Reachability Analysis of Hybrid Systems Goran Frehse Universit - PowerPoint PPT Presentation

Reachability Analysis of Hybrid Systems Goran Frehse Université Grenoble 1 Joseph Fourier Verimag, France CPS Summer School, Grenoble, 2014 1

A Biased Overview from... � Grenoble – Oded Maler – Thao Dang – Antoine Girard (LJK) – Colas Le Guernic (now DGA, France) – Alexandre Donzé (now UC Berkeley) � Carnegie Mellon – Bruce Krogh � Dortmund – Sebastian Engell – Stefan Kowalewski (now RWTH Aachen) – Olaf Stursberg (now U Kassel) � missing related work : – Varaiya, Kurzhanski (ellipsoids) – Althoff (zonotopes) – Sankaranarayanan (Taylor models) 2

Example: Tunnel Diode Oscillator & ( ) 1 V I ( V ) I = � + C d C L C Tunnel V d & ( ) 1 I V RI V = � � + Diode L C L in L Dang, Donze, Maler, FMCAD’ 04 � What are good parameters? – startup conditions – parameter variations – disturbances 3

Example: Tunnel Diode Oscillator R=0.20 � � Oscillation I L [mA] initial states Time [µs] V C [V] 4

Example: Tunnel Diode Oscillator R=0.24 � � Stable equilibrium I L [mA] initial states Time [µs] V C [V] 5

Example: Tunnel Diode Oscillator � Jitter measurement – add clock that is reset at zero crossing 1.0 jitter measurement I L [mA] 14.90 12.75 0.0 0.0 t [ µ s] V d [V] time 0 0.5 6

Example: Tunnel Diode Oscillator Analog/Mixed Signal Circuit Tunnel Diode & ( ) V 1 I ( V ) I = � + C C d C L Formal Model & ( ) I 1 V RI V = � � + L L C L in I L [mA] Reachability Analysis V C [V] • Oscillation Guaranteed Safety Property • Jitter • … 7

Outline � Modeling with Hybrid Automata � Reachability versus Simulation � Reachability Algorithms – piecewise constant dynamics – piecewise affine dynamics � SpaceEx Tool Platform � Bibliography 8

Modeling with Hybrid Automata � 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 9

Part I – Free Fall � Condition for Free Fall x � 0 – ball above ground: x F g � First Principles (physical laws) 0 • gravitational force : F g = � mg g = 9 . 81m / s 2 • Newton's law of motion : m ¨ x = F g 10

Part I – Free Fall F g = � mg m ¨ x = F g x � Obtaining 1 st Order ODE System F g • ordinary differential equation ˙ x = f ( x ) 0 • transform to 1st order by introducing variables for higher derivatives • here: v = ˙ x : x ˙ = v � g v ˙ = 11

Part II – Bouncing � Conditions for “Bouncing” • ball at ground position: x = 0 • downward motion: v < 0 � Action for “Bouncing” • velocity changes direction • loss of velocity (deformation, friction) • v := � cv , 0 � c � 1 12

Combining Part I and II � Free Fall • while x � 0 , continuous dynamics x ˙ = v x ˙ = f ( x ) v ˙ = � g � Bouncing discrete dynamics • if x = 0 and v < 0 x � G � cv v := x := R ( x ) 13

Hybrid Automaton Model initial conditions x = x 0 v = 0 location freefall label x � 0 bounce invariant guard x = 0 � v < 0 x ˙ = v v := � cv � g v ˙ = reset flow discrete transition 14

ODEs with Switching � Continous/Discrete Behaviour – evolution with time according to ODE dynamics – dynamics can switch (instantaneous) – state can jump (instantaneous) x � ( t ) x � ( t ) x � ( t ) 15

Example: Bouncing Ball � States over Time x � x � ( t ) x � ( t ) position x x � ( t ) x � ( t ) x � ( t ) 0 time t 0 velocity v v � ( t ) v � ( t ) v � ( t ) v � ( t ) v � ( t ) time t 16

Example: Bouncing Ball � States over States = State-Space View position x x � x � ( t ) behavior from x � ( t ) single initial state x � ( t ) 0 velocity v 17

Example: Bouncing Ball � Reachability in State-Space position x behaviors from set of initial states = reachable states velocity v 18

Outline � Modeling with Hybrid Automata � Reachability versus Simulation � Reachability Algorithms – piecewise constant dynamics – piecewise affine dynamics � SpaceEx Tool Platform � Bibliography 19

