Polynomial chaos in simulation and engineering applications

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Polynomial chaos in simulation and engineering
applications
VTB Users and Developers Conference 2005
Columbia, SC September 20-21, 2005

• F. Ponci, A.Monti, T.Lovett, A.Smith
• Dept. of Electrical Engineering
• University of South Carolina

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Outline of the presentation

• Overview of Polynomial Chaos Theory
• Current research
• Simulation of uncertain system
• Evaluation and propagation of measurement
  uncertainty
• Control of uncertain systems
• Future Research

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Methods to handle uncertainty in engineering
modeling

• Traditionally only deterministic mathematical
  models
• Problems with parameters or inputs equal to
  random variables
• Monte Carlo method (or other statistical methods,
  expensive especially for already large
  deterministic systems)
• Taylor expansion of the random field (or other
  methods limited to small variances, higher order
  expansion impractical)
• Interval Arithmetic for true worst-case analysis
• Artificial intelligence for qualitative analysis
• Polynomial Chaos expansion

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The Polynomial Chaos Theory
Brown University
General framework of PCT Application to
differential equations
Polynomial Chaos Theory (PCT)
Wiener
GhanemSpanos

Parameters are random variables with given
probability function
Stochastic solution
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Generalized Polynomial Chaos

• The key ingredient of the chaos expansion is to
  express the random process through a complete and
  orthogonal polynomial basis in terms of random
  variables. A second-order random process can be
  represented as

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PCT Steps

• Given the uncertain variables in the system with
  given Probability Density Function PDF, pick a
  polynomial base
• Describe the PDFs in the chosen base
• All the variables are described in the new base
• Apply Galerkin projection
• Calculate process output
• From PCT coefficients reconstruct the PDF of the
  output variable

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Selection of the base
e.g. uncertain resistance with gaussian
distribution
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Stochastic ODE

• Let us consider the following equation
• Where k is a random parameter with a given
  distribution and mean value

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Stochastic ODE

• Apply the generalized polynomial chaos to
  variable y(t) and parameter k
• And substituting

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Stochastic ODE

• Projecting on the random space and applying
  orthogonality condition
• where

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PCT model

• A model in PCT format is described by a set of
  deterministic differential/algebraic equations
• The total number of equations is larger than that
  of the deterministic problem
• In general

u number of uncertain parameters k order of
polynomial expansion n number of state variables
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PCT for simulation

• Computational advantage
• instead or running a deterministic simulation
  thousands of time, we run a larger dimensional
  simulation only once
• Rich simulation result
• an analytical expression for the PDF of all the
  variables is found
• Design advantage
• it easily supports incremental elimination of
  uncertainty in the incremental prototyping

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PCT for simulation
Uncertain capacitor voltage
Uncertain inductor current
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PCT Application to measurements

• Provides an advanced method to combine
  measurement uncertainties
• Provides an advanced method for uncertainty
  propagation in algorithms

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Uncertainty in measurements

• The uncertainty of a measured variable is a
  combination of uncertainties
• The uncertainty of an indirect measurement
  requires the propagation of measurement
  uncertainty through the measurement algorithm
• The knowledge of measurement uncertainty is
  critical in the measurement-based decision making
  processes

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Example combining different distributions

• Due to the characteristics of the PDFs, the
  resulting uncertainty may not be well represented
  by a Gaussian distribution (central limit
  theorem)
• The resulting uncertainty is computed assuming
  the central limit theorem holds and with the PCT

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Example combining different distributions, case 1

• Case 1 central limit theorem holds

Gaussian
PCT
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Example combining different distributions, case 2

• Case 2 central limit theorem does not hold

Gaussian
PCT
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PCT in control applications

• May be used in control of uncertain systems
• Procedures may be developed to design controls
  with limited sensitivity to system parameter
  uncertainty
• The optimal control of a buck converter has been
  designed as a case study
• Resistive load and input voltage are assumed
  uncertain

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Control of Uncertain Systems buck converter

• Let us adopt an average model for the converter
• The state equations look like

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Control Testing

• Let us now assume to adopt a linear optimal
  control approach
• A possible cost function can be

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Applying PCT to the control synthesis

• Let us now suppose to adopt the same cost
  function but to define as set of state variables,
  the extended set in the PCT
• By using the coefficient of the matrix Q we can
  give an higher cost to the coefficients related
  to uncertainty

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Experimental results of the buck converter control
Traditional Case
PCT Case
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Further developments

• Application for monitoring and diagnostics
  PCT-based observer, DDDAS
• Extension of the control analysis Game Theory
• More complicated cases for the definition of
  uncertainty in measurements
• Development of new VTB models
• Application on engineering design tolerances and
  full design space