Reachability Analysis of Nonlinear and Hybrid Systems using Zonotopes May 7, / 56

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

2 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 Reachability Analysis of Nonlinear and Hybrid Systems using Zonotopes May 7, / 56

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

4 Introduction Safety Verification Using Reachable Sets exemplary trajectory unsafe set x 2 x 1 initial set overapproximated reachable set System is safe, if no trajectory enters the unsafe set. Overapproximated system is safe real system is safe. Reachability Analysis of Nonlinear and Hybrid Systems using Zonotopes May 7, / 56

5 Linear Systems Linear Systems with Uncertain Parameters Reachability analysis is performed for the following system class: System Model ẋ = Ax + 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). Example: [ ] [ 1.05, 0.95] [ 4.05, 3.95] A A= [3.95, 4.05] [ 1.05, 0.95] [ ] 1 u(t) U = [ 0.1, 0.1] 1 Reachability Analysis of Nonlinear and Hybrid Systems using Zonotopes May 7, / 56

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

7 Linear Systems Preliminaries: Interval Arithmetic The interval matrix exponential e Ar is computed based on the addition and multiplication rule: Given are the intervals a =[a, a] andb =[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], c = a(b +1)=[ 4, 0], not SUE overapproximated SUE exact Reachability Analysis of Nonlinear and Hybrid Systems using Zonotopes May 7, / 56

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

9 Linear Systems Overview of Reachable Set Computation ➀ Compute reachable set ˆR h (r) attimer (without input). ➁ Obtain convex hull of ˆR(0) and ˆR h (r). ➂ Enlarge reachable set to guarantee enclosure of all trajectories. ➀ {}}{ ˆR([0, r]) = CH(ˆR(0), e Ar ˆR(0)) + F }{{} ˆR(0) + ˆR i ([0, r]) }{{} ➁ ➂ F : Error interval due to the curvature of trajectories within t [0, r]. ˆR i ([0, r]) : Reachable set of the input (inhomogeneous solution). ˆR(0) ˆR h (r) convex hull of ˆR(0), ˆR h (r) ➀ ➁ ➂ ˆR([0, r]) enlargement Reachability Analysis of Nonlinear and Hybrid Systems using Zonotopes May 7, / 56

10 Linear Systems Representation of Reachable Sets Definition of a zonotope Z { Z = x R n x p } = c + β i g (i), 1 β i 1, c, g (i) R n 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) ) l 1 c l 1 l 2 c l (a) c + l 1 (b) c + l 1 + l 2 (c) c + l 1 + l 2 + l l 1 l 2 c Reachability Analysis of Nonlinear and Hybrid Systems using Zonotopes May 7, / 56

11 Linear Systems Operations on Zonotopes Given are Z 1 =(c 1, g (1...p) )andz 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  =[ S, S] andã, S Rn n, it follows that AZ 1 =(à + Â)Z 1 ÃZ 1 + ÂZ 1 ÃZ 1 + Âbox(Z 1 ), box() : generates over-appr. axis-aligned box. Reachability Analysis of Nonlinear and Hybrid Systems using Zonotopes May 7, / 56

12 Linear Systems Operations on Zonotopes Given are Z 1 =(c 1, g (1...p) )andz 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 Example: Zonotope with a single generator: x 2 0 exact solution original zonotope x 1 Reachability Analysis of Nonlinear and Hybrid Systems using Zonotopes May 7, / 56

13 Linear Systems Input Solution (Inhomogeneous Solution) Exact Solution The set of all input solution is: r } x p (r) {e Ar e At u(t) dt A A, u(t) U Over-approximative Solution The integral can be over-approximated as follows: (proof omitted) ˆR i ([0, r]) = 0 r e Aτ Udτ 0 m ( A i r i+1 ) (i +1)! U + E(r) ru. i=0 Reachability Analysis of Nonlinear and Hybrid Systems using Zonotopes May 7, / 56

14 Numerical Example (1) Linear Systems»» [ 1.05, 0.95] [ 4.05, 3.95] 1 ẋ = x + [ 0.1, 0.1] [3.95, 4.05] [ 1.05, 0.95] 1 {z } {z } A U x initial set 0.5 reachable set exemplary trajectories x 1 Reachability Analysis of Nonlinear and Hybrid Systems using Zonotopes May 7, / 56

