Robust Analysis and Synthesis for Linear Parameter Varying Systems Peter Seiler University of Minnesota A EROSPACE E NGINEERING AND M ECHANICS
Research Areas Jen Annoni Bin Hu Parul Singh Inchara Lakshminarayan Shu Wang Raghu Venkataraman Masanori Honda Wind Energy Safety Critical Systems Hard Disk Drives Harald Pfifer Daniel Ossmann Marcio Lacerda 2 A EROSPACE E NGINEERING AND M ECHANICS
Research Areas: Aeroservoelasticity Abhineet Gupta Sally Ann Keyes Aditya Kotikalpudi Adrià Serra Moral Gary Balas Brian Taylor (UAV Lab Director) Harald Pfifer (9/27/60 – 11/12/14) Chris Regan Julian Theis 3 A EROSPACE E NGINEERING AND M ECHANICS
Outline • Linear Parameter Varying (LPV) Systems • Applications • Flexible Aircraft • Wind Farms • Theory for LPV Systems • Robustness Analysis • Model Reduction 4 A EROSPACE E NGINEERING AND M ECHANICS
Outline • Linear Parameter Varying (LPV) Systems • Applications • Flexible Aircraft • Wind Farms • Theory for LPV Systems • Robustness Analysis • Model Reduction 5 A EROSPACE E NGINEERING AND M ECHANICS
Modeling for Aircraft Control u(t) y(t) Altitude Rockwell B-1 Lancer (Photo: US Air Force) 35000 ft Nonlinear ODE 30000 ft 25000 ft 20000 ft 15000 ft 10000 ft 5000 ft Mach 0.5 0.6 0.7 0.8 Flight Envelope 6 A EROSPACE E NGINEERING AND M ECHANICS
Modeling for Aircraft Control u(t) y(t) Altitude Rockwell B-1 Lancer (Photo: US Air Force) 35000 ft Nonlinear ODE 30000 ft 25000 ft 20000 ft Flight Condition 15000 ft r =(0.7,10000ft) 10000 ft Equilibrium Condition 5000 ft Mach 0.5 0.6 0.7 0.8 Flight Envelope 7 A EROSPACE E NGINEERING AND M ECHANICS
Modeling for Aircraft Control u(t) y(t) Altitude Rockwell B-1 Lancer (Photo: US Air Force) 35000 ft 30000 ft Linear Time Invariant (LTI) 25000 ft Linearize near 20000 ft one equilibrium 15000 ft r =(0.7,10000ft) 10000 ft where 5000 ft Mach 0.5 0.6 0.7 0.8 Use for linear control design Flight Envelope 8 A EROSPACE E NGINEERING AND M ECHANICS
Modeling for Aircraft Control u(t) y(t) Altitude Rockwell B-1 Lancer (Photo: US Air Force) 35000 ft 30000 ft 25000 ft Linearize near Parameterized LTI set of (fixed) 20000 ft equilibria 15000 ft 10000 ft where 5000 ft Mach 0.5 0.6 0.7 0.8 Flight Envelope 9 A EROSPACE E NGINEERING AND M ECHANICS
Modeling for Aircraft Control u(t) y(t) Altitude Rockwell B-1 Lancer (Photo: US Air Force) 35000 ft 30000 ft 25000 ft Linearize near Parameterized LTI set of (fixed) 20000 ft equilibria 15000 ft 10000 ft 5000 ft Gain-Scheduling Mach Design controllers at many flight 0.5 0.6 0.7 0.8 conditions and “stitch” together. Flight Envelope 10 A EROSPACE E NGINEERING AND M ECHANICS
Modeling for Aircraft Control u(t) y(t) Altitude Rockwell B-1 Lancer (Photo: US Air Force) 35000 ft Linearize around set 30000 ft of varying equilibria 25000 ft Linear Parameter Varying (LPV) 20000 ft 15000 ft 10000 ft where 5000 ft Mach 0.5 0.6 0.7 0.8 Flight Envelope 11 A EROSPACE E NGINEERING AND M ECHANICS
Outline • Linear Parameter Varying (LPV) Systems • Applications • Flexible Aircraft • Wind Farms • Theory for LPV Systems • Robustness Analysis • Model Reduction 12 A EROSPACE E NGINEERING AND M ECHANICS
Aeroservoelasticity (ASE) Efficient aircraft design • Lightweight structures • High aspect ratios 13 A EROSPACE E NGINEERING AND M ECHANICS
