Optimization and Engineering

, Volume 18, Issue 1, pp 155–178 | Cite as

Overview of estimation methods for industrial dynamic systems

  • John D. HedengrenEmail author
  • Ammon N. Eaton


Measurement technology is advancing in the oil and gas industry. Factors such as wireless transmitters, reduced cost of measurement technology, and increased regulations that require active monitoring tend to increase the number of available measurements. There is a clear opportunity to distill the recent flood of measurements into relevant and actionable information. Common methods to do this include a filtered bias update, implicit dynamic feedback, Kalman filtering, and moving horizon estimation. The purpose of these techniques is to validate measurements and align imperfect mathematical models to the actual process. Additionally, they can determine a best-estimate of the current state of the process and any potential disturbances. These methods allow potential improvements in earlier detection of disturbances, process equipment faults, and improved state estimates for optimization and control.


Dynamic data reconciliation Wired drillpipe Industrial data Moving horizon estimation Kalman filter 

List of symbols

\(\alpha \)

Filter factor for additive bias


Predicted covariance matrix


Predicted state vector

\(\Delta P_v\)

Differential pressure


Prior values of the parameters or disturbances


Vector of prior model values at the sampling times (\({\hat{y}}_0,\ldots ,{\hat{y}}_n\))\(^T\)

\(\Phi \)

Objective function value

\(\sigma _q\)

Standard deviation of state noise

\(\sigma _r\)

Standard deviation of measurement noise

\(\tau \)

Time constant

\(\tau _I\)

Integral time constant for IDF

\(\tilde{\delta }\)

Innovation: comparison of model to measurements


State matrix


Control matrix


Additive model bias


Observation matrix


Slack variables to penalize model value changes below the prior value


Slack variables to penalize model value changes above the prior value


Constant relating valve position to flow


Model parameter or disturbance vector


Slack variables to penalize model values below the measurement dead-band


Slack variables to penalize model values above the measurement dead-band


Differential equation residuals


Valve lift function


Output function residuals


Specific gravity


Inequality constraint residuals


Integral term in IDF


Kalman gain: moderate the measurement correction


Proportional tuning constant for IDF


Sampling time index


Process variable


Estimated process error covariance


Flow rate (ton/h)


Weighting matrix on changes of the disturbance variables


Inverse of the measurement error covariance


Estimated measurement error covariance


Innovation covariance: comparison of real error to prediction




Model input vector


Vector of weights on the model values outside a measurement dead-band


Vector of weights to penalize deviation from the prior solution


Model state vector


Vector of initial states


Vector of model values with corresponding measurements


Vector of measurements


Implicit dynamic feedback


Advanced process control


Bottom hole assembly


Extended Kalman filter


Moving horizon estimation


Model predictive control


Managed pressure drilling


Nonlinear programming


Proportional integral controller


Real time optimization


Single input-single output


Unscented Kalman filter



The authors would like to acknowledge the financial and technical assistance of National Oilwell Varco (NOV) and SINTEF in projects related to modeling and control design for automated drilling systems.


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Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Brigham Young UniversityProvoUSA

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