15 Numerical Example (2) Five dimensional example: Linear Systems 1 initial set 1 x x initial set 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 CPU-time [sec] Reachability Analysis of Nonlinear and Hybrid Systems using Zonotopes May 7, / 56

16 Further Work Linear Systems Compute with Matrix zonotopes instead of interval matrices: Matrix Zonotope A(p) =Â(0) + κ i=1 p(i) Â (i), ˆp (i) [ 1, 1]. Example: ẋ = ( k k [0, 1]. [ ] (1 k) [ ]) x + u(t), Corresponding Interval Matrix: [ ] [ 1.1, 0.9] [ 4.1, 3.9] ẋ = x + u(t). [3.9, 4.1] [ 1.1, 0.9] Reachability Analysis of Nonlinear and Hybrid Systems using Zonotopes May 7, / 56

17 Nonlinear Systems Nonlinear Systems with Uncertain Parameters Reachability analysis is performed for the following system class: System Model ẋ = 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,theparametersetP 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). Reachability Analysis of Nonlinear and Hybrid Systems using Zonotopes May 7, / 56

18 Nonlinear Systems Basic Ideas, Properties ➀ Efficient Computation Embed efficient computation of reachable sets for linear systems using zonotopes. Linearize the system dynamics. ➁ Over-approximation Compute linearization error bounds and add them to the set of uncertain inputs U. Reachable set of the nonlinear system is over-approximative. ➂ Constrain Linearization Error Control linearization error bounds by splitting reachable sets. Allows a tradeoff between accuracy and efficiency. Reachability Analysis of Nonlinear and Hybrid Systems using Zonotopes May 7, / 56

19 Overall Algorithm Nonlinear Systems Initial set X 0, input set U, timestepl =1 Linearize System Compute reachable set R lin without linearization error l := l +1 Obtain set of linearization errors L based on R lin and L (L: set of admissible linearization errors) L L? No Split reachable set Yes Compute reachable set R err due to the linearization error L R = R lin + R err Reachability Analysis of Nonlinear and Hybrid Systems using Zonotopes May 7, / 56

20 Nonlinear Systems Overall Algorithm: Animation R(0) = X 0 Linearize System Reachability Analysis of Nonlinear and Hybrid Systems using Zonotopes May 7, / 56

21 Nonlinear Systems Overall Algorithm: Animation R lin (t 1 ) Compute reachable set R lin without linearization error Reachability Analysis of Nonlinear and Hybrid Systems using Zonotopes May 7, / 56

22 Nonlinear Systems Overall Algorithm: Animation R lin ([t 0, t 1 ]) Compute reachable set R lin without linearization error Reachability Analysis of Nonlinear and Hybrid Systems using Zonotopes May 7, / 56

23 Nonlinear Systems Overall Algorithm: Animation R lin ([t 0, t 1 ]) + R err, R err : reachable set due to L Obtain set of linearization errors L based on R lin + R err Reachability Analysis of Nonlinear and Hybrid Systems using Zonotopes May 7, / 56

24 Nonlinear Systems Overall Algorithm: Animation R([t 0, t 1 ]) R([t 0, t 1]) = R lin ([t 0, t 1]) + R err([t 0, t 1]) Reachability Analysis of Nonlinear and Hybrid Systems using Zonotopes May 7, / 56

25 Nonlinear Systems Overall Algorithm: Animation L L! Reachability Analysis of Nonlinear and Hybrid Systems using Zonotopes May 7, / 56

26 Nonlinear Systems Overall Algorithm: Animation R lin (t n ) Reachability Analysis of Nonlinear and Hybrid Systems using Zonotopes May 7, / 56

27 Nonlinear Systems Overall Algorithm: Animation R encl R lin (t n ) Reachability Analysis of Nonlinear and Hybrid Systems using Zonotopes May 7, / 56

28 Nonlinear Systems Overall Algorithm: Animation Split reachable set Reachability Analysis of Nonlinear and Hybrid Systems using Zonotopes May 7, / 56

