1. Verification and Synthesis of Hybrid Systems Thao Dang October 10, 2000

2. Plan 1- Algorithmic Verification of Hybrid Systems 2- Reachability Analysis of Continuous Systems 3- Safety Verification of Hybrid Systems 4- Safety Controller Synthesis for Hybrid Systems 5- Implementation

3. Plan 1- Algorithmic Verification of Hybrid Systems 2- Reachability Analysis of Continuous Systems 3- Safety Verification of Hybrid Systems 4- Safety Controller Synthesis for Hybrid Systems 5- Implementation

4. Hybrid systems • Hybrid systems: systems which combine • continuous-time dynamics and discrete-event dynamics Continuous processes Digital controllers, (e.g., chemical reactions) switches, gears.. • Arisen virtually everywhere (due to the increasing use of computers)

5. Analysis of Hybrid Systems • Formal verification: prove that the system satisfies a given property • Controller synthesis: design controllers so that the controlled system satisfies a desired property • We concentrate on invariance properties: all trajectories of the system stay in a subset of the state space • Hybrid systems are difficult to analyze • No existing general method

6. Illustrative Example: A Thermostat on off • Verification problem: prove that the temperature x[a,b] • Characterize all behaviorsReachability Analysis

7.  The Thermostat Example (cont’d) x max 0 min 0 t • Two-phase behavior • Non-deterministic behavior • Set of initial states How to characterize and represent“tubes” of trajectories of continuousdynamics in order to treat discrete transitions??

8. Algorithmic Analysis of Hybrid Systems • Exact symbolic methods • applicable for restricted classes of hybrid systems • Our objective: verification method for generalhybrid systems in any dimension

9. Algorithmic Verification of Hybrid Systems What do we need?? a reachability technique which  is applicable for arbitrary continuous systems  can be extended to hybrid systems approximate reachability techniques represent reachable sets by orthogonal polyhedra

10. Approximations by Orthogonal Polyhedra Non-convexorthogonal polyhedra (unions of hyperrectangles) Motivations  canonical representation, efficient manipulation in any dimension easy extension to hybrid systems  termination can be guaranteed Under-approximation Over-approximation

11. Plan 1- Approach to Algorithmic Verification of Hybrid Systems 2-Reachability Analysis of Continuous Systems  Abstract Reachability Algorithm  Algorithm for Linear Continuous Systems  Algorithm for Non-Linear Continuous Systems 3- Safety Verification of Hybrid Systems 4- Safety Controller Synthesis for Hybrid Systems 5- Implementation

12. Plan 1- Approach to Algorithmic Verification of Hybrid Systems 2-Reachability Analysis of Continuous Systems  Abstract Reachability Algorithm  Algorithm for Linear Continuous Systems  Algorithm for Non-Linear Continuous Systems 3- Safety Verification of Hybrid Systems 4- Safety Controller Synthesis for Hybrid Systems 5- Implementation

13. Reachability Analysis of Continuous Systems x(0)F, set of initial states Problem Find an orthogonal polyhedronover-approximating the reachable set from F

14. [0,r](F) Successor Operator r(F) F Reachable set from F: (F) = [0,)(F)

15. Abstract Algorithm for Calculating (F) P0 := F ; repeat k = 0, 1, 2 .. Pk+1 := Pk [0,r](Pk) ; until Pk+1 = Pk r : time step • Use orthogonal polyhedra to • represent Pk • approximate [0,r]

16. Plan 1- Algorithmic Verification of Hybrid Systems 2-Reachability Analysis of Continuous Systems  Abstract Reachability Algorithm Algorithm for Linear Continuous Systems  Algorithm for Non-Linear Continuous Systems 3- Safety Verification of Hybrid Systems 4- Safety Controller Synthesis for Hybrid Systems 5- Implementation

17. Reachability of Linear Continuous Systems F is the set of initial states r(F) = eArF F is a convex polyhedron: F = conv{v1,..,vm} r(vi)=eArvi vi r(F) = conv{r(v1),.., r(vm)} F

18. r(v2) r(v1) X1 X1 Cb1 C1 v1  X0=F X0 v2 X1= r(X0) C1=conv{X1,X0} X2 X2 G2 X1 P1=G1 P2 [r,2r](F) G2 [0,r](F) G1 [0,2r](F) P2 = G1G2 Over-Approximating the Reachable Set Extension to under-approximations

19. Example

20. r(F) i i(r) yi F yi*(r) Extension to Linear Systems with Uncertain Input u1 u2  Computation of r(F) [Varaiya 98]  Bloating amount (Maximum Principle)

21. Example [Kurzhanski and Valyi 97] Advantage: time-efficiency

22. Plan 1- Algorithmic Verification of Hybrid Systems 2-Reachability Analysis of Continuous Systems  Abstract Reachability Algorithm  Algorithm for Linear Continuous Systems Algorithm for Non-Linear Continuous Systems 3- Safety Verification of Hybrid Systems 4- Safety Controller Synthesis for Hybrid Systems 5- Implementation

23.  The initial set F is a convex polyhedron The boundary of F: union of its faces Principle of the Reachability Technique x(0)F, set of initial states  ‘Face lifting’ technique, inspired by [Greenstreet 96] F y  Continuity of trajectories  compute from the boundary of F x

24. fe : projection of f on the outward normal to face e : maximum of fe over the neighborhood N(e) of e e1 H’(e) H(e) r N(e) Over-Approximating [0,r](F) Step 1: rough approximation N(F) Step 2: more accurate approximation N(F) F e

25. Computation Procedure F • Decompose F into non-overlapping hyper-rectangles • Apply the lifting operation to each hyper-rectangle (faces on the boundary of F) • Make the union of the new hyper-rectangles

