Reachability of Hybrid Systems using Support Functions over Continuous Time

Transcription

1 Reachability of Hybrid Systems using Support Functions over Continuous Time Goran Frehse, Alexandre Donzé, Scott Cotton, Rajarshi Ray, Olivier Lebeltel, Rajat Kateja, Manish Goyal, Rodolfo Ripado, Thao Dang, Oded Maler Université Grenoble 1 Joseph Fourier / CNRS Verimag, France Colas Le Guernic DGA, France Antoine Girard Laboratoire Jean Kuntzmann, France GIPSA Seminar, Grenoble, April 22,

2 Outline!! Hybrid Systems and Reachability!! Reachability with Support Functions! Computing with High-Dimensional Sets!! Approximation in Space-Time! Reducing the Number of Sets! Measuring the Approximation Error!! SpaceEx Development Platform 2

3 Hybrid Systems - Semantics!! Continous/Discrete Behaviour! evolution with time according to ODE dynamics! dynamics can switch (instantaneous)! state can jump (instantaneous) x (t) x (t) x (t) 3

4 Modeling Hybrid Systems!! Example: Bouncing Ball! ball with mass m and position x in free fall! bounces when it hits the ground at x = 0! initially at position x and at rest x F g 0 4

5 Hybrid Automaton Model location freefall invariant flow 5

6 !! States over Time Example: Bouncing Ball position x x 0 x (t) x (t) x (t) x (t) x (t) time t velocity v 0 v (t) v (t) v (t) v (t) v (t) time t 6

7 Example: Bouncing Ball!! States over States = State-Space View position x x x (t) behavior from single initial state x (t) x (t) 0 velocity v 7

8 Example: Bouncing Ball!! Reachability in State-Space position x behaviors from set of initial states = reachable states velocity v 8

9 Application: Reachability in Model Based Design Plant Model Controller Synthesis Simulation Reachability Deployment 9

10 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 S. Skogestad and I. Postlethwaite, Multivariable Feedback Control: Analysis and Design. John Wiley & Sons,

11 Simulation vs Reachability!! Simulation! approximative sample of single behavior! over finite time!! Reachability! over-approximative set-valued cover of all behaviors! over finite or infinite time simulation run vertical speed reachable states over time 11

12 Simulation vs Reachability!! Simulation! deterministic! resolve nondet. using Monte Carlo etc.! scalable for nonlinear dyn.!! Reachability! nondeterministic! continuous disturbances...! implementation tolerances...! scalable for linear dynamics 1000 simulations vertical speed Reachable set equiv. >2 28 corner case simulations 12

13 Example: Controlled Helicopter!! Comparing two controllers subject to continuous disturbance 13

14 Outline!! Hybrid Systems and Reachability!! Reachability with Support Functions!! Approximation in Space-Time!! SpaceEx Development Platform 14

15 Hybrid Automata with Affine Dynamics!! linear differential equations!! can be highly nondeterministic:! additive inputs u,wmodel continuous disturbances (noise etc.) Key: find approximation that is efficient but accurate for a large number of continuous variables 15

16 Time Elapse Computation!! Continuous time elapse for affine dynamics! efficient, scalable! approximation without accumulation of approximation error (wrapping effect)!! Much heritage from prior work! Chutinan, Krogh. HSCC 99! Asarin, Bournez, Dang, Maler. HSCC 00! Girard. HSCC 05! Le Guernic, Girard. HSCC 06, CAV 09 16

17 Affine Dynamics!! linear terms plus inputs U:!! solution: matrix exponential factors influence of inputs (stable system forgets the past) 17

18 From Time-Discretization to Reach!! States in discrete time: X 3"#... integral over inputs X "# X 2"# X 0 Reach(X 0 ) " 2" 3" 0 t need to cover also states in between! 18

19 From Time-Discretization to Reach!! Cover in discrete time: $ [2",3"]... $ [0,"] $ [",2"] X 0 Reach(X 0 ) 0 " 2" 3" t Minkowski sum = pointwise sum of sets 19

20 From Time-Discretization to Reach!! 1st order Taylor approximation!! different bounds on the remainder actual trajectory CAV 11: Complex Polytope CAV 11 LeGuernic 10 LeGuernic,Girard 09 20

21 Operations on Convex Sets Polyhedra Operators Constraints Vertices Zonotopes Support F. Convex hull Affine transform +/ Minkowski sum Intersection Le Guernic, Girard. CAV 09 21

22 !! Support Function R n R Support Functions! direction d position of supporting halfspace! exact set representation x * support vector P d 0 22

23 Support Functions!! black box representation of a convex set!! implementation: function objects direction vector Support Function tangent hyperplane 23

24 Support Functions!! black box representation of a convex set!! implementation: function objects direction vector Support Function 24

25 Support Functions!! black box representation of a convex set!! implementation: function objects direction vector Support Function Support Function 25

26 Computing with Support Functions!! Simple Operations (Le Guernic, Girard. CAV 09)! Linear Transform:! O(n 2 ) incl. existential quantification! Minkowski sum:! O(1)! Convex hull:! O(1) 26

27 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 27

28 Example: Switched Oscillator!! Low number of directions sufficient?! here: 6 state variables 12 box constraints (axis directions) 72 octagonal constraints (± x i ± x j ) 28

29 Example: Switched Oscillator!! Scalability Measurements:! fixpoint reached in O(nm 2 ) time! box constraints: O(n 3 )! octagonal constraints: O(n 5 ) 29

30 Outline!! Hybrid Systems and Reachability!! Reachability with Support Functions!! Approximation in Space-Time!! SpaceEx Development Platform 30

