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Research Papers

Convergence Properties of the Kalman Inverse Filter

[+] Author and Article Information
B. S. Petschel

School of Mechanical and Mining Engineering,
The University of Queensland,
Brisbane 4072, Australia

K. Soltani Naveh

School of Mechanical and Mining Engineering,
The University of Queensland,
Brisbane 4072, Australia
e-mail: k.soltaninaveh@uq.edu.au

P. R. McAree

Professor
School of Mechanical and Mining Engineering,
The University of Queensland,
Brisbane 4072, Australia
e-mail: p.mcaree@uq.edu.au

1Corresponding author.

Contributed by the Dynamic Systems Division of ASME for publication in the JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL. Manuscript received August 5, 2013; final manuscript received October 19, 2017; published online December 19, 2017. Editor: Joseph Beaman.

J. Dyn. Sys., Meas., Control 140(6), 061001 (Dec 19, 2017) (15 pages) Paper No: DS-13-1302; doi: 10.1115/1.4038267 History: Received August 05, 2013; Revised October 19, 2017

The Kalman filter has a long history of use in input deconvolution where it is desired to estimate structured inputs or disturbances to a plant from noisy output measurements. However, little attention has been given to the convergence properties of the deconvolved signal, in particular the conditions needed to estimate inputs and disturbances with zero bias. The paper draws on ideas from linear systems theory to understand the convergence properties of the Kalman filter when used for input deconvolution. The main result of the paper is to show that, in general, unbiased estimation of inputs using a Kalman filter requires both an exact model of the plant and an internal model of the input signal. We show that for unbiased estimation, an identified subblock of the Kalman filter that we term the plant model input generator (PMIG) must span all possible inputs to the plant and that the robustness of the estimator with respect to errors in model parameters depends on the eigenstructure of this subblock. We give estimates of the bias on the estimated inputs/disturbances when the model is in error. The results of this paper provide insightful guidance in the design of Kalman filters for input deconvolution.

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Figures

Grahic Jump Location
Fig. 2

The KIF, including the open-loop system governed by A1 and A2, and the closed-loop feedback system governed by Â1 and A3

Grahic Jump Location
Fig. 4

Response of PIG and PMIG for step disturbance in double mass-spring-damper system

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Fig. 3

The double mass-spring-damper system

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Fig. 5

Response of PIG and PMIG for combined sinusoids disturbance in double mass-spring-damper system

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Fig. 6

The block diagram of the mass spring damper system

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Fig. 7

Response of PIG and PMIG for combined step and sinusoid disturbance in double mass-spring-damper system with imperfect model parameters

Grahic Jump Location
Fig. 8

‖E[u−û]‖/‖E[x]‖, bias relative to state versus upper bound calculated by ‖δDM‖·‖X‖+(‖DM‖+‖δDM‖)‖δX‖

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