Root sensitivity to parameter uncertainties: a statistical approach

Abstract A statistical approach to calculating the sensitivity of characteristic roots to plant parameter uncertainties is presented. Means and variances of the eigenvalues are obtained in terms of the statistical moments of the system parameters, the sensitivity of an eigenvalue being defined by its relative standard deviation. The method allows for correlation among the system parameters and therefore among the elements of the plant matrix. The approach is a first-order random perturbation analysis that implicitly assumes gaussian random variables. The results derived by this statistical procedure are shown to be a generalization of the norm sensitivity results previously obtained in the literature by calculating the norm of the sensitivity matrix. Indeed, the two approaches are shown to be equivalent for uncorrelated and identically distributed plant matrix elements. The statistical measure of root sensitivity presented here is believed to be more representative than the usual norm sensitivity measure.

[1]  Leonard Meirovitch,et al.  Computational Methods in Structural Dynamics , 1980 .

[2]  R. Fox,et al.  Rates of change of eigenvalues and eigenvectors. , 1968 .

[3]  L. C. Rogers.  Derivatives of eigenvalues and eigenvectors , 1970 .

[4]  Franz Rellich,et al.  Perturbation Theory of Eigenvalue Problems , 1969 .

[5]  Athanasios Papoulis,et al.  Probability, Random Variables and Stochastic Processes , 1965 .

[6]  Masanobu Shinozuka,et al.  Random Eigenvalue Problems in Structural Analysis , 1971 .

[7]  T. T. Soong,et al.  On the natural frequencies of a disordered linear chain of N degrees of freedom , 1963 .

[8]  Tosio Kato.  Perturbation theory for linear operators , 1966 .

[9]  Robert A. Singer,et al.  Selecting state variables to minimize eigenvalue sensitivity of multivariable systems , 1969, Autom..

[10]  P. N. Paraskevopoulos,et al.  Eigenvalue-eigenvector sensitivity analysis of linear time-invariant singular systems , 1984 .

[11]  J. D. Collins,et al.  The eigenvalue problem for structural systems with statistical properties. , 1969 .

[12]  B. Porter,et al.  Eigenvalue and eigenvector sensitivities in linear systems theory , 1969 .

[13]  R. Courant,et al.  Methods of Mathematical Physics , 1962 .

[14]  P. Morse,et al.  Methods of theoretical physics , 1955 .

[15]  R. Ibrahim.  Structural Dynamics with Parameter Uncertainties , 1987 .

[16]  Arthur E. Bryson,et al.  Applied Optimal Control , 1969 .

[17]  Robert E. Skelton,et al.  MINIMAL ROOT SENSITIVITY IN LINEAR SYSTEMS , 1983 .

[18]  Raymond H. Plaut,et al.  Derivatives of eigenvalues and eigenvectors in non-self-adjoint systems. , 1973 .

[19]  Jr. B. Morgan.  Sensitivity analysis and synthesis of multivariable systems , 1966 .

[20]  J. Scheidt,et al.  Random Eigenvalue Problems , 1983 .

[21]  R. Alden,et al.  Eigenvalue tracking due to parameter variation , 1985 .

[22]  J. H. Wilkinson.  The algebraic eigenvalue problem , 1966 .

[23]  Patrick E. Mantey.  Eigenvalue sensitivity and state-variable selection , 1968 .

[24]  Kurt Friedrichs,et al.  Perturbation of Spectra in Hilbert Space , 1967 .

[25]  T. W. Anderson.  An Introduction to Multivariate Statistical Analysis , 1959 .

[26]  J. N. Yang,et al.  Free Vibration of a Disordered Periodic Beam , 1974 .