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

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Next-Generation Lagrangian Reachability Sophie Grnbacher, 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

SLIDE 1

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

SLIDE 2

Safety of Cyber-Physical Systems

Risk of Blackout Risk of Crash

Sophie Grünbacher Lagrangian Reachability

SLIDE 3

Safety of Cyber-Physical Systems

Neural Network Control Systems

Sophie Grünbacher Lagrangian Reachability

States +x, -x, +!, -! Controls + ̇ #, - ̇ #

SLIDE 4

Lagrangian Reachtube (LRT)

Nonlinear dynamical System

Sophie Grünbacher Lagrangian Reachability

Initial Set t" t"+T x y Bound for Reachset ̇ $ = & $ Unsafe Region Unsafe Region

SLIDE 5

Problem: Wrapping Effect

avoid: Interval Arithmetic

N. Nedialkov, K. Jackson, and G. Corliss. Validated solutions of initial value problems for
rdinary differential equations. Applied Mathematics and Computation, 105(1):21 – 68, 1999.

Sophie Grünbacher Lagrangian Reachability x y ̇ " = $% & ' "

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Wanted: tight conservative ellipse

Minimize volume of enclosing ellipse

!"#(%#, ' ( )* )

Sophie Grünbacher Lagrangian Reachability

x y x y t- t. /

13(/-)

452(/-, 6-)

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Main idea: Use sensitivity analysis

Sophie Grünbacher Lagrangian Reachability

To every nonlinear, (time-dependent) ODEs ! x = f( x), x0 = x(0) There is an associated variational equation: ! F = −J( x0,t )F, F( x0,0) = I

χ(x0 + dx,t ) χ(x0 + dx,0) χ(x0,t ) χ(x0,0)

F = dχ( x0,t ) dx

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Stretching Factor

Sophie Grünbacher Lagrangian Reachability

‖(#$%

$ (&'),#$% $ (&)) ‖

) *+ ≤ ‖(-(.) ‖ ) *%,+ 0 ‖(&', &)) ‖ ) *% 1 23 24

567(87, 1 0 23 )

x y x y t' t: &

#$%

$;(&')

<*%(&', =') &

)

#$%

$;(&))

F = dχ( x0,t ) dx

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Lohner’s QR method

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.

QR decomposition

Change to coordinate system Q

Sophie Grünbacher Lagrangian Reachability

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Work in progress

Wrapping of Reachset Choosing tightest ellipse

Sophie Grünbacher Lagrangian Reachability y x y x

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Sophie Grünbacher Lagrangian Reachability

Choosing optimal metric

x y x y !

"#$

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Sophie Grünbacher Lagrangian Reachability

Choose tightest ellipse

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Wrapping of Reachset - Trick #1

QR decomposition

Change to coordinate system Q

Sophie Grünbacher Lagrangian Reachability

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Tighter enclosing box (used in QR)

Sophie Grünbacher Lagrangian Reachability

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Wrapping of Reachset - Trick #2

Sophie Grünbacher Lagrangian Reachability

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Sophie Grünbacher Lagrangian Reachability

Effects of intersection trick #2

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Comparison with other tools

Sophie Grünbacher Lagrangian Reachability

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Conclusion & Future Work

Scale up LRT Verify Neural Network Control Systems

Sophie Grünbacher Lagrangian Reachability

States +x, -x, +!, -! Controls + ̇ #, - ̇ #

SLIDE 19

Thank you for your attention!

Sophie Grünbacher Lagrangian Reachability sophie.gruenbacher@tuwien.ac.at

Next-Generation Lagrangian Reachability

̇ " = $ "