Phd Defense 2007

PhD Final Defense – Claudio Siviero, Feb. 21, 2007

Nonlinear Black-Box Models of

Digital Integrated Circuits

via System Identification

Claudio Siviero

Politecnico di Torino, Italy

claudio.siviero@polito.it

http://www.emc.polito.it/

Introduction

System-level simulation of high-performance electronic equipments

prediction of signals propagation on interconnects waveforms distortion immunity radiation

dig LCDdigRF

System-level simulation (i)

nonlineardevices

discontinuities(linear junctions: connectors, vias, packages,…)

transmission lines(linear bus)

Decompose the signal path into a cascade of subsystems

dig LCDdig

System-level simulation (ii)

Describe any subsystem by means of suitable numerical models (macromodel)

Interconnect the macromodels to represent the entire structure

Solver(SPICE, VHDL-AMS,…)

System-level simulation (iii)

devices

discontinuities

transmission lines

Frequency

Motivation

linear interconnects (lumped & distributed) well-developed studies [1,2]

devices

discontinuities transmission lines

Macromodeling resources

many open issues need for a systematic study

nonlinear devices preliminary results [1]

[1] F.G.Canavero, et A., "Linear and Nonlinear Macromodels for System-Level Signal Integrity and EMC Assessment" , IEICE Transactions on Communications , August, 2005[2] R. Achar, M.S. Nakhla, “Simulation of High-Speed Interconnects”, Proceedings of the IEEE, May 2001

Outline

• IC macromodels• Parametric modeling • Model representations• Assessment of models• Application example• Conclusion

Macromodels

A macromodel is a set of equations relating the port variables and describing the subsystem behavior "seen from the outside"

v1 v2 v1 v2

iOUT(t) = F( vIN(t),vOUT(t),d/dt )

Time-domain macromodels needed for nonlinear behavior

Macromodels for active devices

RF devices (e.g. Power Amplifiers) [3]

Digital Integrated Circuits (ICs)

Suitable modeling strategies must be devised to

model different devices

[3] C. Siviero, P.M. Lavrador, J.C. Pedro, "A Frequency Domain Extraction Procedure of Low-Pass Equivalent Behavioral Models of Microwave PAs" , 2006 European Microwave Integrated Circuits Conference (EuMIC2006), Manchester (UK), pp. 253-256, September 10-13, 2006

IC macromodels (i)

vi1ii2

set of equations or circuits describing I/O buffer behavior

nonlinear terminations (buffers, receivers)

IC macromodels (ii)

- IP protection (do not disclose technology details)

- Accuracy (include higher order effects )

- Efficiency (speed-up simulation)

- Implementation (plug models in any commercial tool)

Requirements

Available methodologies

• Physical modeling

• Behavioral modeling

reproduce the internal structure

reproduce the external electrical behavior

Physical modeling

Transistor-level models

- disclose IP

- the most accurate

- lack of efficiency

- not easily portable

Behavioral modeling (i)

Simplified equivalent circuits from port transient waveformse.g., IBIS [4] (I/O Buffer Information Specification)

- protect IP

- efficient, sometime complicated

- not so accurate

- supported by all commercial simulators

[4] http://www.eigroup.org/ibis/

Behavioral modeling (ii)

Parametric models & black-box techniques

- protect IP

- efficient

- accurate

- potentially compatible with commercial simulators

i=F(,v,d/dt)

nonlinear mathematical relation

v(t)F()

parameters estimated from port transient waveforms

Parametric models

But...

v(t)F()e.g., i(k)= 2exp(-0.1v(k)) - 0.5[v(k)-v(k-1)]

Typical parametric models are discrete-time relations

Recently applied to ICs: many open research issues

Resources: system identification theory provides methodologies for developing effective parametric models of any unknown nonlinear systems

→ some addressed in this PhD thesis

results from control theory, identification of mechanical systems [5] ,…

[5] L. Ljung, “System Identification: Theory for the User,” Prentice-Hall, 1987.

Outline

(A) Model selection

Select a suitable mathematical representation

v(t) i(k) = F( , v(k) , v(k-1), … )

Many representations are available to describe any nonlinear dynamics [6]

- neural networks (radial, sigmoidal, splines, wavelet, hyperplane,…)- polynomials (Wiener, Volterra,…) - kernel estimators- fuzzy models- composite local linear models- …

[6] J. Sjoberg et al., “Nonlinear Black-Box Modeling in System Identification: a Unified Overview,” Automatica, Vol. 31, No. 12, pp. 1691-1724, 1995.

