 'Beaver' aircraft as a function of angle of attack a and side slip angle P for three different values of Ap / i p V2, as measured at a

Reynolds number of 0.47 * 1 06 on a 1:11 scale model in the

O 0.5 1.0 a--8° a = 8 _{a = 8°} 0 i, 8 12 — ♦ a l0)

Fig. 2-1(b): Lateral aerodynamic force and moment coefficients as a function of sideslip angle (3, for three different values of Ap /|pV2.

a

al •al

p--0°

öp=0° -0.04 -0.08

X

p=o°

öp--0°

Ipv

2 (3=0° 5p--0°

— » » a r )

0 4 8 12 0 | 1 1 1 -►al ■0.04 -0.08

ï

p=o°

6p=0° Ap, 0 4 8 12 -0.10 -0.20 T 1 1 (3 = 0° 5p = 0° ^ 0 Cn - 0 . 5 - 1 . 0 öa

t t

Apt

ipv

2 0.08 ►0.04= Ap. 4 8 —►al0

p=o°

6p-0° 1 ^v2 \ J f0 -{0.5 l1.0

F i g . 2 - l ( c ) : Lateral a i l e r o n and rudder c o n t r o l d e r i v a t i v e s as a function of angle of attack o, at t h r e e d i f f e r e n t values of Ap / ^ p V .

Ao / I P V2. 6 , 6 and 6 . Some of the results are shown in Fig. 2-1. It is shown

Ft e a r

in Chapter 6 that results like these can be used in a systematic way in the aerodynamic model identification procedure. Here it suffices to remark that the wind tunnel results indicate that the fit of the first order models as postu­

lated in (2-2.2) and (2-2.3) to actual dynamic flight test measurements will certainly not be perfect. For example, it follows from Fig. 2-l(a) that C.. and C depend in a nonlinear way on a. Further, a pronounced lateral to longitudinal

m

aerodynamic cross coupling exists in the sense that C depends also on (3.

Fig. 2-1(b) shows that while the Cy-p and C.-p relations are approximately linear, this is certainly not true for the relation C -p. Analogous to the

n

lateral to longitudinal aerodynamic cross coupling, there exists also a rather pronounced longitudinal-to-lateral aerodynamic cross coupling in the form of a strong dependence of C„, C. and C on Ap /$pV2 and <*•

From Fig. 2-l(c) it follows that the control derivatives with respect to 6 depend on Ap./ipV2. This is not surprising because at least part of the vertical

tailplane is in the slipstream of the propeller.

From the discussion in this Section it should be clear that in general aero­ dynamic models can only be expected to approximate reality. Furthermore, models as given in (2.2-2) and (2.2-3) describe only the deterministic components of the aerodynamic force and moment coefficients. Not included are stochastic contributions as generated by turbulent boundary layers, turbulence in the propeller slipstream and local flow separations. The effect of such random fluctuations on aircraft motion is discussed in Ref. 23.

2.2.2 Linearized aerodynamic models

For small deviations from a stationary rectilinear flight condition dimensional models of aerodynamic forces along- and aerodynamic moments about the axes of the stability reference frame F„ may be linearized.

Subsequently, these models may again be nondimensionalized; see Ref. 10. The resulting linear nondimensional models of the longitudinal force coefficients

C and C , and longitudinal moment coefficient C may be written as:

ra c

x

c

x

c

x

c

x«

c

x

x u a q a 6^ c

z

c

z

c

z

c

z .

c

z

A u a q a o^ C C C C C m m„ in m» m , u a , q a 0^

ü/V,

qc/v,

ac/V. (2.2-4) in which:

°X

* P n_{O O} VoS

ip v

2

s

ro o (2.2-5) M m ±P„_{o o} VoSc

In (2.2-4) C„ , C„ , etc. denote the longitudinal so-called stability- and control derivatives in the body fixed reference frame F . In the stability

a

reference frame F,, (2.2-4) is written as:

-c

x

c

ms „ S3 u„ m m. c

x

c

x .

%

a

s

c

z V

c c

m, uS/ Vo

V'

V

o

ac/V (2.2-6)

The linear nondlmensional models of the lateral force coefficient CY and the

lateral moment coefficients C0 and C may be written as: ■*. n

°Y

°Y °Y CY CY« CY °Y

P P. r. *P \

I

6

r p p r 8 *6 6 r r a r C C C C C C n0 n n ns nE nc 8 p r B o Ö r a r

P

pb/2V c rb/2V Pb/V (2.2-7) in which:

s =

*p„

v

;i

s o o ip V2Sb o o (2.2-8) C = n

|p Vjsb

o o

In (2.2-7) CY , Cy , etc. denotes the lateral stability- and control derivatives

P P

in the body fixed reference frame F . In the stability reference frame F„ (2.2-7) is written as: \

c

n

s

_ =

°Y

F

°Y,

n. n, PS b / 2 Vo rSb'2 Vo

pb/v^

(2.2-9)

2.3 Observation models

2.3.1 Nonlinear observation models

In this section the models are derived for observations of air speed V, angle of attack o, side slip angle p, altitude variations and geographical position measurements.

