In this dissertation, a new area of research identified as high-rate state estimation is established along with its associated research challenges, and a path for a solution is provided. High-rate dynamic systems are defined as systems being exposed to highly dynamic environments that are comprised of high-rate and high-amplitude events (greater than 100 g for a duration under 100 ms). Engineering systems experiencing high-rate dynamic events, including airbag, debris detection, and active blast protection systems, could benefit from real-time observability for enhanced performance. This task of high-rate state estimation is particularly challenging for real-time applications, where the rate of an observer's convergence needs to be in the microsecond range. On the other hand, the benefits include a high potential to reduce economic loss and save lives.

The problem is discussed in-depth addressing the fundamental challenges of high-rate systems. A survey of applications and methods for estimators that have the potential to produce accurate estimations for a complex system experiencing highly dynamic events is presented. It is argued that adaptive observers are important to this research. In particular, adaptive data-driven observers are found to be advantageous due to their adaptability to complex problems and lack of dependence on system model.

An adaptive neuro-observer is designed to examine the particular challenges in selecting an appropriate input space for high-rate state estimation to increase convergence rates of adaptive observers. It is found that the choice of inputs has a more significant influence on the observer's performance for high-rate dynamics when compared against a lower rate environment. Additionally, misrepresentation of a system dynamics through incorrect input spaces produces large errors in the estimation, which could potentially trick the decision making process in a closed-loop system in making bad judgments.

A novel adaptive wavelet neural network (WNN)-based approach to compress data into a combination of low- and high-resolution surfaces is proposed to automatically detect concrete cracks and other forms of damage. The adaptive WNN is designed to sequentially self-organize and self-adapt in order to construct an optimized representation. The architecture of the WNN is based on a single-layer neural network consisting of Mexican hat wavelet functions. The approach was verified on four cracked concrete specimens.

A variable input space concept is proposed for incorporating data history of high-rate dynamics, with the objective to produce an optimal representation of the system of interest minimizing convergence times of adaptive observers. Using the embedding theory, the algorithm sequentially selects and adapts a vector of inputs that preserves the essential dynamics of the high-rate system. The variable input space is integrated with a WNN, which constitutes a variable input observer. The observer is simulated using experimental data from a high-rate system. Different input space adaptation methods are studied and the performance is compared against an optimized fixed input strategy. The variable input observer is further studied in a hybrid model-/data-driven formulation, and results demonstrate significant improvement in performance gained from the added physical knowledge.

An experimental test bed, developed to validate high-rate structural health monitoring (SHM) methods in a controllable and repeatable laboratory environment, is modeled as a clamped-pinned-free beam with mass at the free end. The Euler-Bernoulli beam theory is applied to this unique configuration to develop analytical solutions of the system. The transverse vibration of a clamped-pinned-free beam with a point mass at the free end is discussed in detail. Results are derived for varying pin locations and mass values. Eigenvalue plots of the first five modes are presented along with their respective mode shapes. The theoretical calculations are experimentally validated and discussed.