(B) Estimation signals

Excite the real device with suitable stimuli and record the device output responses

VDDMultilevel signals to capture information on both static & dynamic behavior

v(t)+-

(C) Parameter estimation

Estimate by minimizing an error function E between the device response and the model response to the excitation

E() = || i – F()||2

= arg min E()

several methods available

v(t)+-

F() v(t)+-

i(t)=F()

- gradient-based- genetic- extended Kalman- simulated annealing- …

(D) Model validation

(D) Validate the model by comparing the device and the model responses to different excitations

v(t)+-

Accuracy

Stability

Efficiency (small size)

Model selection criteria

F() v(t)+-

i(t)=F()

(E) Macromodel implementation

1. Direct equation description/implementation (e.g., VHDL-AMS)F.vhd

2. Circuit interpretation & SPICE-like Implementation

Discrete-time

Continuous-time

e.g. i(k) = 2exp(-0.1v(k)) - 0.5[v(k)-v(k-1)]

i(t) = 2exp(-0.1v(t)) - 0.5T dv(t)/dt

v(t)2exp(-0.1v(t)) 0.5T

Specific contributions of this study

(C) Parameter estimation assess the performance of different algorithms

(A) Model selection assess the performance of different representations

first application to IC modeling

(B) Estimation signals multilevel excitations have been proven to be effective

(E) Macromodel implementation well-established procedure

(D) Model validation address stability issue

Outline

• IC macromodels• Parametric modeling• Model representations• Assessment of models• Application example• Conclusion

Model representations

Several parametric representations for i = F(,v,d/dt)

unknown nonlinearstate-space equation

[5] L. Ljung, “System Identification: Theory for the User,” Prentice-Hall, 1987.

[7] I.Rivals and L.Personnaz, “Black-Box Modeling with State-Space Neural Networks”, in Neural and Adaptive Control Technology,1996

i(k) = F ( aTφ(k) )

φ(k) = [ v(k),v(k-1),…,i(k-1),…]T

i(k) = F (Az(k) + bv(k) )

Nonlinear Input-Output [5]

Nonlinear State-Space [7]

regressors vector

“virtual” state vector

I/O vs. SS

I/O SS

successfully applied to real modeling problems

involves input-outputmeasurable variables only

well-established estimation methods

multiple inputs

stability

most obvious choice fordynamic systems

needs for virtual statevariables

estimation methods under study

multiple inputs

stability

Literature search

Investigated model representations

i = F(,v,d/dt)

Universal approximators of nonlinear dynamical relations

representation

Echo State Networks (ESN) [8]

Local Linear State-Space (LLSS) [9]

Sigmoidal Basis Functions (SBF) [6]

F structure

large size SS

weighted composition of LTI models

I/0 e.g., Σ tanh

estimation

random + heuristic & linear least squares-based

iterative gradient-based, pseudo-random init.

iterative gradient-based, deterministic init.

[6] J. Sjoberg et al., “Nonlinear Black-Box Modeling in System Identification: a Unified Overview,” Automatica, 1995.[8] H. Jaeger, “The Echo State Approach to Analyzing and Training Recurrent Neural Networks,” GMD Report, 2001.

[9] V.Verdult, “Nonlinear Systems Identification: A State-Space Approach”, Ph.D. Thesis, 2002.

Outline

Assessment of models (i)Synthetic nonlinear dynamic one-port test device

reference responses: Matlab ODE implementation

representative of IC buffers

f1 = 0 → receiver operationf1 ≠ 0 → driver operation @ fixed state

nonlinear dynamic behavior defined by f2 (“clamp”) and L

f2(v2)+

+ v2(t) -

→ tuned to have a stiff example (BENCHMARK)

f1(v) = a1+a2exp(a3v)+a4vf2(v2) = b1exp(b2(v2-VDD))

Assessment of models (ii)Estimation setup

Device response exhibits a slightly different static & dynamic behavior

vs(t)+

test device

estimation signals

0.40.60.8

11.21.4 v(t), V

0 5 10 15 20-50

i(t), mA

For validation, a different vs(t)is employed

Stability

What do we mean for stability?

