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Presented by: Tyler A. Soderstrom

Improved Simultaneous Data Reconciliation, Bias Detection and Identification Using Mixed Integer Optimization Methods. Presented by: Tyler A. Soderstrom. Presentation Overview. Background MILP Method Extension to Nonlinear Problems Inclusion of Statistical Tests as Constraints

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Presented by: Tyler A. Soderstrom

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  1. Improved Simultaneous Data Reconciliation, Bias Detection and Identification Using Mixed Integer Optimization Methods Presented by: Tyler A. Soderstrom

  2. Presentation Overview • Background • MILP Method • Extension to Nonlinear Problems • Inclusion of Statistical Tests as Constraints • Multiple System Models as Constraints • Correlated Data • Conclusions

  3. Background • Data Reconciliation • Optimal estimates for noisy measurements • Bias Detection / Identification • Determine presence and location of bias • Problems are closely related • Presence of bias skews reconciliation results • Common techniques require reconciliation residuals

  4. MILP Method • Type of systems considered • Process data matrix

  5. MILP Method • Problem Formulation

  6. MILP Method • Realizable form

  7. d l - x × - x × U U U U l l l l l l MILP Method • Bias constraint region

  8. Tuning Issues • Horizon Length • Binary variable weighting • Bias bounding and thresholding

  9. Extension to Nonlinear Problems • Straightforward constraint replacement • with • Modified problem is a MINLP • Tougher class of problem • Solution technology is not as mature • Global solution is not guaranteed

  10. Solution Methods • Outer Approximation / Equality Relaxation • DICOPT (J. Viswanathan and I. E. Grossman) • Random Search • Genetic Algorithm (J. Holland) • Meta-Heuristics • Tabu Search (F. Glover)

  11. Basic Method Extensions • Make use of Past Information • Formulate Extensions as Additional Problem Constraints • Incorporate Common Statistical Tests • Include Empirical Data Model • Compensate for Non-Ideal Process Data • May Require Modifications to Objective

  12. Using Previous Estimates • Moving Horizon Estimation Problem • Previous Estimates of Process Variables and Biases Can be Made Available • Including Past Information In Current Problem Execution Improves Stability and Convergence of Estimates • Objective is Pathway to Past Information

  13. Objective Modifications • Add to Objective Φ • Where and are Estimates from Previous Execution • Convert to Realizable Form

  14. Realizable Form • Objective Φ • Additional Constraints

  15. Bias Penalty Term • Inclusion Depends on Objectives • Most Will be Zero • May Delay Identification of Bias • If Bias is Persistent, Improve Estimate Convergence • Not Important for Optimization Engine • Previous Solution Warm Starts Current Run

  16. Statistical Tests as Constraints • Tests are based on Hypothesis Testing • Test Statistic is Proposed • Statistic is Calculated with Current Data • If Statistic Exceeds a Threshold Value (Related to Level of Confidence) Bias is Present • Statistic is defined as a problem variable • Definition added as a problem constraint • Constraint bounding Statistic below threshold Forces the no bias conclusionat solution

  17. Mathematical Description • Hypothesis Testing • H0 there is no bias in the process data • H1 there is at least 1 bias in the process data • Choice Depends on value of test statistic at a given level of significance • Test Statistic Z • Add the Following Constraints to Problem Definition Constraint Z= h(y) Null Enforcement Constraint |Z| < Za

  18. Objective Modifications • Previous Description may be Infeasible • Define new Constraint Violation Variable and Change Constraint Null Enforcement Constraint: |Z|< Za + y • Penalize Variable in Objective Objective:  = old + wiy

  19. MILP Example

  20. Embedded PC Test • Principal Component Test • Form Matrix s.t. it contains eigenvectors of • ye contains principle component scores • scores are normal zero mean, unit variance

  21. Embedded PC Test • Principal Component Test • Threshold Value at confidence level  • Perform Test on Averaged Measurements • Enhance Power of Test • Single set of additional constraints • is used in the formation of

  22. Embedded PC Test • Additional Constraints

  23. Performance Measures • Average number of Type I Errors • Overall Power

  24. Simulation Results

  25. Simulation Results

  26. Simulation Results

  27. Simulation Results

  28. Discussion of Results • No Benefit to OP • Estimates from basic method usually pass tests without specific enforcement • Test Enforcement Increases AVTI • Other biases forced to become active to lower statistics • Usually Global Statistical Tests Used First • require nonlinear equations

  29. Non-Ideal Data Compensation • Serially Correlated Data • Requires New Measurement Noise Model • Error Sequence, , Forms a Stationary Process • Assume no Cross Correlation

  30. Statistical Tests When Data are Serially Correlated • Tests on Individual Vectors are Unaffected • Statistical Tests are Often Used on Vectors of Averaged Measurement • Increase Power of the Test • Autocorrelation Invalidates Test Assumptions • Procedure Must be Modified

  31. Statistical Tests When Data are Serially Correlated • Most Test Statistics Require • Covariance of N Averaged Measurements: • Time Independent Data: • Autocorrelated Data:

  32. Methods of Dealing With Serially Correlated Data • Correcting the Variance • Requires all Autocorrelation Coefficients • Coefficients can be calculated analytically if noise model is known (e.g. time series) • Otherwise coefficients can be estimated • Prewhitening • Filtering Approach • Requires expression of noise model

  33. Methods of Dealing With Serially Correlated Data • Prewhitening (cont.) • Calculate a “approximately independent” sequence • Apply tests designed for independent data

  34. Implementing Compensation Within MIP Framework • Correcting the Variance • Unknown Correlation Model • Estimate with bias free data • Include , calculated using as parameters in MIP program • Known Correlation Model • calculated analytically are used as parameters in MIP program • Include Modified Test as Set of Constraints

  35. Implementing Compensation Within MIP Framework • Prewhitening • Uncorrelated residuals written as functions of noise model parameters and measurements • Equations included as constraints • Tests on uncorrelated residuals included as constraints

  36. Conclusions • MIP bias detection / identification performs better than several other methods • High Power / Low Occurrence of False Identification • Straightforward implementation • Method Enhancements • Consider Past Information • Include Statistical Tests in Constraints • Univariate Tests Do Not Improve Performance • Global Tests May Help, but require nonlinear equations • Handling Autocorrelated Data

  37. Future Work • Investigate Additional Constraints with Nonlinear Models • Nonlinear Statistical Tests • Improve sensitivity to small bias • Compare solution methods on larger nonlinear models • Extend Method to Dynamic Models • Discrete vs. Continuous • Linear and Nonlinear • Computational Issues

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