Interval Reachability Analysis using Second-Order Sensitivity Pierre-Jean Meyer , Murat Arcak IFAC 2020 Pierre-Jean Meyer (UC Berkeley) Interval Reachability Analysis using Second-Order Sensitivity IFAC 2020 1 / 12
Reachability analysis Continuous-time system: ˙ x = f ( x ) Objective : finite-time reachability analysis from initial interval Over-approximation Reachable Initial states set Exact computation of the reachable set: impossible → over-approximation by a multi-dimensional interval Pierre-Jean Meyer (UC Berkeley) Interval Reachability Analysis using Second-Order Sensitivity IFAC 2020 2 / 12
Motivations Discrete-time mixed-monotonicity 1 Sampled-data mixed-monotonicity x = f ( x ) ˙ x + = F ( x ) x + = x ( T ; x 0 ) System Requirement Bounded Jacobian Bounded sensitivity ∂ F ( x ) ∂ x ( T ; x 0 ) ∈ [ J , J ] ∈ [ S , S ] ∂ x ∂ x 0 Main challenge : compute sensitivity bounds with a tunable complexity/accuracy tradeoff 1 Meyer, Coogan and Arcak, IEEE Control Systems Letters , 2018 Pierre-Jean Meyer (UC Berkeley) Interval Reachability Analysis using Second-Order Sensitivity IFAC 2020 3 / 12
Jacobian and Sensitivity definitions Jacobian matrices of the continuous-time system ˙ x = f ( x ): J xx ( x ) = ∂ J x ( x ) J x ( x ) = ∂ f ( x ) ∂ x ∂ x First-order sensitivity Second-order sensitivity S xx ( t ; x 0 ) = ∂ S x ( t ; x 0 ) S x ( t ; x 0 ) = ∂ x ( t ; x 0 ) ∂ x 0 ∂ x 0 S x = J x ∗ S x S xx = J x ∗ S xx + J xx ∗ ( S x ⊗ S x ) ˙ ˙ S x (0; x 0 ) = I n S xx (0; x 0 ) = 0 Pierre-Jean Meyer (UC Berkeley) Interval Reachability Analysis using Second-Order Sensitivity IFAC 2020 4 / 12
Overview of the 4-step reachability procedure [ J x , J x ] [ J xx , J xx ] S x = J x ∗ S x ˙ Interval analysis OA of S x ([0 , T ]; X 0 ) 3 steps to bound S x [ S x RT , S x Interval analysis 2 on linear RT ] S xx = J x ∗ S xx + J xx ∗ ( S x ⊗ S x ) ˙ systems of S x and S xx Interval analysis OA of S xx ( T ; X 0 ) Grid-based sampling on S x [ S xx , S xx ] S x ( T ; x 0 ) = ∂x ( T ; x 0 ) Sampling { y 1 , . . . , y N } ⊆ X 0 ∂x 0 Evaluations S x ( T ; y i ) OA of S x ( T ; X 0 ) Last step for discrete-time [ S x , S x ] mixed-monotonicity on sampled-data system Discrete-time x = f ( x ) ˙ x + = x ( T ; x 0 ) mixed-monotonicity OA of x ( T ; X 0 ) 2 Althoff, Stursberg and Buss, CDC 2007 Pierre-Jean Meyer (UC Berkeley) Interval Reachability Analysis using Second-Order Sensitivity IFAC 2020 5 / 12
Step 1: First-order sensitivity tube S x = J x ∗ S x with initial condition S x (0) = I n Linear system ˙ If J x is constant, solutions are known: S x ( t ) = e J x ∗ t If J x ∈ [ J x , J x ]: S x ( t ) ∈ e [ J x , J x ] ∗ t Interval matrix exponential evaluated using Taylor expansion and interval arithemtics: + ∞ ([ J x , J x ] ∗ t ) i e [ J x , J x ] ∗ t ⊆ � i ! i =0 Truncate Taylor expansion and over-approximate remainder Reachable tube S x ([0 , T ]) Interval hull of initial S x (0) = I n and final solutions S x ( T ) ∈ e [ J x , J x ] ∗ T Enlarge hull to guarantee over-approximation Pierre-Jean Meyer (UC Berkeley) Interval Reachability Analysis using Second-Order Sensitivity IFAC 2020 6 / 12
