Reachability Analysis of Nonlinear and Hybrid Systems using - PowerPoint PPT Presentation

Reachability Analysis of Nonlinear and Hybrid Systems using Zonotopes Matthias Althoff Carnegie Mellon Univ. May 7, 2010 Matthias Althoff (Carnegie Mellon Univ.) Reachability Analysis of Nonlinear and Hybrid Systems using Zonotopes May 7, 2010 1 / 56

Introduction Overview of the Talk Main Talk: Reachability Analysis Linear Systems with Uncertain Parameters Nonlinear Systems Hybrid Systems Stochastic Reachability Analysis of Linear Systems Basic Idea Examples Transregional Collaborative Research Center 28 Cognitive Automobiles Safety Assessment of Autonomous Cars Basic Idea Examples Matthias Althoff (Carnegie Mellon Univ.) Reachability Analysis of Nonlinear and Hybrid Systems using Zonotopes May 7, 2010 2 / 56

Introduction Safety Verification Using Reachable Sets unsafe set exemplary trajectory x 2 reachable set initial set x 1 System is safe, if no trajectory enters the unsafe set. Matthias Althoff (Carnegie Mellon Univ.) Reachability Analysis of Nonlinear and Hybrid Systems using Zonotopes May 7, 2010 3 / 56

Introduction Safety Verification Using Reachable Sets unsafe set exemplary trajectory x 2 overapproximated initial set x 1 reachable set System is safe, if no trajectory enters the unsafe set. Overapproximated system is safe → real system is safe. Matthias Althoff (Carnegie Mellon Univ.) Reachability Analysis of Nonlinear and Hybrid Systems using Zonotopes May 7, 2010 3 / 56

Linear Systems Linear Systems with Uncertain Parameters Reachability analysis is performed for the following system class: System Model x = A x + u ( t ) , ˙ x (0) ∈ X 0 ⊂ R n , u ( t ) ∈ U ⊂ R n , A ∈ A ⊂ R n × n where A is a matrix of intervals and U is a zonotope (specified later). � [ − 1 . 05 , − 0 . 95] � [ − 4 . 05 , − 3 . 95] A ∈ A = [ − 1 . 05 , − 0 . 95] [3 . 95 , 4 . 05] Example: � 1 � u ( t ) ∈ U = [ − 0 . 1 , 0 . 1] 1 Matthias Althoff (Carnegie Mellon Univ.) Reachability Analysis of Nonlinear and Hybrid Systems using Zonotopes May 7, 2010 4 / 56

Linear Systems Initial State Solution (Homogeneous Solution) Exact Solution (no uncertainties) x ( r ) = e A r x (0) . Exact Solution (uncertain system matrix) � � � e A r x (0) � x ( r ) ∈ � A ∈ A The set of exponential matrices is written in short as e A r . How to compute a tight over-approximation? Matthias Althoff (Carnegie Mellon Univ.) Reachability Analysis of Nonlinear and Hybrid Systems using Zonotopes May 7, 2010 5 / 56

Linear Systems Preliminaries: Interval Arithmetic The interval matrix exponential e A r is computed based on the addition and multiplication rule: Given are the intervals a = [ a , a ] and b = [ b , b ]: a + b =[ a + b , a + b ] ab =[min( ab , ab , ab , ab ) , max( ab , ab , ab , ab )] Interval arithmetic is only exact for single-use-expressions (SUE). Example ( a = [ − 2 , − 1] , b = [ − 1 , 1]): c = ab + a = [ − 4 , 1] , not SUE → overapproximated c = a ( b + 1) = [ − 4 , 0] , SUE → exact Matthias Althoff (Carnegie Mellon Univ.) Reachability Analysis of Nonlinear and Hybrid Systems using Zonotopes May 7, 2010 6 / 56

