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Technical Briefs

Parameter Uncertainty Modeling Using the Multidimensional Principal Curves

[+] Author and Article Information
M. Sepasi1

Department of Mechanical Engineering, University of British Columbia, Vancouver, V6T 1Z4, BC, Canadasepasi@mech.ubc.ca

F. Sassani

Department of Mechanical Engineering, University of British Columbia, Vancouver, V6T 1Z4, BC, Canadasassani@mech.ubc.ca

R. Nagamune

Department of Mechanical Engineering, University of British Columbia, Vancouver, V6T 1Z4, BC, Canadanagamune@mech.ubc.ca

1

Corresponding author.

J. Dyn. Sys., Meas., Control 132(5), 054501 (Aug 11, 2010) (7 pages) doi:10.1115/1.4001791 History: Received March 16, 2009; Revised April 21, 2010; Published August 11, 2010; Online August 11, 2010

This paper proposes a technique to model uncertainties associated with linear time-invariant systems. It is assumed that the uncertainties are only due to parametric variations caused by independent uncertain variables. By assuming that a set of a finite number of rational transfer functions of a fixed order is given, as well as the number of independent uncertain variables that affect the parametric uncertainties, the proposed technique seeks an optimal parametric uncertainty model as a function of uncertain variables that explains the set of transfer functions. Finding such an optimal parametric uncertainty model is formulated as a noncovex optimization problem, which is then solved by a combination of a linear matrix inequality and a nonlinear optimization technique. To find an initial condition for solving this nonconvex problem, the nonlinear principal component analysis based on the multidimensional principal curve is employed. The effectiveness of the proposed technique is verified through both illustrative and practical examples.

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Copyright © 2010 by American Society of Mechanical Engineers
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Figures

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Figure 1

Five different dynamical systems with same structure

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Figure 2

(a) Parameter set and (b) linear PCA calculate a manifold approximating the parameter set

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Figure 3

A robust control configuration

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Figure 4

Distance dl for a set of five samples

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Figure 5

A machine toll with a ball screw

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Figure 6

Frequency responses of the ball screw machine (dotted line), the estimated rational form of transfer function (solid line) and the model set calculated by applying linear PCA (dashed line). The results for r≥2 almost overlap the solid lines.

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Figure 7

Convergence rate of algorithms for r=3 and m=2

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