Reachability Analysis for AMS Verification using Hybrid Support Function and SMT- based Method Honghuang Lin, Peng Li Dept. of ECE, Texas A&M University {linhh,

2 FAC 2013 Reachability Analysis for AMS Verification using Hybrid Support Function and SMT-based Method Motivation  Mixed-signal systems –Analog + Digital –Nonlinearity + Digital effects  Reachability Analysis –Formal method for AMS verification –Capable for PLL lock time checking DCO Digital logic TDC Digitally-Intensive PLL [G. Yu et al JLPE’10]

3 FAC 2013 Reachability Analysis for AMS Verification using Hybrid Support Function and SMT-based Method Motivation  Challenges –Digital effects increase the complexity of the reachability analysis –Reachability analysis is expensive for nonlinear systems  Questions –Q1: How to model the two types of signals (especially digital) efficiently for verification? –Q2: How to accelerate reachability analysis?

4 FAC 2013 Reachability Analysis for AMS Verification using Hybrid Support Function and SMT-based Method Q1: Model  Linearization –TDC resolution effect –Complex transition –Digital  Linear analog –Staircase  Linear transition TDC

5 FAC 2013 Reachability Analysis for AMS Verification using Hybrid Support Function and SMT-based Method Q1: Model  Variable Reduction –IIR finite word length –Need state variables for internal nodes –Digital  Linear analog –Reduce # state variables by 8 Second order IIR Multiplication: round-off error Addition: overflow

6 FAC 2013 Reachability Analysis for AMS Verification using Hybrid Support Function and SMT-based Method Q2: Reachability Analysis Acceleration  Support function based method –[A. Girard et al IFAC2008] –Initial space to reachable space –Support function representation –Efficient for linear systems –Unable to solve nonlinear systems Reachable space Support function

7 FAC 2013 Reachability Analysis for AMS Verification using Hybrid Support Function and SMT-based Method Q2: Reachability Analysis Acceleration  Simulation-assisted SMT based method –[L.Yin et al ICCAD2012] –Generic method for nonlinear systems –Discretize  Simulation  Approximation  SAT  Conservative –Suffers from resolution and dimension explosion  Our goal: accelerate this generic method by leveraging support function

8 FAC 2013 Reachability Analysis for AMS Verification using Hybrid Support Function and SMT-based Method Proposed Method AMS Digital  Analog Pure analog model Linear subsystem Nonlinear subsystem Support function based method SMT- based method Reachability analysis Reachable space of the full system

9 FAC 2013 Reachability Analysis for AMS Verification using Hybrid Support Function and SMT-based Method Outline  Motivation  Overview  Pure Analog Model with KRR  Hybrid Reachability Analysis  Experimental Results  Summary

10 FAC 2013 Reachability Analysis for AMS Verification using Hybrid Support Function and SMT-based Method Conservative Model AMS System XaXa XdXd FaFa FdFd Pure analog model X F EuEu ElEl Regression Conservative? Upper bound E u Lower bound E l X a : Analog variables X d : Digital variables F a : Analog transition F d : Digital transition X: Analog variables F: Pure analog transition function E u : Upper bound of errors between F and F a/d E l : Lower bound of errors between F and F a/d

11 FAC 2013 Reachability Analysis for AMS Verification using Hybrid Support Function and SMT-based Method Error Estimation with KRR Min. structural risk

12 FAC 2013 Reachability Analysis for AMS Verification using Hybrid Support Function and SMT-based Method Error Estimation with KRR  Error Estimation of the Model  Next: Hybrid Reachability Analysis X(t) AMS system Pure analog model E i (t+1) Error on the i-th state variable KRR Prediction + Confidence Intervals Features Targets

13 FAC 2013 Reachability Analysis for AMS Verification using Hybrid Support Function and SMT-based Method Partition the Pure Analog Model –Linear: –Nonlinear: –Variables on the boundary: Linearized

14 FAC 2013 Reachability Analysis for AMS Verification using Hybrid Support Function and SMT-based Method Hybrid Method Support function method NL-SMT with support function Combine the two reachable spaces with different dimension

15 FAC 2013 Reachability Analysis for AMS Verification using Hybrid Support Function and SMT-based Method Support Function Based Method  Support function –Definition: –Intersection of half spaces:  Tight polyhedral over approximation of a reachable space –A list of vector l i –Corresponding support function values ρ( l i ) – E.g. represent oval with pentagon

16 FAC 2013 Reachability Analysis for AMS Verification using Hybrid Support Function and SMT-based Method Support Function Based Method  Reachability analysis in linear subsystems –[A. Girard et al IFAC2008] –For the linear subsystem: –A useful property of support function: –The reachable space AU can be easily computed by the initial space U U AU Polyhedral over approximation A AU+ err U AU

17 FAC 2013 Reachability Analysis for AMS Verification using Hybrid Support Function and SMT-based Method SMT Based Method

18 FAC 2013 Reachability Analysis for AMS Verification using Hybrid Support Function and SMT-based Method NL-SMT with Support Function –Another Property of support functions: U V

19 FAC 2013 Reachability Analysis for AMS Verification using Hybrid Support Function and SMT-based Method Intersection of Reachable Subspaces –XY: Reachable space of linear subsystem –XZ: Nonlinear subsystem X X X X X YY Y Y Z Z Z Z Linear Nonlinear

20 FAC 2013 Reachability Analysis for AMS Verification using Hybrid Support Function and SMT-based Method  Error interval of modeling –KRR tool: Dlib-ml [Davis E. King, 2009] –Error intervals of phase difference and the output of loop filter for different word length –Reflects the error between digital implementation and ideal analog characteristic Experiment Results

21 FAC 2013 Reachability Analysis for AMS Verification using Hybrid Support Function and SMT-based Method  Speed up of the hybrid method –Compare single SAT solver running –Overhead in linear subsystem: sec –At least 76X speedup Experiment Results # var. in SAT AMS system>20 >40 Pure analog model5510 Partitioned nonlinear subsystem 235 # SAT constraintsRuntime(sec) Without partition With partition

22 FAC 2013 Reachability Analysis for AMS Verification using Hybrid Support Function and SMT-based Method Hybrid Reachability Analysis of PLL  Lock time < 0.25 us –Resolution determined by the error interval prediction

23 FAC 2013 Reachability Analysis for AMS Verification using Hybrid Support Function and SMT-based Method Conclusions  Model with KRR –AMS  Pure Analog –KRR  Hybrid Reachability Analysis –Model partition –Respective reachability analysis in the linear and nonlinear subsystems –Reachable subspaces intersections  Experiment Results –Model error –Speedup –DI-PLL lock time verification

24 FAC 2013 Reachability Analysis for AMS Verification using Hybrid Support Function and SMT-based Method Thanks  Questions?