Linear Systems Interval Matrix Exponential Taylor series of e A t 3! ( A t ) 3 + . . . e A t A t + 1 2! ( A t ) 2 1 = I + + � �� � � �� � � m i ! ( A t ) i + E ( t ) 1 e A t ⊂ + W ( t ) + I i =3 W is computed exactly: Interval arithmetic (SUE) & Analytical minimum and maximum for non-SUE elements. � m i ! ( A t ) i is overapproximated with interval arithmetic (not SUE). 1 i =3 E ( t ) is a standard approximation for the matrix exponential remainder extended to interval matrices. Matthias Althoff (Carnegie Mellon Univ.) Reachability Analysis of Nonlinear and Hybrid Systems using Zonotopes May 7, 2010 7 / 56

Linear Systems Overview of Reachable Set Computation ➀ Compute reachable set ˆ R h ( r ) at time r (without input). ➁ Obtain convex hull of ˆ R (0) and ˆ R h ( r ). ➂ Enlarge reachable set to guarantee enclosure of all trajectories. ➀ � �� � e A r ˆ R ([0 , r ]) = CH (ˆ ˆ + F ˆ R (0) + ˆ R i ([0 , r ]) R (0) , R (0)) � �� � � �� � ➁ ➂ F : Error interval due to the curvature of trajectories within t ∈ [0 , r ]. ˆ R i ([0 , r ]) : Reachable set of the input (inhomogeneous solution). ˆ R h ( r ) convex hull of R (0), ˆ ˆ R h ( r ) ˆ R ([0 , r ]) ˆ R (0) enlargement ➁ ➀ ➂ Matthias Althoff (Carnegie Mellon Univ.) Reachability Analysis of Nonlinear and Hybrid Systems using Zonotopes May 7, 2010 8 / 56

Linear Systems Representation of Reachable Sets Definition of a zonotope Z p � � x ∈ R n � � c , g ( i ) ∈ R n β i g ( i ) , � − 1 ≤ β i ≤ 1 Z = � x = c + , i =1 Interpretation: Minkowski sum of line segments l i = [ − 1 , 1] g ( i ) . Zonotopes are centrally symmetric to c . Short notation: Z = ( c , g (1 ... p ) ). 3 3 2 l 1 l 2 l 2 l 1 l 1 2 2 1 1 1 c c c 0 0 l 3 0 −1 −1 0 1 2 −1 0 1 2 3 −2 0 2 4 (a) c + l 1 (b) c + l 1 + l 2 (c) c + l 1 + l 2 + l 3 Matthias Althoff (Carnegie Mellon Univ.) Reachability Analysis of Nonlinear and Hybrid Systems using Zonotopes May 7, 2010 9 / 56

Linear Systems Operations on Zonotopes Given are Z 1 = ( c 1 , g (1 ... p ) ) and Z 2 = ( c 2 , d (1 ... p ) ). Addition Z 1 + Z 2 := { x + y | x ∈ Z 1 , y ∈ Z 2 } = ( c 1 + c 2 , g (1 ... p ) , d (1 ... u ) ) Matrix Multiplication LZ 1 := { Lx | x ∈ Z 1 } = ( Lc , Lg (1 ... p ) ) , L ∈ R n × n Interval Matrix Multiplication After defining ˆ A = [ − S , S ] and ˜ A , S ∈ R n × n , it follows that A Z 1 = (˜ A + ˆ A ) Z 1 ⊆ ˜ AZ 1 + ˆ A Z 1 ⊆ ˜ AZ 1 + ˆ A box ( Z 1 ) , box () : generates over-appr. axis-aligned box. Matthias Althoff (Carnegie Mellon Univ.) Reachability Analysis of Nonlinear and Hybrid Systems using Zonotopes May 7, 2010 10 / 56

Linear Systems Operations on Zonotopes Given are Z 1 = ( c 1 , g (1 ... p ) ) and Z 2 = ( c 2 , d (1 ... p ) ). Addition Z 1 + Z 2 := { x + y | x ∈ Z 1 , y ∈ Z 2 } = ( c 1 + c 2 , g (1 ... p ) , d (1 ... u ) ) Matrix Multiplication LZ 1 := { Lx | x ∈ Z 1 } = ( Lc , Lg (1 ... p ) ) , L ∈ R n × n 0.5 overapproximation exact solution Example: x 2 0 Zonotope with a single original generator: zonotope −0.5 −1 −0.5 0 0.5 1 x 1 Matthias Althoff (Carnegie Mellon Univ.) Reachability Analysis of Nonlinear and Hybrid Systems using Zonotopes May 7, 2010 10 / 56

