Computational Framework for Analysis of Aircraft Computational Framework for Analysis of Aircraft Nonlinear Dynamics & Control Design Based on Nonlinear Dynamics & Control Design Based on

Qualitative MethodsQualitative Methods

Mikhail Goman, Andrew Khramtsovsky, De Montfort University, UKEugene Kolesnikov, Bombardier Aerospace, Montreal, Canada

• GARTEUR FM(AG17), Stockholm, FOI, 21-22 March 2006

GARTEUR FM(AG17) FOI, Stockholm 21-22 March 2006 2

ContentsContents

• A bit about flight dynamics

• Computational framework for qualitative analysis of nonlinear aircraft dynamics (methods & tools)

• Attainable equilibrium sets (AES) for ADMIRE airframe

• Closed-loop dynamics & regions of attraction (RA) for ADMIRE+NDI

• Manoeuvre limitation based on analysis of AES & RA

• Summary

GARTEUR FM(AG17) FOI, Stockholm 21-22 March 2006 3

Kinematics of VelocityKinematics of Velocity--Vector Roll Vector Roll ManoeuvreManoeuvre

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Velocity-Vector Roll Manoeuvre Visualization

Fa

Fa

ww

VV

Fa

VV

GARTEUR FM(AG17) FOI, Stockholm 21-22 March 2006 5

Kinematics of Velocity Vector Roll Manoeuvre

“ …The velocity vector roll is defined as an angular rotation of an airplane about its instantaneous velocity vector, constrained to be performed at constant angle-of-attack, no sideslip, and constant velocity…”

Wayne C. Durham, Frederick H. Lutze, William Mason, December 1993

Available definitions of velocity vector roll Available definitions of velocity vector roll manoeuvremanoeuvre like:like:

are valid only at very high rates of rotation as the radius of hare valid only at very high rates of rotation as the radius of helical elical

trajectory is inversely proportional to angular ratetrajectory is inversely proportional to angular rate WW

GARTEUR FM(AG17) FOI, Stockholm 21-22 March 2006 6

Kinematics of Velocity Vector Roll Manoeuvre

“ …The velocity vector roll is defined as an angular rotation of an airplane about its instantaneous velocity vector, constrained to be performed at constant angle-of-attack, no sideslip, and constant velocity…”

Wayne C. Durham, Frederick H. Lutze, William Mason, December 1993

Available definitions of velocity vector roll Available definitions of velocity vector roll manoeuvremanoeuvre like:like:

are valid only at very high rates of rotation as the radius of hare valid only at very high rates of rotation as the radius of helical elical

trajectory is inversely proportional to angular ratetrajectory is inversely proportional to angular rate WW

GARTEUR FM(AG17) FOI, Stockholm 21-22 March 2006 7

Kinematics of Velocity Vector Roll ManoeuvreKinematics of Velocity Vector Roll Manoeuvre

XX

VV

aabb

ww

WW

YYZZ

);,( dxfx =&

);,( dxfx =&Trqpx ),,,,( ba=

TTVTVcrEE rightleft

),,,,,(yq

ddddddd =

Natural kinematic parameters for Velocity Vector Roll

((a,b,Wa,b,W))

VVr

rr )( ×=W

wwhere

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Methods & Tools for Qualitative AnalysisMethods & Tools for Qualitative Analysis

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Computational Framework for Qualitative AnalysisComputational Framework for Qualitative Analysis

, convex optimization (controllability regions for linear unstable systems with limited control inputs), …

Controllability local and global

Continuation & systematic search methods Monte Carlo method, phase portrait investigation (iterative & incremental method), …

Multiple-attractors (critical flight regimes)

Jacoby matrix (eigenvalues, eigenvectors), Lyapunov function methods, computing invariant manifolds (region attraction boundary), 2-dim slices of region of attraction (RA), …

Stability local

Stability global - region of attraction (RA)

Newton’s method, gradient descent methods, predictor-corrector continuation algorithm, direct & inverse simulation methods, attainable equilibrium sets (ASE)

Trimming & Linearization

Computational AlgorithmsAnalysis Method

nBAABBrk n =- ),,,( 1K

GARTEUR FM(AG17) FOI, Stockholm 21-22 March 2006 10

Simulation, Trimming & Local StabilitySimulation, Trimming & Local Stability

);,( dxFdtdx

=m

R

mD

nM

RURURXx

ÌÎ

ÌÎ

ÌÎ

d

d&

Direct simulationDirect simulation)(td

),( dxFx =&)(tx

Inverse simulationInverse simulation)(td

),( dxFx =&)(tx

Direct trimmingDirect trimming

Ed ),(0 dxF= ExInverse trimmingInverse trimming

Ed),(0 dxF=Ex

0det,

=¶¶

EExxF

d

Bifurcation analysis:Bifurcation analysis: Bifurcation free:Bifurcation free: 0det,

¹¶¶

EEx

F

dd

Local stability:Local stability: 0det,

=úúû

ù

êêë

é-

¶¶ E

xF

EEx

ld

eigenvalueseigenvalues

eigenvectorseigenvectors

AESAES

GARTEUR FM(AG17) FOI, Stockholm 21-22 March 2006 11

Some Formal DefinitionsSome Formal DefinitionsAttainable Equilibrium Sets (AES):Attainable Equilibrium Sets (AES):

þýü

îíì ÌÎ== m

DEEE RUxFxAES dd ,0),(:

Regions of Attraction (RA):Regions of Attraction (RA):

þýü

îíì £¥®®Î= )(,),(:),( 00 TtortasxxxRxxRA EE

nEE dd

Controllability Region (CR):Controllability Region (CR):

