TIRA: Toolbox for Interval Reachability Analysis Pierre-Jean Meyer , - PowerPoint PPT Presentation

TIRA: Toolbox for Interval Reachability Analysis Pierre-Jean Meyer , Alex Devonport, Murat Arcak April 18 th 2019 April 18 th 2019 Pierre-Jean Meyer (UC Berkeley) TIRA: Toolbox for Interval Reachability Analysis 1 / 8

Reachability analysis Discrete-time system x + = F ( t , x , p ) Reachability problem Initial time: t 0 Initial states [ x , x ] Input bounds [ p , p ] Reachable set in one discrete step { F ( t 0 , x , p ) | x ∈ [ x , x ] , p ∈ [ p , p ] } x x April 18 th 2019 Pierre-Jean Meyer (UC Berkeley) TIRA: Toolbox for Interval Reachability Analysis 2 / 8

Reachability analysis Continuous-time system Discrete-time system x = f ( t , x , p ) ˙ x + = F ( t , x , p ) Reachability problem Reachability problem Time range: [ t 0 , t f ] Initial time: t 0 Initial states [ x , x ] Initial states [ x , x ] Input bounds [ p , p ] Input bounds [ p , p ] Reachable set at final time t f Reachable set in one discrete step { x ( t f ; t 0 , x 0 , p ) | x 0 ∈ [ x , x ] , { F ( t 0 , x , p ) | x ∈ [ x , x ] , p ∈ [ p , p ] } p : [ t 0 , t f ] → [ p , p ] } x x April 18 th 2019 Pierre-Jean Meyer (UC Berkeley) TIRA: Toolbox for Interval Reachability Analysis 2 / 8

Over-approximations Flexible set representations Intervals polytopes easy to manipulate ellipsoids defined with only 2 state vectors level sets intersection is still an interval interval pavings Good scalability Accurate approximations Lower accuracy Low scalability April 18 th 2019 Pierre-Jean Meyer (UC Berkeley) TIRA: Toolbox for Interval Reachability Analysis 3 / 8

Architecture Core architecture Over-approximation hub [ R, R ] ← TIRA ([ t 0 , t f ] , [ x, x ] , [ p, p ]) [ R, R ] ← TIRA ( t 0 , [ x, x ] , [ p, p ]) Contraction/ growth bound Continuous-time mixed-monotonicity Sampled-data mixed-monotonicity Discrete-time mixed-monotonicity April 18 th 2019 Pierre-Jean Meyer (UC Berkeley) TIRA: Toolbox for Interval Reachability Analysis 4 / 8

Architecture System description Problem definition x = f ( t, x, p ) ˙ t 0 , ( t f ), [ x, x ], [ p, p ] x + = F ( t, x, p ) Core architecture Over-approximation hub [ R, R ] ← TIRA ([ t 0 , t f ] , [ x, x ] , [ p, p ]) User-inputs [ R, R ] ← TIRA ( t 0 , [ x, x ] , [ p, p ]) Required Contraction/ growth bound Continuous-time mixed-monotonicity Sampled-data mixed-monotonicity Discrete-time mixed-monotonicity April 18 th 2019 Pierre-Jean Meyer (UC Berkeley) TIRA: Toolbox for Interval Reachability Analysis 4 / 8

Architecture System description Problem definition x = f ( t, x, p ) ˙ t 0 , ( t f ), [ x, x ], [ p, p ] x + = F ( t, x, p ) Core architecture Over-approximation hub [ R, R ] ← TIRA ([ t 0 , t f ] , [ x, x ] , [ p, p ]) User-inputs [ R, R ] ← TIRA ( t 0 , [ x, x ] , [ p, p ]) Required Contraction/ growth bound Recommended Continuous-time Additional information: mixed-monotonicity - Jacobian bounds Sampled-data - contraction matrix mixed-monotonicity Discrete-time mixed-monotonicity April 18 th 2019 Pierre-Jean Meyer (UC Berkeley) TIRA: Toolbox for Interval Reachability Analysis 4 / 8

