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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.

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Funding k transitions

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.

Aid nonconvex proof search (Lyap fcn coefficients) with constraints from simulation

  • 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

Δ

  • Uncertain Dynamics

    • parametric uncertainty

    • dynamic uncertainty

δ2

M

δ1

FUNDING ($K)

TRANSITIONS

Stability Region Analysis using polynomial and composite polynomial Lyapunov functions and SOS Programming, IEEE-TAC, Oct., 2007.

Local stability analysis using simulations and sum-of-squares programming, to appear, Automatica.

Stability region analysis for uncertain nonlinear systems, to appear, IEEE-TAC

Robust region-of-attraction estimation, submitted IEEE-TAC, July 2008

STUDENTS, POST-DOCS

Ufuk Topcu (PhD July 08), Weehong Tan (PhD Jan 06), Wheeler

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

  • 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

  • Tangible benefits of employing simulations

  • Pragmatic approach to parameter uncertainty

  • Local small-gain theorems

  • Trivial parallelization of some computations


Funding k transitions

Provable ROA with unmodeled dynamics

  • Approach #1

    • Parametrize fixed-structure system representing unmodeled dynamics

    • Compute certified invariant subset of region-of-attraction valid for all parameter values

  • Approach #2

    • Introduce perturbation inputs/outputs as in standard linear robustness theory (μ-analysis)

    • Compute certified L2→L2 gain (|| ||2/2) across channel, valid over a ball of inputs.

    • Apply local version of small-gain theorem

plant

controller

plant

controller

xp

xp

xc

xc

plant

controller

  • Approach #2, results

    • For all nonlinear operators Δ, mapping w= Δz, with || Δ||2/2≤0.6, and all initial conditions satisfying

    • Restriction: Δ must start from “rest”,but conclusion holds for high dynamic complexity (not just 1-state system in Approach #1), linear and nonlinear Δ

  • Approach #1, results

    • Family (over uncertain parameter space) of quartic Lyapunov functions certifies stability of equilibrium point (x=0) for all allowable β, γ and all initial conditions satisfying

    • Restriction: Conclusion is only known to hold for the special, 1-state uncertainty block as drawn.

initial conditions from certified regions, Δ from its allowable class

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