robust analysis and synthesis for linear parameter varying...
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AEROSPACE ENGINEERING AND MECHANICS
Robust Analysis and Synthesis for Linear Parameter Varying Systems
Peter SeilerUniversity of Minnesota
AEROSPACE ENGINEERING AND MECHANICS
Gary J. Balas (Sept. 27. 1960 – Nov. 12, 2014)
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AEROSPACE ENGINEERING AND MECHANICS
Gary and Andy Packard
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AEROSPACE ENGINEERING AND MECHANICS
Spreading the Word
MUSYN Robust Control Theory Short Course (Start: 1989)
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Software Development
µ-Analysis and Synthesis (µ-Tools) Matlab Toolbox (1990)
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µ-Tools merged with the Matlab Robust Control Toolbox (2004)
AEROSPACE ENGINEERING AND MECHANICS
LPVTools: Matlab Toolbox for LPV Systems
• Developed by MuSyn: Balas, Packard, Seiler, Hjartarson
• Funded by NASA SBIR contract #NNX12CA14C
• Contract Monitor: Dr. Martin J. Brenner, NASA Armstrong.
• Goal: Unified framework for grid/LFT based LPV• Modeling, Synthesis, Analysis, and Simulation
• Compatible with Control Toolbox, Robust Control Toolbox, & Simulink using Matlab object-oriented programming.
• Full documentation (manual, command line, Matlab “doc”)
• LPVTools is freely available under a GNU Affero GPL
• Google Search: SeilerControl
• www.aem.umn.edu/~SeilerControl/software.shtml
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AEROSPACE ENGINEERING AND MECHANICS
Aeroservoelasticity
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Brian Taylor (UAV Lab Director)
Chris Regan
Abhineet Gupta
Aditya Kotikalpudi
Sally Ann Keyes
Adrià Serra Moral
Harald Pfifer
Julian Theis
Gary Balas
(9/27/60 – 11/12/14)
AEROSPACE ENGINEERING AND MECHANICS
Performance Adaptive Aeroelastic Wing (PAAW)
• Goal: Suppress flutter, control wing shape and alter shape to optimize performance
• Funding: NASA NRA NNX14AL36A
• Technical Monitor: Dr. John Bosworth
• Two years of testing at UMN followed by two years of testing on NASA’s X-56 Aircraft
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Schmidt &Associates
LM/NASA X-56UMN Mini-Mutt
LM BFF
AEROSPACE ENGINEERING AND MECHANICS
Outline
• Linear Parameter Varying (LPV) Systems
• Applications
• Flexible Aircraft
• Wind Farms
• Theory for LPV Systems
• Robustness Analysis
• Model Reduction
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AEROSPACE ENGINEERING AND MECHANICS
Outline
• Linear Parameter Varying (LPV) Systems
• Applications
• Flexible Aircraft
• Wind Farms
• Theory for LPV Systems
• Robustness Analysis
• Model Reduction
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Modeling for Aircraft Control
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Flight Envelope
Mach
Altitude
0.6 0.7 0.8
5000 ft
10000 ft
0.5
15000 ft
20000 ft
25000 ft
30000 ft
35000 ftRockwell B-1 Lancer (Photo: US Air Force)
u(t) y(t)
Nonlinear ODE
AEROSPACE ENGINEERING AND MECHANICS
Modeling for Aircraft Control
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Flight Envelope
Mach
Altitude
0.6 0.7 0.8
5000 ft
10000 ft
0.5
15000 ft
20000 ft
25000 ft
30000 ft
35000 ftRockwell B-1 Lancer (Photo: US Air Force)
u(t) y(t)
Flight Condition
r=(0.7,10000ft)
Equilibrium Condition
Nonlinear ODE
AEROSPACE ENGINEERING AND MECHANICS
Modeling for Aircraft Control
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Flight Envelope
Mach
Altitude
0.6 0.7 0.8
5000 ft
10000 ft
0.5
15000 ft
20000 ft
25000 ft
30000 ft
35000 ftRockwell B-1 Lancer (Photo: US Air Force)
u(t) y(t)
Linearize near
one equilibrium
Linear Time Invariant (LTI)
wherer=(0.7,10000ft)
Use for linear control design
AEROSPACE ENGINEERING AND MECHANICS
Modeling for Aircraft Control
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Flight Envelope
Mach
Altitude
0.6 0.7 0.8
5000 ft
10000 ft
0.5
15000 ft
20000 ft
25000 ft
30000 ft
35000 ftRockwell B-1 Lancer (Photo: US Air Force)
u(t) y(t)
Parameterized LTILinearize near
set of (fixed)
equilibria
where
AEROSPACE ENGINEERING AND MECHANICS
Modeling for Aircraft Control
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Flight Envelope
Mach
Altitude
0.6 0.7 0.8
5000 ft
10000 ft
0.5
15000 ft
20000 ft
25000 ft
30000 ft
35000 ftRockwell B-1 Lancer (Photo: US Air Force)
u(t) y(t)
Parameterized LTILinearize near
set of (fixed)
equilibria
Gain-Scheduling
Design controllers at many flight
conditions and “stitch” together.