Step 2: Second-order sensitivity set S xx = J x ∗ S xx + J xx ∗ ( S x ⊗ S x ) with initial condition S xx (0) = 0 Affine system ˙ Similar approach defining an interval affine system Need bounds on J x and J xx ∗ ( S x ⊗ S x ) for all t ∈ [0 , T ] Bounds on J x and J xx assumed to be provided This is why step 1 computed the reachable tube of S x Denote J xx ∗ ( S x ⊗ S x ) ∈ [ B , B ] Second-order sensitivity reachable set : � T e [ J x , J x ] ∗ t dt ∗ [ B , B ] S xx ( T ) ∈ 0 Pierre-Jean Meyer (UC Berkeley) Interval Reachability Analysis using Second-Order Sensitivity IFAC 2020 7 / 12
Step 3: First-order sensitivity set Guaranteed over-approximation of S x ( T ; [ x , x ]) from sampling Uniform grid sampling of interval of initial states ( a samples per dimension) Evaluation of S x ( T ; x 0 ) for each sample x 0 Interval hull of sampled sensitivity evaluations ∗ ( I n ⊗ ( 1 n ∗ � x − x � ∞ � | S xx | , | S xx | � Expand hull by M = max )) 2 a x M S x ( T ; x 0 ) x − x a − M x − x S x ( T ; [ x, x ]) 2 a x Pierre-Jean Meyer (UC Berkeley) Interval Reachability Analysis using Second-Order Sensitivity IFAC 2020 8 / 12
Step 4: Reachable set of the continuous-time system Discrete-time mixed-monotonicity 2 applied to sampled-data system x + = x ( T ; x 0 ) using first-order sensitivity bounds S x ( T ; x 0 ) ∈ [ S x , S x ] (centered on S x ∗ ) Auxiliary function g : X × X → X g i ( x , y ) = x i ( T ; z i ) + α i ( x − y ) with state z i = [ z i n ] ∈ R n and row vector α i = [ α i n ] ∈ R 1 × n 1 ; . . . ; z i 1 , . . . , α i � ( x j , max(0 , − S x if S x ∗ ij )) ≥ 0 , ( z i j , α i ij j ) = if S x ∗ ( y j , max(0 , S x ij )) < 0 . ij Theorem (Final reachable set) x ( T ; [ x , x ]) ⊆ [ g ( x , x ) , g ( x , x )] 2 Meyer, Coogan and Arcak, IEEE Control Systems Letters , 2018 Pierre-Jean Meyer (UC Berkeley) Interval Reachability Analysis using Second-Order Sensitivity IFAC 2020 9 / 12
Comparisons Alternative 1-step approaches 3 to bound S x ( T , [ x , x ]) S x = J x ∗ S x Interval analysis on ˙ Random sampling of S x ( T , x 0 ) without bounds on S xx Interval analysis Sampling 3-step approach OA guarantees yes no yes Conservativeness large small tunable Complexity low high tunable [ J x , J x ] [ J x , J x ], [ J xx , J xx ] Requirements none 3 Meyer, Coogan and Arcak, IEEE Control Systems Letters , 2018 Pierre-Jean Meyer (UC Berkeley) Interval Reachability Analysis using Second-Order Sensitivity IFAC 2020 10 / 12
Simulation results Unicycle with constant uncertainties v cos( x 3 ) + x 4 v sin( x 3 ) + x 5 ω + x 6 x = ˙ 0 0 0 Interval analysis Random sampling 3-step approach Samples N - 64 1 64 729 Computation time for [ S x , S x ] 0 . 44 s 3 . 1 s 0 . 35 s 3 . 2 s 36 s Pierre-Jean Meyer (UC Berkeley) Interval Reachability Analysis using Second-Order Sensitivity IFAC 2020 11 / 12
Conclusions Interval over-approximation for the reachable set of continuous-time systems Based on discrete-time mixed-monotonicity 3-step bounding of sensitivity matrix, with tunable complexity/conservativeness Future work Include this new method to the Matlab toolbox TIRA 4 Contact : pjmeyer@berkeley.edu 4 Toolbox for Interval Reachability Analysis: https://gitlab.com/pj_meyer/TIRA Pierre-Jean Meyer (UC Berkeley) Interval Reachability Analysis using Second-Order Sensitivity IFAC 2020 12 / 12
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