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Overview of estimation methods for industrial dynamic systems


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.

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\(\alpha \) :

Filter factor for additive bias

\(\bar{P}\) :

Predicted covariance matrix

\(\bar{x}\) :

Predicted state vector

\(\Delta P_v\) :

Differential pressure

\(\hat{d}\) :

Prior values of the parameters or disturbances

\({\hat{y}}\) :

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

A :

State matrix

B :

Control matrix

b :

Additive model bias

C :

Observation matrix

\(c_L\) :

Slack variables to penalize model value changes below the prior value

\(c_U\) :

Slack variables to penalize model value changes above the prior value

\(C_v\) :

Constant relating valve position to flow

d :

Model parameter or disturbance vector

\(e_L\) :

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

\(e_U\) :

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

f :

Differential equation residuals


Valve lift function

g :

Output function residuals

\(g_s\) :

Specific gravity

h :

Inequality constraint residuals

I :

Integral term in IDF

K :

Kalman gain: moderate the measurement correction

\(K_c\) :

Proportional tuning constant for IDF

n :

Sampling time index

PV :

Process variable

Q :

Estimated process error covariance

q :

Flow rate (ton/h)

\(Q_d\) :

Weighting matrix on changes of the disturbance variables

\(Q_y\) :

Inverse of the measurement error covariance

R :

Estimated measurement error covariance

S :

Innovation covariance: comparison of real error to prediction

SP :


u :

Model input vector

\(w_m\) :

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

\(w_p\) :

Vector of weights to penalize deviation from the prior solution

x :

Model state vector

\(x_0\) :

Vector of initial states

y :

Vector of model values with corresponding measurements

z :

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


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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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Correspondence to John D. Hedengren.

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Hedengren, J.D., Eaton, A.N. Overview of estimation methods for industrial dynamic systems. Optim Eng 18, 155–178 (2017).

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