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AEROSPACE ENGINEERING AND MECHANICS

Robust Analysis and Synthesis for Linear Parameter Varying Systems

Peter Seiler University of Minnesota

AEROSPACE ENGINEERING AND MECHANICS

Research Areas

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Jen Annoni Parul Singh Shu Wang Wind Energy Bin Hu Inchara Lakshminarayan Raghu Venkataraman Safety Critical Systems Masanori Honda Hard Disk Drives Harald Pfifer Marcio Lacerda Daniel Ossmann

AEROSPACE ENGINEERING AND MECHANICS

Research Areas: 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

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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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 ft

Rockwell 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 ft

Rockwell 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 ft

Rockwell B-1 Lancer (Photo: US Air Force)

u(t) y(t)

Linearize near

• ne equilibrium

Linear Time Invariant (LTI) where

r=(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 ft

Rockwell B-1 Lancer (Photo: US Air Force)

u(t) y(t)

Parameterized LTI

Linearize 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 ft

Rockwell B-1 Lancer (Photo: US Air Force)

u(t) y(t)

Parameterized LTI

Linearize 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 ft

Rockwell B-1 Lancer (Photo: US Air Force)

u(t) y(t)

Linear Parameter Varying (LPV) where

Linearize around set

• f 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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AEROSPACE ENGINEERING AND MECHANICS

Aeroservoelasticity (ASE)

Efficient aircraft design

• Lightweight structures
• High aspect ratios

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AEROSPACE ENGINEERING AND MECHANICS

Flutter

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Source: NASA Dryden Flight Research

AEROSPACE ENGINEERING AND MECHANICS

Classical Approach

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Frequency Aeroelastic Modes Rigid Body Modes Frequency Separation Controller Bandwidth Flutter Analysis Flight Dynamics, Classical Flight Control

AEROSPACE ENGINEERING AND MECHANICS

Flexible Aircraft Challenges

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Frequency Aeroelastic Modes Rigid Body Modes Increasing wing flexibility

AEROSPACE ENGINEERING AND MECHANICS

Flexible Aircraft Challenges

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Frequency Rigid Body Modes Integrated Control Design Coupled Rigid Body and Aeroelastic Modes Aeroelastic Modes

AEROSPACE ENGINEERING AND MECHANICS

Body Freedom Flutter

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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-56 UMN Mini-Mutt LM BFF

AEROSPACE ENGINEERING AND MECHANICS

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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AEROSPACE ENGINEERING AND MECHANICS

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
• Model Reduction

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AEROSPACE ENGINEERING AND MECHANICS

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 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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d x f x a a x      ) ( ) ( 

) (

2 1 2 1

x f w x a w d w w ax x         

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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AEROSPACE ENGINEERING AND MECHANICS

IQC Example: Passive System

Pointwise Quadratic Constraint

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AEROSPACE ENGINEERING AND MECHANICS

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.

AEROSPACE ENGINEERING AND MECHANICS

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

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, submitted 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, submitted to TAC)

• Motivated by optimization analysis with ρ-hard IQCs (Lessard, Recht, & Packard)

5. Nonlinear analysis using SOS techniques

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

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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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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 Dynamic Mode Decomposition
• Combine subspace ID with techniques from fluids (POD/DMD).
• No need for adjoint models. Can reconstruct full-order state.
• 2. Parameter-Varying Oblique Projection
• 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.
• 1C. Singh & Seiler, Model Reduction using Frequency Domain Input-Output Dynamic Mode Decomposition, sub. to ‘16 ACC.
• 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 Dynamic Mode Decomposition
• Combine subspace ID with techniques from fluids (POD/DMD).
• No need for adjoint models. Can reconstruct full-order state.
• 2. Parameter-Varying Oblique Projection
• 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.
• 1C. Singh & Seiler, Model Reduction using Frequency Domain Input-Output Dynamic Mode Decomposition, sub. to ‘16 ACC.
• 2. Theis, Seiler, & Werner, Model Order Reduction by Parameter-Varying Oblique Projection, submitted to 2016 ACC.

AEROSPACE ENGINEERING AND MECHANICS

Higher-Fidelity – Large Eddy Simulation (LES)

• Simulator for On/Offshore Wind Farm Applications
• 3D unsteady spatially filtered Navier-Stokes equations
• Simulation time (wall clock): 48 hours

Churchfield, Lee https://nwtc.nrel.gov/SOWFA

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AEROSPACE ENGINEERING AND MECHANICS

Linearized discrete-time Navier-Stokes

Problem Setup

k k k k k k

Du Cx y Bu Ax x    

1

Updated velocity Linearized Navier-Stokes Input operator Inputs : β, τg : indicates location of turbines Outputs Wind speed at nacelle Power Loads Output operator Influence of input

m/s

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AEROSPACE ENGINEERING AND MECHANICS

Linearized discrete-time Navier-Stokes

Problem Setup

k k k k k k

Du Cx y Bu Ax x    

1

Millions of states

m/s

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AEROSPACE ENGINEERING AND MECHANICS

Linearized discrete-time Navier-Stokes

Problem Setup

k k k k k k

Du Cx y Bu Ax x    

1

Millions of states

k k k k k k

u D x C y u B x A x ~ ~ ~ ~ ~ ~ ~

1

   



