The advances in computational and experimental modeling in the area of structural mechanics have stimulated research in a class of hybrid problems that require both of these modeling capabilities to be combined to achieve certain objectives. Real-time substructure (RTS) testing, structural system identification (SSI) and active control techniques fall in the category of hybrid problems that need efficient tools in both computational and experimental phases for their successful implementation. RTS is a hybrid testing method, which aims to overcome the scaling problems associated with the conventional dynamic testing methods (such as shake table test, effective force test and pseudo dynamic test) by testing the critical part of the structure experimentally with minimum compromise on spatio-temporal scaling, while modeling the remaining part numerically. The problem of SSI constitutes an important component within the broader framework of problems of structural health monitoring where, based on the in-situ measurements on the loading and a subset of critical responses of the structure, the system parameters are estimated with a view to detecting damage/degradation. Active control techniques are employed to maintain the functionality of important structures, especially under extreme dynamic loading. The work reported in the present thesis contributes to the areas of RTS, SSI and active control of nonlinear systems, the main focus being the computational aspects, i.e., in developing numerical strategies to address some of the unsolved issues, although limited efforts have also been made to undertake laboratory experimental investigations in the area of nonlinear SSI. The thesis is organized into seven chapters and five appendices. The first chapter contains an overview of the state of the art techniques in dynamic testing, SSI and structural control. The topics covered include effective force test, pseudo dynamic test, RTS test, time and frequency domain methods of nonlinear system identification, dynamic state estimation techniques with emphasis on particle filters, Rao-Blackwellization, structural control methods, control algorithms and active control of nonlinear systems. The review identifies a set of open problems that are subsequently addressed, to an extent, in the thesis. Chapter 2 focuses on the development of a time domain coupling technique, involving combined computational and experimental modeling, for vibration analysis of structures built-up of linear/nonlinear substructures. The numerical and experimental substructures are allowed to interact in real-time. The equation of motion of the numerical substructure is integrated using a step-by-step procedure that is formulated in the state space. For systems with nonlinear substructures, a multi-step transversal linearization method is used to integrate the equations of motion; and, a multi-step extrapolation scheme, based on the reproducing kernel particle method, is employed to handle the time delays that arise while accounting for the interaction between the substructures. Numerical illustrations on a few low dimensional vibrating structures are presented and these examples are fashioned after problems of seismic qualification testing of engineering structures using RTS testing techniques. The concept of substructuring is extended in Chapter 3 for implementing Rao-Blackwellization, a technique of combining particle filters with analytical computation through Kalman filters, for state and parameter estimations of a class of nonlinear dynamical systems with additive Gaussian process/observation noises. The strategy is based on decomposing the system to be estimated into mutually coupled linear and nonlinear substructures and then putting in place a rational framework to account for coupling between the substructures. While particle filters are applied to the nonlinear substructures, estimation of linear substructures proceeds using a bank of Kalman filters. Numerical illustrations for state/parameter estimations of a few linear and nonlinear oscillators with noise in both the process and measurements are provided to demonstrate the potential of the Rao-Blackwellized particle filter (RBPF) with substructuring. In Chapter 4, the concept of Rao-Blackwellization is extended to handle more general nonlinear systems, using two different schemes of linearization. A semi-analytical filter and a conditionally linearized filter, within the framework of Monte Carlo simulations, are proposed for state and parameter estimations of nonlinear dynamical systems with additively Gaussian process/observation noises. The first filter uses a local linearization of the nonlinear drift fields in the process/observation equations based on explicit Ito-Taylor expansions to transform the given nonlinear system into a family of locally linearized systems. Using the most recent observation, conditionally Gaussian posterior density functions of the linearized systems are analytically obtained through the Kalman filter. In the second filter, the marginalized posterior distribution of an appropriately chosen subset of the state vector is obtained using a particle filter. Samples of these marginalized states are then used to construct a family of conditionally linearized system of equations to obtain the posterior distribution of the states using a bank of Kalman filters. The potential of the proposed filters in state/parameter estimations is demonstrated through numerical illustrations on a few nonlinear oscillators. The problem of active control of nonlinear structural dynamical systems, in the presence of both process and measurement noises, is considered in Chapter 5. The focus of the study is on the exploitability of particle filters for state estimation in feedback control algorithms for nonlinear structures, when a limited number of noisy output measurements are available. The control design is done using the state dependent Riccati equation (SDRE) method. The Bayesian bootstrap filter and another based on sequential importance sampling are employed for state estimation. Numerical illustrations are provided for a few typically nonlinear oscillators of interest in structural engineering. The experimental validation of the RBPF using substructuring (developed in Chapter 3) and the conditionally linearized Monte Carlo filter (developed in Chapter 4), for parameter estimation, is reported in Chapter 6. Measured data available through laboratory experiments on simple building frame models subjected to harmonic base motions is processed using the proposed algorithms to identify the unknown parameters of the model. A brief summary of the contributions made in this thesis, together with a few suggestions for future research, are presented in Chapter 7. Appendix A provides an account of the multi-step transversal linearization method. The derivation of the reproducing kernel shape functions are presented in Appendix B. Appendix C provides the details of the stochastic Taylor expansion and derivation of the covariance structure of Gaussian MSI-s. The performance of a particle filtering algorithm (bootstrap filter) and Kalman filter in the state estimation of a linear system is provided in Appendix D and Appendix E contains the theoretical details of the Rao-Blackwellized particle filter.

The estimation problem of stochastic nonlinear parametric models is recognized to be very challenging due to the intractability of the likelihood function. Recently, several methods have been developed to approximate the maximum likelihood estimator and the optimal mean-square error predictor using Monte Carlo methods. Albeit asymptotically optimal, these methods come with several computational challenges and fundamental limitations. The contributions of this thesis can be divided into two main parts. In the first part, approximate solutions to the maximum likelihood problem are explored. Both analytical and numerical approaches, based on the expectation-maximization algorithm and the quasi-Newton algorithm, are considered. While analytic approximations are difficult to analyze, asymptotic guarantees can be established for methods based on Monte Carlo approximations. Yet, Monte Carlo methods come with their own computational difficulties; sampling in high-dimensional spaces requires an efficient proposal distribution to reduce the number of required samples to a reasonable value. In the second part, relatively simple prediction error method estimators are proposed. They are based on non-stationary one-step ahead predictors which are linear in the observed outputs, but are nonlinear in the (assumed known) input. These predictors rely only on the first two moments of the model and the computation of the likelihood function is not required. Consequently, the resulting estimators are defined via analytically tractable objective functions in several relevant cases. It is shown that, under mild assumptions, the estimators are consistent and asymptotically normal. In cases where the first two moments are analytically intractable due to the complexity of the model, it is possible to resort to vanilla Monte Carlo approximations. Several numerical examples demonstrate a good performance of the suggested estimators in several cases that are usually considered challenging. / <p>QC 20171128</p>