Approximate Abstraction for Verification of Continuous and Hybrid Systems

1. VERIMAG Approximate Abstraction for Verification of Continuous andHybrid Systems Antoine Girard Antoine.Girard@imag.fr Guest lectureESE601: Hybrid Systems03/22/2006

2. Hybrid Systems • General modeling framework for complex systems : • - continuous dynamics (ode, pde, sde) • - discrete dynamics (automata, Markov processes) • Several applications including embedded systems : • - design : computer = automata, continuous environment • - implementation : integrated circuits, analogical et numerical components • These systems are generally : • - structured (hierarchical modeling/architecture) • - large scale systems(numerous continuous variables) • - safety critical (plane, subway, nuclear power plant)

3. Algorithmic Verification • Algorithmic proof of the safety of a system: • No trajectory of the system can reacha set of unsafe states. • Initially on the software part [1980 - …] : - verification of discrete systems, Model Checking • - for some properties, one cannot ignore the continuous dynamics • Verification of continuous and hybrid systems [1995 - …] : - exhaustive simulation of systems using set valued computations techniques. - central notion reachable set : subset of the state space, reachable by the trajectories of the system from a subset of initial states.

4. Reachability Analysis Reach Init Unsafe • Computation of the reachable set : • - exactly for some very simple classes of systemsPiecewise constant differential inclusions, some linear systems • - approximately for other classes (over-approximation algorithms) • Over-approximation algorithms • Set-based simulation + numerical errors: • - Polytopes[Asarin, Dang, Maler; Krogh et.al.; Girard] • - Ellipsoids[Kurzhanski, Varayia]

5. Complexity Barrier Dimension of the continuous state space 100 10 Model Complexity Linear systems Piecewise affine systems Nonlinear systems Hybrid systems Computational cost of the reachable set is a major issue ! Complex system

6. Abstraction • Notion of system approximation : • S2 is an abstraction of S1 iffevery trajectory of S1 is also a trajectory of S2. • Hybridization : Approximation of complex continuous dynamics by simpler hybrid dynamics.[Asarin, Dang, Girard; Lefebvre, Gueguen; Frehse] • Dimension reduction [Pappas et.al.; van der Schaft] • If S2 is safe then S1 is safe :

7. Analysis of complex systems Dimension of the continuous state space 100 10 Model complexity Linear systems Piecewise affine systems Nonlinear systems Hybrid system Abstraction methods for complexity reduction of systems. Complex system Dimension reduction Hybridization Abstraction

8. Outline 1. Abstraction and Approximation : - Simulation relation - Approximate simulation relation 2. Approximate simulation relations for continuous systems. 3. Approximate simulation relations for hybrid systems.

9. Simulation Relations • Local characterization of trajectories inclusion. • Simulation relation R  X1 x X2 : • If for all initial state x1 of S1 there exists an initial state x2 of S2 such that (x1,x2)  R then S2 is an abstraction of S1.

10. From Abstraction to Approximation • Trajectories inclusion is well suited to discrete systems. • For continuous and hybrid systems, it is restrictive : • Natural topology on the state space •  • Distance between the trajectories seems more appropriate • Thus, S2 is an approximate abstraction or approximation of S1 if • For every trajectory of S1, there exists a trajectory of S2 such that the distance between the trajectories remains bonded by  •  is the precision of the approximation ( = 0, abstraction).

11. A Useful Notion for Verification • If S2 is an approximation of S1 of precision: • Therefore, • The safety of S1 can be proved using an approximation S2.

12. Approximate Simulation Relation • Local characterization of the notion of approximation. • Approximate simulation relation of precision , R  X1 x X2 : • If for every initial state x1 of S1 there exists an initial state x2 of S2 such that (x1,x2)  R, then S2 is an approximation of S1 of precision . - A. Girard, G.J. Pappas, Approximation metrics for discrete and continuous systems, IEEE TAC, accepted 2006.

13. Outline 1. Abstraction and Approximation : - Simulation relation - Approximate simulation relation 2. Approximate simulation relations for continuous systems. 3. Approximate simulation relations for hybrid systems.