Flutter Source: NASA Dryden Flight Research 14 A EROSPACE E NGINEERING AND M ECHANICS
Classical Approach Controller Bandwidth Frequency Separation Aeroelastic Rigid Body Modes Modes 0 Frequency Flight Dynamics, Flutter Analysis Classical Flight Control 15 A EROSPACE E NGINEERING AND M ECHANICS
Flexible Aircraft Challenges Increasing wing flexibility Aeroelastic Rigid Body Modes Modes 0 Frequency 16 A EROSPACE E NGINEERING AND M ECHANICS
Flexible Aircraft Challenges Integrated Control Design Rigid Body Aeroelastic Modes Modes 0 Frequency Coupled Rigid Body and Aeroelastic Modes 17 A EROSPACE E NGINEERING AND M ECHANICS
Body Freedom Flutter 18 A EROSPACE E NGINEERING AND M ECHANICS
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 LM BFF Schmidt & Associates LM/NASA X-56 UMN Mini-Mutt 19 A EROSPACE E NGINEERING AND M ECHANICS
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 20 A EROSPACE E NGINEERING AND M ECHANICS
Modeling and Control for Wind Farms 1. Parameter Dependent Dynamics • Models depend on windspeed due to structural/aero interactions Eolos: http://www.eolos.umn.edu/ • LPV is a natural framework. 2. Model Reduction • High fidelity CFD/CSD models have many (millions) of states. Saint Anthony Falls: http://www.safl.umn.edu/ 3. Model Uncertainty • Use of simplified low order models OR reduced high fidelity models Simulator for Wind Farm Applications, Churchfield & Lee http://wind.nrel.gov/designcodes/simulators/SOWFA 21 A EROSPACE E NGINEERING AND M ECHANICS
Outline • Linear Parameter Varying (LPV) Systems • Applications • Flexible Aircraft • Wind Farms • Theory for LPV Systems • Robustness Analysis • Model Reduction 22 A EROSPACE E NGINEERING AND M ECHANICS
LPV Analysis Gridded LPV System Induced L 2 Gain 23 A EROSPACE E NGINEERING AND M ECHANICS
(Standard) Dissipation Inequality Condition Theorem Proof: 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 24 A EROSPACE E NGINEERING AND M ECHANICS
Uncertainty Modeling • 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). x ( a a ) x f ( x ) d Nominal LTI, G x ax w w d 1 2 w a x 1 w f ( x ) 2 Perturbation, 25 A EROSPACE E NGINEERING AND M ECHANICS
Robustness Analysis for LPV Systems • 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. 26 A EROSPACE E NGINEERING AND M ECHANICS
IQC Example: Passive System Pointwise Quadratic Constraint 27 A EROSPACE E NGINEERING AND M ECHANICS
General (Time Domain) IQCs General IQC Definition: 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 * M Y . • Multiple IQCs can be used to specify behavior of . 28 A EROSPACE E NGINEERING AND M ECHANICS
IQC Dissipation Inequality Condition Theorem Proof: Comment • Dissipation inequality can be expressed/solved as LMIs. • Extends standard D/G scaling but requires selection of basis functions for IQC. 29 A EROSPACE E NGINEERING AND M ECHANICS
Less Conservative IQC Result 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 * M Y . • 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. 30 A EROSPACE E NGINEERING AND M ECHANICS
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. 31 A EROSPACE E NGINEERING AND M ECHANICS
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