Trajectory-Directed Discrete State Space Modeling for Formal Verification of Nonlinear Analog...
-
Upload
clement-cox -
Category
Documents
-
view
222 -
download
2
Transcript of Trajectory-Directed Discrete State Space Modeling for Formal Verification of Nonlinear Analog...
![Page 1: Trajectory-Directed Discrete State Space Modeling for Formal Verification of Nonlinear Analog Circuits Presented by Valeriy Balabanov.](https://reader035.fdocuments.net/reader035/viewer/2022062515/56649ceb5503460f949b7577/html5/thumbnails/1.jpg)
Trajectory-Directed Discrete State Space Modeling for Formal Verification of
Nonlinear Analog Circuits
Presented by Valeriy Balabanov
![Page 2: Trajectory-Directed Discrete State Space Modeling for Formal Verification of Nonlinear Analog Circuits Presented by Valeriy Balabanov.](https://reader035.fdocuments.net/reader035/viewer/2022062515/56649ceb5503460f949b7577/html5/thumbnails/2.jpg)
Outline
Introduction
Problem description
Main algorithm
Experimental validation
Summary and discussion
![Page 3: Trajectory-Directed Discrete State Space Modeling for Formal Verification of Nonlinear Analog Circuits Presented by Valeriy Balabanov.](https://reader035.fdocuments.net/reader035/viewer/2022062515/56649ceb5503460f949b7577/html5/thumbnails/3.jpg)
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
![Page 4: Trajectory-Directed Discrete State Space Modeling for Formal Verification of Nonlinear Analog Circuits Presented by Valeriy Balabanov.](https://reader035.fdocuments.net/reader035/viewer/2022062515/56649ceb5503460f949b7577/html5/thumbnails/4.jpg)
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):
![Page 5: Trajectory-Directed Discrete State Space Modeling for Formal Verification of Nonlinear Analog Circuits Presented by Valeriy Balabanov.](https://reader035.fdocuments.net/reader035/viewer/2022062515/56649ceb5503460f949b7577/html5/thumbnails/5.jpg)
Discrete analog transition structure
![Page 6: Trajectory-Directed Discrete State Space Modeling for Formal Verification of Nonlinear Analog Circuits Presented by Valeriy Balabanov.](https://reader035.fdocuments.net/reader035/viewer/2022062515/56649ceb5503460f949b7577/html5/thumbnails/6.jpg)
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
![Page 7: Trajectory-Directed Discrete State Space Modeling for Formal Verification of Nonlinear Analog Circuits Presented by Valeriy Balabanov.](https://reader035.fdocuments.net/reader035/viewer/2022062515/56649ceb5503460f949b7577/html5/thumbnails/7.jpg)
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
![Page 8: Trajectory-Directed Discrete State Space Modeling for Formal Verification of Nonlinear Analog Circuits Presented by Valeriy Balabanov.](https://reader035.fdocuments.net/reader035/viewer/2022062515/56649ceb5503460f949b7577/html5/thumbnails/8.jpg)
Main algorithm (example of partition)
![Page 9: Trajectory-Directed Discrete State Space Modeling for Formal Verification of Nonlinear Analog Circuits Presented by Valeriy Balabanov.](https://reader035.fdocuments.net/reader035/viewer/2022062515/56649ceb5503460f949b7577/html5/thumbnails/9.jpg)
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
![Page 10: Trajectory-Directed Discrete State Space Modeling for Formal Verification of Nonlinear Analog Circuits Presented by Valeriy Balabanov.](https://reader035.fdocuments.net/reader035/viewer/2022062515/56649ceb5503460f949b7577/html5/thumbnails/10.jpg)
Main algorithm
![Page 11: Trajectory-Directed Discrete State Space Modeling for Formal Verification of Nonlinear Analog Circuits Presented by Valeriy Balabanov.](https://reader035.fdocuments.net/reader035/viewer/2022062515/56649ceb5503460f949b7577/html5/thumbnails/11.jpg)
Main algorithm
• Mapping the trajectory-directed partitioning to DATS
![Page 12: Trajectory-Directed Discrete State Space Modeling for Formal Verification of Nonlinear Analog Circuits Presented by Valeriy Balabanov.](https://reader035.fdocuments.net/reader035/viewer/2022062515/56649ceb5503460f949b7577/html5/thumbnails/12.jpg)
Main algorithm
![Page 13: Trajectory-Directed Discrete State Space Modeling for Formal Verification of Nonlinear Analog Circuits Presented by Valeriy Balabanov.](https://reader035.fdocuments.net/reader035/viewer/2022062515/56649ceb5503460f949b7577/html5/thumbnails/13.jpg)
Experimental validation
• TDD (trajectory-directed discretization method) vs HBD (hyper box discretization)
![Page 14: Trajectory-Directed Discrete State Space Modeling for Formal Verification of Nonlinear Analog Circuits Presented by Valeriy Balabanov.](https://reader035.fdocuments.net/reader035/viewer/2022062515/56649ceb5503460f949b7577/html5/thumbnails/14.jpg)
Experimental validation
![Page 15: Trajectory-Directed Discrete State Space Modeling for Formal Verification of Nonlinear Analog Circuits Presented by Valeriy Balabanov.](https://reader035.fdocuments.net/reader035/viewer/2022062515/56649ceb5503460f949b7577/html5/thumbnails/15.jpg)
Experimental validation
• Model checking case study
• Has been tested and widely manufactured• Only lately found that under certain conditions
has critical behaviour
![Page 16: Trajectory-Directed Discrete State Space Modeling for Formal Verification of Nonlinear Analog Circuits Presented by Valeriy Balabanov.](https://reader035.fdocuments.net/reader035/viewer/2022062515/56649ceb5503460f949b7577/html5/thumbnails/16.jpg)
Experimental validation
![Page 17: Trajectory-Directed Discrete State Space Modeling for Formal Verification of Nonlinear Analog Circuits Presented by Valeriy Balabanov.](https://reader035.fdocuments.net/reader035/viewer/2022062515/56649ceb5503460f949b7577/html5/thumbnails/17.jpg)
Experimental validation
![Page 18: Trajectory-Directed Discrete State Space Modeling for Formal Verification of Nonlinear Analog Circuits Presented by Valeriy Balabanov.](https://reader035.fdocuments.net/reader035/viewer/2022062515/56649ceb5503460f949b7577/html5/thumbnails/18.jpg)
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
![Page 19: Trajectory-Directed Discrete State Space Modeling for Formal Verification of Nonlinear Analog Circuits Presented by Valeriy Balabanov.](https://reader035.fdocuments.net/reader035/viewer/2022062515/56649ceb5503460f949b7577/html5/thumbnails/19.jpg)
Thank you!