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Engineers have applied knowledge gained in certain areas of science in order to develop control systems. Physics is needed in the development of mathematical models of dynamical systems so that we may analyze and test our adaptive controllers. | Stable Adaptive Control and Estimation for Nonlinear Systems Neural and Fuzzy Approximator Techniques. Jeffrey T. Spooner Manfredi Maggiore Raul Ordonez Kevin M. Passino Copyright 2002 John Wiley Sons Inc. ISBNs 0-471-41546-4 Hardback 0-471-22113-9 Electronic Part I Foundations Stable Adaptive Control and Estimation for Nonlinear Systems Neural and Fuzzy Approximator Techniques. Jeffrey T. Spooner Manfredi Maggiore Raul Ordonez Kevin M. Passino Copyright 2002 John Wiley Sons Inc. ISBNs 0-471-41546-4 Hardback 0-471-22113-9 Electronic Chapter Mathematical Foundations Overview Engineers have applied knowledge gained in certain areas of science in order to develop control systems. Physics is needed in the development of mathematical models of dynamical systems so that we may analyze and test our adaptive controllers. Throughout this book we will assume that a mathematical model of the system is provided so we will not cover the physics required to develop the model. We do however require an understanding of background material from mathematics and thus it is the primary focus of this chapter. In particular mathematical foundations are presented in this chapter to establish the notation used in this book and to provide the reader with the background necessary to construct adaptive systems and analyze their resulting dynamical behavior. Here we overview some ideas from vector matrix and signal norms and properties function properties and stability and boundedness analysis. The reader who already understands all these topics should quickly skim this chapter to get familiar with the notation. For the reader who is unfamiliar with all or some of these topics or for those in need of a review of these topics we recommend doing a variety of the examples throughout the chapter and some of the homework problems at the end of it. Vectors Matrices and Signals Norms and Properties Norms measure the size of elements in a set S. In general given two elements x y 5 a norm .