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Research Papers

Optimal Parameter Estimation for Long-Term Prediction in the Presence of Model Mismatch

[+] Author and Article Information
Ryan Sangjun Lee

Houston Pressure and Sampling, Schlumberger, Sugar Land, TX 77478-3135rlee36@slb.com

Gregery T. Buzzard

Department of Mathematics,  Purdue University, West Lafayette, IN 47907-2067

Peter H. Meckl

 School of Mechanical Engineering, Purdue University, West Lafayette, IN 47907-2088

J. Dyn. Sys., Meas., Control 134(4), 041010 (May 07, 2012) (16 pages) doi:10.1115/1.4005497 History: Received April 29, 2011; Revised November 23, 2011; Published May 04, 2012; Online May 07, 2012

For nonlinear multi-input multi-output (MIMO) systems such as multilink robotic manipulators, finding a correct, physically derived model structure is almost impossible, so that significant model mismatch is nearly inevitable. Moreover, in the presence of model mismatch, the use of least-squares minimization of the one-step-ahead prediction error (residual error) to estimate unknown parameters in a given model structure often leads to model predictions that are extremely inaccurate beyond a short time interval. In this paper, we develop a method for optimal parameter estimation for accurate long-term prediction models in the presence of significant model mismatch in practice. For many practical cases, where a correct model and the correct number of degrees of freedom for a given model structure are unknown, we combine the use of long-term prediction error with frequency-based regularization to produce more accurate long-term prediction models for actual MIMO nonlinear systems.

Copyright © 2012 by American Society of Mechanical Engineers
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References

Figures

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Figure 1

Predicted output, ŷ(k) of a model using actual outputs (parameter estimation using residual error) and simulated outputs (parameter estimation using simulation error)

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Figure 2

Flow diagram of OPE using simulation error

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Figure 3

Simulation result of the second-order ARMAX model identified by the linear least squares fit to minimize residual error

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Figure 4

Simulation result of the second-order ARMAX model in Fig. 3 from 5.5 to 8 s

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Figure 5

Simulation result of the second-order ARMAX model identified by the OPE to minimize simulation error

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Figure 6

Simulation result of the second-order ARMAX model in Fig. 5 from 5.5 to 8 s

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Figure 7

Simulation result of the eighth-order ARMAX model identified by the linear least squares fit to minimize residual error

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Figure 8

Simulation result of the eighth-order ARMAX model in Fig. 7 from 0 to 0.3 s

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Figure 9

Simulation result of the eighth-order ARMAX model identified by the OPE to minimize simulation error

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Figure 10

Simulation results of the second-order ARMAX model in Fig. 5 and the eighth-order ARMA model in Fig. 9 from 6.5 to 8 s

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Figure 11

Simulation result of the continuous model identified by the linear least squares fit to minimize residual error

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Figure 12

Simulation results of the continuous model in Fig. 1 from 0 to 3.5 s

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Figure 13

Simulation result of the continuous model identified by the OPE to minimize simulation error

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Figure 14

Simulation results of the continuous model in Fig. 1 from 0 to 3.5 s

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Figure 15

Square penalty function

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Figure 16

Simulation result of the eighth order ARMAX model identified by the cost function with SSSE only

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Figure 17

Simulation result of the eighth order ARMAX model identified by the composite cost function with ωc  = 8 Hz and ρ* = 20

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Figure 18

System configuration of the two-link flexible-joint robot (side view)

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Figure 19

Photograph of the two-link flexible joint robotic manipulator [31]

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Figure 20

Schematic diagram of the system with physical parameters [5]

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Figure 21

Simulation results of the Lagrangian model starting at 0 deg angle of θ2

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Figure 22

Simulation results of the Lagrangian model obtained by unconstrained OPE. Some of the identified parameter values turned out not to have physical meanings (J2  = −0.0308, J6  = −0.0297 in Table 5).

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Figure 23

Simulation results of the Lagrangian model obtained by OPE with parameter constraints. The same upper and lower bounds for parameter values were used as in the constrained linear least squares fitting in Sec. 4.

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Figure 24

Simulation results of the twelfth order MIMO NARMAX model identified by the linear least squares fit

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Figure 25

Simulation results of the twelfth order MIMO NARMAX model identified by the OPE with the composite cost function (ωc  = 10 Hz and ρ* = 2)

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Figure 26

Simulation result of θ2 of the model obtained with SSSE

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Figure 27

Simulation result of θ2 of the model obtained with the composite cost

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Figure 28

Validation results of the MIMO NARMAX model starting at 50 deg of initial θ2 angle

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Figure 29

Validation results of the MIMO NARMAX model starting at −30 deg of initial θ2 angle

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