26. Example: Airplane Safety [Lygeros et al. 98] P = [Vmin,Vmax][min,max]

27. Plan 1- Algorithmic Verification of Hybrid Systems 2- Reachability Analysis of Continuous Systems  Abstract Reachability Algorithm  Algorithm for Linear Continuous Systems  Algorithm for Non-Linear Continuous Systems 3-Safety Verification of Hybrid Systems 4- Safety Controller Synthesis for Hybrid Systems 5- Implementation

28. Hybrid Systems • Hybrid automata • continuous dynamics: linear with uncertain input,non-linear • staying and switching conditions: convex polyhedra • reset functions :affineof the formRqq’ (x) = Dqq’x + Jqq’ switching condition reset function discrete state q1 q0 continuous dynamics staying condition

29. Reachability of Hybrid Automata • The state(q, x) of the system can change in two ways: • continuous evolution: q remains constant, and xchanges continuously according to the diff. eq. at q • discrete evolution (by making a transition): qchanges, and xchanges according to the reset function. • Reachability analysis • continuous-successors • discrete-successors •  approximations byorthogonal polyhedra

30. [0,r](F)P F Hq Over-approximating Continuous-Successors • Use the reachability algorithms forcontinuous systems • Take into account the staying conditions

31. Rqq’(b) b Fg  FGqq’ Over-approximating Discrete-Successors qq’(q, F) = (q’, Rqq’(F  Gqq’) Hq’) Hq’ Fg Gqq’ F

32. q1 q0 q0 q1 q0 Example

33. Plan 1- Algorithmic Verification of Hybrid Systems 2- Reachability Analysis of Continuous Systems  Abstract Reachability Algorithm  Algorithm for Linear Continuous Systems  Algorithm for Non-Linear Continuous Systems 3- Safety Verification of Hybrid Systems 4-Safety Controller Synthesis for Hybrid Systems 5- Implementation

34. q3 Switching Controller Synthesis: Introduction Discrete Switching Controller q1 q2 q1 q2 q x f1 q3 f2 Mode selection f3 Plant

35. The Safety Synthesis Problem Given a hybrid automaton A and a set F  How to restrict the guards and the staying conditions of A so that all trajectories of the resulting automatonA*stay inF Solution: Compute the maximal invariant set (set of ‘winning’ states)

36. Operator  Given F={(q, Fq) | qQ},(F) consists of states from which all trajectories • stay indefinitely in Fwithout switchingOR • stay in F for some time and then make a transition to another discrete state and still in F x3 Fq x2 Gqq’Fq’ x1

37. Calculation of the Maximal Invariant Set P0 := F; repeat k = 1, 2, .. Pk+1 := Pk (Pk) ; untilPk+1 = Pk P*= Pk ; P* :maximal invariant set A* : H* =HP*,G* =GP*

38. Effective Approximate Synthesis Algorithm To approximate the maximal invariant set: • Use our reachability techniques for hybrid automata to approximate (F) • Under-approximations  Effective approximate synthesis algorithm for hybrid systems with linear continuous dynamics

39. F1 G10F0 F0 G01F1 F0 F1 G10 G01=[-0.2,-0.01] [-0.2,-0.01] G01 G10=[0.01,0.32] [-0.01,0.1]

40. Plan 1- Approach to Algorithmic Verification of Hybrid Systems 2- Reachability Analysis of Continuous Systems  Abstract Reachability Algorithm  Algorithm for Linear Continuous Systems  Algorithm for Non-Linear Continuous Systems 3- Safety Verification of Hybrid Systems 4- Safety Controller Synthesis for Hybrid Systems 5-Implementation

41. The tool d/dt Three types of automatic analysis for hybrid systems with linear differential inclusions  Reachability Analysis: compute an over-approximation of the reachable set from a given initial set  Safety Verification: check whether the system reaches a set of bad states  Safety Controller Synthesis: synthesize a switching controller so that the controlled system always remains inside a given set

42. Implementation d/dt Interface Verification Algorithms Controller Synthesis Algorithms OpenGL LEDA Numerical Integration CVODE Geometric Algorithms Qhull, Polka, Cubes Orthogonal Approximations

43. The tool d/dt

44. Conclusions Generality of Systems  Complexity of continuous and discrete dynamics  High dimensional systems Variety of Problems SafetyVerification and Synthesis Applications collision avoidance (4 continuous variables, 1 discrete state) double pendulum (3 continuous variables, 7 discrete states) freezing system (6 continuous variables, 9 discrete states)

45. More classes of problems • - more properties to verify, more synthesis criteria • - controller synthesis for more general systems, e.g linear diff. games Perspectives • More efficient analysis techniques • - Combining with analytic/qualitative methods • - Adapting existing techniques for discrete/timed systems • Tool • - more interactive analysis, simulation features • - experimentation: real-life problems

46. Related Work • Reachability Analysis • Polygonal Projections [Greenstreet and Mitchell 99] • Ellipsoidal Techniques [Kurzhanski and Varaiya 00] • Approximations via Parallelotopes [Kostoukova 99] • Verification • CheckMate [Chutinan and Krogh 99] • HyperTech [Henzinger et al. 00] • VeriShift [Botchkarev and Tripakis 00] • Symbolic Method [Lafferriere, Pappas, and Yovine 99] • Synthesis • Synthesis for timed automata [Asarin, Maler, Pnueli, and Sifakis 98] • Hamilton Jacobi Partial Diff. Eq. [Lygeros, Tomlin, and Sastry 98] • Computer Algebra [Shakernia, Pappas, and Sastry 00]

47. FinMerci