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.
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

2. 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