Systems Research in the Aerospace Engineering and Mechanics at the University of Minnesota

Gary J. Balas Aerospace Engineering and Mechanics University of Minnesota Minneapolis, MN [email protected]

SAE Aerospace Controls and Guidance Meeting 19 October 2005 1

University of Minnesota Aerospace Engineering and Mechanics Systems Faculty

• William Garrard, Department Head – Modeling, flight control, parachutes • Yiyuan Zhao – Optimization, air traffic control, rotorcraft • Demoz Gebre-Egziabher – Navigation, GPS, sensor fusion • Gary Balas – Robust control, real-time embedded systems, flight control • Bernard Mettler (starts Jan 2006) – Real-time control, planning, rc helicopters and planes 2

Current Research

• “Control Reconfiguration and Fault Detection and Isolation Using Linear, Parameter Varying Techniques,” NASA Langley Research Center, NASA Aviation Safety Program, Dr. Christine Belcastro Technical Monitor • “Stability and Control of Supercavitating Vehicles,” ONR, Dr. Kam Ng Program Manager – Special Session planned for the 2006 American Control Conference entitled “Modeling and Control of High-Speed Underwater Vehicles” • Local Arrangements Chair, 2006 American Control Conference, 14-16 June 2006, Minneapolis, MN 3

Control of Projectiles

Using control thruster firings, the projectile maneuvers to the optimum angle of attack • Tradeoff between many small maneuvers and wider spaced, large maneuvers • Controllability of projectile given a finite number of impulses • Optimal control of a number of thrusters.

• Effect of – Burn time – Impulse size and number – Achievable performance 4

Development of Analysis Tools for Certification of Flight Control Laws - AFOSR Andy Packard (UC Berkeley), Pete Seiler (Honeywell) Initial focus is on nonlinear robustness analysis – Region-of-attraction – Disturbance-to-error gains – Inner and Outer Bounds Connection to MilSpecs 5

Quantitative Nonlinear Analysis Initial focus – Region of attraction estimation – – L L 2 2   L L 2  induced norms induced norms for – finite-dimensional nonlinear systems, with • polynomial vector fields • parameter uncertainty (also polynomial) Main Tools: – Lyapunov/HJI formulation – Sum-of-squares proofs to ensure nonnegativity – Semidefinite programming (SDP), Bilinear Matrix Inequalities • Optimization interface: YALMIP and SOSTOOLS • SDP solvers: Sedumi • BMIs: using PENBMI (academic license from www.penopt.com

) 6

Estimating Region of Attraction Dynamics, equilibrium point

x

 

f

(

x

),

f

(

x

)  0 User-defined function whose sub-level sets are to be in region-of-attraction 

x

:

p

(

x

)     ROA x

V

 1

p

 3

p

 2

p

 1

x

By choice of positive-definite

V

, maximize  so that 

x

:

p

(

x

)    

x:V(x)

 1 

compact



x

:

x



x

,

V

(

x

)  1  

x: dV dx f

 0

dV dx f

 0 7