M.Goman, A.Khramtsovsky (2006) - Analysis of Aircraft Nonlinear Dynamics Based on Parametric Continuation and Phase Portrait Investigation

June 21, 2006June 21, 2006 ICNPAAICNPAA--2006: Mathematical Problems in Engineering and Aerospace Science2006: Mathematical Problems in Engineering and Aerospace Sciencess 11
Analysis of Aircraft Nonlinear Dynamics
Based on Parametric Continuation and
Phase Portrait Investigation
Dr MikhailDr Mikhail GomanGoman and Dr Andrewand Dr Andrew KhramtsovskyKhramtsovsky
DeDe MontfortMontfort University, Leicester, the UKUniversity, Leicester, the UK

ContentsContents
nn Goals and tools of applied bifurcation analysisGoals and tools of applied bifurcation analysis
nn Permanent retention of computed data in a databasePermanent retention of computed data in a database
structure: new algorithms, new possibilitiesstructure: new algorithms, new possibilities
nn Phase portrait analysisPhase portrait analysis
nn ExamplesExamples
nn ConclusionsConclusions

Typical goals of the BifurcationTypical goals of the Bifurcation
Analysis (BA)Analysis (BA)
nn In the parameter space:In the parameter space:
–– to find boundaries between the regions withto find boundaries between the regions with
different types of the nonlinear systemdifferent types of the nonlinear system’’s dynamicss dynamics
nn In the state space:In the state space:
–– To find out all the steadyTo find out all the steady--state solutionsstate solutions
(equilibrium, periodic, strange attractors) and to(equilibrium, periodic, strange attractors) and to
study their evolution with respect to thestudy their evolution with respect to the
parameters,parameters,
–– To understand each type of the nonlinear systemTo understand each type of the nonlinear system’’ss
dynamicsdynamics

Aircraft Rigid Body DynamicsAircraft Rigid Body Dynamics
Equations of Motion
State Variables
Control Variables
w
w
w
ww
d
d
d
d
d
d
d
d
t
t
t
t
I + Ix = Ma+Mc
V
m + V = F+T + Gx( ) a
R
=
=
C( )
)
VQ
Q
Q E(
Q q f y
w
a
a
b
b
b
d d d d d d
h h z
R =
=
=
(X Y Z )g g g
(
(
)
)
V
= p q r
T
T
T
T
T
V
V
V
cos cos
cos
sin
sin
e er l
l l
=
=
(
(
)
)
a r c...
T T Tr r

Numerical MethodsNumerical Methods
for Qualitative Analysisfor Qualitative Analysis
nn Continuation algorithm:Continuation algorithm:
–– branching and limit pointsbranching and limit points
processing;processing;
–– systematic search for all solutionssystematic search for all solutions
of nonlinear system at fixedof nonlinear system at fixed
parameters;parameters;
–– bifurcation points identificationbifurcation points identification
and collectionand collection
nn Regions of attraction:Regions of attraction:
–– reconstruction of stability regionreconstruction of stability region
boundary;boundary;
–– computation of twocomputation of two--dimensionaldimensional
cross sectionscross sections
nn Numerical simulation:Numerical simulation:
–– perturbations in particularperturbations in particular
manifolds of trajectories;manifolds of trajectories;
detF = 0
det = 0F
x
s
x
c
limit point
branching point
parameter variation

SteadySteady--state solutionsstate solutions
and their evolutionand their evolution
nn Computationally intensive, but relativelyComputationally intensive, but relatively
straightforward taskstraightforward task
nn Main tool is the continuation technique:Main tool is the continuation technique:
–– Bifurcation points and alternative branches areBifurcation points and alternative branches are
determineddetermined ““on the flyon the fly””
–– All the data (the branches, stability data,All the data (the branches, stability data,
bifurcation and other special points) arebifurcation and other special points) are
recorded in the permanent databaserecorded in the permanent database

1D bifurcations diagrams: equilibriums1D bifurcations diagrams: equilibriums
(computed using(computed using KritKrit package)package)
Legend:Legend:
nn StableStable
nn OscillatoryOscillatory
unstableunstable
nn AperiodicallyAperiodically
unstableunstable
nn UnstableUnstable
(other types)(other types)
BranchingBranching
pointspoints
TestTest ““KubichekKubichek--88””

1D bifurcations diagrams:1D bifurcations diagrams:
periodic solutionsperiodic solutions
nn For the closedFor the closed
orbits the minimumorbits the minimum
and maximumand maximum
values of X(1) statevalues of X(1) state
vector componentvector component
are plottedare plotted vsvs thethe
parameter valueparameter value
nn Both timeBoth time--advanceadvance
andand PoincarePoincare
mappings weremappings were
used during theused during the
computationscomputations

1D bifurcations diagrams: complex1D bifurcations diagrams: complex
attractors and jumpsattractors and jumps
nn Continuation of the complexContinuation of the complex
attractors:attractors:
–– fixed increments of the parameterfixed increments of the parameter
–– IntegrationIntegration
nn Jumps: Analysis of the systemJumps: Analysis of the system’’ss
behavior near the bifurcation pointsbehavior near the bifurcation points

Example: Lorenz modelExample: Lorenz model
andand rr -- varvar
Parameter values are:Parameter values are:

Equilibrium and periodic solutionsEquilibrium and periodic solutions

Lorenz strange attractorLorenz strange attractor

Continuation of the complex attractorContinuation of the complex attractor

Stability of the complex attractor:Stability of the complex attractor:
LyapunovLyapunov indicesindices
nn One index shouldOne index should
always be equalalways be equal
to zeroto zero
nn Stable complexStable complex
attractor has oneattractor has one
positivepositive LyapunovLyapunov
indexindex

