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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

1

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

2

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) 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
X ( ) 

a
j0



j



j

  ( ) 

5

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

VTB solver takes care
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Selection of the base
e. g .

Random variable Continuous Gaussian Gamma Beta Uniform Discrete Poisson Binomial Negative Binomial Hypergeometric

Polynomials Hermite Laguerre Jacobi Legendre Charlier Krawtchouk Meixner Hahn

Support -inf, +inf 0,+inf [a,b] [a,b] {0,1,….} {0,1,….N} {0,1,….} {0,1,….N}

 0    1  1      2      1
2

...

e.g. uncertain resistance with gaussian distribution
R  R 0  0    R1  1     R 0  R1 R 2  0 , R 3  0 ,...

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Stochastic ODE
• Let us consider the following equation:
dy ( t ) dt   ky ( t ), y (0)  y 0 ,

• Where k is a random parameter with a given distribution and mean value

k  k 0  k 1

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Stochastic ODE
• Apply the generalized polynomial chaos to variable y(t) and parameter k
y (t ) 



M

y i ( t )  i ( ),

k 

i0

k
i i0

M

i

( )

• And substituting:


i0

M

dy i ( t ) dt

 i ( )   

M

i0 j0

k

M

i

y j ( t )  i ( )  j ( ).

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Stochastic ODE
• Projecting on the random space and applying orthogonality condition:
dy k ( t ) dt  1 
2 k

k
i0 j0

M

M

i

y j ( t ) eijk ,

k  0,

,M ,

• where
eijk   i  j  k

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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:

P  ( u * k  1) * n

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
A, expected= 4.997634 A, standard deviation= 0.295948 B1, expected = -0.002656 B1, standard deviation = 0.575690 B2, expected = 0.001992 B2, standard deviation = 0.5758

Gaussian
coverage factor 1, coverage factor 1.645, coverage factor 1.658, coverage factor 2, coverage factor 3, confidence confidence confidence confidence confidence  68.2689%  90 %  90.2681%  95.4499%  99.7300%
coverage factor 1, coverage factor 1.645, coverage factor 1.658, coverage factor 2, coverage factor 3,

PCT
confidence confidence confidence confidence confidence  66.0190%  89.7047  89.9837%  96.2179%  99.9980%  90%

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Example: combining different distributions, case 2
• Case 2: central limit theorem does not hold
Gaussian
 x   0.7166   x   0.8465

2

PCT
 x   0.6641   x   0.8149

 68.2688%  90.0029%  95.4498%  99.73 %
2

coverage factor 1, coverage factor 1.645, coverage factor 2, coverage factor 3,

confidence confidence confidence confidence

coverage factor 1, coverage factor 1.645, coverage factor 2, coverage factor 3,

confidence confidence confidence confidence

 65.0379%  89.1376%  96.5808%  99.9999 %

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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:
dil dt dv c dt     vc L vc RC  d V cc L il C

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

J  x (  )Q x (  )  u (  ) R u (  )

T

T

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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

1 Q  0

0  1

Q PCT

1  0  0   0 0  0 

0 100 0 0 0 0

0 0 100 0 0 0

0 0 0 1 0 0

0 0 0 0 100 0

0   0  0   0  0   100  
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Experimental results of the buck converter control

Output voltage
7 6 5

Output voltage (chaos case)
7 6 5

vout [V]

4 3 2 1 0 -0.004 -0.002 0 0.002 t [s] 0.004 0.006 0.008

vout [V]

4 3 2 1 0 -0.004 -0.002 0 0.002 t [s] 0.004 0.006 0.008

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

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