29 Nonlinear Systems Linearization and Lagrange Remainder Original system: ẋ = f (z(t), p(t)), with z T := [x T, u T ]: Taylor series ẋ i f i (z, p)+ f i(z, p) z (z z ) + z=z }{{} 1 st ordertaylorseriesˆ=a(p)x+b(p)u+f i (z,p) 1 2 (z z ) T 2 f i (ξ,p) z 2 ) (z z ), ξ = z +[0, 1](z z ) z=z }{{} Lagrange remainderl i In case of parameter uncertainties: A(p) A, B(p) Bare bounded by interval matrices A, B. Linearization error is obtained from the Lagrange remainder using interval arithmetic enclose reachable set by multidimensional interval. Reachability Analysis of Nonlinear and Hybrid Systems using Zonotopes May 7, / 56

30 Nonlinear Systems Control of the Linearization Error Linearization error not enclosed by set of admissible linearization errors (L L) Split reachable set (reduces search space for L). 1. Problem: How to split a zonotope, such that the resulting sets are zonotopes? Split of a zonotope A zonotope Z =(c, g (1...p) ) is split into Z 1 and Z 2 such that Z 1 Z 2 = Z, Z 1 Z 2 = Z by splitting a single generator: Z 1 =(c 1 2 g (j), g (1...j 1) 1, 2 g (j), g (j+1...p) ) Z 2 =(c g (j), g (1...j 1) 1, 2 g (j), g (j+1...p) ) Z =(c, g (1...j 1), g (j+1...p) ) Reachability Analysis of Nonlinear and Hybrid Systems using Zonotopes May 7, / 56

31 Nonlinear Systems Splitting of Zonotopes: Overlapping vs. Overapproximation 2. Problem: Proposed split is not effective if zonotope is composed by many generators. Zonotope is over-approximated by less generators. Tradeoff between over-approximation and effectiveness. g (j) red 1 g (j) 2 1 g (j) 2 red g (j) Z Z red Z 2 Z 1 Z 2 red Z 1 red (a) Z, Z red (b) Z 1, Z 2 (c) Z 1 red, Z 2 red 3. Problem: Which generator should be split? Brute force approach is applied. Reachability Analysis of Nonlinear and Hybrid Systems using Zonotopes May 7, / 56

32 Van-der-Pol Oscillator Nonlinear Systems ẋ 1 = x 2, ẋ 2 =(1 x 2 1 )x 2 x 1 x initial set x 1 Number of computed sets per time step time t Computational time: 19 sec (Matlab, AMD Athlon processor). Reachability Analysis of Nonlinear and Hybrid Systems using Zonotopes May 7, / 56

33 Nonlinear Systems Water Tank System: Set Up The states x i are the water levels of each tank and u is the water flow into the first tank. The differential equation for the i th tank is ẋ i = 1 A i (k i 1 2gxi 1 k i 2gxi ). The uncertain parameters are k i [0.0148, 0.015] and the inflow disturbance is v [ 0.005, 0.005]. x 1 u x 2 x 3 Reachability Analysis of Nonlinear and Hybrid Systems using Zonotopes May 7, / 56

34 Nonlinear Systems Water Tank System: Reachable Sets 4 initial set 6 5 x 2 3 x 6 4 initial set 2 k i =0.015 k i [0.0148, 0.015] x 1 (a) Projection onto x 1, x x 1 (b) Projection onto x 1, x 6. Dimension n CPU-time [sec] CPU-time [sec] (uncertain param.) Reachability Analysis of Nonlinear and Hybrid Systems using Zonotopes May 7, / 56

35 Nonlinear Systems Water Tank System: Reachable Sets x initial set x initial set k i =0.015 k i [0.0148, 0.015] x 3 x 5 (a) Projection onto x 3, x 4. (b) Projection onto x 5, x 6. Dimension n CPU-time [sec] CPU-time [sec] (uncertain param.) Reachability Analysis of Nonlinear and Hybrid Systems using Zonotopes May 7, / 56