Evolution diagram:Evolution diagram:
useful tool for recovery studiesuseful tool for recovery studies
nn The diagramThe diagram
shows how toshows how to
enter into andenter into and
to recoverto recover
from certainfrom certain
stable regimestable regime
nn During thisDuring this
analysis newanalysis new
solutionssolutions
could becould be
foundfound

2D bifurcations diagrams2D bifurcations diagrams

Permanent database gives rise to aPermanent database gives rise to a
new classes of algorithmsnew classes of algorithms
nn Permanent memory makes the algorithms trulyPermanent memory makes the algorithms truly
intelligent: they can capitalize on the previousintelligent: they can capitalize on the previous
resultsresults
nn Some possibilities:Some possibilities:
–– Faster search for the steadyFaster search for the steady--state solutions by usingstate solutions by using
known solutions as initial guesses,known solutions as initial guesses,
–– Possibility to stop simulation if stable steadyPossibility to stop simulation if stable steady--statestate
solution is approached,solution is approached,
–– Possibility to identify statePossibility to identify state--space structures, or tospace structures, or to
check completeness of the bifurcation diagrams,check completeness of the bifurcation diagrams,
–– And many moreAnd many more ……

1D Permanent database contains:1D Permanent database contains:
nn Branches of the steadyBranches of the steady--state solutionsstate solutions
including stability data and trajectories ofincluding stability data and trajectories of
periodic solutions and strange attractors,periodic solutions and strange attractors,
nn Starting points, bifurcation points, error pointsStarting points, bifurcation points, error points
etc.etc.
nn Branching information computed for bifurcationBranching information computed for bifurcation
pointspoints
nn Data on relationships between bifurcationData on relationships between bifurcation
points and branchespoints and branches

Database Structure for Nonlinear AutonomousDatabase Structure for Nonlinear Autonomous
System: UML descriptionSystem: UML description
Class diagram for the autonomous nonlinear dynamical systemClass diagram for the autonomous nonlinear dynamical system

Example: completeness checkExample: completeness check
((““AdvisorAdvisor”” module)module)
nn Test KubicheckTest Kubicheck--88
nn At the end theAt the end the
database containsdatabase contains
dozens of specialdozens of special
points and continuationpoints and continuation
trajectories.trajectories.
nn ““ManualManual”” analysis isanalysis is
difficultdifficult

nn inconsistenciesinconsistencies
in the databasein the database
are identifiedare identified
nn inconsistenciesinconsistencies
can be visualizedcan be visualized
nn recoveryrecovery
probability isprobability is
evaluatedevaluated
nn autoauto--repairrepair
featurefeature

Phase portrait analysisPhase portrait analysis
nn SystemSystem’’s dynamics is studied at a parameters dynamics is studied at a parameter
valuesvalues ““typicaltypical”” for a certain type of it.for a certain type of it.
nn The following is computed:The following is computed:
a.a. SteadySteady--state solutions and their stability (may bestate solutions and their stability (may be
extracted from the permanent memory),extracted from the permanent memory),
b.b. Special trajectories (incoming to or outgoing from aSpecial trajectories (incoming to or outgoing from a
steadysteady--state solution along the eigenvectors)state solution along the eigenvectors)
c.c. SimulationsSimulations
d.d. CrossCross--sections of the regions of attraction of stablesections of the regions of attraction of stable
steadysteady--state solutionsstate solutions
b.b.--d. may result in finding new steadyd. may result in finding new steady--state solutionsstate solutions

Phase Portrait GUI and databasePhase Portrait GUI and database
KubicheckKubicheck--88
problemproblem
nn The databaseThe database
contains multiplecontains multiple
equilibriumequilibrium
solutions andsolutions and
specialspecial
trajectoriestrajectories

Partial automationPartial automation
of the phase portrait analysisof the phase portrait analysis
Phase Portrait GUI (KRIT package):Phase Portrait GUI (KRIT package):
nn SteadySteady--state solutions and stability data arestate solutions and stability data are
extracted from 1D Permanent databaseextracted from 1D Permanent database
nn Special trajectories are computedSpecial trajectories are computed
nn Optionally points belonging to a separating surfaceOptionally points belonging to a separating surface
between two stable attractors are computed. Thisbetween two stable attractors are computed. This
often leads to the finding of unstable periodicoften leads to the finding of unstable periodic
solutions.solutions.
nn UserUser--controlled simulation and computation of thecontrolled simulation and computation of the
crosscross--sections of asymptotic stability regionssections of asymptotic stability regions

FF--18, 818, 8thth
order: Computation of ASRorder: Computation of ASR

FF--18 with FCS: Computation of ASR18 with FCS: Computation of ASR

ConclusionsConclusions
nn Continuation and mapping technique areContinuation and mapping technique are
powerful tools. Comprehensive database of thepowerful tools. Comprehensive database of the
results obtained multiplies their power andresults obtained multiplies their power and
allows to construct new algorithms.allows to construct new algorithms.
nn GUIs make continuation and phase portraitGUIs make continuation and phase portrait
computations flexible. More automation iscomputations flexible. More automation is
necessary.necessary.
nn Advances in the computing of regions ofAdvances in the computing of regions of
attraction are highly desirable.attraction are highly desirable.

AcknowledgementsAcknowledgements
nn This research was funded byThis research was funded by
QinetiQQinetiQ/DERA (Bedford, the UK)/DERA (Bedford, the UK)

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