The observation model for V follows directly from its definition as the resultant of the air velocity components u, v and w along the axes of F :

V - (u2 + v2 + w2) * . (2.3-1)

Per definition, the angle of attack is:

a = arctan ^ , (2.3-2)

which Is different from a , the angle of attack measured by an angle of attack vane. This is due to aircraft-induced air velocity components and the rotation of F„ about the 3L, and Y_ axes, and results In:

w " x q + y p

« . a r ctan - + C a + Ca , (2.3-3)

o

if it is assumed that the aircraft induced part of the measured angle of attack depends linearly on a; see Ref. 27. In practice, the actual upwash may also depend on engine power settings.

The side slip angle is defined as:

p - arctan - r , (2.3-4)

(u2 + w2) *

which is different again from what a side slip vane would measure, as shown in Fig. 2-2. When the vane axis of rotation Is parallel to the ZB axis and the

effects of an aircraft induced side velocity components and the rotation of F„ about the X„ and Z„ axes are taken into account, the side slip vane angle Is:

8 = a r c t a n v v + xRr - zftp

^ - ^

+ c

si

e

+

' p

<

o ( 2 . 3 - 5 ) see a g a i n Ref. 27.

In (2.3-5), CQ accounts for the vane being positioned outside the aircraft's

PQ

geometrical plane of symmetry as well as for asymmetry of the air flow due to, for instance, rotation in the propeller slipstream. The aircraft-induced part of the measured side slip angle is, analogous to the assumption mode in (2.3-3) with respect to the induced angle of attack, assumed to be a linear function of 8.

Flight altitude changes can, in quasi-steady flight conditions, accurately be measured with differential pressure transducers. The corresponding observation model is:

Ah = - z, (2.3-6)

Fig. 2-2: Definition of side slip angle 3 and side slip vane angle 8

In principle any navigation system (e.g. inertial platform, doppler radar, OMEGA or DME) may be used for the measurement of geographical position. In the case of a flat earth approximation it is often convenient to express geographical

position in terms of coordinates x_ and y„ in a vertical earth-fixed reference

frame F„.

2.3.2 Linearized observation models

In the s t a b i l i t y reference frame, (2.3-1) i s w r i t t e n a s :

V - ( u | + v | + w|)* . (2.3-7)

The corresponding linearized form is:

*v *%*

V = u

s

. (2.3-8)

Linearization of (2.3-3), the observation model of the angle of attack vane, for

small deviations from the nominal stationary and rectilinear flight condition

results in:

w 1 w x y

«N* / O \ f CL rsj CL tSê\ - - *s'v

a + a = arctan (—) + ( q + —i p) + C (a + a ) + C =

v v u

' o ^u u u ' up o a

o o w * o o o o

1 + W

v

u

o x

ff

y

a

"

(1 + C

up>

a

o

+ C

a

+

<!

+

V " " V-* *S

+

V ^ PS ' <

2

'

3

"

9

>

o o o

In the nominal flight condition, the vane angle is:

a = (1 + C ) a + C .

v up' o a

o o

Subtraction of a from both sides of (2.3-9) results in the following

linear-o

ized observation model:

S ~ S

a

v

= C

a ,

a

~ V —

q

S

+

? -

p

S • (2.3-10)

1 o o

in which C

a

- (1 + C

u p

) .

The observation model of the side slip vane (2.3-5) can be linearized in a

similar way resulting in:

p

+

\ - arctan £ ) + L _ ( L + i ? -'Ifi p)

+ C (

p + p)

+

C

o o v

z

o o o o

1 + (-£)

o

V Xg „ Z

- — + - E - r - -

E

- p + C . p + C „ ,

u u u

r

si

r

8 '

o o o o

since v = p = 0 . Substitution of u = V cos o and transformation of x

Q

, z

Q

,

o o o o o p p

r and p from F to F results in:

P

v

+

K

= C P l

?

+

V cos a ^S - V cos a ^S

+ C

p » t

2

'

3

"

1 1

)

o 1 o o o o o

in which C. = + C .

p, cos a si

1 o

In principle it is impossible to determine Cg in flight. This can be seen as

follows. Assume first a stationary rectilinear flight condition with roll angle

equal to zero. Then, for a strictly symmetrical air flow condition, the side

slip must be zero. For the present test aircraft, however, the airflow cannot be

assumed to be symmetrical due to the rotation in the propeller slipstream.

Consequently, a stationary rectilinear flight with zero roll angle does no

longer imply a zero side slip angle. Let the side slip then be equal to P and

the vane indicate a value P in this condition of zero roll angle. In the

o

nominal flight condition with zero roll angle mentioned above it follows from

(2.3-11) that:

and because p is unknown, C„ is unknown also. The consequence of this is that

the linearized observation model of the side slip vane, comprises an unknown

constant C

Q

according to:

P

o

x8 Z 6

^v = C P l P + V cos a ' s " V cos' a PS + Cp <2-3~12>