36 Hybrid Systems Hybrid Systems Hybrid Automaton HA =(Z, z 0, X, X 0, inv, T, g, j, flow) the set of locations Z with initial location z 0, the continuous state space X R n with initial state set X 0, the invariant inv and guard sets g of each location z which are modeled as polytopes, the set of discrete transitions T Z Z, the linear jump function j such that x = C g x + d g (x : state after jump) the linear flow function ẋ = A z x + u(t) invariant initial set reachable set jump etc. x 2 x 1 z 1 guards z 2 Reachability Analysis of Nonlinear and Hybrid Systems using Zonotopes May 7, / 56

37 Hybrid Systems Combined use of Zonotopes and Polytopes Representation of continuous evolution by zonotopes: Representation of the intersection with guards, invariants by polytopes: P = { x R n Cx d }, C R q n, d R q Alternative definition: Intersection of halfspaces S i. S 4 S 1 S 3 S 2 Reachability Analysis of Nonlinear and Hybrid Systems using Zonotopes May 7, / 56

38 Hybrid Systems Reachable Set Computation for Guard Set Intersection ➀ Compute reachable set using zonotopes. ➁ Transform zonotopes to overapproximated polytopes. ➂ Intersect polytopes with the guard set. ➃ Overapproximate intersected polytopes by a single zonotope Continue computation within the invariant of the next discrete state. guard set reachable set overappr. polytopes overappr. zonotope (a) Step ➀ (b) Step ➁ (c) Step ➂ and ➃ Alternative for guards modeled by hyperplanes: A. Girard, C. Le Guernic (HSCC 08) Reachability Analysis of Nonlinear and Hybrid Systems using Zonotopes May 7, / 56

39 Hybrid Systems Representation of Zonotopes by Halfspaces Overview: Zonotopes are a special case of polytopes: Exact Conversion Conversion of parallelotopes Conversion of zonotopes The change of representation is computationally expensive: Overapproximative Conversion Overapproximate zonotopes by parallelotopes Order reduction of zonotopes Overapproximate zonotopes by several parallelotopes intersection Reachability Analysis of Nonlinear and Hybrid Systems using Zonotopes May 7, / 56

40 Hybrid Systems Preliminaries and Notations ➀ Parallelotope is a zonotope of order 1: Number of generators p equals the dimension n. ➁ Facets are spanned by n 1 generators. Matrix of generators with the i th generator missing: G i =[g (1),...,g (i 1), g (i+1),...,g (n) ]. g (3) g (2) C + 1 c g (1) Reachability Analysis of Nonlinear and Hybrid Systems using Zonotopes May 7, / 56

41 Hybrid Systems Preliminaries and Notations ➀ Parallelotope is a zonotope of order 1: Number of generators p equals the dimension n. ➁ Facets are spanned by n 1 generators. Matrix of generators with the i th generator missing: G i =[g (1),...,g (i 1), g (i+1),...,g (n) ]. ➂ Normal vector C + i of the i th facet is perpendicular to all generators in H := G i : C + i = nx (H) :=[...,( 1) k+1 det(h [k] ),...] T, where H [k] is the matrix H whose k th row is removed. Reachability Analysis of Nonlinear and Hybrid Systems using Zonotopes May 7, / 56

42 Hybrid Systems Conversion from Generator to Halfspace Representation of Parallelotopes Halfspace representation Cx d C = ˆC + C + T d = ˆd + d T C + i = nx(g i ) T / nx(g i ) 2 d + i = C + i c +Δd i, d i = C + i c +Δd i Δd i = C + i g (i) Δd 1 = C + i g (i) C + 1 C + i c d + 1 c g (1) d 1 x 2 g (2) C + 1 x 1 Reachability Analysis of Nonlinear and Hybrid Systems using Zonotopes May 7, / 56

43 Hybrid Systems Conversion from Generator to Halfspace Representation of Parallelotopes Halfspace representation Cx d C = ˆC + C + T d = ˆd + d T C + i = nx(g i ) T / nx(g i ) 2 d + i = C + i c +Δd i, d i = C + i c +Δd i Δd i = C + i g (i) Δd 1 = C + i g (i) C + 1 C + i c d + 1 c g (1) d 1 x 2 g (2) C + 1 x 1 Reachability Analysis of Nonlinear and Hybrid Systems using Zonotopes May 7, / 56

