A Polynomial Chaos Approach to Simulation Under Uncertainty

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A Polynomial Chaos Approach to Simulation Under
Uncertainty
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• Teems E. Lovett
• September 15, 2004

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Uncertainty Simulation Methods

• Monte Carlo simulation
• Multiple simulation runs
• Under-estimation of uncertainty
• Full statistics (PDF)
• Interval analysis
• Over-estimation of uncertainty
• Bounds - Max and Min

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Polynomial Chaos

• Homogeneous Chaos by N.Wiener in 1938
• Multi-dimensional representation of variables and
  their relationships (equations)
• Polynomial (Originally Hermite)
• L2 - complete space
• Individual values are now represented by a
  function of random variables
• Extended to Wiener-Askey polynomial chaos
• Askey scheme of orthogonal polynomials

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Resistive Companion Equation

• Model representation
• Independent variables
• G(h) - Conductance
• B(t-h) - Source
• Dependent variables
• V(t) - Across
• I(t) - Through
• B(t-h) - Dependent source
• Nodal analysis
• System representation

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Resistive Companion (Expanded)
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Polynomial Choice

• Askey scheme of polynomials (System choice)
• Hermite (Gaussian Distribution) (Most likely
  value)
• Legendre (Uniform Distribution) (Design)
• Laguerre (Exponential Distribution) (Threshold)
• Individual uncertainties are have their own
  polynomial
• Systems with multiple uncertainties
• Space represented by a multi-variable
  polynomial basis
• Contains cross terms

Multiple variable system
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Polynomial Order

• Single variable polynomial order
• Independent
• Known
• Dependent
• Unknown
• V(t) I(t)
• Output Results
• PDF (Number of random points)
• Statistics (Min, 25, Mean, 75, Max)
  (Time domain)

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Concepts Review

• User input
• Polynomial type
• Single variable polynomial order
• Number of random points
• Expanded resistive companion
• Output (PDF) (Statistics)

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Achievements

• Extended resistive companion
• Component architecture
• Stand alone models
• Model development tools
• Polynomial generator
• Conductance expander
• Matrix equations
• PDF generator

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RC Circuit Example

• V1 10.0 V
• RSRC 1.0 mO
• RC 20.0 O
• C1 0. 1 mF
• Tolerance on C1
• Hermite-Chaos 10
• Laguerre-Chaos 10

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3rd Order Hermite-Chaos
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3rd Order Laguerre-Chaos
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RLC Circuit Example

• V1 10.0 V
• RSRC 1.0 mO
• L1 1.0 mH
• RL 0.5 O
• C1 10.0 uF
• RC 0.5 O
• RO 20.0 O
• Legendre-Chaos
• Uniform distribution

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50 Tolerance on C1 and 5 Tolerance on RC, RL,
L1, and RO
Legendre-Chaos Monte Carlo analysis
PSPICE
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PDFs of Inductor Current at 0.0004 sec.
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VTB Uncertainty Solver (Under Development)
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Review and Future Work

• Complete statistics in a single simulation
• Closed loop simulation
• Condensed results
• Data storage
• Complicated input distributions
• System simulation to system design
• Future work (Non-linear) (Signal extension)