Linear Systems Input Solution (Inhomogeneous Solution) Exact Solution The set of all input solution is: � r � � � e A r e − A t u ( t ) dt � x p ( r ) ∈ � A ∈ A , u ( t ) ∈ U 0 Over-approximative Solution The integral can be over-approximated as follows: � r ˆ R i ([0 , r ]) = e A τ U d τ 0 � A i r i +1 m � � ⊆ + E ( r ) r U . ( i + 1)! U i =0 (proof omitted) Matthias Althoff (Carnegie Mellon Univ.) Reachability Analysis of Nonlinear and Hybrid Systems using Zonotopes May 7, 2010 11 / 56

Linear Systems Numerical Example (1) » [ − 1 . 05 , − 0 . 95] – » 1 – [ − 4 . 05 , − 3 . 95] x = ˙ x + [ − 0 . 1 , 0 . 1] [3 . 95 , 4 . 05] [ − 1 . 05 , − 0 . 95] 1 | {z } | {z } A U 1.5 1 0.5 initial set x 2 0 −0.5 reachable set exemplary trajectories −1 −1 0 1 x 1 Matthias Althoff (Carnegie Mellon Univ.) Reachability Analysis of Nonlinear and Hybrid Systems using Zonotopes May 7, 2010 12 / 56

Linear Systems Numerical Example (2) Five dimensional example: 1 initial set 1 0.5 x 3 x 5 0.5 initial set 0 0 −0.5 −0.5 0 0.5 1 1.5 0 0.5 1 x 2 x 4 (a) Projection on x 2 and x 3 (b) Projection on x 4 and x 5 Computation with systems of higher dimensions for 125 time intervals: Dimension n 5 10 20 50 100 CPU-time [sec] 0.14 0.20 0.35 1.72 7.96 Matthias Althoff (Carnegie Mellon Univ.) Reachability Analysis of Nonlinear and Hybrid Systems using Zonotopes May 7, 2010 13 / 56

Linear Systems Further Work Compute with Matrix zonotopes instead of interval matrices: Matrix Zonotope A (0) + � κ p ( i ) ∈ [ − 1 , 1] . A ( p ) = ˆ i =1 p ( i ) ˆ A ( i ) , ˆ Example: � � − 1 . 1 � � − 0 . 9 �� − 4 . 1 − 3 . 9 x = ˙ k · + (1 − k ) x + u ( t ) , − 1 . 1 − 0 . 9 3 . 9 4 . 1 k ∈ [0 , 1] . Corresponding Interval Matrix: � [ − 1 . 1 , − 0 . 9] � [ − 4 . 1 , − 3 . 9] x = ˙ x + u ( t ) . [3 . 9 , 4 . 1] [ − 1 . 1 , − 0 . 9] Matthias Althoff (Carnegie Mellon Univ.) Reachability Analysis of Nonlinear and Hybrid Systems using Zonotopes May 7, 2010 14 / 56

Nonlinear Systems Nonlinear Systems with Uncertain Parameters Reachability analysis is performed for the following system class: System Model x = f ( x ( t ) , u ( t ) , p ) , ˙ x (0) ∈ X 0 ⊂ R n , u ( t ) ∈ U ⊂ R m , p ∈ P ⊂ I o and u ( t ) is Lipschitz continuous. Representations of the initial set X 0 , the parameter set P and the input set U : Initial state set X 0 , input set U : represented by a zonotope. Parameter set P : represented by an o-dimensional interval ( I is the set of real valued intervals). Matthias Althoff (Carnegie Mellon Univ.) Reachability Analysis of Nonlinear and Hybrid Systems using Zonotopes May 7, 2010 15 / 56