44 Hybrid Systems Conversion from Generator to Halfspace Representation of Parallelotopes Halfspace representation Cx d C = ˆC + C + T d = ˆd + d T C + i = nx(g i ) T / nx(g i ) 2 d + i = C + i c + Δd i, d i = C + i c + Δd i Δd i = C + i g (i) Δd 1 = C + i g (i) C + 1 C + i c d + 1 c g (1) d 1 x 2 g (2) C + 1 x 1 Reachability Analysis of Nonlinear and Hybrid Systems using Zonotopes May 7, / 56

45 Hybrid Systems Conversion from Generator to Halfspace Representation of Zonotopes Extension is straightforward: n 1 generators selected from p generators for each non-parallel facet 2 ( p n 1) facets. Facet obtained by cancelling p n + 1 generators from the G -matrix which is denoted by G γ,...,η. Halfspace representation Cx d C = ˆC + C + T d = ˆd + d T C + i = nx(g γ,...,η ) T / nx(g γ,...,η ) 2 d + i = C + i c +Δd i, d i = C + i c +Δd i Δd i = P p υ=1 C + i g (υ) Complexity with respect to the number p of generators is O( ( p n 1) p) linear in the number of facets. Reachability Analysis of Nonlinear and Hybrid Systems using Zonotopes May 7, / 56

46 Hybrid Systems Overapproximation of a Zonotope by a Parallelotope Basic Procedure ➀ Choose n generators that should represent the parallelotope. ➁ Stretch chosen generators such that the zonotope is enclosed. The overapproximated parallelotope Ψ is generated as follows: Ψ=Γ box(γ 1 Z ) where Γ R n n is the matrix of n generators g (i) taken out of all p generators, box(z ) returns the axis-oriented bounding box of a zonotope Z. Reachability Analysis of Nonlinear and Hybrid Systems using Zonotopes May 7, / 56

47 Hybrid Systems Overapproximation of a Zonotope by a Parallelotope Basic Procedure ➀ Choose n generators that should represent the parallelotope. ➁ Stretch chosen generators such that the zonotope is enclosed. x g (2) Z Γ 1 Γ box(γ 1 Z) Γbox(Γ 1 Z) 4 10 Γ 1 Z Z 2 5 x 2 Γ 1 g (2) x 2 g (2) g (1) Γ 1 g (1) g (1) x 1 x 1 x 1 Remaining question: How to choose Γ? Reachability Analysis of Nonlinear and Hybrid Systems using Zonotopes May 7, / 56

48 Hybrid Systems Overapproximation of a Zonotope by a Parallelotope Choose Γ such that a metric is minimized: Proposed Metric Θ=(vol(W Z red )/vol(w Z)) 1/n, W = diag(w). Z: original zonotope, Z red : reduced zonotope, W : normalizes coordinate axes. W = 1: Determines the ratio of the edge length of two cubes, in which the volume of the reduced zonotope and the original zonotope fit. Candidates for Γ have to pass two tests: ➀ Length test: longest generators (2-norm) pass. ➁ Pseudo volume test: generator combination spanning the largest volume Θ = det[g (i1),...,g (in) ] 1 pass (no stretching considered). For the remaining generator combinations, the best performance index Θ wins (sufficient to compute vol(w Z red )). Reachability Analysis of Nonlinear and Hybrid Systems using Zonotopes May 7, / 56

49 Hybrid Systems Overapproximation of a Zonotope by a Parallelotope Choose Γ such that a metric is minimized: Proposed Metric Θ=(vol(W Z red )/vol(w Z)) 1/n, W = diag(w). Z: original zonotope, Z red : reduced zonotope, W : normalizes coordinate axes. W = 1: Determines the ratio of the edge length of two cubes, in which the volume of the reduced zonotope and the original zonotope fit. dimension order mean mean [min,max] variance of t [sec]: ofθ: ofθ: ofθ: [1.0030, ] [1.0369, ] [1.0343, ] [1.2143, ] [1.0779, ] Reachability Analysis of Nonlinear and Hybrid Systems using Zonotopes May 7, / 56