Reachability in Model Based Design Plant Model Controller Synthesis Simulation Reachability Deployment 20

Example: Overhead Crane � State variables x,v u – position x , speed v – line angle y , angle rate w � Feedback controller y,w – state estimated by observer � Goals – validate observer for y,w – validate swing 21

Overhead Crane – Observer angle � Validation of rate actual observer quality � Standard: – Simulation of “representative estimated trajectories” time angle � Reachability: rate error – Error bounds over range of initial states & inputs angle error 22

Overhead Crane - Controller � Evaluation of swing (angle range) angle angle setpoint setpoint position position over small initial range over full operating range over small initial range over full operating range [-0.17,0.12] [-0.17,0.17] [-0.17,0.12] [-0.17,0.17] 23

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 24 S. Skogestad and I. Postlethwaite, Multivariable Feedback Control: Analysis and Design. John Wiley & Sons, 2005.

Simulation vs Reachability � Simulation � Reachability – approximative – over-approximative sample set-valued cover of single behavior of all behaviors – over finite time – over finite or infinite time simulation run vertical speed reachable states over time 25

Simulation vs Reachability � Simulation � Reachability – deterministic – nondeterministic • resolve nondet. using • continuous disturbances... Monte Carlo etc. • implementation tolerances... – scalable for nonlinear dyn. – scalable for linear dynamics 1000 simulations vertical speed Reachable set equiv. Reachable set equiv. >2 28 corner case simulations >2 28 corner case simulations 26 Frehse et al. "SpaceEx: Scalable verification of hybrid systems." Computer Aided Verification. Springer, 2011.

Example: Controlled Helicopter � Comparing two controllers subject to continuous disturbance 27 Frehse, G., et al. "SpaceEx: Scalable verification of hybrid systems." Computer Aided Verification. Springer, 2011.

Outline � Modeling with Hybrid Automata � Reachability versus Simulation � Reachability Algorithms – piecewise constant dynamics – piecewise affine dynamics � SpaceEx Tool Platform � Bibliography 28

Computing Reachable States � Computing One-Step Successors � Fixpoint computation • Initialization: R 0 = Ini • Recurrence: R k +1 = R k � Post d ( R k ) � Post c ( R k ) • Termination: R k +1 = R k � Reach = R k . 29

Computing Reachable States � Set-based integration can answer many interesting questions about a system – safety, bounded liveness,… � Problems – in general termination not guaranteed – set-based integration of ODEs is hard � Solution – piecewise constant approximations – piecewise linear approximations – math tricks (implicit set representations,...) 30

Piecewise Constant Dynamics � A very simple class of hybrid systems: Linear Hybrid Automata – trajectories are straight lines � Exact computation of successor states possible – reachability is nonetheless undecidable . 31

Linear Hybrid Automata � Continuous Dynamics • piecewise constant: ˙ x = 1 • intervals: ˙ x � [1 , 2] • conservation laws: ˙ x 1 + ˙ x 2 = 0 • general form: conjunctions of linear constraints a � Z n , b � Z , � a · ˙ � � { <, � } . x � � b, = convex polyhedron over derivatives 32

Linear Hybrid Automata � Discrete Dynamics • affine transform: x := ax + b • with intervals: x 2 := x 1 ± 0 . 5 • general form: conjunctions of linear constraints (new value x � ) a · x + a � · x � � a, a � � Z n , b � Z , � � � { <, � } � b, = convex polyhedron over x and x ’ 33

Linear Hybrid Automata � Invariants, Initial States • general form: conjunctions of linear constraints a � Z n , b � Z , � a · x � � b, � � { <, � } , = convex polyhedron over x 34

Linear Hybrid Automata (source: wikipedia) � model complex behavior – discrete jump maps can model discrete-time linear control systems (widely used in industry) source: mathworks.com 35

Linear Hybrid Automata � chaos – even with 1 variable, 1 location, 1 transition (tent map) – observed in actual production systems [Schmitz,2002] states of the Tent map brewery and chaotic throughput [Schmitz,2002] source: wikipedia Schmitz, J. P. M., D. A. Van Beek, and J. E. Rooda. "Chaos in discrete production systems?." Journal of Manufacturing Systems 21.3 36 (2002): 236-246.c

Compute time elapse states Post c ( S ) � arbitrary trajectory iff straight line exists (convex invariant) [Alur et al.] Inv � time elapse along straight line can be computed as projection along cone [Halbwachs et al.] derivatives projection cone 37