Shaping Nonlinearity in Reset Control Systems to Realize Complex-Order Controllers: Application in Precision Motion Control

This dissertation addresses the demand for faster, more precise, and robust controllers in the precision motion industry. Traditional linear controllers have limitations due to the waterbed effect and Bode’s phase-gain relationship. To overcome these limitations, complex-order controllers are explored in this study. The dissertation focuses on shaping nonlinearities in reset controllers to realize complex-order behavior. Various methods and approaches are investigated, each contributing to the understanding and improvement of reset control systems for linear time-invariant systems. The dissertation demonstrates that reset controllers exhibit first-order harmonic behavior, which can be advantageous in achieving complex-order behavior and enhancing controller performance compared to linear controllers. However, higher-order harmonics resulting from nonlinearities play a significant role and should not be neglected. The study explores methods to shape and manipulate these higher-order harmonics for improved performance. Different approaches are categorized into two main categories: methods based on shaping the reset phase (ψ) and continuous reset (CR) methods. In the first category, ψ-shaping methods focus on manipulating ψ to achieve desirable non-linear behaviors. This includes the introduction of elements such as fractional-order lag elements and filters to shape ψ and suppress higher-order harmonics. The dissertation presents practical frameworks for analyzing and utilizing ψ-shaping concepts. The second category, continuous reset methods, addresses both transient and steadystate performance. By introducing lead and lag elements as pre- and post-filters for reset elements, improvements in transient response are achieved. Additionally, CR methods preserve the first-order harmonic behavior while reducing higher-order harmonics across the entire frequency range. The dissertation highlights the advantages and tradeoffs between ψ-shaping and CR methods, providing insights for selecting the appropriate approach based on application requirements. Practical implementation aspects are also considered throughout the dissertation. Challenges such as noise amplification caused by lead elements in CR architectures are addressed, offering solutions through increased filter orders or observer-based filtering techniques. The dissertation demonstrates the effectiveness of the proposed approaches through implementation in industrial precision motion stages, showcasing the superiority of complex-order reset controllers over their linear counterparts. Overall, this dissertation contributes to the understanding and practical implementation of reset controllers for realizing complex-order behavior in precision motion control. It provides insights into shaping nonlinearities, optimizing steady-state and transient performance, and selecting suitable architectures based on specific application needs. The findings and guidelines presented in this study offer valuable contributions to the precision motion industry and pave the way for further advancements in controller design and performance.