Behavioral Metrics for Simulation-based Circuit Validation

1. Behavioral Metrics forSimulation-based Circuit Validation Antoine Girard VAL-AMS Project MeetingApril 2007

2. Transient dynamics analysis: Desired performance characteristics: Maximum overshoot Rise time Delay time Settling time Constraints on input/states Response sensitivity Use Linear or Metric Temporal Logic Time Domain Properties of Circuits

3. Property: Time Domain Properties of Circuits System: Step input(t > 0): Steady state at t=0-: from Zhi Han’s PhD Thesis 2005

4. Model based validation of time domain properties of circuits and systems: - Specifications: Temporal Logic Formula. - For a set of possible initial states, inputs and parameters. Testing: - Simulate a (large) number of trajectories. - Does each trajectory satisfies the specification ? - No validation proof: notion of coverage. Reachability based verification: - Compute the (infinite) set of all possible trajectories. - Does each trajectory satisfies the specification ? - Formal proof. Intermediate approach: - Can we build a formal proof from a finite number of trajectories ? Computer Aided Techniques forCircuit Validation

5. Following the approach presented in: Fainekos, Girard and Pappas, Temporal logic verification using simulation, FORMATS 2006. Behavioral metrics: reachable set covering Can I compute a finite number of trajectories y1,…,yN and parameters e1,…,eN such that Quantitative interpretation of temporal logic formula: robustness degree [](y) How much can I perturb the trajectory y and the property  remains true / false ? Circuit validation: Verification using Simulation

6. Behavioral metrics. Quantitative interpretation of temporal logics Algorithms for circuits validation. Outline of the Talk

7. Discrete time dynamical system with continuous/discrete inputs. Distance between trajectories starting from neighbour states, for neighbour sequences of inputs, remains small. Notion of behavioral metrics a.k.a. - Contraction metrics(Slotine) -  - ISS Lyapunov functions(Angeli) - Bisimulation functions(Girard & Pappas) Behavioral Metrics

8. Behavioral metric: function V: Rn× Rn R+ such thatwith 0<<1 and 0. Intuitively the function V :- bounds the distance between observations. - decreases under the evolution of the system. Behavioral Metrics

9. Behavioral metric exists: Behavioral Metrics - Example ITransmission Line Model

10. Behavioral metric exists: Behavioral Metrics - Example IIBoost DC/DC Converter

11. Behavioral metrics for LCS ? Use results on Lyapunov stability of LCS: Behavioral Metrics - Example IIIElectrical oscillator with half-wave rectifier

12. Assume for simplicity that I={x0}, sample the set of inputs U : Build the simulation tree : Reachable Set Covering

13. Then for any trajectory y0,...yN, of S, there exists a path q0,...,qN in the simulation tree such that Good point: any accuracy can be achieved by choosing  fine enough ! Bad point: number of points in the simulation tree is exponential in time horizon N ! - Solution: construction of the tree guided by the property to be verified. Reachable Set Covering

14. Behavioral metrics. Quantitative interpretation of temporal logics Algorithms for circuits validation. Outline of the Talk

15. Metric Temporal Logic (MTL) • Syntax: • Boolean Semantics: I can be of any bounded or unbounded interval of N. i.e. I = [0,+), I = [2,9] Fainekos, Pappas: Robustness of Temporal Logic Specifications, 2006

16. But the Boolean truth value is not enough … MTL Spec: ((x-10)  2(x10)) Fainekos, Pappas: Robustness of Temporal Logic Specifications, 2006

17. Robust Semantics for MTL • Syntax: • Robust Semantics: I can be of any bounded or unbounded interval of N. i.e. I = [0,+), I = [2,9] Fainekos, Pappas: Robustness of Temporal Logic Specifications, 2006

18. Proposition: Let Φbe an MTLformula and T be a signal, then Robust and Boolean Semantics for MTL Theorem:Let Φbe an MTLformula and T be a signal, then N Fainekos, Pappas: Robustness of Temporal Logic Specifications, 2006

19. Behavioral metrics. Quantitative interpretation of temporal logics Algorithms for circuits validation. Outline of the Talk

20. Let us define the robustness of the property Φ over the trajectories of S : Build the simulation tree with sampling parameter : for any trajectory y0,...yN, of S, there exists a path q0,...,qN in the simulation tree The property is verified / falsified if The number of nodes in the simulation tree is Circuit Validation

21. Property guided Simulation • The previous algorithm allows to sample uniformly the reachable set • When interested in property verification, we can adapt locally the sampling to increase efficiency. • e.g. for safety property: • - use coarse sampling when far from the unsafe set • - use fine sampling when near the unsafe set • This multiresolutionsampling of the reachable set is obtained by the procedure: • - start with a coarse simulation graph • - refine adaptively in regions where it is needed

22. Property guided Simulation • Multiresolution simulation graph :

23. Property guided Simulation • Mark the unsafe states :

24. Property guided Simulation • Refinement procedure:

25. Property guided Simulation • Refinement procedure:

26. Property guided Simulation • Refinement procedure:

27. Property guided Simulation • until you can conclude.

28. Three-dimensional linear system: Example Unsafe = {x2  -7.4} Unsafe = {x2  -7} Unsafe = {x2  -6.2} Unsafe = {x2  -5.8}

29. Conclusions • Verification of infinite state systems using simulation • Based on the notion of behavioral metrics • Computational cost related to the robustness of the system • - the more robust, the easier the computation • - for very robust system, verification requires one simulation • Future work (in VAL-AMS project) - computation of behavioral metrics for LCS - interface with SICONOS - algorithms for computing “smartly” the simulation tree. - deeper analysis of the computational cost.