The Theory of Linear Systems

The Theory of Linear Systems

1st Edition - January 1, 1971

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  • Author: J. E. Rubio
  • eBook ISBN: 9781483219875

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The Theory of Linear Systems presents the state-phase analysis of linear systems. This book deals with the transform theory of linear systems, which had most of its success when applied to time-invariant systems. Organized into nine chapters, this book begins with an overview of the development of some properties of simple differential systems that are mostly of a nonalgebraic nature. This text then presents a brief treatment of vector spaces, matrices, transformations, norms, and inner products. Other chapters deal with the inductive process used to define dynamical systems. This book discusses as well the existence and uniqueness theorem for the solutions of a homogeneous linear differential system. The final chapter deals with the abstract concept of a dynamical system and derives properties of these systems. This book is a valuable resource for advanced graduate students in areas such as economics and bioengineering. Engineers engaged in systems design will also find this book useful.

Table of Contents

  • Preface


    1. The Search for an Internal Structure

    1.1 Introduction

    1.2 The State of a System: An Example

    1.3 Transfer Functions and State

    1.4 Some Conclusions and a Program

    Supplementary Notes and References


    2. An Introduction to the Theory of Linear Spaces

    2.1 Introduction

    2.2 Set and Operations on Sets

    2.3 Functions

    2.4 Linear Spaces

    2.5 Basis and Dimension

    2.6 Linear Transformations and Matrices

    2.7 Inverse Operators and Inverse Matrices

    2.8 Metric Properties of Linear Spaces. Norms

    2.9 Inner Products. Adjoint Operators

    2.10 An Introduction to Matrix Algebra

    Supplementary Notes and References


    3. Differential Systems, I

    3.1 Introduction

    3.2 The Homogeneous System

    3.3 Fundamental Matrices

    3.4 The Transition Matrix

    3.5 The Adjoint System

    3.6 The General Solution of Nonhomogeneous Systems

    3.7 A Summary and Some Conclusions

    Supplementary Notes and References


    4. Differential Systems, II

    4.1 Introduction

    4.2 Similarity, Eigenvalues and Eigenvectors

    4.3 Reduction

    4.4 Diagonalization

    4.5 More on Reduction

    4.6 Functions of Matrices

    Supplementary Notes and References


    5. Controllability, Observability

    5.1 Introduction

    5.2 Controllability

    5.3 A Necessary and Sufficient Condition for Controllability

    5.4 A Necessary and Sufficient Condition for the Controllability of Constant Systems

    5.5 Transformations and Controllability

    5.6 Controllability and Transfer Functions

    5.7 A Condition for the Controllability of Time-Varying Systems

    5.8 Observability

    5.9 The Controllability and Observability of Composite Systems

    5.10 Normal Systems

    5.11 Output Controllability

    5.12 Total Controllability. Total Observability

    5.13 The Attainable Set

    5.14 The Phase-Canonical Form

    Supplementary Notes and References


    6. Synthesis

    6.1 Introduction

    6.2 Some Results on the Synthesis of Weighting Patterns

    6.3 Introduction to Synthesis

    6.4 A Method of Synthesis

    6.5 Analytic Systems

    6.6 The Reduction to Minimal Order in the General Case

    6.7 The Observer

    Supplementary Notes and References


    7. Difference Systems

    7.1 Introduction

    7.2 The Homogeneous System

    7.3 The Nonhomogeneous System

    7.4 A Summary

    7.5 Controllability, Observability

    Supplementary Notes and References


    8. Stability

    8.1 Introduction

    8.2 The Stability of Constant Systems

    8.3 Exponential Stability

    8.4 External Stability, Controllability, Observability and Exponential Stability

    8.5 The Stability of Difference Systems

    Supplementary Notes and References


    9. Infinite Dimensions

    9.1 Introduction

    9.2 Some Further Mathematical Topics

    9.3 Dynamical Systems

    9.4 Semigroups and Infinitesimal Generators

    9.5 Infinitesimal Generators and Resolvents

    9.6 Homogeneous and Nonhomogeneous Systems

    9.7 Systems with Boundary Controls

    9.8 Hilbert Spaces

    9.9 Controllability

    9.10 Conclusion

    Supplementary Notes and References


    Appendix. An Introduction to Computational Methods

    A.1 Elementary Row and Column Operations

    A.2 The Computation of Eigenvalues, Eigenvectors and of the Matrix (λIn - Τ)-1

    A.3 The Computation of the Transition Matrices of Constant Systems

    A.4 Positive Definite Matrices


    Author Index

    Subject Index

Product details

  • No. of pages: 344
  • Language: English
  • Copyright: © Academic Press 1971
  • Published: January 1, 1971
  • Imprint: Academic Press
  • eBook ISBN: 9781483219875

About the Author

J. E. Rubio

About the Editors

Henry G. Booker

Nicholas Declaris

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