AEROSPACE ENGINEERING AND MECHANICS
Modeling for Aircraft Control
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Flight Envelope
Mach
Altitude
0.6 0.7 0.8
5000 ft
10000 ft
0.5
15000 ft
20000 ft
25000 ft
30000 ft
35000 ftRockwell B-1 Lancer (Photo: US Air Force)
u(t) y(t)
Linear Parameter Varying (LPV)
where
Linearize around set
of varying equilibria
AEROSPACE ENGINEERING AND MECHANICS
Outline
• Linear Parameter Varying (LPV) Systems
• Applications
• Flexible Aircraft
• Wind Farms
• Theory for LPV Systems
• Robustness Analysis
• Model Reduction
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Aeroservoelasticity (ASE)
Efficient aircraft design
• Lightweight structures
• High aspect ratios
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Flutter
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Source: NASA Dryden Flight Research
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Classical Approach
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Frequency
AeroelasticModes
Rigid BodyModes
0
Frequency
Separation
Controller Bandwidth
Flutter AnalysisFlight Dynamics,
Classical Flight Control
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Flexible Aircraft Challenges
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Frequency
AeroelasticModes
Rigid BodyModes
0
Increasing
wing flexibility
AEROSPACE ENGINEERING AND MECHANICS
Flexible Aircraft Challenges
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Frequency
Rigid BodyModes
0
Integrated Control Design
Coupled Rigid Body and
Aeroelastic Modes
AeroelasticModes
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Body Freedom Flutter
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Modeling and Control for Flex Aircraft
1. Parameter Dependent Dynamics
• Models depend on airspeed due to structural/aero interactions
• LPV is a natural framework.
2. Model Reduction
• High fidelity CFD/CSD models have many (millions) of states.
3. Model Uncertainty
• Use of simplified low order models OR reduced high fidelity models
• Unsteady aero, mass/inertia & structural parameters
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Modeling and Control for Wind Farms
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Simulator for Wind Farm Applications, Churchfield & Lee
http://wind.nrel.gov/designcodes/simulators/SOWFA
Saint Anthony Falls: http://www.safl.umn.edu/
Eolos: http://www.eolos.umn.edu/
1. Parameter Dependent Dynamics
• Models depend on windspeed due to structural/aero interactions
• LPV is a natural framework.
2. Model Reduction
• High fidelity CFD/CSD models have many (millions) of states.
3. Model Uncertainty
• Use of simplified low order models OR reduced high fidelity models
AEROSPACE ENGINEERING AND MECHANICS
Outline
• Linear Parameter Varying (LPV) Systems
• Applications
• Flexible Aircraft
• Wind Farms
• Theory for LPV Systems
• Robustness Analysis (Pfifer, Wang, Hu, Lacerda, Venkataraman)
• Model Reduction
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LPV Analysis
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Gridded LPV System
Induced L2 Gain
AEROSPACE ENGINEERING AND MECHANICS
(Standard) Dissipation Inequality Condition
Comments
• Dissipation inequality can be expressed/solved using LMIs.• Finite dimensional LMIs for LFT/Polytopic LPV systems
• Parameterized LMIs for Gridded LPV (requires basis functions, gridding, etc)
• Condition is IFF for LTI systems but only sufficient for LPV
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Theorem (Wu, 1995)
Proof:
AEROSPACE ENGINEERING AND MECHANICS
• Goal: Assess the impact of model uncertainty/nonlinearities
• Approach: Separate nominal dynamics from perturbations• Pert. can be parametric, LTI dynamic, and/or nonlinearities (e.g. saturation).