þýü

îíì Îή®$Î= +

RDEEmn

EE UtUtatxxtxtstRuxxCR ),(,),(),(..,),(:),(),( 000000 dddddddd &

GARTEUR FM(AG17) FOI, Stockholm 21-22 March 2006 12

Attainable Equilibrium Sets (AES)

TrEE rightleft),,( ddd

TTTVTVLEcrEE rightleft

),,(),,,,,,( WÛ badddddddyq

Direct trimmingDirect trimming

Inverse trimmingInverse trimming

T),,( Wba

AES sliceAES slice

),( Wb

GARTEUR FM(AG17) FOI, Stockholm 21-22 March 2006 13

Connection with Bifurcation Analysis

TrEE rightleft),,( ddd T),,( Wba

•• AESAES•• Control SpaceControl Space

bifurcation diagram

bifurcation tailoring

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ADMIRE Bare Airframe AnalysisADMIRE Bare Airframe Analysis

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Attainable Equilibrium Sets and Local Stability Maps

Iml

Rel

Iml

Rel

Iml

Rel

Im

Re

l

l

Iml

Rel

•• Classification of Classification of eigenvalueseigenvalues

Iml

Rel

GARTEUR FM(AG17) FOI, Stockholm 21-22 March 2006 16

ADMIRE AES: (a,b) and (a,W) slices

GARTEUR FM(AG17) FOI, Stockholm 21-22 March 2006 17

ADMIRE AES: (a,b) and (W,b) slices

M=0.4M=0.4

aa=5 deg=5 deg

aa=15 deg=15 deg

aa=30 deg=30 deg

GARTEUR FM(AG17) FOI, Stockholm 21-22 March 2006 18

ADMIRE AES: (a,b) slices at different M

M=0.4M=0.4 M=0.8M=0.8 M=1.0M=1.0 M=1.2M=1.2

GARTEUR FM(AG17) FOI, Stockholm 21-22 March 2006 19

AES figure saves controls & eigenvalues

((b,Wb,W) AES M=0.4, ) AES M=0.4, aa=15 deg=15 deg

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ADMIRE ClosedADMIRE Closed--Loop DynamicsLoop Dynamics

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Feed-forward transformation

NDIBS or FL based

+ Reference Modelfor HQ

Control Allocation

(Newton-Raphsonbased)

Aircraftactuators,airframe,sensors

Signal conditioning

CVcom Mc CVd

CV = [a, b, W]’ - control variables; CVcom = [a, b, W]’ - control inputs;

d = [DEL,DER,DR]’ - control effector commands

Mc – control moment demand

max

maxmin

dd

ddd&& £

££

NDI-based control laws

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NDI blockNDI block--functional diagramsfunctional diagrams

J-1(-wxJw + M) M

a a* - k1a(s)(a-a*) - k2a(s)(a-a*)b = Aw + Aw + E = b* - k1b(s)(b-b*) - k2b(s)(b-b*)W W* - k1W(s)(W-W*)

desired dynamics, stability, robustness etc.

..

..

.. . .

..

..

.

..

Feedback linearization (FL) NDI

ab = A(a,b)w + E(a,b,V,n,q,f)W

.

.

w = J-1(-wxJw + M).

a a* - ka(s) (a-a*)b = Aw + E = b* - kb(s) (b-b*)W W*

.

.

w = w* - Kw(s)(w-w*). .

M

Backstepping (BS) NDI

J-1(-wxJw + M)

.

.

desired dynamics... 1st level

desired dynamics…. 2nd level

a*,b*,W*

a*,b*,W*

abW

reference model for handling qualities

, kWs + kW

wa,b2

s2 + 2xa,bwa,bs + wa,b2

com

Equations of motion

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openopen--loop loop AESAES

Nose Pointing Manoeuvre

closedclosed--loop loop AESAES

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openopen--loop AESloop AES

Velocity-Vector Roll Manoeuvre

closedclosed--loop AESloop AES

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Departure at Velocity-Vector Roll Manoeuvre

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Open-loop system

Regions of Attraction for NDI closedRegions of Attraction for NDI closed--loop system:loop system:critical external disturbancescritical external disturbances

NDI closed-loop system

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Convergence and Departure DynamicsConvergence and Departure Dynamics

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Effect of Rate SaturationEffect of Rate Saturation

sdeg/50£d& sdeg/500£d&

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AES of open-loop system

Regions of Attraction at Regions of Attraction at VelocityVelocity--Vector Roll Vector Roll ManoeuvreManoeuvre

Critical BoundariesCritical Boundaries

AES of NDI closed-loop system

Regions of AttractionRegions of Attraction

GARTEUR FM(AG17) FOI, Stockholm 21-22 March 2006 30

Manoeuvre limitation based on analysis of AES & RAManoeuvre limitation based on analysis of AES & RAAllowable NDI control inputs acom,bcom,Wcom

DD

- allowable inputs

critical boundariescritical boundaries

DD –– safety margin selected safety margin selected based on a size of RAbased on a size of RA

Next step Next step -- avoid rate saturation!avoid rate saturation!How?How?Why?Why?

GARTEUR FM(AG17) FOI, Stockholm 21-22 March 2006 31

Summary

• Computational framework based on numerical implementation of qualitative methods is feasible and effective approach to flightdynamics applications.

• Matlab tools for computation of AES for velocity-vector roll manoeuvre and regions of attraction RA for the closed-loop system allow the effective analysis of aircraft dynamics and manoeuvrability.

• Assignment of allowable NDI control inputs can be made using analysis of ASE and RA.

• The plan for future work: a) avoiding rate saturation in NDI control (?); b) control allocation and inversion at the presence of saturated control; c) …

• “… Noah’s Ark was built by amateurs and Titanic by professionals…”