DAE System Parameter Identification with Goptimal Criterion Design Of Experiments

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DAE System Parameter Identification with
G-optimal Criterion Design Of Experiments

• Yang Zhang and Thomas F. Edgar
• Department of Chemical Engineering
• University of Texas at Austin
• Fall 07, TWCCC (Madison, WI)

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Presentation Outline

• Design of Experiments (DOE)
• DOE Algorithm and Different Criterion Comparison
• PCA Algorithm
• G-optimal Criteria
• Case Studies (Algebraic and Differential Equation
  Examples)
• Conclusions

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Model-Based DOE Short History

• Box and Hunter Nonlinear Algebraic Model.
  (1960s)
• Zullo, Espie and Macchietto Differential
  Algebraic Equation (DAE) Systems. (1990s)
• Macchietto et al. Parallel Experiments Design.
  (2007)
• Applications in Chemical Industry
• Reaction kinetics biological networks
  fermentation processes adsorption kinetics

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Newtons Law Example
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Newtons Law Example (Parameter Estimation)

• Suppose three distance measurements (S1 S2 and
  S3) are taken at t1, t2, and t3 and the Newtons
  law calculation results are ft1 , ft2 and ft3

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Newtons Law Example (DOE)

• Improve the estimation by choosing a better
  sampling time.
• Confidence region derived from the parameter
  covariance matrix (V?) (proportional to the
  diagonal term of V?)

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Newtons Law Example (DOE)

• Parameter covariance matrix (V?) is 2 by 2.
  D-optimal is used with the aim of shrinking the
  size of V? with an initial guess on unknown
  parameters

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Newtons Law Example (DOE)

• Initial parameter guess
• u0,guess 1 m/s m0,guess 3 kg.
• DOE result
• tD-optimal 5, 10, 10 s
• S 27.111, 102.22, 100.66 m.
• Lease Square Estimation
• u0,est 0.70045 4.5159 m/s utrue 0 m/s
• mest 5.2947 0.86312 kg mtrue 5 kg

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Criteria in Minimizing V?

• DOE for all parameters simultaneously
• DOE for part of the parameters
• in which M is the information matrix defined
  above and Ms is defined as

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Principal Component Analysis

• Data matrix X(nm) is decomposed into score
  T(nl) and loading P(ml) matrices
• The first few columns of T and P capture most of
  the variance in cov(X) and l can be calculated
  by

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G-optimal Criterion for DOE

• Mapping between DOE and PCA
• Eigenvalues and eigenvectors of M are ?i (in
  descending order) and pi, respectively.
• Argument if ?1 gtgt ?2, it is not necessary to
  optimize ?1 . If we want to estimate ?2, it is
  better to select p1 instead of p2.

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G-optimal Criterion for DOE

• G-optimal objective function
• m is the number of parameters in the model and l
  is the retained number of principal components.
  pji2 describes the estimated variance of ?j
  captured by ?i. ai is the eigenvalue of V? (ai
  1/ ?i).
• pji is orthogonal and normalized, which
  guarantees
• and the value of pi indicates
  which parameters are affected the most by ?i.

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Relationship with Available Criteria

• Improve the precision of all parameters (j 1m)
  and one PC is retained by PCA

• Improve the precision of all parameters (j 1m)
  and all the PCs are retained by PCA

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G-optimal Algorithm Advantages

• For large scale DAE system cases, it is always
  easy to reduce the scale of the DOE problem by
  grouping the estimated parameters (loading
  matrix).
• G-criterion is a general form of most widely-used
  criteria such as D- and E- optimality.
• By introducing PCA, the ill-conditioned M is
  avoided.
• The criterion can be easily embedded in parallel
  and sequential DOE routines.

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Algebraic Equation Example

• Michaelis-Menten (M-M) algebraic model
• a and b are between 0 and 150 and true values
  are 100 and 15. Three experimental points for
  each batch. The sensitivity matrix is

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Algebraic Equation Example

• Case I a is accurately known. Perform DOE for b
• Case II b is accurately known. Perform DOE for
  a

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DOE under different criteria
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M-M Parameter Estimation
Parameter estimation results by least squares
(areal 100 breal 15)
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Fed-batch Example

• Mathematical Model
• X biomass concentration
• S substrate concentration
• u1 dilution factor (0.05 0.20 h-1)
• u2 substrate concentration in the feed (5 35
  g/l)

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Fed-batch Example

• Design vector

• Initial condition

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Fed-batch Example

• Initial Guess ?0 0.5 0.5 0.5 0.5
• Information matrix with initial experiment design
  and parameter guessing

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Control Inputs from DOE
Control inputs suggested by different DOE
criteria. Left upper D-optimal Left lower
E-optimal Right upper G-optimal
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DOE Results by Different Criteria
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Parameter Estimation Results
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Sequential DOE by G-optimal (?2)
? 0.321 0.0312 0.1776 0.0924
0.582 0.0459 0.0571 0.01883
t 29.48 5.51 36.3 8.670
Control inputs and sampling points calculated by
sequential Generalized optimal criterion.
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Control Inputs from DOE

• Bad Initial Guess (70 off the true value) ?0
  0.527 0.054 0.935 0.015
• Control inputs suggested by D-optimal (left) and
  G-optimal (right)

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Parameter Estimation Results
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Sequential Design Comparison
Sequential Design control inputs suggested by
D-optimal (left) and G-optimal (right) .
G-optimal is focusing on improving the precision
of ?2.
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Sequential Estimation Results
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Conclusions

• G-optimal is proposed by combining PCA with
  sensitivity matrix analysis.
• The main advantages of G-optimal criteria
  include
• 1. Easy to rescale a large DAE system into small
  components according to their sensitivity
  behavior.
• 2. Easy to focus on improving a specific subset
  of parameters.
• 3. Robust to parameter initial guess.
• 4. Easy to implement into DAE systems and further
  parallel experiment design.

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Acknowledgment

• Emerson Process Management