Approximate analysis of a class of linear stochastic systems with colored noise parameters

Abstract In this article, linear stochastic systems containing both additive and multiplicative noise parameters are considered. A method for the approximate evaluation of the moments, in particular the mean and variance, is presented. Techniques from matrix calculus and Kronecker algebra are employed in the investigation, and the results derived are easily amenable to computational procedures. Possible applications of the results are illustrated by two engineering examples. An attempt is made to explore the interplay of additive and multiplicative random elements.

[1]  T. K. Caughey,et al.  On the monte carlo methodology for cumulative damage , 1983 .

[2]  R. Brockett.  Lie Theory and Control Systems Defined on Spheres , 1973 .

[3]  H. Neudecker.  Some Theorems on Matrix Differentiation with Special Reference to Kronecker Matrix Products , 1969 .

[4]  G. J. Hahn,et al.  Statistical models in engineering , 1967 .

[5]  F. Ma.  Stability Theory of Stochastic Difference Systems , 1983 .

[6]  Michael D. Dettinger,et al.  First order analysis of uncertainty in numerical models of groundwater flow part: 1. Mathematical development , 1981 .

[7]  Humberto Contreras,et al.  The stochastic finite-element method , 1980 .

[8]  Masanobu Shinozuka,et al.  Monte Carlo solution of structural dynamics , 1972 .

[9]  E. Coddington,et al.  Theory of Ordinary Differential Equations , 1955 .

[10]  R. Freeze.  A stochastic‐conceptual analysis of one‐dimensional groundwater flow in nonuniform homogeneous media , 1975 .

[11]  R. Wilcox.  Exponential Operators and Parameter Differentiation in Quantum Physics , 1967 .

[12]  W. Vetter.  Matrix Calculus Operations and Taylor Expansions , 1973 .

[13]  J. L. Willems,et al.  Stability criteria for stochastic systems with colored multiplicative noise , 1975 .

[14]  Joe Brewer,et al.  Kronecker products and matrix calculus in system theory , 1978 .

[15]  W. Vetter.  Derivative operations on matrices , 1970 .

[16]  D. Ludwig.  Persistence of Dynamical Systems under Random Perturbations , 1975 .

[17]  George F. Pinder,et al.  Fundamental concepts in the numerical solution of differential equations , 1985 .

[18]  R. Brockett.  Lie Algebras and Lie Groups in Control Theory , 1973 .

[19]  T. Apostol.  Mathematical Analysis , 1957 .

[20]  M. C. Pease,et al.  Methods of Matrix Algebra , 1965 .

[21]  Alfredo Ang H.-S.,et al.  Probability concepts in engineering planning and design, vol i : basic principles , 1979 .

[22]  T. K. Caughey,et al.  On the stability of linear and non-linear stochastic transformations , 1981 .

[23]  T. Caughey.  Nonlinear Theory of Random Vibrations , 1971 .

[24]  R. Bellman.  Limit theorems for non-commutative operations. I. , 1954 .

[25]  W. Vetter.  Correction to "Derivative operations on matrices" , 1971 .

[26]  Frank Kozin,et al.  A survey of stability of stochastic systems , 1969, Autom..

[27]  Wilson H. Tang,et al.  Probability concepts in engineering planning and design , 1984 .

[28]  Alexander Graham,et al.  Kronecker Products and Matrix Calculus: With Applications , 1981 .

[29]  Lamberto Cesari,et al.  Asymptotic Behavior and Stability Problems in Ordinary Differential Equations , 1963 .

[30]  Richard Bellman,et al.  Introduction to Matrix Analysis , 1972 .