Uncertainty Modeling
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dxfxaax )()(
)(2
1
21
xfw
xaw
dwwaxx
Nominal LTI, G
Perturbation,
AEROSPACE ENGINEERING AND MECHANICS
• Goal: Extend analysis tools to LPV uncertainty for an
• Approach:
• Use Integral Quadratic Constraints to model input/output behavior (Megretski & Rantzer, TAC 1997).
• Extend dissipation inequality approach for robustness analysis
• Results for Gridded Nominal system
• Parallels earlier results for LFT nominal system by Scherer, Veenman, Köse, Köroğlu.
Robustness Analysis for LPV Systems
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IQC Example: Passive System
Pointwise Quadratic Constraint
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General (Time Domain) IQCs
General IQC Definition:
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Comments:• Megretski & Rantzer (‘97 TAC) has a library of IQCs for various
components.
• IQCs can be equivalently specified in the freq. domain with a multiplier P
• A non-unique factorization connects P=Y*MY.
• Multiple IQCs can be used to specify behavior of .
AEROSPACE ENGINEERING AND MECHANICS
IQC Dissipation Inequality Condition
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Theorem
Proof:
Comment
• Dissipation inequality can be expressed/solved as LMIs.
• Extends standard D/G scaling but requires selection of basis functions for IQC.
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Less Conservative IQC Result
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Theorem
Technical Result
• Positive semidefinite constraint on V and time domain IQC constraint can be dropped.
• These are replaced by a freq. domain requirement on P=Y*MY.
• Some energy is “hidden” in the IQC.
Refs:P. Seiler, Stability Analysis with Dissipation Inequalities and Integral Quadratic Constraints, IEEE TAC, 2015.
H. Pfifer & P. Seiler, Less Conservative Robustness Analysis of Linear Parameter Varying Systems Using Integral Quadratic Constraints, submitted to IJRNC, 2015.
AEROSPACE ENGINEERING AND MECHANICS
Less Conservative IQC Result
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Theorem
Key Idea:
Refs:P. Seiler, Stability Analysis with Dissipation Inequalities and Integral Quadratic Constraints, IEEE TAC, 2015.
H. Pfifer & P. Seiler, Less Conservative Robustness Analysis of Linear Parameter Varying Systems Using Integral Quadratic Constraints, submitted to IJRNC, 2015.
We only need the sum of the boxed terms to be ≥ 0, i.e. each term
individually need not be ≥ 0.
AEROSPACE ENGINEERING AND MECHANICS
Time-Domain Dissipation Inequality Analysis
Summary: Under some technical conditions, the frequency-domain conditions in (M/R, ’97 TAC) are equivalent to the time-domain dissipation inequality conditions.
Applications:
1. LPV robustness analysis (Pfifer, Seiler, IJRNC)
2. General LPV robust synthesis (Wang, Pfifer, Seiler, accepted to Aut)
3. LPV robust filtering/feedforward (Venkataraman, Seiler, in prep)• Robust filtering typically uses a duality argument. Extensions to the time domain?
4. Exponential rates of convergence (Hu,Seiler, accepted to TAC)• Motivated by optimization analysis with ρ-hard IQCs (Lessard, Recht, & Packard)
5. Nonlinear analysis using SOS techniques
6. Discrete-time IQC analysis (Hu, Lacerda, Seiler, submitted to IJRNC)
Item 1 has been implemented in LPVTools. Items 2 & 3 parallel results by (Scherer, Köse, and Veenman) for LFT-type LPV systems.
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AEROSPACE ENGINEERING AND MECHANICS
Outline
• Linear Parameter Varying (LPV) Systems
• Applications
• Flexible Aircraft
• Wind Farms
• Theory for LPV Systems
• Robustness Analysis
• Model Reduction (Annoni, Theis, Singh)
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AEROSPACE ENGINEERING AND MECHANICS
LPV Model Reduction
• Both flexible aircraft and wind farms can be modeled with high fidelity fluid/structural models.
• LPV models can be obtained via Jacobian linearization:
• State dimension can be extremely large (>106)
• LPV analysis and synthesis is restricted to ≈50 states.
• Model reduction is required.
