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MODEL VALIDATION & IDENTIFICATION

A Polynomial Chaos-Based Kalman Filter Approach for Parameter Estimation of Mechanical Systems

[+] Author and Article Information
Emmanuel D. Blanchard1

Department of Mechanical Engineering, Virginia Polytechnic Institute and State University, 3103 Commerce Street, Blacksburg, VA 24061eblancha@vt.edu

Adrian Sandu

Department of Computer Science, Virginia Polytechnic Institute and State University, 2224 Knowledge Works, Blacksburg, VA 24061sandu@cs.vt.edu

Corina Sandu

Department of Mechanical Engineering, Virginia Polytechnic Institute and State University, 104 Randolph Hall, Blacksburg, VA 24061csandu@vt.edu

1

Corresponding author.

J. Dyn. Sys., Meas., Control 132(6), 061404 (Nov 09, 2010) (18 pages) doi:10.1115/1.4002481 History: Received September 20, 2008; Revised April 13, 2010; Published November 09, 2010; Online November 09, 2010

Mechanical systems operate under parametric and external excitation uncertainties. The polynomial chaos approach has been shown to be more efficient than Monte Carlo for quantifying the effects of such uncertainties on the system response. Many uncertain parameters cannot be measured accurately, especially in real time applications. Information about them is obtained via parameter estimation techniques. Parameter estimation for large systems is a difficult problem, and the solution approaches are computationally expensive. This paper proposes a new computational approach for parameter estimation based on the extended Kalman filter (EKF) and the polynomial chaos theory for parameter estimation. The error covariances needed by EKF are computed from polynomial chaos expansions, and the EKF is used to update the polynomial chaos representation of the uncertain states and the uncertain parameters. The proposed method is applied to a nonlinear four degree of freedom roll plane model of a vehicle, in which an uncertain mass with an uncertain position is added on the roll bar. The main advantages of this method are an accurate representation of uncertainties via polynomial chaos, a computationally efficient update formula based on EKF, and the ability to provide a posteriori probability densities of the estimated parameters. The method is able to deal with non-Gaussian parametric uncertainties. The paper identifies and theoretically explains a possible weakness of the EKF with approximate covariances: numerical errors due to the truncation in the polynomial chaos expansions can accumulate quickly when measurements are taken at a fast sampling rate. To prevent filter divergence, we propose to lower the sampling rate and to take a smoother approach where time-distributed observations are all processed at once. We propose a parameter estimation approach that uses polynomial chaos to propagate uncertainties and estimate error covariances in the EKF framework. Parameter estimates are obtained in the form of polynomial chaos expansion, which carries information about the a posteriori probability density function. The method is illustrated on a roll plane vehicle model.

Copyright © 2010 by American Society of Mechanical Engineers
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Figures

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Figure 1

Polynomial chaos-based EKF versus traditional EKF using linear propagation histograms: (a) forecast state for EKF with linear propagation, (b) forecast state for polynomial chaos-based EKF, (c) assimilated state for EKF with linear propagation, and (d) assimilated state for polynomial chaos-based EKF

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Figure 2

Four degree of freedom roll plane model (adapted from the model used in Ref. 66)

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Figure 3

Beta (2,2) distribution: (a) for value of the mass and (b) for value of the position of the CG of the mass

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Figure 4

Road profile—speed bump

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Figure 5

Observed states—displacements and velocities: (a) measured and (b) for nominal values (ξ1=0, ξ2=0)

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Figure 6

EKF estimation (one-time-step-at-a-time) for speed bump input with ten time points (noise=1%): (a) mass in the form of PDF, (b) distance in the form of PDF, (c) mass for each term index, and (d) distance for each term index

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Figure 7

EKF estimation (one-time-step-at-a-time) for speed bump input with 100 time points (noise=1%): (a) mass in the form of PDF, (b) distance in the form of PDF, (c) mass for each term index, and (d) distance for each term index

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Figure 8

Absolute error for the estimated parameters ξ1 and ξ2 with the nonlinear half-car model for the speed bump with respect to (a) the number of time points and (b) the length of the time step

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Figure 9

EKF estimation (whole-set-of-data-at-once) for speed bump input with ten time points (noise=1%): (a) mass in the form of PDF and (b) distance in the form of PDF

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Figure 10

EKF estimation (whole-set-of-data-at-once) for speed bump input with 100 time points (noise=1%): (a) mass in the form of PDF and (b) distance in the form of PDF

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Figure 11

Road input at 1 Hz

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Figure 12

EKF estimation (one-time-step-at-a-time) at 1 Hz with ten time points (noise=1%): (a) mass in the form of PDF, (b) distance in the form of PDF, (c) mass at each time index, and (d) distance at each time index

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Figure 13

EKF estimation (whole-set-of-data-at-once) at 1 Hz with ten time points (noise=1%): (a) mass in the form of PDF and (b) distance in the form of PDF

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Figure 14

EKF estimation (one-time-step-at-a-time) at 1 Hz with 100 time points (noise=1%): (a) mass in the form of PDF, (b) distance in the form of PDF, (c) mass at each time index, and (d) distance at each time index

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Figure 15

EKF estimation (whole-set-of-data-at-once) at 1 Hz with 100 time points (noise=1%): (a) mass in the form of PDF and (b) distance in the form of PDF

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Figure 16

Covariance of EN after convergence with no model error (Q=0, B=0): (a) R=0.0001, Mu=0.0005, and a=−1; (b) R=0.0001, Mu=0.0050, and a=−1

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Figure 17

Covariance of EN after convergence for R=0.0001, Mu=0.0050, a=−1, Q=0.01, and B=0 (i.e., model error, but with no bias): (a) covariance due to model errors and (b) covariance due to measurement noise

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Figure 18

Covariance of EN after convergence for R=0.0001, Mu=0.0050, a=−1, Q=0.01, and B=1 (i.e., model error, but with bias): (a) error due to model errors and (b) covariance of error due to measurement noise

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