Architecture System description Problem definition x = f ( t, x, p ) ˙ t 0 , ( t f ), [ x, x ], [ p, p ] x + = F ( t, x, p ) Core architecture Over-approximation hub Method [ R, R ] ← TIRA ([ t 0 , t f ] , [ x, x ] , [ p, p ]) choice User-inputs [ R, R ] ← TIRA ( t 0 , [ x, x ] , [ p, p ]) Parameters Required Contraction/ growth bound Recommended Continuous-time Additional information: Optional mixed-monotonicity - Jacobian bounds Sampled-data - contraction matrix mixed-monotonicity Discrete-time mixed-monotonicity April 18 th 2019 Pierre-Jean Meyer (UC Berkeley) TIRA: Toolbox for Interval Reachability Analysis 4 / 8

Architecture System description Problem definition x = f ( t, x, p ) ˙ t 0 , ( t f ), [ x, x ], [ p, p ] x + = F ( t, x, p ) Core architecture Over-approximation hub Method [ R, R ] ← TIRA ([ t 0 , t f ] , [ x, x ] , [ p, p ]) choice User-inputs [ R, R ] ← TIRA ( t 0 , [ x, x ] , [ p, p ]) Parameters Required Contraction/ growth bound Recommended Continuous-time Additional information: Optional mixed-monotonicity - Jacobian bounds Sampled-data - contraction matrix mixed-monotonicity Extensible architecture Discrete-time mixed-monotonicity New methods provided by user April 18 th 2019 Pierre-Jean Meyer (UC Berkeley) TIRA: Toolbox for Interval Reachability Analysis 4 / 8

Over-approximation methods User inputs Internal functions Methods and limitations Growth bound Contraction matrix/scalar Contraction/growth function definition Continuous-time, x = f ( t, x ) + p ˙ Continuous-time mixed-monotonicity Discrete-time Jacobian bounds mixed-monotonicity Interval arithmetics Sampled-data mixed-monotonicity Sampling/Falsification Continuous-time, constant input p April 18 th 2019 Pierre-Jean Meyer (UC Berkeley) TIRA: Toolbox for Interval Reachability Analysis 5 / 8

Method comparison State x ∈ R n Method Tightness conditions Complexity Contraction/growth bound 1 CT mixed-monotonicity monotone 2 DT mixed-monotonicity sign-stable Jacobian 2 n SD mixed-monotonicity (IA) sign-stable sensitivity ≥ O (2 n ) O (2 n ) SD mixed-monotonicity (SF) sign-stable sensitivity Complexity ∼ number of successor evaluations April 18 th 2019 Pierre-Jean Meyer (UC Berkeley) TIRA: Toolbox for Interval Reachability Analysis 6 / 8

Illustration: traffic network x 6 x 4 x 2 p x 1 x 3 x 5 x 7 Piecewise affine model: x = f ( x , p ) ˙ x ∈ R n : road section densities p : constant input to section 1 Method n = 3 n = 99 Contraction/growth bound 0 . 13 s 0 . 37 s CT mixed-monotonicity 0 . 05 s 4 . 4 s SDMM (Interval arithmetics) 0 . 28 s 338 s SDMM (Sampling/falsification) 7 s − April 18 th 2019 Pierre-Jean Meyer (UC Berkeley) TIRA: Toolbox for Interval Reachability Analysis 7 / 8

Conclusions Matlab library of 4 interval reachability methods covering most nonlinear systems both continuous-time and discrete-time systems good scalability Toolbox architecture easily extensible can choose most suitable method https://gitlab.com/pj_meyer/TIRA pjmeyer@berkeley.edu April 18 th 2019 Pierre-Jean Meyer (UC Berkeley) TIRA: Toolbox for Interval Reachability Analysis 8 / 8

Comparison with DynIbex 1 Protein interactions 2 � − x 1 + x 2 � ˙ � � x 1 = x 2 x 2 ˙ 1 2 − 0 . 3 x 2 1+ x 2 Initial set: [0 , 0 . 3] 2 Time range: [0 , 10] s Method Time TIRA (CTMM) 26 ms DynIbex 1 . 9 s 1 Julien Alexandre and Alexandre Chapoutot, Validated explicit and implicit Runge-Kutta methods . Reliable Computing v. 22, 2016. 2 Lee A. Segel, Modeling dynamic phenomena in molecular and cellular biology . Cambridge University Press, 1984. April 18 th 2019 Pierre-Jean Meyer (UC Berkeley) TIRA: Toolbox for Interval Reachability Analysis 9 / 8