1. Trajectory-Directed Discrete State Space Modeling for Formal Verification of Nonlinear Analog Circuits Presented by Valeriy Balabanov

2. Outline  Introduction  Problem description  Main algorithm  Experimental validation  Summary and discussion

3. Introduction Digital circuits vs Analog circuits – Digital circuits Operate with discrete signals High level of automation Many model and equivalence checking tools – Analog circuits Operate with continuous state space Model and equivalence checking still needed Deal with nonlinear differential–algebraic equations (DAE) – Analytical approaches are not feasible – Good discretization methods are needed – Discretization error

4. Analog state space First order nonlinear DAE – x – vector of variables – x’ – first derivative (vector of velocity vectors) – u – input variables State space is spanned by a linearly independent subset z Extended state space Candidates for state space variables can be identified in the DAE by their occurrence as first-order time derivatives – Example (capacitor):

5. Discrete analog transition structure

6. Problem description State space need to be partitioned (discretized) User specified bounds Partition of state space Z into R 1.. R k Maximum length error Number of partitions Overall mean-out degree error Overall mean successor relation error Maximum direction error

7. Main algorithm Discretization shall be rotation invariant – State space intersections cannot be axis-parallel Over-approximation of the successor relation significantly weakens expressiveness of verification algorithms – Geometric structure of partitions should follow the flow of state space dynamics – Intersections should be either parallel or orthogonal to the state space trajectories Use time step control algorithm to ensure homogeneity of the enclosed state space dynamics -> Trajectory directed discrete modeling algorithm

8. Main algorithm (example of partition)

9. Main algorithm Coordinate transformation to centralize/normalize vectors Random starting point that is not a DC-operating-point Gram-Schmidt procedure Control discretization error Find new points by Control the structure of the new points in order to avoid overlapping with existing points

10. Main algorithm

11. Mapping the trajectory-directed partitioning to DATS

12. Main algorithm

13. Experimental validation TDD (trajectory-directed discretization method) vs HBD (hyper box discretization)

14. Experimental validation

15. Model checking case study Has been tested and widely manufactured Only lately found that under certain conditions has critical behaviour

16. Experimental validation

18. Summary and discussion Paper presents a completely new approach for state space discretization New algorithm outperforms existing one in partitioning strength There are many application in modern mixed (analog-digital) designs The material in paper is well presented No visible improvements are needed

19. Thank you!