Nonlinear Model Reduction for Flexible Aircraft Control Des - PowerPoint Presentation

Slide1
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Nonlinear Model Reduction for Flexible Aircraft Control Design

A. Da Ronch, K. J. BadcockUniversity of Liverpool, Liverpool, U.K.Y. Wang, A. Wynn, and R. PalaciosImperial College, London, U.K.

AIAA Paper 2012-

4404

Minneapolis, 13 August 2012

andreadr@liverpool.ac.ukSlide3

Objectives

Physics-based simulation of very flexible aircraft gust interaction large amplitude/low frequency modes coupled rigid body/structural dynamics nonlinearities from structure/fluid (& control) Nonlinear model reduction for control design system identification methods manipulation of full order residual (more)- Control design for FCS of flexible aircraftProject webpage:http://www.cfd4aircraft.com/Slide4

Global Hawk type

CFD grid: ~ 6 million grid points Control surfaces on wing & V-tail Structural modelSlide5

Model Reduction

- Nonlinear full order model (FOM)- Nonlinear reduced order model (ROM)

from

n

~ 10

6

unknowns to

m

~ 10 (few dominant modes)

Eigenvalue

problem of large dimension system is difficult

Schur complement eigenvalue solver

Badcock et al., “

Transonic

Aeroelastic

Simulation for Envelope Searches and Uncertainty Analysis

”,

Progress in Aerospace Sciences

; 47(5): 392-423, 2011Slide6

Model Reduction

Taylor expansion around equilibrium point

Nonlinear terms

B

&

C

with matrix-free products

need extended order arithmetic with CFD

2

nd

/3

rd

Jacobian operators for NROMBadcock et al., “Transonic Aeroelastic Simulation for Envelope Searches and Uncertainty Analysis”, Progress in Aerospace Sciences; 47(5): 392-423, 2011Slide7

Model Reduction

Taylor expansion around equilibrium point

How to introduce gust into CFD?

field-velocity approach

dependence on control, gust, etc.Slide8

Model Reduction

Linear ROMNonlinear ROMSlide9

Application Examples

1. Pitch-plunge aerofoil with strip aero FOM  NROM generation  control design model problem to test methods2. Geometrically-exact nonlinear beam with strip aero static deflection + small/large deformation set of FOM/ROM analyses

3. Pitch-plunge aerofoil with CFD

how to solve large sparse system?

how to calculate gust terms?Slide10

1. Pitch-Plunge Aerofoil

Struct model: linear/nonlinear cubic stiffness in plunge K=K(1+β3 ξ 3)Aero model: strip flap motion (Wagner) gust encounter (Küssner)IDEs

to ODEs by adding 8 aero states

Total of 12 states 

model problemSlide11

Aeroelastic

rα = 0.5µ = 100ωξ/ωα = 0.2Liu et al., “Application of the Centre Manifold Theory in Non-Linear Aeroelasticity”, Journal of Fluids and Structures; 11(3): 225-246, 1997mode tracing: U*L = 6.285 eigenspectrum at U*/U*L = 0.95FOM validation213Slide12

Gust:

“sin”hg = 20w0 = 10-3Simulation:dt = 0.10FOM/ROM gust response - linear structural modelSlide13

Full order model

gust responseGust:“sin”hg = 40w0 = 10-1Simulation:dt = 0.10βξ3 = 3Slide14

NFOM/NROM

gust response - nonlinear structural modelGust:“sin”hg = 40w0 = 10-1Simulation:dt = 0.10βξ3 = 3Slide15

Closed-loop

gust responseGust:“sin”hg = 40w0 = 10-1Simulation:dt = 0.10 H∞ designed on ROM to minimize pitch response NROM to be used for nonlinear control techniques? parametric study on control/noise weightsSlide16

2. Geometrically-Exact Nonlinear Beam

Geometrically-exact nonlinear equations for unrestrained body Research code at Imperial College Two-noded displacement-based elements Coupled with strip aerodynamics (UVLM and CFD ongoing) Set of linear/nonlinear analyses for FOM/ROM

Palacios et al., “Structural and Aerodynamic Models in Nonlinear Flight Dynamics of Very Flexible Aircraft”,

AIAA Journal

; 48(11): 2648-2659, 2010Slide17

HALE Wing

Flexible HALE wing Stability analysis at ρ∞ = 0.0889 kg/m3 (h = 20000 m): UL = 102 m/s ωL = 69.7 rad

/s

DoF

: 16*(12+8) =

320

Murua

et al., “Stability and Open-Loop Dynamics of Very Flexible Aircraft Including Free-Wake Effects”,

AIAA paper 2011-1915

span

16 m

chord

1

m

e.a. &

c.g.

50% chord from LE

bending rigidity

1·10

5

N·m

2

torsional

rigidity

2·10

5

N·m

2

beam elements

16Slide18

beam span [m]

vertical displacement [m] Slide19

FOM/ROM

response to follower force - small deflectionsFA = 10 N, ω = 2 rad/sFOM DoF: 320ROM/NROM DoF: 3Slide20

large deformation around large (static) deflection:

F0 = 800 NFA = 200 Nω = 2 rad/sFOM DoF: 320ROM/NROM DoF: 3FOM/ROM response to follower force - large deflectionsSlide21

NFOM/NROM

gust responseGust:“1-cos”w0 = 10-2U∞ = 10 m/s, α∞ = 10 degU∞ = 60 m/s, α∞ = 2.5 degControl is made possible using ROMsFOM DoF: 320ROM/NROM DoF: 8Slide22

3. Pitch-Plunge Aerofoil using CFD

Struct model: linear/nonlinear polynomial form for stiffness K=K(1+β3 α3+ β5 α5)Aero model: CFD Euler equations point distribution, 7974 points“Heavy” case

Aeroelastic

r

α

=

0.539

µ =

100

ω

ξ

/

ωα = 0.343Badcock et al., “Hopf Bifurcation Calculations for a Symmetric Airfoil in Transonic Flow”, AIAA Journal; 42(5): 883-892, 2004Slide23

Parallel

Meshless- Research code, University of Liverpool Simulation of complex geometries in relative motion Cloud of points Euler, laminar and RANS Schur complement eigenvalue solverKennett et al., “An Implicit Meshless Method for Application in Computational Fluid Dynamics”, International Journal for Numerical Methods in Fluids; 2012Slide24

FOM/ROM

free responseSimulation:U* = 2M = 0.6α0 = 1 degdt = 0.10FOM DoF: approx. 32000ROM Dof: 2Slide25

FOM/ROM

gust responseGust:“1-cos”hg = 12.5w0 = 10-2 Worst-case gust searchFOM DoF: approx. 32000ROM Dof: 2Slide26

Future Work

Need to have extended order arithmetic for NROM with CFD Coupling CFD with nonlinear beam More on control studiesso that flexible aircraft gust interaction using CFD, nonlinear beam with controls many ROM calculationsProject webpage:http://www.cfd4aircraft.com/Slide27

Conclusion

Systematic approach to model reduction detailed scalable to large order problems 3 application examples with CFD, nonlinear beams and control FOM calcs, ROM generation for control Schur complement for large sparse systems How to calculate gust terms with CFD- Good performance with nonlinearities at reduced costProject webpage:http://www.cfd4aircraft.com/