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This dissertation introduces novel methods for solving highly challenging modeling and control problems, motivated by advanced aerospace systems. Adaptable, robust and computationally effcient, multi-resolution approximation algorithms based on Radial Basis Function Network and Global-Local Orthogonal Mapping approaches are developed to address various problems associated with the design of large scale dynamical systems. The main feature of the Radial Basis Function Network approach is the unique direction dependent scaling and rotation of the radial basis function via a novel Directed Connectivity Graph approach. The learning of shaping and rotation parameters for the Radial Basis Functions led to a broadly useful approximation approach that leads to global approximations capable of good local approximation for many moderate dimensioned applications. However, even with these refinements, many applications with many high frequency local input/output variations and a high dimensional input space remain a challenge and motivate us to investigate an entirely new approach. The Global-Local Orthogonal Mapping method is based upon a novel averaging process that allows construction of a piecewise continuous global family of local least-squares approximations, while retaining the freedom to vary in a general way the resolution (e.g., degrees of freedom) of the local approximations. These approximation methodologies are compatible with a wide variety of disciplines such as continuous function approximation, dynamic system modeling, nonlinear signal processing and time series prediction. Further, related methods are developed for the modeling of dynamical systems nominally described by nonlinear differential equations and to solve for static and dynamic response of Distributed Parameter Systems in an effcient manner. Finally, a hierarchical control allocation algorithm is presented to solve the control allocation problem for highly over-actuated systems that might arise with the development of embedded systems. The control allocation algorithm makes use of the concept of distribution functions to keep in check the "curse of dimensionality". The studies in the dissertation focus on demonstrating, through analysis, simulation, and design, the applicability and feasibility of these approximation algorithms to a variety of examples. The results from these studies are of direct utility in addressing the "curse of dimensionality" and frequent redundancy of neural network approximation.
Unifying the most important methodology in this field, Multi-Resolution Methods for Modeling and Control of Dynamical Systems explores existing approximation methods as well as develops new ones for the approximate solution of large-scale dynamical system problems. It brings together a wide set of material from classical orthogonal function
This contributed volume presents some of the latest research related to model order reduction of complex dynamical systems with a focus on time-dependent problems. Chapters are written by leading researchers and users of model order reduction techniques and are based on presentations given at the 2019 edition of the workshop series Model Reduction of Complex Dynamical Systems – MODRED, held at the University of Graz in Austria. The topics considered can be divided into five categories: system-theoretic methods, such as balanced truncation, Hankel norm approximation, and reduced-basis methods; data-driven methods, including Loewner matrix and pencil-based approaches, dynamic mode decomposition, and kernel-based methods; surrogate modeling for design and optimization, with special emphasis on control and data assimilation; model reduction methods in applications, such as control and network systems, computational electromagnetics, structural mechanics, and fluid dynamics; and model order reduction software packages and benchmarks. This volume will be an ideal resource for graduate students and researchers in all areas of model reduction, as well as those working in applied mathematics and theoretical informatics.
Modern complex large-scale dynamical systems exist in virtually every aspect of science and engineering, and are associated with a wide variety of physical, technological, environmental, and social phenomena, including aerospace, power, communications, and network systems, to name just a few. This book develops a general stability analysis and control design framework for nonlinear large-scale interconnected dynamical systems, and presents the most complete treatment on vector Lyapunov function methods, vector dissipativity theory, and decentralized control architectures. Large-scale dynamical systems are strongly interconnected and consist of interacting subsystems exchanging matter, energy, or information with the environment. The sheer size, or dimensionality, of these systems necessitates decentralized analysis and control system synthesis methods for their analysis and design. Written in a theorem-proof format with examples to illustrate new concepts, this book addresses continuous-time, discrete-time, and hybrid large-scale systems. It develops finite-time stability and finite-time decentralized stabilization, thermodynamic modeling, maximum entropy control, and energy-based decentralized control. This book will interest applied mathematicians, dynamical systems theorists, control theorists, and engineers, and anyone seeking a fundamental and comprehensive understanding of large-scale interconnected dynamical systems and control.
The essential introduction to the principles and applications of feedback systems—now fully revised and expanded This textbook covers the mathematics needed to model, analyze, and design feedback systems. Now more user-friendly than ever, this revised and expanded edition of Feedback Systems is a one-volume resource for students and researchers in mathematics and engineering. It has applications across a range of disciplines that utilize feedback in physical, biological, information, and economic systems. Karl Åström and Richard Murray use techniques from physics, computer science, and operations research to introduce control-oriented modeling. They begin with state space tools for analysis and design, including stability of solutions, Lyapunov functions, reachability, state feedback observability, and estimators. The matrix exponential plays a central role in the analysis of linear control systems, allowing a concise development of many of the key concepts for this class of models. Åström and Murray then develop and explain tools in the frequency domain, including transfer functions, Nyquist analysis, PID control, frequency domain design, and robustness. Features a new chapter on design principles and tools, illustrating the types of problems that can be solved using feedback Includes a new chapter on fundamental limits and new material on the Routh-Hurwitz criterion and root locus plots Provides exercises at the end of every chapter Comes with an electronic solutions manual An ideal textbook for undergraduate and graduate students Indispensable for researchers seeking a self-contained resource on control theory
Data-driven dynamical systems is a burgeoning field?it connects how measurements of nonlinear dynamical systems and/or complex systems can be used with well-established methods in dynamical systems theory. This is a critically important new direction because the governing equations of many problems under consideration by practitioners in various scientific fields are not typically known. Thus, using data alone to help derive, in an optimal sense, the best dynamical system representation of a given application allows for important new insights. The recently developed dynamic mode decomposition (DMD) is an innovative tool for integrating data with dynamical systems theory. The DMD has deep connections with traditional dynamical systems theory and many recent innovations in compressed sensing and machine learning. Dynamic Mode Decomposition: Data-Driven Modeling of Complex Systems, the first book to address the DMD algorithm, presents a pedagogical and comprehensive approach to all aspects of DMD currently developed or under development; blends theoretical development, example codes, and applications to showcase the theory and its many innovations and uses; highlights the numerous innovations around the DMD algorithm and demonstrates its efficacy using example problems from engineering and the physical and biological sciences; and provides extensive MATLAB code, data for intuitive examples of key methods, and graphical presentations.
Dynamics of Coupled Structures, Volume 4: Proceedings of the 35th IMAC, A Conference and Exposition on Structural Dynamics, 2017, the fourth volume of ten from the Conference brings together contributions to this important area of research and engineering. The collection presents early findings and case studies on fundamental and applied aspects of the Dynamics of Coupled Structures, including papers on: Experimental Nonlinear Dynamics Joints, Friction & Damping Nonlinear Substructuring Transfer Path Analysis and Source Characterization Analytical Substructuring & Numerical Reduction Techniques Real Time Substructuring Assembling & Decoupling Substructures & Boundary Conditions