50 Hybrid Systems Overapproximation of a Zonotope by a Reduced Zonotope Basic Procedure ➀ Split zonotope into a part Ž which is unchanged and a part Z reduced to a parallelotope (Z = Ž + Z). ➁ Selected generators of unchanged part Ž are the longest generators (2-norm). The reduced zonotope Z red is generated as follows: Z red = Ž +Ψ, Ψ=Γbox(Γ 1 Z). Result: Improvements are marginal; computation time for halfspace conversion is drastically increased. Reachability Analysis of Nonlinear and Hybrid Systems using Zonotopes May 7, / 56

51 Hybrid Systems Overapproximation of a Zonotope by Intersected Parallelotopes Basic Procedure ➀ Compute several enclosing parallelotopes with the best performance indices. ➁ Intersect obtained parallelotopes. dim. order inters. mean mean [min,max] variance of t [sec]: ofθ: ofθ: ofθ: [1.0343, ] [1.0019, ] [1.2143, ] [1.0808, ] [1.0779, ] [1.0251, ] Reachability Analysis of Nonlinear and Hybrid Systems using Zonotopes May 7, / 56

52 Hybrid Systems Overapproximation of a Zonotope by Intersected Parallelotopes Basic Procedure ➀ Compute several enclosing parallelotopes with the best performance indices. ➁ Intersect obtained parallelotopes. (a) No intersection. (b) 4 intersections. Reachability Analysis of Nonlinear and Hybrid Systems using Zonotopes May 7, / 56

53 Hybrid Systems Overapproximation of a Set of Polytopes Polytopes have only a few facets computation of vertices is feasible. Task: Enclose points in R n Possible methods are, e.g.: ➀ Oriented rectangular hulls based on singular value decomposition (O. Stursberg, B. Krogh: HSCC 2003). ➁ Compute axis-oriented box where one of these generators is replaced by the flow direction. ➂ Compute in parallel with several enclosures. Reachability Analysis of Nonlinear and Hybrid Systems using Zonotopes May 7, / 56

54 Hybrid Systems Over-Approximation of a Set of Polytopes x guard set reachable set x over-appr. halfspace representation x ➀ 1 x Z encl 10 f before 15 x ➁ guard set reachable set x 2 f before Z encl 1 guard set vertices x ➂ x ➃ 1 Reachability Analysis of Nonlinear and Hybrid Systems using Zonotopes May 7, / 56

55 Hybrid Systems Room Heating Example 6 rooms with heaters in room 1 and 6 are considered. The heaters are switched on if the temperature is below 20 degree celsius and switched off when the temperature exceeds 24 degree. The temperature dynamics of room i is: ẋ i = c i h i + b i (u x i )+ i j a ij (x j x i ) with room specific constant parameters a ij, b i and c i heater Reachability Analysis of Nonlinear and Hybrid Systems using Zonotopes May 7, / 56

56 Reachable Sets Hybrid Systems 24 initial set 24 x 2 23 reachable set simulations x x 1 (a) Projection of x 1, x x 1 initial set (b) Projection of x 1, x 6 Computation time: 16.8 sec on an AMD Athlon processor (single core) in Matlab. Reachability Analysis of Nonlinear and Hybrid Systems using Zonotopes May 7, / 56

57 Reachable Sets Hybrid Systems initial set x x x 3 (c) Projection of x 3, x initial set x 5 (d) Projection of x 5, x 6 Computation time: 16.8 sec on an AMD Athlon processor (single core) in Matlab. Reachability Analysis of Nonlinear and Hybrid Systems using Zonotopes May 7, / 56

58 Hybrid Systems Conclusions Linear Systems: Uncertain parameters within specified intervals. No wrapping-free implementation as for LTI systems. Nonlinear Systems: Efficient computation for high dimensional nonlinear systems with uncertain parameters. Algorithm is best suited for systems with lower nonlinearity measure. In case of highly nonlinear systems, the current implementation may get stuck due to numerical problems. Hybrid Systems: Zonotopes allow efficient computations for the cont. evolution. Efficient conversion from zonotopes to polytopes and vice versa possible; drawback: introduced overapproximation. However: Reachable sets computed by polytopes also generate an overapproximation when intersected by guards. Reachability Analysis of Nonlinear and Hybrid Systems using Zonotopes May 7, / 56

