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Risk Assessment Via Monte Carlo Simulation: Tolerances Versus Statistics

B. Ross BarmishECE Department

University of Wisconsin, MadisonMadison, WI 53706

barmish@engr.wisc.edu

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• A. C. Antoniades, UC Berkeley

• A. Ganesan, UC Berkeley

• C. M. Lagoa, Penn State University

• H. Kettani, University of Wisconsin/Alabama

• M. L. Muhler, DLR, Oberfaffenhofen

• B. T. Polyak, Moscow Control Sciences

• P. S. Shcherbakov, Moscow Control Sciences

• S. R. Ross, University of Wisconsin/Berkeley

• R. Tempo, CENS/CNR, Italy

COLLABORATORS

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Overview

• Motivation

• The New Monte Carlo Method

• Truncation Principle

• Surprising Results

• Conclusion

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Monte Carlo Simulation

• Used Extensively to Assess System Safety

• Uncertain Parameters with Tolerances

• Generate “Thousands” of Sample Realizations

• Determine Range of Outcomes, Averages, Probabilities etc.

• How to Initialize the Random Number Generator

• High Sensitivity to Choice of Distribution

• Unduly Optimistic Risk Assessment

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It's Arithmetic Time

• Consider

1 + 2 + 3 + 4 + 5 + ! + 20 = 210

• Data and Parameter Errors

• Classical Error Accumulation Issues

(1 ± 1) + (2 ± 1) + (3 ± 1) + ! (20 ± 1) = 210 ± 20

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Two Results

Actual Result Alternative Result

190 · SUM · 230

190 195 200 205 210 215 220 225 230

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x

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Circuit Example

+_

10 Ω

10 Ω

10 Ω5 Ω

R

R

R

R

2

3

I1I 2

I3

I 4

x

1

1 VVout+ _

Output Voltage

• Range of Gain Versus Distribution

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More Generally

k1

b

k2

2b

k3

3b

k4

4

m =12m =11 m =13 m =14

b1

y

F

• Desired Simulation

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Two Approaches

• Interval of Gain

• Monte Carlo

0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.90

100

200

300

400

500

600

700

800

900

1000

Student Version of MATLAB

• How to Generate Samples?

Gain Histogram: 20,000 Uniform Samples

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Key Issue

What probability distribution should be used?Conclusions are often sensitive to choice of distribution.

• Intractability of Trial and Error

• Combinatoric Explosion Issue

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Key Idea in New Research

Classical Monte Carlo New Monte Carlo

RandomNumber

Generator

F

Samples of q

System Model

Samples of F(q)

Statistics of q

RandomNumber

GeneratorFSamples of q

System Model

Samples of F(q)

Statistics of q

NewTheory

Bounds on q

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The Central Issue

What probability distribution to use?

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Manufacturing Motivation

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Distributions for Capacitor

-20 -15 -10 -5 0 5 10 15 200

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

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Interpretation

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Interpretation (cont.)

RandomNumber

GeneratorFSamples of q

System Model

Samples of F(q)

Statistics of q

NewTheory

Bounds on q

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Truncation Principle

− r1 − t1 r1t1 − t2 t 2− r2 r2x1 x2

f x1 1( ) f x2 2( )

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Example 1: Ladder Network

...

...

+

-

VoutVin

R1

R3

R2

R4

R6

R5

Rn-2

Rn

Rn-1

• Study Expected Gain

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Illustration for Ladder Network

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Example 2: RLC Circuit

Consider the RLC circuit

1

L

I

+

-

CC R V1 2 021R

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Solution Summary For RLC

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5x 10 -7

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

x 10 -6

t1

t2

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Current and Further Research

• The Optimal Truncation Problem

• Exploitation of Structure

• Correlated Parameters

• New Application Areas; e.g., Cash Flows