Reduction of Nonlinear Models for Control Applications
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Reduction of Nonlinear Models for Control Applications
A. Da Ronch, N.D. Tantaroudas, and K.J. BadcockUniversity of Liverpool, U.K.
AIAA Paper 2013-1941Boston, MA, 08 April 2013
email: A.Da-Ronch@[liverpool, soton].ac.uk
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MotivationPhysics-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
Control design for FCS of flexible aircraft
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Testcase
Global Hawk type (P. Hopgood of DSTL)• beam/plate FE model of composite material
• control surfaces
• structural model > 1,300 points
• CFD grid > 6 mio points
- stability augmentation (SAS)
- worst case gust search
- gust load alleviation (GLA)
- performance optimization
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Model Reduction
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Model Reduction
Eigenvalue problem of large order system is difficult
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Badcock et al., “Transonic Aeroelastic Simulation for Envelope Searches and Uncertainty Analysis”, Progress in Aerospace Sciences; 47(5): 392-423, 2011
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Model Reduction
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2nd/3rd Jacobian operators for NROM
Da Ronch et al., “Nonlinear Model Reduction for Flexible Aircraft Control Design”, AIAA paper 2012-4404; AIAA Atmospheric Flight Mechanics, 2012
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Model Reduction
How to add gust term in CFD-based ROM?
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control surfaces,gust encounter, speed/altitude
Da Ronch et al., “Model Reduction for Linear and Nonlinear Gust Loads Analysis”, AIAA paper 2013-1942; AIAA Structural Dynamics and Materials, 2013
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Model ReductionSystematic generation of Linear/Nonlinear ROMs• independent of large order model
• control design done in the same way
xfuBAxdtdx
u nlnLin/Nln ROM dynamics
Linear Aerodynamics Linear Structure
Nonlinear Structure
Nonlinear Aerodynamics Linear Structure
(Nonlinear Structure)
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Pitch-Plunge AerofoilStructure: linear/nonlinear
- cubic stiffness, K = K(1+βx x 3)
Aerodynamics: 2D strip- flap motion (Wagner)
- gust encounter (Küssner)
IDEs to ODEs by adding 8 aero states
Total of 12 states model problem Ts
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Stability Analysis
“heavy case”
ωξ/ωα = 0.343µ = 100ah = -0.2xα = 0.2rα = 0.539
U*L = 4.6137
U* = 4.6 (99.7% of U*L)
ROM basis
2 modes (related to pitch & plunge)
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ROM evaluation: free response dξ0/dτ = 0.01
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Von Kármán spectrum
ROM evaluation: gust response
“1 minus cosine”
Hg = 25
wg = 0.05 U*
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Worst-case gust search “1 minus cosine”wg = 0.05 U*
Search for Hg < 100
1,000 design sites
ROM-based Kriging
Cost (based on CFD)
ROM-based search: 31 min
FOM-based search: 43 h
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Closed loop response: worst-case gust
H∞ control on ROM
measured input: pitch
control input: flap
“1 minus cosine”Hg = 41wg = 0.05 U*
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Nonlinear structure: worst-case gust “1 minus cosine”Hg = 41wg = 0.05 U*
βξ = 3βα = 3
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Nonlinear ROM evaluation: worst-case gust “1 minus cosine”Hg = 41wg = 0.05 U*
βξ = 3βα = 3
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Closed loop response: worst-case gust “1 minus cosine”Hg = 41wg = 0.05 U*
βξ = 3βα = 3
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UAV ModelStructure: geometrically non-linear beam (from Imperial College)
- Nastran plate model to 1D beam- model tuning
Aerodynamics: 2D strip- control surface, gust encounter- 20 states for each beam node
FOM of 540 states (20 wing + 4 tail + 3 fuse)
Nastran [Hz] Beam [Hz]
1st bending 3.56 3.48
2nd bending 7.75 6.99
1st torsion 14.9 12.2
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ROM convergence M = 0.08h = 20,000 mα∞ = 5 deg
“1 minus cosine”Hg = 26wg = 0.06 U∞
From 540 states to 16 states
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ROM evaluation: continuous gust M = 0.08h = 20,000 mα∞ = 5 deg
Von Kármán
Cost
FOM time domain: 55 s
ROM time domain: 5.4 s
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Closed loop response: continuous gust
H∞ control on ROM
measured input: wing tip deflection
control input: flap
M = 0.08h = 20,000 mα∞ = 5 deg
Von Kármán
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M = 0.08h = 2,900 mα∞ = 10 deg
Von Kármán
Nonlinear effects: continuous gust
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M = 0.08h = 2,900 mα∞ = 10 deg
Von Kármán
Closed loop response: continuous gust
H∞ control on ROM
measured input: wing tip deflection
control input: flap
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ConclusionSystematic approach to model reduction
• applicable to any (fluid/structure/flight) systems
• control design done in the same way
Nonlinear ROM is straightforward
• include nonlinearity in small size system
• assess limitations from using linear models
Enabling 10,000 calcs
• worst case search, GLA
• SAS, performance optimization
... demonstration of UAV model with CFD, gust and flight dynamics
email: A.Da-Ronch@[liverpool, soton].ac.uk web: http://www.cfd4aircraft.com/