59 Stochastic Systems Stochastic Safety Verification exemplary trajectory unsafe set x 2 x 1 probability density function initial set Probability of being in an unsafe set can be computed from the probability density function (pdf). Reachability Analysis of Nonlinear and Hybrid Systems using Zonotopes May 7, / 56

60 Stochastic Systems Stochastic Safety Verification exemplary trajectory unsafe set x 2 x 1 initial set over-approximated probability density function? Probability of being in an unsafe set can be computed from the probability density function (pdf). Is the exact probability density function computable? Is an over-approximation computable and how is it defined? Reachability Analysis of Nonlinear and Hybrid Systems using Zonotopes May 7, / 56

61 Stochastic Systems Two Different Definitions Probability of entering an unsafe set X unsafe : P(reach tf )=P( t [0, t f ], x(t) X unsafe ) P(reach )=P( t 0, x(t) X unsafe ) Applied methods: Monte Carlo simulation, Markov chain abstraction, reformulation as stochastic optimal control problem,... Except for Monte Carlo simulation, methods suffer under the curse of dimensionality (usually exponential complexity in number of continuous state variables). Probability of being in an unsafe set X unsafe : P(x X unsafe, t) = f X (x, t) dx. X unsafe Equivalent to above definition when unsafe set is absorbing. Reachability Analysis of Nonlinear and Hybrid Systems using Zonotopes May 7, / 56

62 Stochastic Systems Definition of the Considered System Class Considered System Class Ẋ = AX(t)+u(t)+Cξ(t), X (0) f X (x, t =0), u(t) U, ξˆ=white noise where A and C are matrices of proper dimension and A has full rank. X (t) is a stochastic process, f X (x, t) its probability density function. There are two kinds of inputs: u(t): can take values from a set U; no stochastic information given. Cξ(t): white noise input with multivariate Gaussian distribution. Reachability Analysis of Nonlinear and Hybrid Systems using Zonotopes May 7, / 56

63 Stochastic Systems Enclosing Hull of Probability Distributions Input trajectory u(t) is known: exact solution has Gaussian distribution: X (t) N (μ(t), Σ(t)) with mean value μ(t) and covariance matrix Σ(t). Input trajectory u(t) is unknown: Enclosing hulls required: f X (x, t = r) =sup{f X (x, t = r) X (t) is a stochastic process, u(t) U, f X (x, 0) = f 0 } f X (x) Enclosing hull f X (x, t = r) Exemplary probability density function x Reachability Analysis of Nonlinear and Hybrid Systems using Zonotopes May 7, / 56

64 Stochastic Systems Enclosing Hull for Time Intervals Numerical examples: f (0), f (r): probability distribution at time t =0andt = r, f ([0, r]): enclosing probabilistic hull for t [0, r]. f ([0, r]) f (r) f (0) x (e) One dimensional example. 0.8 f ([0, r]) 0.4 f (r) 0 f (0) 5 5 x x 1 (f) Two dimensional example. Uncertain mean is modeled by a zonotope computational methods from previous slides can be applied. Reachability Analysis of Nonlinear and Hybrid Systems using Zonotopes May 7, / 56

65 Stochastic Systems Two-dimensional example ẋ = [ ] 1 4 x [ ] [ ] [ 0.01, 0.01] ξ. [ 0.01, 0.01] (a) Simulation examples. Γ(R(0)) (b) Enclosing probabilistic hulls. Reachability Analysis of Nonlinear and Hybrid Systems using Zonotopes May 7, / 56

66 Stochastic Systems Five-dimensional example A = ẋ = Ax + u +0.5 I ξ, T [ 0.1, 0.1], u U =. [ 0.1, 0.1] Unsafe set B Γ(R(0)) Γ(R(0)) (a) Projection on x 2, x 3. (b) Projection on x 4, x 5. Reachability Analysis of Nonlinear and Hybrid Systems using Zonotopes May 7, / 56

