FUNDING ($K) TRANSITIONS

1. 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

2. 1.5 1 0.5 x 2 0 -0.5 -1 -1.5 -1 -0.5 0 0.5 1 1.5 x 1 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