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2025 | Book

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Data-Driven, Nonparametric, Adaptive Control Theory

Authors: Andrew J. Kurdila, Andrea L'Afflitto, John A. Burns

Publisher: Springer Nature Switzerland

Book Series : Lecture Notes in Control and Information Sciences

Part of: Springer Professional "Business + Economics & Engineering + Technology" , Springer Professional "Engineering + Technology" , Springer Professional "Business + Economics"

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About this book

Data-Driven, Nonparametric, Adaptive Control Theory introduces a novel approach to the control of deterministic, nonlinear ordinary differential equations affected by uncertainties. The methods proposed enforce satisfactory trajectory tracking despite functional uncertainties in the plant model. The book employs the properties of reproducing kernel Hilbert (native) spaces to characterize both the functional space of uncertainties and the controller's performance. Classical control systems are extended to broader classes of problems and more informative characterizations of the controllers’ performances are attained.

Following an examination of how backstepping control and robust control Lyapunov functions can be ported to the native setting, numerous extensions of the model reference adaptive control framework are considered. The authors’ approach breaks away from classical paradigms in which uncertain nonlinearities are parameterized using a regressor vector provided a priori or reconstructed online. The problem of distributing the kernel functions that characterize the native space is addressed at length by employing data-driven methods in deterministic and stochastic settings.

The first part of this book is a self-contained resource, systematically presenting elements of real analysis, functional analysis, and native space theory. The second part is an exposition of the theory of nonparametric control systems design. The text may be used as a self-study book for researchers and practitioners and as a reference for graduate courses in advanced control systems design. MATLAB® codes, available on the authors’ website, and suggestions for homework assignments help readers appreciate the implementation of the theoretical results.

