Approximate Abstraction for Verification of Continuous and Hybrid Systems Antoine Girard Guest lecture ESE601: Hybrid Systems 03/22/2006 VERIMAG

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)

Algorithmic Verification Algorithmic proof of the safety of a system: No trajectory of the system can reach a set of unsafe states. Initially on the software part [ …] : - verification of discrete systems, Model Checking - for some properties, one cannot ignore the continuous dynamics Verification of continuous and hybrid systems [ …] : - 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.

Reachability Analysis Computation of the reachable set : - exactly for some very simple classes of systems Piecewise 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] Reach Init Unsafe

Complexity Barrier Computational cost of the reachable set is a major issue ! Linear systemsPiecewise affine systems Nonlinear systemsHybrid systems Model Complexity Dimension of the continuous state space Complex system

Abstraction Notion of system approximation : S 2 is an abstraction of S 1 iff every trajectory of S 1 is also a trajectory of S 2. 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 S 2 is safe then S 1 is safe :

Analysis of complex systems Abstraction methods for complexity reduction of systems Linear systemsPiecewise affine systems Nonlinear systemsHybrid system Model complexity Dimension of the continuous state space Complex system Abstraction Dimension reduction Hybridization

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

Simulation Relations Local characterization of trajectories inclusion. Simulation relation R  X 1 x X 2 : If for all initial state x 1 of S 1 there exists an initial state x 2 of S 2 such that (x 1,x 2 )  R then S 2 is an abstraction of S 1.

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, S 2 is an approximate abstraction or approximation of S 1 if For every trajectory of S 1, there exists a trajectory of S 2 such that the distance between the trajectories remains bonded by   is the precision of the approximation (  = 0, abstraction).

A Useful Notion for Verification If S 2 is an approximation of S 1 of precision  : Therefore, The safety of S 1 can be proved using an approximation S 2.

Approximate Simulation Relation Local characterization of the notion of approximation. Approximate simulation relation of precision , R  X 1 x X 2 : If for every initial state x 1 of S 1 there exists an initial state x 2 of S 2 such that (x 1,x 2 )  R, then S 2 is an approximation of S 1 of precision . - A. Girard, G.J. Pappas, Approximation metrics for discrete and continuous systems, IEEE TAC, accepted 2006.

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

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

Simulation Functions Simulation functions define approximate simulation relations: Particularly, Let then S 2 is an approximation of S 1 of precision . - A. Girard, G.J. Pappas, Approximation metrics for discrete and continuous systems, IEEE TAC, accepted 2006.

Simulation function: Example

Indeed, and Then, Since Reach(S 2 ) = (-1,8.5], Example

Linear Systems is a simulation function if

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

Truncated Quadratic Functions Universal for stable linear systems : Two stable linear systems are 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 subsystems are approximations of each other.

MATISSE 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 Metrics for Approximate TransItion Systems Simulation and Equivalence

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

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

Hybrid Systems Hybrid automaton H 1 of the type:

Approximation of Hybrid Systems Approximation H 2 of the hybrid automaton H 1 : Metrics on the set of observations H 1 et H 2 have the same discrete structure - same underlying automaton - approximation of the continuous dynamics

Approximation of Hybrid Systems H 2 approximation of H 1 of the form:

Approximation of the Continuous Dynamics For each mode l  L, the continuous dynamics of H 1 is approximated. We compute a simulation function We define a notion of neighborhood

Simulation relation of the form : of precision δ=max(δ 1, …, δ |L| ). Sufficient conditions : If then H 2 is an approximation of H 1 of precision δ=max(δ 1, …, δ |L| ). Approximate Simulation Relations for Hybrid Systems A. Girard, A.A. Julius, G.J. Pappas, Approximate simulation relations for hybrid systems, ADHS 2006, submitted.

Example

The first dynamics (dimension 4) is approximated by a 2 dimensional dynamics. Original systemApproximation

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 A.A. Julius, A. Girard, G.J. Pappas, Approximate bisimulation for a class of stochastic hybrid systems, ACC A.A. Julius, Approximate abstraction of stochastic hybrid automata, HSCC 2006.

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. Conclusion