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Development of Analysis Tools for Certification of Flight Control Laws UC Berkeley, Andrew Packard, Honeywell, Pete Seiler, U Minnesota, Gary Balas. Region-of-attraction Disturbance-to-error gain Verify set containments in state-space with SOS proof certificates. Long-Term PAYOFF

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slide1

Development of Analysis Tools for Certification of

Flight Control Laws

UC Berkeley, Andrew Packard,

Honeywell, Pete Seiler, U Minnesota, Gary Balas

.

Region-of-attraction

Disturbance-to-error gain

Verify set containments in state-space with SOS proof certificates.

  • Long-Term PAYOFF
  • Direct model-based analysis of nonlinear systems
  • OBJECTIVES
  • Develop robustness analysis tools applicable to certification of flight control laws: quantitative analysis of locally stable, uncertain systems
  • Complement simulation with Lyapunov-based proof techniques, actively using simulation
  • Connect Lyapunov-type questions to MilSpec-type measures of robustness and performance

Convex outer bound

Aid nonconvex proof search (Lyapunov fcn coeffs) with constraints from simulation

FUNDING ($K)

TRANSITIONS

“Stability region analysis using simulations and SOS programming,” 2007 ACC

“Stability region analysis via composite Lyap. functions & SOS programming,” IEEE TAC, 10/07.

“Local stability analysis using simulations & SOS programming,” sub. Automatica (12/06, 4/07)

“Stability region analysis for uncertain nonlinear systems,” to appear 2007 CDC

“Local stability analysis for uncertain nonlinear systems,” submitted IEEE TAC (6/07)

STUDENTS, POST-DOCS

Ufuk Topcu, Tim Wheeler, Weehong Tan

LABORATORY POINT OF CONTACT

Dr. Siva Banda, Dr. David Doman

  • APPROACH/TECHNICAL CHALLENGES
  • Analysis based on Lyapunov/storage fcn method
  • Non-convex sum-of-squares (SOS) optimization
  • Merge info from conventional simulation-based assessment methods to aid in the nonconvex opt
  • Unfavorable growth in computation: state order, vector field degree and # of uncertainties.
  • Reliance on SDP and BMI solvers, which remain under development, unstable and unreliable
  • ACCOMPLISHMENTS/RESULTS
  • Pointwise-max storage functions
  • Tangible benefits of employing simulations
  • Pragmatic approach to parameter uncertainty
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ROA for Uncertain Systems

  • Parameter and vector field uncertainty models
    • find one Lyapunov fcn certifying the ROA at vertices of parameter-space
    • employ simulations, as before
    • Branch & Bound for nonconvex (or “large” parameter space, nonaffine dependence, etc)
  • Properties (superior to ParameterDependent Lyapunov approach)
    • Parameter-space issues trivially parallelize
    • Avoids limitations of parameter-dependent basis choices
    • Better answers than previous & current literature

A feasible path towards attacking problems with, eg., 10 states, 5 uncertainties, and cubic (in state) vector fields

Dynamics have strong dependence on δ

BTree(k).Analysis

Analysis.ParameterDomain

Analysis.VertexDynamics

Analysis.LyapunovCertificate

Analysis.SOSCertificates

Analysis.CertifiedVolume

BTree(k).Children

Certified Robust-Region-of-Attractions

35 partitions of [0 1], quadratic and quartic V, quadratic and quartic SOS multipliers for each partition, <5 minutes on 8-machine PC cluster.

Analysis gives binary tree that partitions parameter-space, with certificates at each subdomain

Best literature estimate from Chesi, et.,al.

Application to flight control, with gain and time delay (Pade) variations

A. Packard/ UC Berkeley, P. Seiler/Honeywell, G. Balas / University of Minnesota