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High Order Model Reduction
Large literature with recent results for LPV and Param. LTI• Antoulas, Amsallem, Carlberg , Gugercin, Farhat, Kutz, Loeve, Mezic, Poussot-
Vassal, Rowley, Schmid, Willcox, …
Two new results for LPV:
1. Input-Output Reduced Order Models (Annoni)• Combine subspace ID with techniques from fluids (POD/DMD).
• No need for adjoint models. Can reconstruct full-order state.
2. Parameter-Varying Oblique Projection (Theis)• Petrov-Galerkin approximation with constant projection space and
parameter-varying test space.
• Constant projection maintains state consistency avoids rate dependence.
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References
1A. Annoni & Seiler, A method to construct reduced-order parameter varying models, submitted to IJRNC, 2015.
1B. Annoni, Nichols, & Seiler, “Wind farm modeling and control using dynamic mode decomposition.” AIAA, 2016.
2. Theis, Seiler, & Werner, Model Order Reduction by Parameter-Varying Oblique Projection, submitted to 2016 ACC.
AEROSPACE ENGINEERING AND MECHANICS
High Order Model Reduction
Large literature with recent results for LPV and Param. LTI• Antoulas, Amsallem, Carlberg , Gugercin, Farhat, Kutz, Loeve, Mezic, Poussot-
Vassal, Rowley, Schmid, Willcox, …
Two new results for LPV:
1. Input-Output Reduced Order Models (Annoni)• Combine subspace ID with techniques from fluids (POD/DMD).
• No need for adjoint models. Can reconstruct full-order state.
2. Parameter-Varying Oblique Projection (Theis)• Petrov-Galerkin approximation with constant projection space and
parameter-varying test space.
• Constant projection maintains state consistency avoids rate dependence.
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References
1A. Annoni & Seiler, A method to construct reduced-order parameter varying models, submitted to IJRNC, 2015.
1B. Annoni, Nichols, & Seiler, “Wind farm modeling and control using dynamic mode decomposition.” AIAA, 2016.
2. Theis, Seiler, & Werner, Model Order Reduction by Parameter-Varying Oblique Projection, submitted to 2016 ACC.
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Application: Large Eddy Simulation (LES)
• Simulator fOr Wind Farm Applications (SOWFA)
• 3D unsteady spatially filtered Navier-Stokes equations
• Simulation time (wall clock): 48 hours
Churchfield, Lee
https://nwtc.nrel.gov/SOWFA
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• Two turbine setup (NREL 5 MW turbines)
• Turbine Diameter D=126m
• Approximately 1.2 million grid points
• 3 velocity components → 3.6 million states
Wind Turbine Array Setup
5D
m/s
Streamwise distance (x/D)
Cro
ssw
ind
dis
tan
ce (
y/D
)
AEROSPACE ENGINEERING AND MECHANICS43
• Two turbine setup (NREL 5 MW turbines)
• Control inputs: Blade pitch angle
• Control outputs: Power at each turbine
• Exogenous Disturbance: Mean wind speed
Wind Turbine Array Setup
Pitch(1) Power(1) Power(2)Pitch(2)
Wind Speed
AEROSPACE ENGINEERING AND MECHANICS
Discrete-Time Direct Subspace ID (Viberg, 95)
• Gather snapshots of inputs, outputs, and state
• Fit a linear state-space model to the data
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Input, u
Output, y
State, x
],...,,[
],...,,[
321
1210
m
m
xxxX
xxxX
],...,,[
],...,,[
1210
1210
m
m
yyyY
uuuU
000
001
DUCXY
BUAXX
0
0
0
1
U
X
Y
X
DC
BA
Computationally Intractable for large systems
AEROSPACE ENGINEERING AND MECHANICS
Reduced Order Model
• Compute SVD of state snapshot data:
• Project state data onto the POD modes:
• Fit a linear state-space model to the reduced data:
• Comments:
• SVD can be done on laptop in a few hours with Tall QR methods.
• This is a variation of DMDc by Proctor, et al, 2014.
• We can approximate the full state as
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TVUX 0
POD modesX0
State
Dim.
# of
Samples0
*
0 XUZ 1
*
1 XUZ
000
001
DUCZY
BUAZZ
0
0
0
1
~
~~
U
Z
Y
Z
DC
BA
kk Uzx
AEROSPACE ENGINEERING AND MECHANICS
Summary
• SVD can be done on laptop in a few hours with Tall QR methods.
• This combines techniques from system ID and fluids (POD/DMD)
• The approach is a variation of DMDc by Proctor, et al, 2014.
