1st Edition

Fundamentals of Linear Systems for Physical Scientists and Engineers

By N.N. Puri Copyright 2010
    900 Pages 16 Color & 202 B/W Illustrations
    by CRC Press

    904 Pages 16 Color & 202 B/W Illustrations
    by CRC Press

    Thanks to the advent of inexpensive computing, it is possible to analyze, compute, and develop results that were unthinkable in the '60s. Control systems, telecommunications, robotics, speech, vision, and digital signal processing are but a few examples of computing applications. While there are many excellent resources available that focus on one or two topics, few books cover most of the mathematical techniques required for a broader range of applications. Fundamentals of Linear Systems for Physical Scientists and Engineers is such a resource.

    The book draws from diverse areas of engineering and the physical sciences to cover the fundamentals of linear systems. Assuming no prior knowledge of complex mathematics on the part of the reader, the author uses his nearly 50 years of teaching experience to address all of the necessary mathematical techniques. Original proofs, hundreds of examples, and proven theorems illustrate and clarify the material. An extensive table provides Lyapunov functions for differential equations and conditions of stability for the equilibrium solutions. In an intuitive, step-by-step manner, the book covers a breadth of highly relevant topics in linear systems theory from the introductory level to a more advanced level. The chapter on stochastic processes makes it invaluable for financial engineering applications.

    Reflecting the pressures in engineering education to provide compact yet comprehensive courses of instruction, this book presents essential linear system theoretic concepts from first principles to relatively advanced, yet general, topics. The book’s self-contained nature and the coverage of both linear continuous- and discrete-time systems set it apart from other texts.

    System Concept Fundamentals and Linear Vector Spaces

    Introduction

    System Classifications and Signal Definition

    Time Signals and Their Representation

    System Input-Output Relations

    Signal representation via Linear Vector Spaces

    Linear Operators and Matrix Algebra

    Introduction

    Introduction to Matrix Algebra - Euclidian Vector Space

    Systems of Linear Algebraic Equations

    Diagonalization - Eigenvalue Decomposition of Matrices

    Multiple Eigenvalues and Jordan Canonical Form

    Determination of the Co-Efficients of the Characteristic Polynomial of a Matrix

    Computation of the Polynomial Function of the Matrix A

    S-N Decomposition of a Non-singular Matrix

    Computation of An without Eigenvectors

    Operator Algebra and Related Concepts (Finite and Infinite Dimensions)

    Addendum

    Ordinary Differential and Difference Equations

    Introduction

    System of Differential and Difference Equations

    Matrix Formulation and Solution of n-th Order Differential Equations

    Matrix Formulation of the k-th Order Difference Equation

    Linear Differential Equations with Variable Coefficients

    Summary

    Complex Variables for Transform Methods

    Introduction

    Theory of Complex Variables and Contour Integration

    Poisson’s Integral on Unit Circle (or disk)

    Positive Real Functions

    Integral Transform Methods

    Introduction

    Fourier Transform Pair Derivation

    Another Derivation of Fourier Transform

    Derivation of Bilateral Laplace Transform Lb

    Another Derivation of the Bilateral Laplace Transform

    Single-Sided Laplace Transform (Laplace Transform)

    Summary of Transform Definitions

    Laplace Transform Properties

    Recovery of the Original Time Function from the given Single-Sided Laplace Transform

    Solution of Linear Constant Coefficient Differential Equations via The Laplace Transform

    Computation of x(t) from X(s) For Causal Processes

    Inverse of Bilateral (Two-Sided) Laplace Transform Fb(s)

    Transfer Function

    Impulse Response

    Time Convolution for Linear Time Invariant System

     Frequency Convolution in Laplace Domain

    Parseval’s Theorem

    Generation of Orthogonal Signals in Frequency Domain

    The Fourier Transform

    Fourier Transform Properties

    Fourier Transform Inverse

    Hilbert Transform

    Application of The Integral Transforms to The Variable Parameter Differential Equations

    Generalized Error Function

    Digital Systems, Z-Transforms and Applications

    Introduction

    Discrete Systems and Difference Equations

    Realization of a general Discrete System

    Z-Transform for the Discrete Systems

    Fundamental Properties of Z-Transforms

    Evaluation of f (n), given its Single Sided Z-Transform

    Solution of Difference Equations using Z-Transforms

    Computation Algorithm for the Sum of the Squares of the Discrete Signal Sequence

    Bilateral Z-Transform f (n) ↔ Fb(z)

    Evaluation of some of the Important Series via Z-Transforms

    Reconstruction of a Continous-Time Band-limited Signal from Uniform Samples

    State Space Description Of Dynamic Systems

    Introduction

    State Space Formulation

    Selection of The State Variables and Formulation of The State Space Equations

    Methods of Deriving State Variable Equations for The Physical System

    State Space Concepts

    Calculus Of Variations

    Introduction

    Calculus of Maxima, Minima and stationary points (Extrema of a Function)

    Extremal of a Function subject to Multiple Constraints

    Extremal of a Definite Integral - Derivation of Euler-Lagrange Equations with variable end points

    Extremal of a Definite Integral with Multiple Constraints

    Mayer Form

    Bolza’s Form

    Variational Principles and Optimal Control

    Hamilton-Jacobi Formulation of Euler-Lagrange Equations

    Pontryagin’s Extremum Principle

    Dynamic Programming

    Stochastic Processes and Linear Systems Response to Stochastic Inputs

    Preliminaries

    Continous Random Variable and probability density function (pdf)

    Random Walk, Brownian Motion and Wiener Process

    Markov Chains, Inequalities and Law of Large Numbers

    Stochastic Hilbert Space

    Random or Stochastic Processes

    Wiener Filters

    Optimal Estimation, Control, Filtering and Prediction - Continuous Kalman Filters

    Biography

    N.N. Puri is Professor of Electrical and Computer Engineering at Rutgers University, New Jersey.