. 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 FY04 FY05 FY06 FY07 FY08 AFOSR Funds 97 141 142 47 Other

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 xδ controller plant controller plant xc xp xc xp controller plant 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. 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 initial conditions from certified regions, Δ from its allowable class A. Packard/ UC Berkeley, P. Seiler/Honeywell, G. Balas / University of Minnesota