Numerical Analysis and Reachability

Transcription

1 Numerical Analysis and Reachability Hybrid Systems: Reachability Goran Frehse February 22, 2017 Ph.D. Course: Hybrid Systems Ph.D. in Information Technology Dipartimento di Elettronica, Informazione e Bioingegneria (DEIB) Politecnico di Milano

2 Overview Numerical Analysis Running Example Solving ODEs Computing Trajectories and Jumps Reachability with Piecewise Affine Dynamics Reachability with Nonlinear Dynamics 2

3 From Numbers to Sets Symbolic state-space exploration can be seen as a generalization of numerical analysis from numbers to sets. Look at numerical analysis to understand the principles see the difficulties 3

4 Running Example: Ball on String L L x 0 m F s x 0 m F g F g (a) extension (b) freefall 4

5 Equations of Motion dynamics in freefall when x 0, with mass m, mẍ = F g = mg. dynamics in extension when x 0, with spring constant k, damping factor d, mẍ = F g + F s = mg kx dẋ. transition when x = L, collision factor c [0, 1], ẋ = cẋ. 5

6 Equations of Motion as 1st order ODEs auxiliary variable v = ẋ, so v = ẍ. dynamics in freefall when x 0, ) ( ) ( ) ( ) (ẋ 0 1 x 0 = + v 0 0 v g dynamics in extension when x 0, ) ( ) ( ) ( ) (ẋ 0 1 x 0 = + v v g k m d m 6

7 Jumps as 1st order Difference Equations transition when x = L, collision factor c [0, 1], ẋ = cẋ. difference equation using auxiliary variable v = ẋ, denoting by x the target value of x: ( ) ( ) ( ) x 1 0 x = v 0 c v 7

8 Hybrid automaton model clip from SpaceEx Model Editor 1 1 G. Frehse, C. L. Guernic, A. Donzé, R. Ray, O. Lebeltel, R. Ripado, A. Girard, T. Dang, and O. Maler, SpaceEx: Scalable verification of hybrid systems, in CAV 11, ser. LNCS, Springer,

9 Behavior over Time 1 x 2 position x 0 x 1 x 3 x 4 x 5 1 x t velocity v 5 0 v 0 v 1 v 2 v 2 v 4 v 5 5 v t 9

10 Behavior in the State Space ξ 0 (δ 0 ) = x 1 5 ξ 3 (δ 3 ) = x 4 ξ 1 (δ 1 ) velocity v 0 x 0 ξ 4 (δ 4 ) = x 5 x 2 5 ξ 2 (δ 2 ) = x position x 10

11 Overview Numerical Analysis Running Example Solving ODEs Computing Trajectories and Jumps Reachability with Piecewise Affine Dynamics Reachability with Nonlinear Dynamics 11

12 Solving ODEs Given an ordinary differential equation (ODE) ẋ = f(x), with initial value x 0, find ξ(t) with ξ(0) = x 0 and ξ(t) = f(ξ(t)) for all t 0. Numerical solution by computing x 0,..., x N such that x i ξ(t i ) at time points t 0,..., t N. 2 Using fixed time step h: t i = ih. 2 R. P. Canale and S. C. Chapra, Numerical methods for engineers, Mc Graw Hill, New York,

13 Euler s Method Compute x 0,..., x N with the sequence x i+1 = x i + f(x i )h. Comparing to Taylor series around x i, (n 1) x i+1 = x i + ẋ i h + ẍi x i 2! h n! hn +, obtain estimate of local error ε a = O(h 2 ). global error ε g = O(h) first-order method accuracy limited by numerical roundoff error O(1/h) 13

14 Ball on String in Extension: Euler s Method 1 h = 0.05 position x 0 h = h = t 14

15 Ball on String in Extension: Euler s Method x 2 x 3 velocity v 5 0 x 1 x h = h = x 15

16 Stability The linear ODE ẋ = ax, converges to zero iff a < 0. Euler s method x i+1 = x i + f(x i )h = x i + ax i h = (1 + ah)x i converges to zero iff 1 + ah < 1 conditionally stable. 16

17 Backwards Euler Method Compute x 0,..., x N with the sequence x i+1 = x i + f(x i+1 )h, solved for x i+1 at each i using root-finding (Newton s method). implicit method Backwards Euler for ẋ = ax, x i+1 = x i + ax i+1 h = 1 1 ah x i converges for all a < 0, h > 0 unconditionally stable. 17

