Structures can get damaged by severe events such as earthquakes and hurricanes or deteriorate over time. Therefore the need to find cost-effective and reliable inspection and monitoring solutions for structures such as bridges, wind turbines, and buildings is important. Structural Health Monitoring (SHM) is the process of using damage detection and characterization techniques to determine whether a structure is in a healthy state or a damaged state.Damage localization and quantification, collectively referred to as damage characterization, can be addressed as a parameter estimation problem. In this setting, the location and extent of damage are inferred from the model parameters that are estimated from features extracted from the measurements. The measurements are collected from the sensors. For success, the features from the measurements must be sensitive to the parameters and have low variability to non-damage-related changes. Eigenvalues can be measured more precisely than eigenvectors and, for this reason, are widely used as features for damage characterization.An issue in using eigenvalues only for parameter estimation is that the number of eigenvalues extracted from the measurements can often be less than the number of model parameters that are candidates for updating, making the problem under-determined. While the number of candidate parameters is large, one expects that only a few will change due to damage. A solution that has only a few non-zeros is called sparse. Sparsity has been added as a constraint to the under-determined parameter estimation problem to obtain a solution that will likely be aligned with what happens in reality. Sparsity has been exploited in the last ten years or so using a linearized approximation. In this dissertation, the error resulting from the linear approximation is examined, and approaches that consider the nonlinearity are presented.Feature selection for parameter estimation is another item that is treated in this dissertation. Output feedback control has been used to increase the sensitivity of the eigenvalues to parameters in the last twenty years. Gains have been designed to obtain closed-loop systems with eigenvalues that have more sensitivity to damage. Applying output feedback to a structure requires that the structure is equipped with controllers, which can be a limitation. Virtual output feedback, however, only requires measuring the open-loop input-output data. The closed-loop matrix is formed offline after system identification. Virtual output feedback can be used for feature selection. It is shown that replacing the open-loop eigenvalues with more sensitive closed-loop eigenvalues will also increase their variability. Notwithstanding, Virtual output feedback can still be used to use multiple sets of closed-loop eigenvalues instead of open-loop eigenvectors. This is shown to provide better conditioning in the parameter estimation problem and more robustness to noise.This dissertation presents a damage detection method based on nonlinear output feedback as the last item. Unlike the virtual output feedback, hardware and controllers are required to perform the tests in this approach. The objective is to announce whether the structure is damaged or non-damaged by observing a feature from the nonlinear system. Using nonlinear output feedback in a linear system will generate a nonlinear closed-loop system. Nonlinear systems have features that do not exist in linear systems. We used the period of a Limit Cycle (LC) for damage detection. The limit cycle is obtained by applying the nonlinear feedback law at the point of actuation in the structure. The sensitivity of the period of the limit cycle is orders of magnitude larger than the change of period in the open-loop setting while showing robustness to non-damaged related variabilities such as noise, environmental changes, and model error.