Trajectory-Directed Discrete State Space Modeling for Formal Verification of

Nonlinear Analog Circuits

Presented by Valeriy Balabanov

Outline

Introduction

Problem description

Main algorithm

Experimental validation

Summary and discussion

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

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):

Discrete analog transition structure

Problem description

• State space need to be partitioned (discretized)

User specified bounds

Partition of state space Z into R1 .. Rk

Maximum length error

Number of partitions

Overall mean-out degree error

Overall mean successor relation error

Maximum direction error

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

Main algorithm (example of partition)

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

Main algorithm

Main algorithm

• Mapping the trajectory-directed partitioning to DATS

Main algorithm

Experimental validation

• TDD (trajectory-directed discretization method) vs HBD (hyper box discretization)

Experimental validation

Experimental validation

• Model checking case study

• Has been tested and widely manufactured• Only lately found that under certain conditions

has critical behaviour

Experimental validation

Experimental validation

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

Thank you!