31 Flowpipe Reachable States over Time y flowpipe relatively easy to compute at discrete time points initial states x 31

32 Approximation with Template Polyhedra y template polyhedron covers time interval x!asarin, Dang, Maler, CAV 02!Girard, HSCC 05!Le Guernic, Girard, CAV 09!Frehse et al. CAV 11 initial states 32

33 Approximation with Template Polyhedra start from convex hull compensate for curvature 33

34 Approximation with Template Polyhedra y template polyhedra cover flowpipe x 34

35 Problem 1: Too many sets y sets intersect with guard guard x 35

36 Problem 1: Too many sets!! Jumps: Exponential increase in the number of sets! case studies: 10x 100x per jump y convex hull clustering (expensive) x 36

37 Problem 2: Quantify error y approximation error x four sources of error... 37

38 Problem 2: Quantify error (2)! bloating error (curvature estimation) (1)! template error (distance to convex hull) (3) convexity error (unknown) 38

39 Problem 2: Quantify error y straight line trajectories non-convex reach set It s not just curvature... x 39

40 Problem 2: Quantify error y t x convex hull over any interval incurs error 40

41 Problem 2: Quantify error y (3)! convexity error x 41

42 Problem 2: Quantify error y (4)! clustering error (hard) x 42

43 Goals!! Reduce the number of convex sets!! Measure the total error 1)! template error 2)! bloating error 3)! convexity error directional error 4)! clustering error 43

44 Approximation in Space-Time y Improve the approximation by adding time... x 44

45 Approximation in Space-Time y x 45

46 Approximation in Space-Time y x 46

47 Approximation in Space-Time y t x approximation constant over time interval 47

48 Approximation in Space-Time y x approximation pointwise in time (convex sets) t 48

49 Support Sampling!! interval bound on support function in template directions interval bound on support function 49

50 Support Sampling!! bounds support function in all directions!! bounds Hausdorff distance bound on approximation error worst-case points within sampling direction with worst error 50

51 Flowpipe Sampling y support in d d x t compute support at discrete time points t 51

52 Flowpipe Sampling - Interpolation!! 1st order Taylor approx. CAV 11 support in d upper bound lower bound t!! independent time scales! per direction!! parallelizeable piecewise linear scalar functions 52

53 Flowpipe Sampling - Convexification y support in d d support sampling at each t x t infinite union of template polyhedra (one for each t) t 53

54 Flowpipe Sampling - Convexification y non-template polyhedron support in d d concave piece x t finite union of non-template polyhedra (one for each concave piece) template, bloating & convexity error measurable t 54

55 Flowpipe Sampling - Convexification support in d linear constraint in space-time bloating & convexity error flowpipe samplings need to be concave for every direction t 55

56 Flowpipe Sampling - Clustering support in d concave envelope bloating, convexity & clustering error convex hull = concave envelope (efficient for pwl functions) t 56

57 Flowpipe Sampling - Clustering!! for given error, minimize number of convex sets! equiv. number of concave pieces upper bound lower bound 57

58 Flowpipe Sampling - Clustering!! for given error, minimize number of convex sets! equiv. number of concave pieces lower bound + error desired error upper bound lower bound 58

59 Flowpipe Sampling - Clustering lower bound + error desired error any function inside gap is -!conservative upper bound -!within desired error upper bound 59

60 Flowpipe Sampling - Clustering lower bound + error upper bound find inflection intervals: every pw concave function must cut pieces here (cannot drive a car from left to right without making two left turns) 60

61 Flowpipe Sampling - Clustering lower bound + error upper bound 2 inflection intervals: at least 3 concave pieces combining inflection intervals over all directions = graph coloring problem 61

62 Example: Bouncing Ball Clustering up to total error 0.1 = 8 pieces 62

63 Example: Bouncing Ball Clustering up to total error 1.0 = 2 pieces 63

64 Example: Bouncing Ball CAV 11 HSCC 13 64

65 !! 28 state variables + clock Example: Helicopter CAV 11: 1440 sets in 5.9s 1440 time steps 65

66 !! 28 state variables + clock Example: Helicopter CAV 11: 32 sets in 15.2s (4.8s clustering) time steps, median

67 Example: Overhead Crane!! 4 state variables, error < 0.05 angular velocity 369 sets, 0.15s 15 sets, 0.28s position 67

68 Example: Overhead Crane!! 4 state variables, error < angular velocity 1108 sets, 0.36s 32 sets, 0.46s position 68

69 Performance Comparison CAV 11 HSCC 13 Time Sets Time Sets Helicopter (29) Helicopter 2 (29) Bouncing Ball (3) Bouncing Ball w. Inputs (3) Three-Tank (3) Overhead Crane (4)

70 Outline!! Hybrid Systems and Reachability!! Reachability with Support Functions!! Approximation in Space-Time!! SpaceEx Development Platform 70

71 SpaceEx Verification Platform!! Analysis of Hybrid Systems! Reachability! Monitoring! Simulation!! Open Source: spaceex.imag.fr! proprietary polyhedra library! number type is templated (substitute your own)! interfaces to linear programming solvers (GLPK,PPL), Parma Polyhedra Library, ode solvers (CVODES) 71

72 SpaceEx Model Editor Networks of Hybrid Automata!real-values variables!odes 72

73 SpaceEx Model Editor Connect components in block diagrams!templates, nesting 73

74 SpaceEx Web Interface Browser-based GUI!2D/3D output!runs remotely 74

75 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 75

76 !! Reachability with 100+ variables Conclusions! convex sets as support functions!! Convexification with semitemplate data structures! total approximation error measurable!! Ongoing Work! abstraction refinement (directions)! extension to nonlinear dynamics spaceex.imag.fr 76