Non trivial issue for nonlinear systems [10]

How can we formulate / implement the stability requirement?

Global stability (complicate, probably too restrictive)

Stability for a certain class of excitations / load conditions

(would be ok, but how can we validate exhaustively a model?)

Local stability (simplest, readily extended from the linear case)

[10] Hassan K.Kalihl, “Nonlinear Systems”, Prentice Hall, 2004.

Local stability

@ each time step :

compute the eigenvalues of the linearized model equation [11]

[11] C.Alippi, V.Piuri, “Neural Modeling of Dynamic Systems with Nonmeasurable State Variables”, TI&M 1999.

F (v(t),d/dt) ≈ F (v(t- Δt),d/dt) + J(t-Δt) [v(t)-v(t-Δt)] + …

stable responses

oscillating or saturating responses

possible eigenvaluesoutside the 1 circle

SBF validationEstimation : Levenberg Marquardt-based algorithms (10 runs)

ref. best model other models

dependence oninitial guess

0 2 4 6 8 10-60

60i(t), mA

t ns -1 -0.5 0 0.5 1

realpossible

instability

ESN validation

accuracy is independent of initialization, but large size...

0 2 4 6 8 10-60

60i(t), mA

ref. model

Estimation : random+heuristic & linear least squares-based algorithm (1 run)

-1 -0.5 0 0.5 1

Stable a priori

LLSS validation

Good accuracyUnique solution

0 2 4 6 8 10-60

60i(t), mA

Estimation : Levenberg Marquardt-based algorithm (1 run)

ref. model

0.85 0.9 0.95 1

Stable a posteriori

Efficiency comparison

Estimationtime

Simulation time

reference

10 ÷ 60 s

Speed-up

Matlab estimation & simulation time

Models comparison

feature/model

stability

efficiency / size

accuracy

SBF ESN LLSS

LLSS is the best solution for the problem at hand

Outline

• IC macromodels• Parametric modeling• Model representations• Assessment of models• Application example• Conclusion

Application example

TASK : evaluate the performance of LLSS macromodels for the simulation of a real NOKIA mobile data link

waveforms distortion ← immunity radiation

dig LCDdigRF

Mobile data link

DATA = 50 bit pseudo-random, Tbit = 5 ns, rise-time = 500 ps

RF-to-Digital interfaceCourtesy of Nokia Research Center, Helsinki (Finland)

Devices: NOKIA single-ended, VDD=1.8V (reference: ELDO transistor-level)

Validation: functional signals

0 50 100 150 200 250-0.5

driver output voltage

0 50 100 150 200 250-1

receiver input voltage

referenceLLSS macromodel

Validation: power & ground noise referenceLLSS macromodel

0 50 100 150 200 250-40

driver ground voltage

0 50 100 150 200 2501.76

driver power supply voltage

Remarks

Model (ELDO)

transistor-level

LLSS macromodel

simulation time

36 min. 26 sec.

1 min. 45 sec.

LLSS macromodels can be effectively used for real applications

Good accuracy

Include power & ground fluctuations

Speed up ~ 30x

Outline

• IC macromodels• Parametric modeling• Model representations• Assessment of models• Application examples• Conclusion

Conclusion: summary

System-level simulation of complex high-performance systems (e.g., high-end digital devices)

Accurate and efficient behavioral models (macromodels) of active components play a key role

→ Systematic study of the application of parametric models and system identification techniques for the behavioral modeling of digital ICs

Conclusion: results

Performance assessment of the performance of different representations (first time applied to IC macromodeling)

→ Local-Linear State-Space (LLSS) models provide the best results

good accuracy unique solution local stability verified a posteriori

Application of LLSS models for the system-level simulation of a real data link

Conclusion: future work

Formulate a tighter criteria to enforce / analyze the stability of a generic nonlinear parametric model.

Further explore LLSS features and capabilities.

Extend the device modeling methodology to account for the enhanced features of devices / applications

overclocked device operation

current models do not include incomplete state-transitions

include additional inputs/effects

e.g., to account for the RF immunity effects of digital ICs

Phd Defense 2007

portable v t i t

behavioral modeling

ic macromodels i

outside i

o1 v dd v o1

nonlinear macromodels

dd v i1

nonlinear behavior

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