Next-Generation Lagrangian Reachability Sophie Grnbacher, Jacek - PowerPoint PPT Presentation

Next-Generation Lagrangian Reachability Sophie Grünbacher, Jacek Cyranka, Md. Ariful Islam, Scott A. Smolka and Radu Grosu IFIP WG 2.2 meeting in Vienna 25 September 2019 funded by FWF project W1255-N23

Safety of Cyber-Physical Systems Risk of Blackout Risk of Crash Sophie Grünbacher Lagrangian Reachability

Safety of Cyber-Physical Systems Neural Network Control Systems States +x, -x, + ! , - ! Controls + ̇ # , - ̇ # Sophie Grünbacher Lagrangian Reachability

̇ Lagrangian Reachtube (LRT) Nonlinear dynamical System Unsafe Region Bound for y Reachset Initial Set Unsafe Region x t " t " +T $ = & $ Sophie Grünbacher Lagrangian Reachability

Problem: Wrapping Effect ̇ " = $ % & ' " y avoid: Interval Arithmetic x N. Nedialkov, K. Jackson, and G. Corliss. Validated solutions of initial value problems for ordinary differential equations. Applied Mathematics and Computation, 105(1):21 – 68, 1999. Sophie Grünbacher Lagrangian Reachability

Wanted: tight conservative ellipse Minimize volume of enclosing ellipse ! " # (% # , ' ( ) * ) y y 4 5 2 (/ - , 6 - ) 1 3 (/ - ) 0 1 2 / - x x t . t - Sophie Grünbacher Lagrangian Reachability

Main idea: Use sensitivity analysis To every nonlinear, (time-dependent) ODEs x = f( x), x 0 = x(0) ! There is an associated variational equation: ! F = − J( x 0 ,t )F, F( x 0 , 0 ) = I χ ( x 0 + dx,t ) F = d χ ( x 0 ,t ) dx χ ( x 0 + dx, 0 ) χ ( x 0 , 0 ) dx χ ( x 0 ,t ) Sophie Grünbacher Lagrangian Reachability

F = d χ ( x 0 ,t ) Stretching Factor dx $ (& ' ) , # $ % $ (& ) ) ‖ ‖(# $ % ) * + ≤ ‖(-(.) ‖ ) * %,+ 0 ‖(& ' , & ) ) ‖ ) * % 2 4 1 2 3 $ ; (& ) ) # $ % & y y ) < * % (& ' , = ' ) $ ; (& ' ) # $ % 5 6 7 (8 7 , 1 0 2 3 ) & ' x x t : t ' Sophie Grünbacher Lagrangian Reachability

Lohner’s QR method QR decomposition Change to coordinate system Q N. Nedialkov, K. Jackson, and G. Corliss. Validated solutions of initial value problems for ordinary differential equations. Applied Mathematics Sophie Grünbacher Lagrangian Reachability and Computation, 105(1):21 – 68, 1999.

Work in progress Wrapping of Reachset Choosing tightest ellipse y y x x Sophie Grünbacher Lagrangian Reachability

Choosing optimal metric ! "#$ y x y x Sophie Grünbacher Lagrangian Reachability

Choose tightest ellipse Sophie Grünbacher Lagrangian Reachability

Wrapping of Reachset - Trick #1 QR decomposition Change to coordinate system Q Sophie Grünbacher Lagrangian Reachability

Tighter enclosing box (used in QR) Sophie Grünbacher Lagrangian Reachability

Wrapping of Reachset - Trick #2 Sophie Grünbacher Lagrangian Reachability

Effects of intersection trick #2 Sophie Grünbacher Lagrangian Reachability

Comparison with other tools Sophie Grünbacher Lagrangian Reachability

Conclusion & Future Work Scale up LRT Verify Neural Network Control Systems States +x, -x, + ! , - ! Controls + ̇ # , - ̇ # Sophie Grünbacher Lagrangian Reachability

Thank you for your attention! Next-Generation Lagrangian Reachability " ̇ = $ " Sophie Grünbacher sophie.gruenbacher@tuwien.ac.at Lagrangian Reachability