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Towards Formal Verification of Analog/Mixed-Signal Systems: „The Algebraic Approach“
Carna Radojicic, Christoph Grimm AG Design of Cyber-Physical Systems
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Digital v.s. AMS Verification
FAC'14, July, 9-10. 2014 Grenoble, France
Digital AMS
Specification
Verification (To be covered)
Finite state automata
Math.functions (multipliers, integrators, differentiators, …, ) + Properties (S/N, Step Response, BIBO stability)
Huge input space No variability
few measurements (not all possible inputs) Uncertainties/deviations challenge
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Agenda
Overview of approaches Related work: Affine Arithmetic Extended Affine Arithmetic for Systems with Discontinuities Benchmark: 3rd Order Sigma-Delta Modulator Conclusion
FAC'14, July, 9-10. 2014 Grenoble, France
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Overview of approaches
FAC'14, July, 9-10. 2014 Grenoble, France
Specification, Verification Plan
Design: Testbenches Modeled by HDS
Simulation
Physical design and fabrication
Formal model of design, PSL
Specific verification tool
Proof engine
Symbolic models
Variations, Uncertainties
Symbolic algebra and Proof engine and testing and verification
Simulation (Monte Carlo, Design of Exp.[Rafalia, Grimm], Worst-Case Analysis)
Formal methods [Hedrich, Henzinger, Little] The algebraic approach
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Agenda
Overview of state of art approaches Related work: Affine Arithmetic (AA) Extended Affine Arithmetic for Systems with Discontinuities Benchmark: 3rd Order Sigma-Delta Modulator Conclusion
FAC'14, July, 9-10. 2014 Grenoble, France
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Related work: Range Arithmetics, AA
Compute a system behavior for the set of variation/uncertain values enclosed within ranges
Ranges modeled using Affine Arithmetic (AA)[Andrade ‘94]
AA Properties
Handles the dependency problem in Interval Arithmetic Exact computation result for affine operations Results of non-affine operations over approximated
FAC'14, July, 9-10. 2014 Grenoble, France
-x1
hb
xo
lb
x1 x2 .. -xn..-x2 xn
AA form
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Control loop
+i(t)
quanti-sationnoise
+
whitenoise
+
Input
)(H csH )(ôR(t)
-
OpAmp:offseterror
+
C. Grimm, W. Heupke, K. Waldschmidt: „Refinement of Mixed-Signal Systems with Affine Arithmetic“, DATE ‘04, 2004.
FAC'14, July, 9-10. 2014 Grenoble, France
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Analog circuits
AAF: 5 sec. 50 runs (M.C.): 50 sec.
D. Grabowski, C. Grimm, E. Barke: „Symbolic Modeling and Simulation of Circuits and Systems“, ISCAS ‘06, 2006.
FAC'14, July, 9-10. 2014 Grenoble, France
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Limitations of AA: Discontinuities
Modeling with AA limited to the analog continuous domain No models for digital components as comparators, quantizers, PLL
phase/frequency detectors, ADCs, etc.
Requires extension to handle discontinuities in a MS system
FAC'14, July, 9-10. 2014 Grenoble, France
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Extended Affine Arithmetic (X-AAF)
System discontinuities are handled with deviation symbols
X-AAF is defined as:
- mean value - deviation from the mean value
FAC'14, July, 9-10. 2014 Grenoble, France
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Why {-1, 1} for ω?
FAC'14, July, 9-10. 2014 Grenoble, France
next time step
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The X-AAF implementation
X-AAF is implemented as Abstract Data Type The same concept can be implemented in any simulator which
supports the use of ADT (by replacing double/int with XAAF) We did this for SystemC AMS In SystemC AMS signals are instantiated with sca_tdf::sca_signal<T> some_signal; T- template parameter specifying the type of a signal value Example: A signal whose value is real number is instantiated with sca_tdf::sca_signal<double> some_signal; Signal with XAAF type value: sca_tdf::sca_signal<XAAF> some_signal;
FAC'14, July, 9-10. 2014 Grenoble, France
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Split and Merge operations for Control Flow Graphs
Merging
FAC'14, July, 9-10. 2014 Grenoble, France
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Computation with XAAF terms
Computation with XAAF terms requires operator overloading Overloaded binary arithmetic operators:
Addition and subtraction operator +, +=, -, -= Multiplication operator *, *= 1. Multiplication with constant 2. Multiplication of two XAAF terms
FAC'14, July, 9-10. 2014 Grenoble, France
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Agenda
Overview of approaches Related work: Affine Arithmetic Extended Affine Arithmetic for Systems with Discontinuities Benchmark: 3rd Order Sigma-Delta Modulator Conclusion
FAC'14, July, 9-10. 2014 Grenoble, France
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Benchmark: 3rd Order Sigma-Delta Modulator
FAC'14, July, 9-10. 2014 Grenoble, France
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Integrator outputs
FAC'14, July, 9-10. 2014 Grenoble, France
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FFT of the modulator output v
FAC'14, July, 9-10. 2014 Grenoble, France
Simulation time + FFT computation took 23 minutes. The number of ω symbols used to cover all possible cases was 236 << 4096
N=4096
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Agenda
Overview of approaches Related work: Affine Arithmetic Extended Affine Arithmetic for Systems with Discontinuities Benchmark: 3rd Order Sigma-Delta Modulator Conclusion
FAC'14, July, 9-10. 2014 Grenoble, France
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Conclusion
Easy integration of the approach into existing design flows and simulators Scalability (Mostly linear) Simulation of MS systems for the set of input conditions, variations…. (Problem coverage increased) Symbolic representation of system response allows efficient sensitivity
analysis
FAC'14, July, 9-10. 2014 Grenoble, France
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Thank you for your attention!
FAC'14, July, 9-10. 2014 Grenoble, France