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Research Papers

Maximizing Sensitivity Vector Fields: A Parametric Study

[+] Author and Article Information
Andrew R. Sloboda

Department of Mechanical Engineering,
University of Michigan,
Ann Arbor, MI 48109
e-mail: asloboda@umich.edu

Bogdan I. Epureanu

Department of Mechanical Engineering,
University of Michigan,
Ann Arbor, MI 48109
e-mail: epureanu@umich.edu

1Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received December 10, 2012; final manuscript received December 27, 2013; published online February 12, 2014. Assoc. Editor: D. Dane Quinn.

J. Comput. Nonlinear Dynam 9(2), 021018 (Feb 12, 2014) (8 pages) Paper No: CND-12-1217; doi: 10.1115/1.4026366 History: Received December 10, 2012; Revised December 27, 2013

Sensitivity vector fields (SVFs) have proven to be an effective method for identifying parametric variations in dynamical systems. These fields are constructed using information about how a dynamical system's attractor deforms under prescribed parametric variations. Once constructed, they can be used to quantify any additional variations from the nominal parameter set as they occur. Since SVFs are based on attractor deformations, the geometry and other qualities of the baseline system attractor impact how well a set of SVFs will perform. This paper examines the role attractor characteristics and the choices made in SVF construction play in determining the sensitivity of SVFs. The use of nonlinear feedback to change a dynamical system with the intent of improving SVF sensitivity is explored. These ideas are presented in the context of constructing SVFs for several dynamical systems.

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Figures

Grahic Jump Location
Fig. 3

SV magnitude varies the evolution time for the ensemble of initial conditions u· = 0,u > 0 on the Corron attractor

Grahic Jump Location
Fig. 5

The length of time the linear assumption for SVs holds for a 1% variation in k. Here the assumption is considered broken when a 5% difference is observed between predictions of the system equations with varied parameters and the variational equations. 100% indicates the assumption holds for a period or longer.

Grahic Jump Location
Fig. 4

Average SV magnitude varies with the evolution time for different regions of the parameter space for the Duffing oscillator of Eq. (15). The subfigures are Lyapunov exponent (a), average SV magnitude for 50% of a period (b), 75% of a period (c), and 100% of a period (d).

Grahic Jump Location
Fig. 6

SV magnitudes are distributed for 10,000 SVs in a SVF for the Duffing oscillator of Eq. (15) with variation in k (A=0.4,b=0.25). The subfigures are for ΔT of 50% of a period (a), and 100% of a period (b).

Grahic Jump Location
Fig. 7

SV are distributed within the SVF field for the Duffing oscillator of Eq. (15) with variation in k (A=0.4,b=0.25). The subfigures are for ΔT of 50% of a period (a), and 100%of a period (b).

Grahic Jump Location
Fig. 8

High sensitivity spline force surface generated using 12 control points and symmetry: angled view

Grahic Jump Location
Fig. 2

Average SV magnitude varies with number of samples around the attractor for the damped, driven harmonic oscillator (k=4,b=1). Variation is in k.

Grahic Jump Location
Fig. 1

SV magnitude varies with evolution time and initial phase for the damped, driven harmonic oscillator (k=4,b=1). Variation is in k.

Grahic Jump Location
Fig. 11

Average SV magnitude over the parameter space for the Duffing oscillator of Eq. (15) reconstructed using discrete data sets and local modeling

Grahic Jump Location
Fig. 9

High sensitivity spline force surface generated using 12 control points and symmetry: top view

Grahic Jump Location
Fig. 12

Initial conditions used in generating SVs via PCA for variations in k(A=0.4,b=0.25). Darker points indicate an initial conditions from which SVs could be constructed.

Grahic Jump Location
Fig. 10

Bifurcations of a fixed-form feedback control found using the genetic search algorithm. The nominal control parameters are [Abckcβcαc]=[0.67160.09020.26600.68470.1076]. The subfigures are kc=0.2560 (a), kc=0.2660 (b), and kc=0.2760 (c).

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