Table of Contents

Frontmatter
Chapter 1. Introduction
Abstract
This chapter introduced the general problem addressed in this book, that is, the design of robust and adaptive control strategies for nonlinear, deterministic systems of ordinary differential equations affected by uncertainties, which are assumed to lie in some native space. The proposed framework is unique not only for its ability to forecast the performance of the controller as an explicit function of key properties of the native space elected to capture the functional uncertainties. The proposed framework is unique for having merged in a unique manner dynamical systems theory, machine learning theory, and approximation theory, and having extracted essential tools from each of these three macro research areas. This chapter is closed by a brief description of the content of this book to allow readers to choose their strategy for addressing the proposed topics.
Andrew J. Kurdila, Andrea L’Afflitto, John A. Burns
Chapter 2. Elements of Real and Functional Analysis
Abstract
This chapter presents some key elements of real and functional analysis employed throughout this book. The scope of this chapter is not to provide a comprehensive discussion on this topic, which is widely covered by a plethora of well-referenced books. The purpose of this chapter is to present all foundational elements of this book succinctly, coherently, and rigorously. This chapter can be studied as a first step toward the main goal of this book, namely the design of control systems and machine learning methods based on reproducing kernel Hilbert spaces (native spaces), or can be consulted by the reader as a compendium of resources for a deeper understanding of the material presented in later chapters. Numerous references are provided throughout this chapter to guide the reader to a deeper understanding of the topics discussed herein.
Andrew J. Kurdila, Andrea L’Afflitto, John A. Burns
Chapter 3. Elements of Native Space Theory
Abstract
After an overview of key elements of functional analysis, this chapter is devoted to a specific topic in the theory of Hilbert spaces, namely reproducing kernel Hilbert spaces (RKHSs), also known as native spaces. This chapter provides the reader with all the tools needed to comprehend the key topics of this book, namely the design of control systems and machine learning methods based on native space theory. Initially, scalar-valued RKHSs are examined and their properties are discussed in detail and a tutorial manner. Successively, scalar-valued RKHSs are examined and special emphasis is given to highlighting connections with results for scalar-valued RKHSs. Two sets of results in RKHS theory are analyzed, namely how to correlate different RKHSs and how to bound projection errors in RKHSs.
Andrew J. Kurdila, Andrea L’Afflitto, John A. Burns
Chapter 4. Elements of Dynamical Systems Theory
Abstract
This chapter introduces key elements of dynamical systems theory with special emphasis on those concepts leveraged in the next chapters for the design of control systems and machine learning methods based on reproducing kernel Hilbert spaces. Special emphasis is given to the notions of stability, boundedness, attractivity, and control of Lyapunov functions. A dedicated section discusses distributed parameter systems.
Andrew J. Kurdila, Andrea L’Afflitto, John A. Burns
Chapter 5. Native Space Embedding Control Methods
Abstract
Leveraging the material presented in the previous chapters, this chapter presents robust and adaptive control techniques for nonlinear plants affected by multiple forms of uncertainties. These results are obtained by extending classical and advanced nonlinear control techniques to account for uncertainties modeled as unknown elements of some operator-valued native space. We begin by designing robust control Lyapunov functions for nonlinear dynamical systems, and, successively, we design backstepping control systems under the assumption that functional uncertainties lie in some RKHS. A large portion of this chapter is dedicated to the design of model reference adaptive control systems (MRAC) that are robust to parametric, matched, unmatched, and functional uncertainties. To this goal, we generalize the dead-zone modification of MRAC, the \(\sigma \)-modification of MRAC, and the use of convex projection operators. Finally, we extend variable structure methods, such as the error bounding control architecture and the adaptive error bounding architecture, to the native embedding setting. The same numerical example is worked out multiple times and the underlying codes are presented in an Appendix to this book.
Andrew J. Kurdila, Andrea L’Afflitto, John A. Burns
Chapter 6. Data-Driven Methods and Adaptive Control: Deterministic Analysis
Abstract
This chapter discusses how deterministic data-driven approaches can be applied to the design of adaptive control systems for nonlinear plants affected by parametric, matched, unmatched, and functional uncertainties. After a brief outlook on the learning problem in general, this chapter presents three specific approaches, namely learn-then-control (LTC), sequential learn-and-control (SLC), and concurrent learn-and-control (CLC). Whereas numerical examples are provided for each of these three techniques, special emphasis is given to the theoretical aspects of LTC and CLC. Consistently with the rest of the book and to better emphasize the role of the results in Chap. 5, this chapter exploits the properties of native spaces to create suitable approximations of the hypothesis space, and hence, of the functional uncertainties affecting the plant dynamics.
Andrew J. Kurdila, Andrea L’Afflitto, John A. Burns
Chapter 7. Data-Driven Methods and Adaptive Control: Stochastic Analysis
Abstract
This chapter describes in detail how stochastic data-driven approaches can be employed to define the centers underlying a native space-based approach to the design of adaptive control systems for deterministic, continuous-time ordinary differential equations. Leveraging some key properties of native spaces, this chapter characterizes the ultimate bounds on the closed-loop trajectory tracking error. These bounds are explicit functions of the dimension of the approximating hypothesis space and the number of samples employed to estimate the matched functional uncertainty. Numerical examples demonstrate the applicability of these approaches in a learn-then-control and a switched learn-and-control framework.
Andrew J. Kurdila, Andrea L’Afflitto, John A. Burns
Chapter 8. Conclusion
Abstract
This chapter summarizes the key achievements of this book and discusses how it extends in a nontrivial manner classical results on robust and adaptive control systems. It is discussed how the material in this book marks the starting point for a new branch in control theory, that is, nonparametric control. This chapter also summarizes some of the limitations of the proposed results and proactively recommends future areas of research and possible approaches. This book is finally closed recommending wholeheartedly that the interested readers contact the authors to explore additional topics from those addressed in this work.
Andrew J. Kurdila, Andrea L’Afflitto, John A. Burns
Backmatter
Metadata
Title
Data-Driven, Nonparametric, Adaptive Control Theory
Authors
Andrew J. Kurdila
Andrea L'Afflitto
John A. Burns
Copyright Year
2025
Electronic ISBN
978-3-031-78003-5
Print ISBN
978-3-031-78002-8
DOI
https://doi.org/10.1007/978-3-031-78003-5