reachability analysis for ams verification using hybrid support function and smt- based method...
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Reachability Analysis for AMS Verification using Hybrid Support Function and SMT-based Method
Honghuang Lin, Peng LiDept. of ECE, Texas A&M University
{linhh, pli} @neo.tamu.edu
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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
Digitallogic
TDC
Digitally-Intensive PLL[G. Yu et al JLPE’10]
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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?
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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
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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
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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 systemsReachable
spaceSupportfunction
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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
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Reachability Analysis for AMS Verification using Hybrid Support Function and SMT-based Method
Proposed Method
AMSDigital Analog Pure analog
model
Linear subsystem
Nonlinear subsystem
Support function based
method
SMT-based
method
Reachability analysis
Reachable space of the full system
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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
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Reachability Analysis for AMS Verification using Hybrid Support Function and SMT-based Method
Conservative ModelA
MS
Sys
tem
Xa
Xd
Fa
Fd
Pu
re a
nal
og
m
od
el
X
F
Eu
El
Reg
ress
ion
Conservative?
Upper boundEu
Lower boundEl
Xa: Analog variablesXd: Digital variablesFa: Analog transitionFd: Digital transition
X: Analog variablesF: Pure analog transition functionEu: Upper bound of errors between F and Fa/d
El: Lower bound of errors between F and Fa/d
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Reachability Analysis for AMS Verification using Hybrid Support Function and SMT-based Method
Error Estimation with KRR Kernel Ridge Regression (KRR)
– [C. Saunders et al 1998; J.A.K. Suykens et al 2002]
– Subject to
– A.k.a Least Squares Support Vector Regression
– Plenty of training data accurate prediction
– Confidence interval computation
– [K. De Brabanter et al 2011]
– Error
– Smoother bias and variance
Min. structural risk
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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
Ei(t+1)
Error on the i-th state
variable
KRRPrediction + Confidence Intervals
Features Targets
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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
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Reachability Analysis for AMS Verification using Hybrid Support Function and SMT-based Method
Hybrid Method
Support function method
NL-SMT with support function
Reachable space with state variables
,,
Reachable space with state variables
,,
Combine the two reachable spaces with
different dimension
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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 li
– Corresponding support function values ρ(li)
– E.g. represent oval with pentagon
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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
AAU+err
U AU
ρ𝑈 ( 𝑙) ρ𝐴𝑈 ( 𝐴𝑇 𝑙)
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Reachability Analysis for AMS Verification using Hybrid Support Function and SMT-based Method
SMT Based Method Nonlinear Satisfiability Modulo Theory(SMT)
based method– [L.Yin et al ICCAD2012]
– Convert verification problems to satisfiability problems composed of boolean combinations of multiple arithmetic constraints(can be nonlinear)
– E.g. can be converted to
SAT solver– iSAT [http://isat.gforge.avacs.org/]
– Davis-Putnam-Logemann-Loveland (DPLL) Algorithm
– Produces an existing solution that satisfies all the constraints or “unsatisfiable”
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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
ρ𝑈 ( 𝑙)
ρ𝑉 (𝑙)
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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
Y Y
Y
YZ
Z
Z
Z
Linear
Nonlinear
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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
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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: 21.163 sec
– At least 76X speedup
Experiment Results
# var. at # var. at # var. in SAT
AMS system >20 >20 >40
Pure analog model 5 5 10
Partitioned nonlinear subsystem
2 3 5
# SAT constraints Runtime(sec)
Without partition 64420 1942.9
With partition 532 4.1868
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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
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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