Hybrid Systems, Iterative Learning Control, and Non-minimum Phase
Creators: name: Spiegel, Isaac
Publisher: name: My University
Year: 2021
Subjects: subject: Hybrid Systems
subject: Iterative Learning Control
subject: Non-minimum phase
subject: Machine Learning
subject: Robotics
subject: Additive Manufacturing
subject: Mechanical Engineering
subject: FOS: Mechanical engineering
subject: Engineering

description: Hybrid systems have steadily grown in popularity over the last few decades because they ease the task of modeling complicated nonlinear systems. Legged locomotion, robotic manipulation, and additive manufacturing are representative examples of systems benefiting from hybrid modeling. They are also prime examples of repetitive processes; gait cycles in walking, product assembly tasks in robotic manipulation, and material deposition in additive manufacturing. Thus, they would also benefit substantially from Iterative Learning Control (ILC), a class of feedforward controllers for repetitive systems that achieve high performance in output reference tracking by learning from the errors of past process cycles. However, the literature is bereft of ILC syntheses from hybrid models. The main thrust of this dissertation is to provide a broadly applicable theory of ILC for deterministic, discrete-time hybrid systems, i.e. piecewise defined (PWD) systems. A type of ILC called Newton ILC (NILC) serves as the foundation for this mission due to its admittance of an unusually broad range of nonlinearities. Preventing the synthesis of NILC from hybrid models is the fact that contemporary hybrid modeling frameworks do not admit closed-form function composition of a single state transition formula capturing the complete hybrid system dynamics. This dissertation offers a new, closed-form PWD modeling framework to solve this problem. However, NILC itself is not without flaw. This dissertation's research reveals that it generally fails to converge when synthesized from models with unstable inverses (i.e. non-minimum phase (NMP) models), a class that includes flexible-link robotic manipulators. Thus, to fulfill the goal of providing the most broadly applicable control theory possible, improvement to NILC must be made to avoid the operation that causes divergence when applied to NMP systems (a particular matrix inversion). Stable inversion---a technique for generating stable state trajectories from unstable systems by decoupling their stable and unstable modes---is identified as a valuable tool in this endeavor. This concept is well-explored for linear time invariant systems, but stable inversion for hybrid systems has not been explored by the prior art. Thus, to focus the research, this dissertation specifically examines piecewise affine (PWA) systems (a subset of PWD systems) for the study of NMP hybrid system control. For PWA systems (and their PWD superset), in addition to a lack of stable inversion, a general, closed-form solution to the conventional inversion problem is also absent from the literature. Having a closed-form conventional inverse model is a prerequisite for stable inversion, but inversion of PWA models is nontrivial because the uniqueness of PWA system inverses is not guaranteed as it is for ordinary affine systems. Therefore, to achieve the first ILC of a hybrid system with an unstable inverse, theory for both conventional inversion and stable inversion must be delivered for PWA systems. In summary, the three main gaps addressed by this dissertation are (1) the lack of compatibility between existing hybrid modeling frameworks and ILC synthesis techniques, (2) the failure of NILC for NMP systems, and (3) the lack of inversion and stable inversion theory for PWA systems. These issues are addressed by (1) developing a closed-form representation for PWD systems, (2) developing a new ILC framework informed by NILC but free of matrix inversion, and (3) deriving conventional and stable model inversion theories for PWA systems.