18 Runge-Kutta Methods Explicit Runge-Kutta methods compute the sequence x i+1 = x i + ϕ(x i, h)h, ϕ(x i, h) = a 1 k 1 + a 2 k a n k n, weights a i,q ij and derivative k j = f(ˆx i j ) at intermediate states ˆx 1 i = x i, ˆx 2 i = x i + q 11 k 1 h, ˆx 3 i = x i + q 21 k 1 h + q 21 k 2 h,. ˆx n i = x i + q (n 1)1 k 1 h + q (n 1)2 k 2 h + + q (n 1)(n 1) k n 1 h 18

19 (Kutta, 1901) Runge-Kutta Methods Runge-Kutta method defined by n and parameters a i,q ij chosen to match first n terms of Taylor series. Remaining degrees of freedom used to optimize, e.g., truncation error O(h n+1 ) and global error O(h n ) for n = 2,..., 5. Ralston s method has the smallest truncation error for n = 2: k 1 = f(ˆx 1 i ), ˆx 1 i = x i, k 2 = f(ˆx 2 i ), ˆx 2 i = x i + 3/4k 1 h, ϕ(x i, h) = 1k k

20 Ball on String in Extension: Ralston s Method position x x 2 ˆx 2 2 x 3 ˆx 2 3 h = 0.05 h = ˆx 2 x t 20

21 Ball on String in Extension: Ralston s Method ˆx 2 1 x 1 ˆx h = 0.05 x 2 h = 0.1 velocity v 0 x 0 x 3 ˆx ˆx 2 4 x x 21

22 Error Estimation Error estimation using difference between results for time steps h/2 and h, results for (n 1)th and nth order solver. Runge-Kutta-Fehlberg (RKF) methods (Fehlberg, 1970) compare (n 1)th and nth order RK, reuse intermediate results, no more evaluations than nth order RK. Popular: RKF 2(3) and RKF 4(5), aka and. 22

23 Adaptive Time Steps Adapt time step using error estimation ε a and desired error ϵ d. Heuristics 3 : h h ϵd /ε a 0.25 if εa < ϵ d h h ϵ d /ε a 0.2 if εa ϵ d 3 W. H. Press, Numerical recipes 3rd edition: The art of scientific computing. Cambridge University Press,

24 Stiff Systems Stiff ODEs have time constants (Eigenvalues) differing by a factor of 1000 or more. Solvers take tiny time steps throughout the entire time horizon Special solvers are available for stiff ODEs, using implicit methods to achieve stability at larger time steps. Ball on String example is stiff for small mass. 24

25 Ball on String: Stiff for Small Mass 0 h = position x velocity v x 0 h = { , t , t > 0.4 h = h = { , t , t > 0.4 h = h = 5 10 v t t 25

26 Overview Numerical Analysis Running Example Solving ODEs Computing Trajectories and Jumps Reachability with Piecewise Affine Dynamics Reachability with Nonlinear Dynamics 26

27 Computing Trajectories and Jumps Numerical simulation of hybrid automata: use ODE solver to approximate trajectories, detect when trajectory enters guard, detect when trajectory leaves invariant. ODE solver offer zero crossing detection using root-finding algorithms. Detecting guards/invariants using root functions is computationally expensive and potentially inaccurate. 27

28 Shortcomings 4 Missed roots violations of invariant or entering guard go undetected. Increased cost ODE solvers reuse intermediate states for increasing time sequence. Lost through back-and-forth of root-finding. Spurious behavior Numerically approximated state may lie slightly outside the guard or invariant, so constraints are relaxed 4 F. Zhang, M. Yeddanapudi, and P. Mosterman, Zero-crossing location and detection algorithms for hybrid system simulation, in IFAC World Congress, 2008, pp

29 Zeno Behavior Zeno behavior occurs if infinitely many events occur in a bounded time interval. 5 Chattering Zeno: zero-time events Genuine Zeno: event times converge towards a fixed point Simulator seems to get stuck as switching times converge. Ball on String example zeno if upside-down (negative gravity) bouncing ball. 5 A. D. Ames and S. Sastry, Characterization of zeno behavior in hybrid systems using homological methods, in ACC 05,

30 Ball on String: Zeno Behavior 1 x 0 position x t velocity v v t 30

31 Accounting for Nondeterminism The biggest challenge is nondeterminism: select initial state and successor states in jump relation; choose between transitions if guards overlap; choose jump time from interval of time; differential inclusions, such as ẋ [ 1, 1], require to pick derivative for each time step; Number of runs increases exponentially with each choice. Simulators like Simulink, Modelica, or Ptolemy use purely deterministic models that jump as soon as possible. 31

32 Overview Numerical Analysis Reachability with Piecewise Affine Dynamics Set Representations SpaceEx (advertisement) EMB Case Study Reachability with Nonlinear Dynamics 32

