efficient modeling and verification of analog/mixed-signal circuits using labeled hybrid petri nets

186 Pages·2008·1.2 MB·English

EFFICIENT MODELING AND VERIFICATION OF
ANALOG/MIXED-SIGNAL CIRCUITS USING
LABELED HYBRID PETRI NETS
by
Scott R. Little
A dissertation submitted to the faculty of
The University of Utah
in partial fulfillment of the requirements for the degree of
Doctor of Philosophy
in
Computer Science
School of Computing
The University of Utah
December 2008
Copyright (cid:13)c Scott R. Little 2008
All Rights Reserved
THE UNIVERSITY OF UTAH GRADUATE SCHOOL
SUPERVISORY COMMITTEE APPROVAL
of a dissertation submitted by
Scott R. Little
This dissertation has been read by each member of the following supervisory committee
and by majority vote has been found to be satisfactory.
Chair: Chris J. Myers
Ganesh Gopalakrishnan
Priyank Kalla
John Regehr
Ken Stevens
THE UNIVERSITY OF UTAH GRADUATE SCHOOL
FINAL READING APPROVAL
To the Graduate Council of the University of Utah:
I have read the dissertation of Scott R. Little in its final form
and have found that (1) its format, citations, and bibliographic style are consistent and
acceptable; (2) its illustrative materials including figures, tables, and charts are in place;
and (3) the final manuscript is satisfactory to the Supervisory Committee and is ready
for submission to The Graduate School.
Date Chris J. Myers
Chair, Supervisory Committee
Approved for the Major Department
Martin Berzins
Chair/Dean
Approved for the Graduate Council
David S. Chapman
Dean of The Graduate School
ABSTRACT
Analog circuit design is traditionally done by expert designers in an ad hoc manner
heavily dependent on simulation. This methodology has worked successfully for many
years,butprocessvariationanddesigncomplexityarepromptingdesignerstoexplorenew
techniques. Formal methods are being used successfully to aid in the complex validation
problem for digital circuits. This dissertation presents formal methods for analog and
mixed-signal (AMS) circuits.
This dissertation describes the development of a formal model, labeled hybrid Petri
nets (LHPNs),appropriateforthemodelingandverificationofAMScircuits. AnLHPNis
a Petri net variant capable of modeling both continuous and discrete quantities. Creating
an LHPN model of an AMS circuit by hand is a complicated and error prone exercise
that requires expert knowledge. This is unacceptable for practical adoption of the LHPN
model and its associated analysis methods. For this reason, this dissertation introduces
an automatic LHPN model generation method. The method uses a set of simulation
traces and a desired system property to generate an LHPN modeling the behavior of the
simulationtraces. ThemodelgeneratorcanalsobeusedtogenerateabstractVerilog-AMS
or VHDL-AMS models suitable for use in system-level simulations.
FormalverificationofapropertyovertheentirestatespaceofanLHPNmodeliscom-
plicatedbytheinfinitestateofthemodel. Forthisreason, theinfinitestatesofthemodel
are grouped into potentially finite groups of equivalent states for verification. Difference
bound matrices (DBMs), arestrictedformofconvexpolygons, areusedtorepresentthese
equivalent classes of infinite states. Reachability analysis using DBMs is very efficient at
the cost of exactness. This dissertation presents algorithms for conservative state space
analysis and verification of LHPNs.
Finally, these methods are demonstrated on several case studies of AMS circuits from
both academia and industry. The formal verification methods demonstrate the ability
to find bugs missed by standard simulations. The abstract modeling methods show the
promise of using automatically generated abstract models by demonstrating up to 40x
speedup for some examples.
To good light and the landscape...
