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On-line estimation

A comparison and evaluation of alternative recursive and batchwise approaches

Tore Lid, On-line estimation

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Outline

• Introduction• The Kalman filter• The Extended Kalman Filter• Moving Horizon Estimator• Simple example• Conclusions

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What is estimation?

Estimation is the calculated approximation of a result which is usable even if input data may be incomplete, uncertain, or noisy.

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Why estimate?

Monitor

Control

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The process model

The process

Measuredoutputs

Measured inputs

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The state space model

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The linear state space model

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The Kalman Filter

A priori estimate

A posteriori estimate

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The Kalman FilterA priori estimate

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The Kalman FilterA posteriori estimate

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The Kalman Filter

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The Kalman Filter

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The Kalman Filter

timet(k) t(k+1)

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Nonlinear state space model

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The Extended Kalman Filter

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The Extended Kalman Filter

Time update Measurement update

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Moving horizon estimator

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Moving horizon estimator

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Moving horizon estimator

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Moving horizon estimator

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ExampleMeasurements:•Mass in Eq. Tank•Mass in Tank 1•Mass in Tank 2•Mass in Tank 3•Waste liquid mass flow

Objective: Estimate possible tank leakage

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ExampleLinear state space model

Simulation Estimation

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Example

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Conclution

• Extended Kalman filter– Has a fixed computational load– Linearization degrades the performance– Does not handle constraints on states and disturbances

• Moving horizon estimator– Handle constraints on states and disturbances

• Should be used with care, may have negative side effects

– No linearization of nonlinear process models– The computation of the arrival cost is still a challenge– High computational load for large systems

• R and Q has to be estimated

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Acknowledgements

• Tor Steinar Schei• Magne Hillestad• Stig Strand• Marius Govatsmark

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References• [1] Tor Steinar Schei, On-line Estimation for Process Control and Optimization

Applications, Presented at DYCOPS June 6-8th 2007, 8th International Symposium on Dynamics and Control of Process Systems

• [2] C.V. Rao and J. B. Rawlings, Constrained Process Monitoring: Moving Horizon Approach, AIChE Journal, 2002, 48, 1, 97-108

• [3] G. Welch and G. Bishop, An Introduction to the Kalman Filter, University of North Carolina at Chapel Hill, Department of Computer Science,TR 95-041

• [4] E. L Haseltine and J. B. Rawlings, A Critical Evaluation of Extended Kalman Filter and Moving Horizon Estimation, Ind. Chem. Eng. Res. 2005, 44, 2451-2460