33 Piecewise Affine Dynamics Hybrid automata with piecewise affine dynamics (PWA) initial states and invariants are polyhedra, flows are affine ODEs ẋ = Ax + Bu, u U, jumps have a guard set and assignments x = Cx + w, w W. 33

34 Exercise: When will Lake Mead go dry? 6 by Serge Melki from Indianapolis, USA [CC BY 2.0], via Wikimedia Commons Lake Mead is fed by the Colorado River r in 15 million acre feet per year (maf/yr). The feed rate is expected to decrease by up to 30% over t dec = 50 years. Water is taken out of the lake at a rate of 14 maf/yr, plus 1.7 maf/yr due to evaporation and infiltration, resulting in a constant loss of r out 15.7 maf/yr. 6 Barnett and Pierce. When will Lake Mead run dry? J. Water Resources Research,

35 Exercise: When will Lake Mead go dry? Switched Affine Model: In the rainy season, April to June, r in,rainy = 30 maf/yr, while in the rest of the year, dry season, r in,dry = 10 maf/yr. a) Model that r in decreases linearly by 30% in t dec years. b) In what time frame could Lake Mead be dry? 35

36 Reachability Discrete Successors Edge e = (l, α, l ) with guard x G and nondeterministic assignment x = Cx + w, w W, post e (l P) = l ( C(P G) W ) Inv(l ). 36

37 Reachability Continuous successors ẋ = Ax + Bu, u U, trajectory ξ(t) from ξ(0) = x 0 for given input signal ζ(t) U: ξ x0,ζ(t) = e At x 0 + t 0 e A(t s) Bζ(s)ds. reachable states from set X 0 for any input signal: Y t = t 0 X t = e At X 0 Y t, t/δ e As BUds = lim e Aδk δbu. δ 0 k=0 37

38 Computing a Convex Cover Ω 2 X 2δ X δ Ω 1 X 0 Ω 0 Compute Ω 0, Ω 1,... such that X t Ω 0 Ω t T 38

39 Time Discretization Ω 2 X 2δ X δ Ω 1 X 0 Semi-group property: (X kδ ) δ = X (k+1)δ Time discretization: X (k+1)δ = e Aδ X kδ Y δ. Ω 0 Given initial approximations Ω 0 and Ψ δ such that X t Ω 0, Y δ Ψ δ, 0 t δ X t is covered by the sequence Ω k+1 = e Aδ Ω k Ψ δ. 39

40 Ball on String: Reachable States (clip from SpaceEx output) 40

41 Initial Approximations X δ Ω 0 X 0 (a) convex hull and pushing facets (b) convex hull and bloating 41

42 Initial Approximations Forward Bloating Bloating based on norms: 7 Ω 0 = chull(x 0 e Aδ X 0 ) (α δ + β δ )B, Ψ δ = β δ B, α δ = µ(x 0 ) (e A δ 1 A δ), β δ = 1 µ(bu) A (e A δ 1), with radius µ(x ) = max x X x and unit ball B. 7 A. Girard, Reachability of uncertain linear systems using zonotopes, in HSCC, 2005, pp

43 Initial Approximations Forward Bloating X 2δ X δ ˆΩ 2 ˆΩ 1 X 0 Forward bloating is tight on X 0 and bloated on X δ. Improvements: intersect forward bloating with backward bloating bloat based on interpolation error (shown before) ˆΩ 0 43

44 Wrapping Effect Appr(e Aδ Appr(e Aδ X 0 )) Appr(e Aδ X 0 ) Appr(e A2δ X 0 ) Appr(e Aδ X 0 ) e Aδ X 0 e Aδ X 0 X 0 X 0 (a) with wrapping effect (b) using a wrapping-free algorithm avoid increasing complexity through approximation ˆΩ k+1 = Appr(e Aδ ˆΩk Ψ δ ). wrapping effect: error accumulation 44

45 Wrapping Effect Solution: Split sequence 8 ˆΨ k+1 = Appr(e Akδ Ψ δ ) ˆΨ k, with ˆΨ 0 = {0}, ˆΩ k = Appr(e Akδ Ω 0 ) ˆΨ k. satisfies ˆΩ k = Appr(Ω k ) (wrapping-free) if e.g., bounding box. Appr(P Q) = Appr(P) Appr(Q), 8 A. Girard, C. L. Guernic, and O. Maler, Efficient computation of reachable sets of linear time-invariant systems with inputs, in HSCC, 2006, pp

46 Polyhedra Ω 2 Ω 1 X 2δ Ω 0 X δ X 0 #ops polyhedra m constr. k gen. convex hull exp 1 Minkowski sum exp nk 2 linear map n 2 m / exp n 2 k intersection 1 exp 46