67 Stochastic Systems Results and Computational Times r =0.02 r =0.04 p p(kr) p([kr, (k +1)r]) 0.6 p p([kr, (k +1)r]) p(kr) time t time t Computational times: Higher order systems with randomly generated matrices A, C computed (Matlab + single core desktop computer (AMD Athlon )). Dimension n CPU-time [sec] Reachability Analysis of Nonlinear and Hybrid Systems using Zonotopes May 7, / 56

68 Conclusions Stochastic Systems Efficient computation of enclosing probabilistic hulls for high dimensional linear systems. Algorithm allows combining Gaussian white noise with disturbances of unknown stochastic properties. Over-approximative approach allows one to consider non-gaussian noise. Possible integration in algorithms for the reachability analysis of nonlinear systems. Reachability Analysis of Nonlinear and Hybrid Systems using Zonotopes May 7, / 56

69 Autonomous Cars Safety Assessment of Autonomous Cars Trajectory planner: Planned trajectory Cycle time 0.5 sec Environment sensors: Road geometry Static obstacles Dynamic obstacles Prediction horizon > Cycle time Prediction has to be faster than real time. Safety Verification: Predict situation for each cycle Crash probabilities Reachability Analysis of Nonlinear and Hybrid Systems using Zonotopes May 7, / 56

70 Autonomous Cars Modeling of Other Traffic Participants Structured Environments: Vehicles follow preferred paths such as traffic lanes. Used for: Normal behavior of traffic participants on a road network. Longitudinal dynamics: ṡ = v, v = f (v, u), s : position, v :velocity, u : input. Longitudinal probability distribution based on this model. Lateral dynamics: Difficult to model (e.g. driver model for lane keeping) Static probability distribution. Reachability Analysis of Nonlinear and Hybrid Systems using Zonotopes May 7, / 56

71 Autonomous Cars Modeling of Other Traffic Participants Structured Environments: Vehicles follow preferred paths such as traffic lanes. Used for: Normal behavior of traffic participants on a road network. path 1 pathsegment car Δs f (s,δ) path 2 δ f (δ) f (s) s Reachability Analysis of Nonlinear and Hybrid Systems using Zonotopes May 7, / 56

72 Autonomous Cars Abstraction to Markov Chains via Reachable Sets Geometric determination of the transition probabilities Φ α ij : Φ α ij = V (Rα j X i ) V (Rj α), V(): Volume j: Initial cell, i: Cell after transition, alpha: Input cell initial cell reachable set R α j initial cell reachable cells v [m/s] v [m/s] s [m] s [m] Reachability Analysis of Nonlinear and Hybrid Systems using Zonotopes May 7, / 56

73 Autonomous Cars Abstraction to Markov Chains via Simulations Counting the number N of final states in cells: Φ α ij = Nα ij. i Nα ij j: Initial cell, i: Cell after transition, alpha: Input cell initial cell simulation results initial cell reachable cells v [m/s] v [m/s] s [m] s [m] Reachability Analysis of Nonlinear and Hybrid Systems using Zonotopes May 7, / 56

74 Overtaking Scenario Autonomous Cars t [0, 0.5] s t [2, 2.5] s t [4, 4.5] s t [6, 6.5] s t [8, 8.5] s other car planned path bicycle autonom. car Reachability Analysis of Nonlinear and Hybrid Systems using Zonotopes May 7, / 56

75 Overtaking Scenario Autonomous Cars probability of crash 8 x Car (right turn) Car (left turn) Bicycle Total time t [sec] Reachability Analysis of Nonlinear and Hybrid Systems using Zonotopes May 7, / 56

76 Test Drive Autonomous Cars left car right car autonomous car initial position of the prediction Reachability Analysis of Nonlinear and Hybrid Systems using Zonotopes May 7, / 56

77 Autonomous Cars Conclusions Traffic is inherently unsafe Stochastic verification. Non-stochastic safety verification is only useful when all vehicles broadcast their plans. Separation into longitudinal & lateral dynamics saves computational time. Things that have not been presented: Adaption of the longitudinal dynamics according to lane curvature, speed limit, interaction with traffic participants, lane changing. Comparison with Monte Carlo simulation: Probability distribution: Markov chain abstraction is better. Crash probability: Monte Carlo simulation is better. Reachability Analysis of Nonlinear and Hybrid Systems using Zonotopes May 7, / 56