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Research Papers

# Parameter Reduction in Estimated Model Sets for Robust Control

[+] Author and Article Information
Ryozo Nagamune

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

Jongeun Choi

Department of Mechanical Engineering and Department of Electrical and Computer Engineering, Michigan State University, East Lansing, MI 48824-1226jchoi@egr.msu.edu

Hereafter, we omit the superscript $M$ of $θ̂M$ for simplicity.

$Eℓ$ is the expectation operator over $ℓ$.

The covariance matrix $Λ=Eℓ{λℓλℓT}$ is a user’s choice. An example of a random process ${λℓ}$ that appears in robust control applications is the uniform distribution, with each vector element $λ(i)$, $i=1,…,q$, having the probability density function $fλ(i)$ and the covariance matrix $Λ$ as $fλ(i)=1/2$, $λ(i)∊[−1,1]$, $Λ=Iq/3$.

$[∇3G(θ)]$ is a three-dimensional matrix $(i,j,k)$ whose entry is given by $∂3G/∂θi∂θj∂θk$.

J. Dyn. Sys., Meas., Control 132(2), 021002 (Feb 02, 2010) (10 pages) doi:10.1115/1.4000661 History: Received November 09, 2008; Revised September 18, 2009; Published February 02, 2010; Online February 02, 2010

## Abstract

This paper proposes two techniques for reducing the number of uncertain parameters in order to simplify robust controller design and to reduce conservatism inherent in robust controllers. The system is assumed to have a known structure with parametric uncertainties that represent plant dynamics variation. An original set of parameters is estimated by nonlinear least-squares (NLS) optimization using noisy frequency response functions. Utilizing the property of asymptotic normality for NLS estimates, the original parameter set can be reparameterized by an affine function of the smaller number of uncorrelated parameters. The correlation among uncertain parameters is detected by the principal component analysis in one technique and optimization with a bilinear matrix inequality in the other. Numerical examples illustrate the usefulness of the proposed techniques.

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Copyright © 2010 by American Society of Mechanical Engineers
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## Figures

Figure 1

Three noisy FRF data

Figure 2

Three samples of θℓ⋆ are distributed in the square support of the probability density function for θℓ⋆. For each sample of θℓ⋆, there is an asymptotic normal distribution of its NLS estimates. Ellipsoids correspond to approximate confidence regions with some probability.

Figure 3

The noisy FRF data (dotted lines) and Bode plots of transfer functions obtained by optimally perturbing one uncertain parameter λ (solid lines). The left two and right two figures, respectively, correspond to parameter reduction based on PCA and BMI.

Figure 4

The noisy FRF data (dotted lines), and Bode plots of transfer functions obtained by optimally perturbing an uncertain parameter vector λ (solid lines). The first resonance mode around 40 rad/s was not considered in the model structure for simplicity.

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