47 Ellipsoids 9 X 2δ X δ ˆΩ 2 ˆΩ 1 X 0 ˆΩ 0 #ops polyhedra ellipsoids m constr. k gen. n n matrix convex hull exp 1 approx Minkowski sum exp nk 2 approx linear map n 2 m / exp n 2 k n 3 intersection 1 exp approx 9 A. B. Kurzhanski and P. Varaiya, Dynamics and control of trajectory tubes. Springer,

48 Zonotopes v 2 v 1 v 4 c v 3 Zonotope with center c R n and generators v 1,..., v k R n P = { c + k i=1 } α i v i α i [ 1, 1]. subclass of centrally symmetric polytopes 48

49 Zonotopes alternative representation: affine image of hypercube P = c MB k. generator matrix: M = [v 1... v k ] unit hypercube in k dimensions: { } B k = x R k x 1. 49

50 Zonotopes convex hull: approximation 10 CH(P e Aδ P) 1 2 { c + e Aδ c + k P i=1 α i (v i + e Aδ v i ) + 2k P i=k P +1 α i (v i e Aδ v i ) } + α 2kP +1(c e Aδ c) α i [ 1, 1]. binary operations: 2n 2 (k + 1) + 2n(k + 2) 10 A. Girard, Reachability of uncertain linear systems using zonotopes, in HSCC, 2005, pp

51 Zonotopes Minkowski sum: add centers and join generators P Q = { c P + c Q + k P i=1 α i v P i + k P +k Q i=k P +1 α i v Q i } α i [ 1, 1]. binary operations: n 51

52 Zonotopes linear map: map center and generators MP = { Mc + k i=1 binary operations: n 2 (k + 1) } α i Mv i α i [ 1, 1]. 52

53 Zonotopes in Reachability reachable states from set X 0 for any input signal: approximation: t/δ X t = e At X 0 lim e Aδk δu. δ 0 k=0 t/δ ˆX t = e At X 0 e Aδk (δu E U,δ ). k=0 with X 0, U zonotopes, ˆX t is a zonotope for any δ > 0 53

54 Zonotopes 11 X 2δ X δ ˆΩ 2 ˆΩ 1 X 0 ˆΩ 0 #ops polyhedra ellipsoids zonotopes m constr. k gen. n n matrix k generators convex hull exp 1 approx n 2 k approx Minkowski sum exp nk 2 approx n linear map n 2 m / exp n 2 k n 3 n 2 k intersection 1 exp approx approx 11 A. Girard, Reachability of uncertain linear systems using zonotopes, in HSCC, 2005, pp

55 Support Functions d 2 ρ P (d) d d 3 d 1 0 d 4 (a) support function in direction d (b) outer approximation support function = linear optimization (efficient!) ρ P (d) = max{d T x x P}. computed values define polyhedral outer approximation P D = d D{ d T x ρ P (d) }. 55

56 Support Functions d 2 ρ P (d) d d 3 d 1 0 d 4 (a) support function in direction d (b) outer approximation linear map: ρ MX (l) = ρ X (M T l), O(mn), convex hull: ρ chull(p Q) (l) = max{ρ P (l), ρ Q (l)}, O(1), Minkowski sum: ρ X Y (l) = ρ X (l) + ρ Y (l), O(1). 56

57 (Le Guernic, Girard, 09)[12] Support Functions X 2δ X 2δ X δ X δ ˆΩ 2 ˆΩ 2 ˆΩ 1 X 0 ˆΩ 1 X 0 ˆΩ 0 ˆΩ 0 support functions: lazy approximation on demand #ops polyhedra ellipsoids zonotopes support f. m constr. k gen. n n matrix k generators convex hull exp 1 approx n 2 k approx 2#ops Minkowski sum exp nk 2 approx n 2#ops linear map n 2 m / exp n 2 k n 3 n 2 k n 2 + #ops intersection 1 exp approx approx opt. / approx 57

58 Example: Switched Oscillator Switched oscillator 2 continuous variables 4 discrete states similar to many circuits (Buck converters, ) plus linear filter m continuous variables dampens output signal affine dynamics total 2 + m continuous variables 58

59 Example: Switched Oscillator Low number of directions sufficient? here: 6 state variables 12 box constraints (axis directions) 72 octagonal constraints (± x i ± x j ) 59

60 Example: Switched Oscillator Scalability Measurements: fixpoint reached in O(nm 2 ) time box constraints: O(n 3 ) octagonal constraints: O(n 5 ) runtime [s] number of variables n 60