14. is a simulation function if Simulation Functions A. Girard, G.J. Pappas, Approximate bisimulations for constrained linear systems, CDC 2005. A.Girard, G.J. Pappas, Approximate bisimulations for nonlinear dynamical systems, CDC 2005.

15. Simulation Functions • Simulation functions define approximate simulation relations: • Particularly, • Let • then S2 is an approximation of S1 of precision . - A. Girard, G.J. Pappas, Approximation metrics for discrete and continuous systems, IEEE TAC, accepted 2006.

16. Example Simulation function:

17. Example Indeed, and Then, Since Reach(S2) = (-1,8.5],

18. is a simulation function if Linear Systems

19. We look for simulation functions of the form Decomposition of the approximation error: transient /asymptotic Characterization Truncated Quadratic Functions For a λ > 0. A. Girard, G.J. Pappas, Approximate bisimulations for constrained linear systems, CDC 2005.

20. Truncated Quadratic Functions • Universal for stable linear systems : • Two stable linear systemsare approximations of each other.(though the precision may be very bad) • Characterisation allows algorithmic computation of simulation functions. • Generalizable to non-stable systems : • Two linear systems with identical unstable subsystemsare approximations of each other.

21. MATISSE Metrics for Approximate TransItion Systems Simulation and Equivalence • MATLAB toolbox • Functionalities: • - Computation of a simulation function between a system and its projection. - Evaluates the precision of the approximation of a system by its projection. • - Finds a good projection of a system (for a given dimension). • - Reachability computations based on zonotopes. • Available from • http://www.seas.upenn.edu/~agirard/Software/MATISSE/index.html

22. MATISSE Metrics for Approximate TransItion Systems Simulation and Equivalence Example of application: safety verification of a 10 dimensional system 10 dimensionaloriginal system 5 dimensionalapproximation 7 dimensional approximation

23. Outline 1. Abstraction and Approximation : - Simulation relation - Approximate simulation relation 2. Approximate simulation relations for continuous systems. 3. Approximate simulation relations for hybrid systems.

24. Hybrid Systems Hybrid automaton H1 of the type:

25. Approximation of Hybrid Systems • Approximation H2 of the hybrid automaton H1: • Metrics on the set of observations • H1 et H2 have the same discrete structure - same underlying automaton - approximation of the continuous dynamics

26. Approximation of Hybrid Systems H2 approximation of H1 of the form:

27. Approximation of the Continuous Dynamics • For each mode lL, the continuous dynamics of H1 is approximated. • We compute a simulation function • We define a notion of neighborhood

28. Approximate Simulation Relationsfor Hybrid Systems • Simulation relation of the form : • of precision δ=max(δ1, … , δ|L|). • Sufficient conditions : • If then H2 is an approximation of H1 of precision δ=max(δ1, … , δ|L|). A. Girard, A.A. Julius, G.J. Pappas, Approximate simulation relations for hybrid systems, ADHS 2006, submitted.

29. Example

30. Example The first dynamics (dimension 4) is approximated by a 2 dimensional dynamics. Original system Approximation

31. Extensions • Methods for the computation simulation functions for continuous nonlinear systems (SOS programs) • Theoretical framework and aglorithms for approximation of stochastic hybrid systems A. Girard, G.J. Pappas, Approximate bisimulations for nonlinear dynamical systems, CDC 2005. A.A. Julius, A. Girard, G.J. Pappas, Approximate bisimulation for a class of stochastic hybrid systems, ACC 2006. A.A. Julius, Approximate abstraction of stochastic hybrid automata, HSCC 2006.

32. Conclusion • Unified (discrete/continuous/hybrid) framework for system approximation. • Approximation as a relaxation of the notion of abstraction:- distance between trajectories rather than an inclusion relation.-allows additional simplifications. • Approach based on simulation functions- Lyapunov-like characterization - Algorithms (LMIs, SOS, Optimization) • Framework suitable for safety verification of complex systems.