M.Goman, A.Khramtsovsky (2006) - Analysis of Aircraft Nonlinear Dynamics Based on Parametric...

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 Mikhail Dr Mikhail GomanGoman and Dr Andrew and Dr Andrew KhramtsovskyKhramtsovskyDe De MontfortMontfort University, Leicester, the UKUniversity, Leicester, the UK


ContentsContents

nn Goals and tools of applied bifurcation analysis Goals and tools of applied bifurcation analysis nn Permanent retention of computed data in a database Permanent retention of computed data in a database

structure: new algorithms, new possibilitiesstructure: new algorithms, new possibilitiesnn Phase portrait analysisPhase portrait analysisnn ExamplesExamplesnn ConclusionsConclusions


Typical goals of the Bifurcation Typical goals of the Bifurcation Analysis (BA)Analysis (BA)

nn In the parameter space:In the parameter space:–– to find boundaries between the regions with to 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 solutions state solutions

(equilibrium, periodic, strange attractors) and to (equilibrium, periodic, strange attractors) and to study their evolution with respect to the study their evolution with respect to the parameters,parameters,

–– To understand each type of the nonlinear systemTo understand each type of the nonlinear system’’s s dynamicsdynamics


Aircraft Rigid Body DynamicsAircraft Rigid Body Dynamics

Equations of Motion

State Variables

Control Variables

w

w

w

ww

d

d

d

d

dd

d

d

t

t

t

tI + Ix = Ma+Mc

Vm + V = F+ T + Gx( ) a

R =

=

C( )

)

VQ

QQ E(

Q q f y

w

a

a

bb

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

VVV

cos cos

cossinsin

e er l

l l

=

=

(

(

)

)

a r c...

T T Tr r


Numerical Methods Numerical Methods for Qualitative Analysisfor Qualitative Analysis

nn Continuation algorithm:Continuation algorithm:–– branching and limit points branching and limit points

processing;processing;–– systematic search for all solutions systematic search for all solutions

of nonlinear system at fixed of nonlinear system at fixed parameters;parameters;

–– bifurcation points identification bifurcation points identification and collectionand collection

nn Regions of attraction:Regions of attraction:–– reconstruction of stability region reconstruction of stability region

boundary;boundary;–– computation of twocomputation of two--dimensional dimensional

cross sectionscross sections

nn Numerical simulation: Numerical simulation: –– perturbations in particular perturbations in particular

manifolds of trajectories;manifolds of trajectories;

detF = 0

det = 0Fx

s

x

c

limit point

branching point

parameter variation


SteadySteady--state solutions state solutions and their evolutionand their evolution

nn Computationally intensive, but relatively Computationally intensive, but relatively straightforward taskstraightforward task

nn Main tool is the continuation technique:Main tool is the continuation technique:–– Bifurcation points and alternative branches are Bifurcation points and alternative branches are

determined determined ““on the flyon the fly””–– All the data (the branches, stability data, All the data (the branches, stability data,

bifurcation and other special points) are bifurcation 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 StableStablenn Oscillatory Oscillatory

unstableunstablenn AperiodicallyAperiodically

unstableunstablenn Unstable Unstable

(other types)(other types)

Branching Branching pointspoints

Test Test ““KubichekKubichek--88””


1D bifurcations diagrams: 1D bifurcations diagrams: periodic solutionsperiodic solutions

nn For the closed For the closed orbits the minimum orbits the minimum and maximum and maximum values of X(1) state values of X(1) state vector component vector component are plotted are plotted vsvs the the parameter valueparameter value

nn Both timeBoth time--advance advance and and PoincarePoincaremappings were mappings were used during the used during the computationscomputations


1D bifurcations diagrams: complex 1D bifurcations diagrams: complex attractors and jumpsattractors and jumps

nn Continuation of the complex Continuation 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’’s s behavior near the bifurcation pointsbehavior near the bifurcation points


Example: Lorenz modelExample: Lorenz model

and and 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 should One index should always be equal always be equal to zeroto zero

nn Stable complex Stable complex attractor has one attractor has one positive positive LyapunovLyapunovindexindex


Evolution diagram: Evolution diagram: useful tool for recovery studiesuseful tool for recovery studies

nn The diagram The diagram shows how to shows how to enter into and enter into and to recover to recover from certain from certain stable regimestable regime

nn During this During this analysis new analysis new solutions solutions could be could be foundfound


2D bifurcations diagrams2D bifurcations diagrams(computed using (computed using KritKrit package)package)


Permanent database gives rise to a Permanent database gives rise to a new classes of algorithmsnew classes of algorithms

nn Permanent memory makes the algorithms truly Permanent memory makes the algorithms truly intelligent: they can capitalize on the previous intelligent: they can capitalize on the previous resultsresults

nn Some possibilities:Some possibilities:–– Faster search for the steadyFaster search for the steady--state solutions by using state 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--state state

solution is approached,solution is approached,–– Possibility to identify statePossibility to identify state--space structures, or to space structures, or to

check completeness of the bifurcation diagrams,check completeness of the bifurcation diagrams,–– And many more And many more ……


1D Permanent database contains:1D Permanent database contains:

nn Branches of the steadyBranches of the steady--state solutions state solutions including stability data and trajectories of including stability data and trajectories of periodic solutions and strange attractors,periodic solutions and strange attractors,

nn Starting points, bifurcation points, error points Starting points, bifurcation points, error points etc.etc.

nn Branching information computed for bifurcation Branching information computed for bifurcation pointspoints

nn Data on relationships between bifurcation Data on relationships between bifurcation points and branchespoints and branches


Database Structure for Nonlinear Autonomous Database 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 the At the end the database contains database contains dozens of special dozens of special points and continuation points and continuation trajectories.trajectories.

nn ““ManualManual”” analysis is analysis is difficultdifficult


nn inconsistencies inconsistencies in the database in the database are identifiedare identified

nn inconsistencies inconsistencies can be visualizedcan be visualized

nn recovery recovery probability is probability is evaluatedevaluated

nn autoauto--repair repair featurefeature


Phase portrait analysisPhase portrait analysis

nn SystemSystem’’s dynamics is studied at a parameter s dynamics is studied at a parameter values values ““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 be state 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 a Special trajectories (incoming to or outgoing from a

steadysteady--state solution along the eigenvectors)state solution along the eigenvectors)c.c. Simulations Simulations d.d. CrossCross--sections of the regions of attraction of stable sections of the regions of attraction of stable

steadysteady--state solutionsstate solutionsb.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--8 8 problemproblem

nn The database The database contains multiple contains multiple equilibrium equilibrium solutions and solutions and special special trajectoriestrajectories


Partial automation Partial 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 are state solutions and stability data are

extracted from 1D Permanent databaseextracted from 1D Permanent databasenn Special trajectories are computedSpecial trajectories are computednn Optionally points belonging to a separating surface Optionally points belonging to a separating surface

between two stable attractors are computed. This between two stable attractors are computed. This often leads to the finding of unstable periodic often leads to the finding of unstable periodic solutions.solutions.

nn UserUser--controlled simulation and computation of the controlled 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 are Continuation and mapping technique are powerful tools. Comprehensive database of the powerful tools. Comprehensive database of the results obtained multiplies their power and results obtained multiplies their power and allows to construct new algorithms.allows to construct new algorithms.

nn GUIs make continuation and phase portrait GUIs make continuation and phase portrait computations flexible. More automation is computations flexible. More automation is necessary.necessary.

nn Advances in the computing of regions of Advances in the computing of regions of attraction are highly desirable.attraction are highly desirable.


AcknowledgementsAcknowledgements

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


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