61 Example: Controlled Helicopter Photo by Andrew P Clarke 28-dim model of a Westland Lynx helicopter 8-dim model of flight dynamics 20-dim continuous H controller for disturbance rejection stiff, highly coupled dynamics 61

62 28 state variables + clock Example: Helicopter CAV 11: 1440 sets in 5.9s 1440 time steps 62

63 Example: Helicopter 28 state variables + clock convex in 29 dimensions! HSCC 13: 32 sets in 15.2s (4.8s clustering) time steps, median

64 Example: Chaotic Circuit piecewise linear Rössler-like circuit Pisarchik, Jaimes-Reátegui. ICCSDS 05 added nondet. disturbances 3 variables, hard! 64

65 Overview Numerical Analysis Reachability with Piecewise Affine Dynamics Set Representations SpaceEx (advertisement) EMB Case Study Reachability with Nonlinear Dynamics 65

66 SpaceEx Verification Platform Browser-based GUI 2D/3D output runs remotely 66

67 SpaceEx Model Editor Components = Hybrid Automata real-values variables ODE, linear DAE 67

68 SpaceEx Model Editor Block diagrams connect components templates, nesting 68

69 SpaceEx Reachability Algorithms PHAVer constant dynamics (LHA) formally sound and exact Support Function Algo many continuous variables low discrete complexity Simulation nonlinear dynamics based on CVODE spaceex.imag.fr 69

70 Overview Numerical Analysis Reachability with Piecewise Affine Dynamics Set Representations SpaceEx (advertisement) EMB Case Study Reachability with Nonlinear Dynamics 70

71 Case Study: Electro-Mechanical Brake 71

72 Case Study: Electro-Mechanical Brake Controller Implementation discrete time fixed-point arithmetic multi-tasking processor: scheduling with uncertain frequency worst-case analysis too conservative 72

73 Case Study: Electro-Mechanical Brake 73

74 Case Study: Electro-Mechanical Brake Typical Worst-Case Execution Time limit missed schedules per time interval 74

75 Case Study: Electro-Mechanical Brake caliper position time 75

76 Case Study: Electro-Mechanical Brake caliper position only failure hard to detect artificial failure case (inconsistent with classical theory) time 76

77 Case Study: Electro-Mechanical Brake current physical properties: maximum impulse on contact (measured via current) time 77

78 Overview Numerical Analysis Reachability with Piecewise Affine Dynamics Reachability with Nonlinear Dynamics 78

79 Nonlinear Dynamics Linearization ẋ = f(x), with f globally Lipschitz continuous. Linearization: choose domain S (partition, sliding window) overapproximate in S with ẋ = Ax + u, u U linearizing f(x) around x 0 S gives a ij = f i x j and b = f(x 0 ) Ax 0. x=x0 U = Appr {f(x) (Ax + b), x S} b. 79

80 Example: Van der Pol Oscillator 12 ẋ = y ẏ = y(1 x 2 ) x hybridization: here triangular partition of size 0.05 partitioning generally doesn t scale well 12 E. Asarin, T. Dang, and A. Girard, Hybridization methods for the analysis of nonlinear systems, Acta Inf., vol. 43, no. 7, pp ,

81 Nonlinear Dynamics Polynomial Approximations Bernstein polynomials for polynomial f(x) polyhedral approximation of successors 13 Taylor models polynomial approximations of Taylor expansion represent sets with polynomials Flow* verification tool [15] 13 T. Dang and R. Testylier, Reachability analysis for polynomial dynamical systems using the bernstein expansion, Reliable Computing, vol. 17, no. 2, pp ,

82 References [2] R. Alur, C. Courcoubetis, N. Halbwachs, T. Henzinger, P.-H. Ho, X. Nicollin, A. Olivero, J. Sifakis, and S. Yovine, The algorithmic analysis of hybrid systems, Theoretical Computer Science, vol. 138, pp. 3 34, [3] T. A. Henzinger, The theory of hybrid automata., in LICS, Los Alamitos: IEEE Computer Society, 1996, pp [8] T. A. Henzinger, P. W. Kopke, A. Puri, and P. Varaiya, What s decidable about hybrid automata? Journal of Computer and System Sciences, vol. 57, pp , [12] C. Le Guernic and A. Girard, Reachability analysis of linear systems using support functions, Nonlinear Analysis: Hybrid Systems, vol. 4, no. 2, pp , [15] X. Chen, E. Ábrahám, and S. Sankaranarayanan, Taylor model flowpipe construction for non-linear hybrid systems, in RTSS, IEEE Computer Society, 2012, pp , ISBN: