Performance measures and optimal design of linear structural systems under stochastic stationary excitation

Abstract The optimal design of linear structural systems under stochastic, stationary dynamic excitation is discussed. Several measures are considered for quantification of the stochastic performance, including the standard H 2 and multi-objective H 2 measures related to the second-order statistics of the response, as well as a measure related to first-passage system reliability. Of these, first-passage reliability is the performance measure most closely aligned with the design of Civil Engineering structures, but it is also the least analytically-tractable. This paper discusses the evaluation and optimization of these performance measures, including an efficient approximation to first-passage reliability, and examines the degree to which the optimal structural design, and its reliability, vary when the optimization is carried out under these different measures. For the reliability-optimal design, the effect of the spatial and temporal correlation of multiple out-crossing events is investigated. An example is presented, which considers the optimal configuration of viscous dampers for protection of a five storey structure under earthquake excitation.

[1]  A. Genz.  Numerical Computation of Multivariate Normal Probabilities , 1992 .

[2]  James L. Beck,et al.  Robust reliability-based design of liquid column mass dampers under earthquake excitation using an analytical reliability approximation , 2007 .

[3]  R. Skelton,et al.  “Convexifying” Linear Matrix Inequality Methods for Integrating Structure and Control Design , 2003 .

[4]  G. Schuëller,et al.  Equivalent linearization and Monte Carlo simulation in stochastic dynamics , 2003 .

[5]  Robert E. Melchers,et al.  Structural Reliability: Analysis and Prediction , 1987 .

[6]  Anil K. Agrawal,et al.  Optimal design of passive energy dissipation systems based on H∞ and H2 performances , 2002 .

[7]  Armen Der Kiureghian,et al.  Joint First-Passage Probability and Reliability of Systems under Stochastic Excitation , 2006 .

[8]  M. Athans,et al.  On the determination of the optimal constant output feedback gains for linear multivariable systems , 1970 .

[9]  Chaouki T. Abdallah,et al.  Linear Quadratic Control: An Introduction , 2000 .

[10]  Phaedon-Stelios Koutsourelakis.  A note on the first-passage problem and VanMarcke's approximation - short communication , 2007 .

[11]  Yoshimi Goda,et al.  Random Seas and Design of Maritime Structures , 1985 .

[12]  Lawrence A. Bergman,et al.  Reliability-Based Approach to Linear Covariance Control Design , 1998 .

[13]  James C. Spall,et al.  Introduction to stochastic search and optimization - estimation, simulation, and control , 2003, Wiley-Interscience series in discrete mathematics and optimization.

[14]  P. Spanos,et al.  Random vibration and statistical linearization , 1990 .

[15]  Costas Papadimitriou,et al.  EFFECTS OF STRUCTURAL UNCERTAINTIES ON TMD DESIGN: A RELIABILITY-BASED APPROACH , 1997 .

[16]  Max Donath,et al.  American Control Conference , 1993 .

[17]  T. T. Soong,et al.  Random Vibration of Mechanical and Structural Systems , 1992 .

[18]  J. Beck,et al.  First excursion probabilities for linear systems by very efficient importance sampling , 2001 .

[19]  E. Vanmarcke.  On the Distribution of the First-Passage Time for Normal Stationary Random Processes , 1975 .

[20]  Emil Simiu,et al.  Wind effects on structures : fundamentals and applications to design , 1996 .

[21]  Karolos M. Grigoriadis,et al.  A unified algebraic approach to linear control design , 1998 .

[22]  Ahsan Kareem,et al.  Feedback–feedforward control of offshore platforms under random waves , 2001 .

[23]  Mahendra P. Singh,et al.  Optimal placement of dampers for passive response control , 2002 .

[24]  Mircea Grigoriu,et al.  Vector-Process Models for System Reliability , 1977 .

[25]  S. Rice.  Mathematical analysis of random noise , 1944 .

[26]  Stephen P. Boyd,et al.  Linear Matrix Inequalities in Systems and Control Theory , 1994 .

[27]  Chaouki T. Abdallah,et al.  Static output feedback: a survey , 1994, Proceedings of 1994 33rd IEEE Conference on Decision and Control.

[28]  H. Horisberger,et al.  Solution of the optimal constant output feedback problem by conjugate gradients , 1974 .

[29]  Richard A. Johnson,et al.  Applied Multivariate Statistical Analysis , 1983 .

[30]  James L. Beck,et al.  Analytical approximation for stationary reliability of certain and uncertain linear dynamic systems with higher‐dimensional output , 2006 .

[31]  James L. Beck,et al.  Reliability‐based control optimization for active base isolation systems , 2006 .

[32]  James L. Beck,et al.  Stochastic Subset Optimization for reliability optimization and sensitivity analysis in system design , 2009 .

[33]  James L. Beck,et al.  Reliability‐based robust control for uncertain dynamical systems using feedback of incomplete noisy response measurements , 2003 .

[34]  C. Allin Cornell,et al.  Energy Fluctuation Scale and Diffusion Models , 1985 .

[35]  S. Sarkani,et al.  Stochastic analysis of structural and mechanical vibrations , 1996 .

[36]  James L. Beck,et al.  Reliability-Based Performance Objectives and Probabilistic Robustness in Structural Control Applications , 2008 .

[37]  Y. Belyaev.  On the Number of Exits Across the Boundary of a Region by a Vector Stochastic Process , 1968 .

[38]  R. Clough,et al.  Dynamics Of Structures , 1975 .

[39]  James L. Beck,et al.  Probabilistic control for the Active Mass Driver benchmark structural model , 1998 .

[40]  Giuseppe Marano,et al.  Constrained reliability-based optimization of linear tuned mass dampers for seismic control , 2007 .