• The method does not require adjoints or solution of Lyapunov Eqns.
• We can approximate the full state from the reduced state:
• The state consistency can be used to extend the approach to LPV model reduction.
• Annoni, Seiler, submitted to IJRNC, ‘16
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],...,,[
],...,,[
321
1210
m
m
xxxX
xxxX
],...,,[
],...,,[
1210
1210
m
m
yyyY
uuuU
TVUX 0
0
*
0 XUZ
1
*
1 XUZ
0
0
0
1
~
~~
U
Z
Y
Z
DC
BA
Excite system &
collect data
Compute spatial
modes (POD)
Project states onto
(low-order) modes
Estimate low-order
state matrices via
least-squares
kkk
kkk
DuzCy
uBzAz
~
~~1 Reduced-order
model
kk Uzx
AEROSPACE ENGINEERING AND MECHANICS
• Two turbine setup (NREL 5 MW turbines)
• D = turbine diameter (126 m)
• Neutral boundary layer
• 7 m/s with 10% turbulence
Wind Turbine Array Setup
5D
m/s
Streamwise distance (x/D)
Cro
ssw
ind
dis
tan
ce (
y/D
)
47
AEROSPACE ENGINEERING AND MECHANICS
• Two turbine setup (NREL 5 MW turbines)
IOROM with SOWFA
5D
m/s
Cro
ssw
ind
dis
tan
ce (
y/D
)
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Blade pitch angle changes from 0⁰ to 4⁰
Streamwise distance (x/D)
AEROSPACE ENGINEERING AND MECHANICS
Reconstructed Flow
• Model constructed using 20 modes
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• Validation case – same setup with a different input
Blade pitch angle changes from 0⁰ to 4⁰
Model applied to Validation Data
m/s
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Video of reconstructed flow
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Acknowledgements
• US National Science Foundation • Grant No. NSF-CMMI-1254129: “CAREER: Probabilistic Tools for High
Reliability Monitoring and Control of Wind Farms.” Prog. Manager: J. Berg.
• Grant No. NSF/CNS-1329390: “CPS: Breakthrough: Collaborative Research: Managing Uncertainty in the Design of Safety-Critical Aviation Systems”. Prog. Manager: D. Corman.
• NASA• NRA NNX14AL36A: "Lightweight Adaptive Aeroelastic Wing for Enhanced
Performance Across the Flight Envelope," Tech. Monitor: J. Bosworth.
• NRA NNX12AM55A: “Analytical Validation Tools for Safety Critical Systems Under Loss-of-Control Conditions.” Tech. Monitor: C. Belcastro.
• SBIR contract #NNX12CA14C: “Adaptive Linear Parameter-Varying Control for Aeroservoelastic Suppression.” Tech. Monitor. M. Brenner.
• Eolos Consortium and Saint Anthony Falls Laboratory• http://www.eolos.umn.edu/ & http://www.safl.umn.edu/
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AEROSPACE ENGINEERING AND MECHANICS
Conclusions
Main Contributions in LPV Theory:
• Robustness analysis tools
• Model reduction methods
Applications to:
• Flexible and unmanned aircraft
• Wind energy
• Hard disk drives
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http://www.aem.umn.edu/~SeilerControl/
AEROSPACE ENGINEERING AND MECHANICS54
AEROSPACE ENGINEERING AND MECHANICS
• Scale 1:750• 4.5 m/s• 10% turbulence intensity
Model Turbines
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0.128 m 96 m
Photo credits: Kevin Howard
AEROSPACE ENGINEERING AND MECHANICS
SAFL Wind Tunnel
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Photo credits: Kevin Howard
AEROSPACE ENGINEERING AND MECHANICS
• Understand the input/output dynamics
• Square waves with varying frequencies: 0.02Hz to 10HzOutput voltage Power
Voltage Measurements
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Input voltage generator torque
Rated
Derated
AEROSPACE ENGINEERING AND MECHANICS
Typical Result
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Turb
ine 1
Turb
ine 2
AEROSPACE ENGINEERING AND MECHANICS
Dynamic Response
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AEROSPACE ENGINEERING AND MECHANICS
Dynamic Park Model
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Upstream
Turbine
Dynamics
Wind
Farm
Model
Time
Delay
Downstream
Turbine
Dynamics
AEROSPACE ENGINEERING AND MECHANICS
Dynamic Park Model
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