CONTENTS
ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv
LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix
LIST OF TABLES. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii
LIST OF ALGORITHMS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiv
ACKNOWLEDGEMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv
CHAPTERS
1. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 AMS Circuit Design and Verification Methodology . . . . . . . . . . . . . . . . . . 1
1.2 AMS Circuit Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2.1 Linear Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.2.2 Nonlinear Systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.3 AMS Circuit Verification. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.3.1 Automated Theorem Proving . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.3.2 Model Checking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.3.3 Equivalence Checking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.3.4 Lightweight Verification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.4 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.5 Dissertation Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2. LABELED HYBRID PETRI NETS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.1 Motivating Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.2 Related Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.2.1 Hybrid Systems Modeling Languages . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.2.2 Automata Based Models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.2.3 Petri Net Based Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.3 Labeled Hybrid Petri Nets: Syntax. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.4 Labeled Hybrid Petri Nets: VHDL-AMS Compiler . . . . . . . . . . . . . . . . . . 34
2.5 Labeled Hybrid Petri Nets: Semantics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.6 LHPN Expansion for Constant Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3. ABSTRACT MODEL GENERATION . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.1 Motivating Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.2 Related Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.3 Abstract Model Generation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.3.1 Assigning the Regions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
3.3.2 Calculating the Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
3.3.3 Extracting Discrete Multivalued Variables . . . . . . . . . . . . . . . . . . . . . 62
3.3.4 LHPN Generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
3.3.5 Pseudo-regions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
3.3.6 Writing Normalized LHPN Models. . . . . . . . . . . . . . . . . . . . . . . . . . . 71
3.3.7 Writing VHDL-AMS Models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
3.3.8 Writing Verilog-AMS Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
3.4 Coverage Metrics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
4. DIFFERENCE BOUND MATRICES . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
4.1 Related Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
4.2 State Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
4.3 Difference Bound Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
4.4 Warping DBMs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
4.5 Updating the State Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
5. DBM-BASED MODEL CHECKER . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
5.1 Related Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
5.2 Reachability Analysis of LHPNs Using DBMs . . . . . . . . . . . . . . . . . . . . . . 100
5.2.1 Initial State Set Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
5.2.2 Finding Possible Events . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
5.2.3 Updating the State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
5.2.4 Firing a Transition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
5.2.5 Advancing Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
5.2.6 Error Trace Generation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
5.3 Reachability Analysis Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
6. RESULTS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
6.1 Hybrid Systems Examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
6.1.1 Water Level Monitor. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
6.1.2 Temperature Controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
6.1.3 Billiards Game . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
6.2 Tunnel Diode Oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
6.3 Switched Capacitor Integrator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
6.4 PLL Phase Detector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
6.5 CMOS Ring Oscillator with Feedforward Inverters. . . . . . . . . . . . . . . . . . . 142
7. CONCLUSIONS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
7.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
7.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
7.2.1 Automated Model Abstraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
7.2.2 Coverage Metrics for AMS Circuits . . . . . . . . . . . . . . . . . . . . . . . . . . 153
7.2.3 Counter-Example Guided Abstraction-Refinement. . . . . . . . . . . . . . . 153
7.2.4 False Negative Detection. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
7.2.5 AMS Property Specification Language . . . . . . . . . . . . . . . . . . . . . . . . 154
7.2.6 Assertion Based Verification for AMS Circuits. . . . . . . . . . . . . . . . . . 155
7.2.7 Automatic Stability Verification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
vii
7.2.8 Embedded Software Verification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
viii
LIST OF FIGURES
1.1 A block diagram describing the analog design process. . . . . . . . . . . . . . . . . . 4
1.2 A block diagram describing LEMA. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.1 A schematic diagram of a switched capacitor integrator circuit. . . . . . . . . . . 19
2.2 A simulation of the switched capacitor integrator circuit under nominal
conditions, an output slew rate of ±20 mV/µs.. . . . . . . . . . . . . . . . . . . . . . . 20
2.3 A simulation of the switched capacitor integrator circuit with an output
slew rate of ±22 mV/µs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.4 A simulation of the switched capacitor integrator circuit with an output
slew rate of ±18 mV/µs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.5 A random simulation of the switched capacitor integrator circuit with a
variable output slew rate. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.6 A worst case simulation of the switched capacitor integrator circuit showing
saturation of the op amp. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.7 A behavioral VHDL-AMS description of the switched capacitor integrator
circuit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.8 A behavioral Verilog-AMS of the the switched capacitor integrator circuit. . 26
2.9 A hybrid automata model of the switched capacitor integrator circuit. . . . . 28
2.10 A THPN model of the switched capacitor integrator circuit. . . . . . . . . . . . . 30
2.11 AnLHPNmodeloftheswitchedcapacitorintegratorcircuit. (a)TheLHPN
controlling the rate of change for variable V . (b) The LHPN changing
out
the Boolean signal V . (c) The LHPN checking for saturation. (d) The
in
initial values for V , its rate of change, and the Boolean signals. . . . . . . . . 33
out
2.12 VHDL-AMS code for the span procedure. . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.13 An example of the piecewise approximation used by LHPN expansion for
nonnegative and nonpositive ranges of rates.. . . . . . . . . . . . . . . . . . . . . . . . . 40
2.14 A template for transforming LHPNs where the range of rates are nonnega-
tive or nonpositive. (a) Original LHPN. (b) Transformed LHPN. . . . . . . . . 41
2.15 AtemplatefortransformingLHPNswheretherangeofratesisnonpositive.
(a) Original LHPN. (b) Transformed LHPN without using absolute value
before comparison. (c) Transformed LHPN using a correct transform.. . . . . 41
2.16 An example of the piecewise approximation used by LHPN expansion for
ranges of rate that span zero. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43