10s to 100s of states

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AEROSPACE ENGINEERING AND MECHANICS

Typical Approaches in Fluids

• Project onto the dominant modes of the system
• Proper orthogonal decomposition (POD)
• Lumley, et. al. 1967
• Dynamic mode decomposition (DMD)
• Schmid, Mezic, Rowley, Kutz, others

Churchfield et. al. “NWTC design codes- SOWFA”

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AEROSPACE ENGINEERING AND MECHANICS

Dynamic Mode Decomposition

• Gather snapshots from simulation or experiments
• Fit a linear operator to the snapshots

Churchfield et. al. “NWTC design codes- SOWFA” 



1X

X A

Intractable for large systems

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] ,..., , [ ] ,..., , [

1 3 2 1 2 1 

 

m m

x x x X x x x X

Fit linear operator to snapshots Gather snapshots

AEROSPACE ENGINEERING AND MECHANICS

Dynamic Mode Decomposition

• Gather snapshots from simulation or experiments
• Fit a linear operator to the snapshots

Churchfield et. al. “NWTC design codes- SOWFA”

] ,..., , [ ] ,..., , [

1 3 2 1 2 1 

 

m m

x x x X x x x X

42

T

V U X  

Compute POD Modes

1 * *

X U X U

r r

Project onto POD Modes



 ) ( ~

* 1 *

X U X U A

r r

Reduced

• rder model

Gather snapshots

AEROSPACE ENGINEERING AND MECHANICS

• Subspace identification
• Fit low-order, “black-box” ODE to input/output data
• Katayama, Larimore, Ljung, van Overschee, de Moor, Viberg,

Verhaegen, others

Typical Approaches in Controls

• Flow domain
• Interactions
• Forcing
• Etc.

input

• utput

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AEROSPACE ENGINEERING AND MECHANICS

Direct Subspace Identification (Viberg, ‘95)

• Gather snapshots from simulation or experiments
• Measurements of inputs and outputs
• Fit a linear operator to the snapshots

] ,..., , [ ] ,..., , [

3 2 1 1 2 1 m m

x x x X x x x X  

 

                  

1

U X Y X D C B A

] ,..., , [ ] ,..., , [

1 2 1 1 2 1  

 

m m

y y y Y u u u U

1

DU CX Y BU AX X    

Intractable for large systems

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AEROSPACE ENGINEERING AND MECHANICS

IODMD

• Project state data onto a subspace
• Obtain a discrete reduced-order model of the system
• Blends direct subspace ID with POD/DMD
• Handles inputs/outputs
• Full state can be reconstructed from reduced state
• Input forcing increases the signal to noise ratio
• Parameter-varying version that maintains state consistency



                  

* 1 *

~ ~ ~ ~ U X U Y X U D C B A

r r

                  

 k k k k

u x D C B A y x ~ ~ ~ ~ ~ ~

1

T

V U X  

POD Modes

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AEROSPACE ENGINEERING AND MECHANICS

• Two turbine setup (NREL 5 MW turbines)
• D = turbine diameter (126 m)
• Neutral boundary layer
• 7 m/s with 6% turbulence

Wind Turbine Array Setup

5D

m/s

Streamwise distance (x/D) Crosswind distance (y/D)

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AEROSPACE ENGINEERING AND MECHANICS

Wind Turbine Array Setup

• Two turbine setup (NREL 5 MW turbines)
• Control inputs: Blade pitch angle, generator torque
• Control outputs: Power at each turbine

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5D

m/s

Streamwise distance (x/D) Crosswind distance (y/D)

AEROSPACE ENGINEERING AND MECHANICS

Wind Turbine Array Setup

• Two turbine setup (NREL 5 MW turbines)
• Approximately 1.2 million grid points
• 3 velocity components → 3.6 million states
• Intractable for control design

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5D

m/s

Streamwise distance (x/D) Crosswind distance (y/D)

AEROSPACE ENGINEERING AND MECHANICS

• Forcing Input to first turbine

IODMD with SOWFA

Blade pitch angle changes from 0⁰ to 4⁰

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5D

m/s

Streamwise distance (x/D) Crosswind distance (y/D)

AEROSPACE ENGINEERING AND MECHANICS

Flow Simulation

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AEROSPACE ENGINEERING AND MECHANICS

Reduced-order model

• Choose 20 modes to construct a reduced-order model
• 3.6 million states projected onto 20 modes
• Tall QR computations can be done on a laptop (hours)
• Retain input-output behavior

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AEROSPACE ENGINEERING AND MECHANICS

• Validation case – same setup with a different input

Model applied to Validation Data

Blade pitch angle changes from 0⁰ to 4⁰

m/s

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AEROSPACE ENGINEERING AND MECHANICS

Model applied to Validation Data

• Input-output behavior is retained on validation data

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AEROSPACE ENGINEERING AND MECHANICS

Flow Simulation

• Reconstruct the full-state using a reduced-order model

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AEROSPACE ENGINEERING AND MECHANICS

Compare Individual Snapshots

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AEROSPACE ENGINEERING AND MECHANICS

56

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/