Browse the Annotated Table of Contents
Computer-aided techniques for analysis and design of systems have come into widespread use in recent years thanks to a proliferation of powerful desktop workstations and the availability of sophisticated software packages that provide easy-to-use ``tools'' for analysis and design which incorporate powerful numerical and graphical procedures. These technological innovations in computer hardware and software have been rapidly deployed in the commercial world; they have also made a major impact on research and graduate-level teaching in systems, control, and signal processing.
The major premise behind this book is that a valuable and interesting approach to the teaching of undergraduate systems and signals material can be based on a set of topics having clear importance in modern computer-aided system analysis and design. In an electrical engineering curriculum, the ``systems and signals'' courses have traditionally provided considerable exposure to important applied mathematics material; it seems very appropriate that these courses reflect the evolution of engineering practice, especially with regard to the recent rapid advances in computer-aided design and analysis technology. The fact that computer-aided methods for electronic circuit analysis have already proved to be beneficial for use in courses starting at an introductory level is taken as further evidence supporting the value of a systems and signal course of the kind for which this book was developed. As for its interest, experience suggests that engineering students are attracted to courses that exploit and complement a high level of computer expertise.
This book provides an introduction to analytical and computational methods of fundamental importance for computer-aided systems analysis and design. The aim is to provide an understanding of basics, and the book is written at an introductory level, in the sense that no substantial sophistication in any application is assumed. A single course could not cover the vast range of applications where computer-aided techniques play an important role, but various specialized courses can build on the material presented here. As in a ``traditional'' systems and signals course, the tools discussed are important and versatile ones. It is essential that students be provided with opportunities (with some training if necessary) for using these tools in practice; this means, typically, an opportunity for gaining familiarity with using one or more software packages while doing analysis and design problems on PC-class workstations.
The book is structured to support several modes of integrating the subject matter into a curriculum. It was developed at Princeton University to support a junior-level Electrical Engineering elective course that sets the stage for a traditional signals and linear systems course which follows. The course is taken concurrently with a course in analog electronic circuits that includes SPICE simulations. The PC-MATLAB software package is used in course assignments. About 80% of the material is covered in one semester. The book could also be used for a course in which a fluency in Laplace and Fourier transforms is assumed as a prerequisite; the material on linear systems can be covered quickly, and more time can be devoted to nonlinear systems and optimization. How best to structure a year long course sequence which combines the material presented in this book with other important analytical methods covered in ``conventional'' systems and signals textbooks is a challenging problem of curriculum design. It is hoped that this book some will suggest some new possibilities for consideration.
Basic mechanical and electrical systems provide many of the examples and problems which are treated in this book. For example, frequency response and time response characteristics of such systems are two aspects of behavior that are often the subject of design and analysis tasks. Examples from a variety of fields are included, and one goal of the presentation is broad appeal. The mathematical and computational tools presented are of fundamental importance in a wide variety of applied problems arising in fields such as electronic circuits, signal processing, telecommunications, dynamics, automation and control, and econometrics. Thus the ideas and methods developed in this book offer essentially unlimited possibilities for use in studying quantitative models of all kinds of dynamical phenomena. In addition to a sophomore-level engineering familiarity with electrical and mechanical systems, a mathematical background including linear algebra and differential equations is assumed. Important background material is introduced, reviewed, and motivated by applications. The discussion of a few relatively sophisticated mathematical ideas relies on intuition and analogies rather than on rigorous formalism. Examples are used throughout to reinforce ideas and to demonstrate applications.
The book does not include any discussion of implementation details at the programming language level. Thus the book is not tied to a particular software development environment. At Princeton, a self-guided introduction to PC-MATLAB is assigned at the beginning of the course, and later assignments provide additional suggestions for implementing programs for specific tasks. The justification of this approach is that the book deals with system analysis and design, not development of reliable numerical software; nor is a main purpose of the book to provide a thorough understanding of the use and limitations of particular software tools. However, a significant appreciation of such issues can be conveyed through hands-on experience in using a computer to solve problems such as those included in the book.
There are four major areas covered by the book.
1.) Mathematical Fundamentals. The coverage of this first topic is intended to illustrate the utility of linear algebra, differential equations, and difference equations in formulating quantitative descriptions of system behavior. Parts of the first chapter will primarily serve as a review of material that students are expected to have seen before. Based on experience with classes of first-semester junior electrical engineering students, covering this material is essential for developing an adequate understanding of topics that provide the foundation for much of what follows in later chapters. Almost all undergraduate readers will find some new material introduced and some new approaches given; instructors will need to tailor the coverage of this material to suit particular audiences. One particular focus of the presentation is to motivate the kinds of numerical techniques that are useful in system analysis and design, e.g. solving systems of linear equations, computing eigenvalues and eigenvectors, etc. The emphasis is on understanding fundamental ideas and their interpretations and applications.
2.) Linear and Nonlinear Systems. Linear models are widely used because they adequately describe a wide variety of dynamical systems, and they are amenable to detailed analysis, including the use of special techniques such as frequency response methods which very much rely on the superposition principle for justification. Various analytical and computational methods are available to handle linear and nonlinear systems; time-domain methods are the focus of the presentation. A variety of topics involving Fourier and Laplace transforms is covered in a ``traditional'' introductory (linear) systems and signals course; in this book, these topics have been replaced by some ``nontraditional'' ones, broadening the scope of the course while keeping the presentation at an introductory level. As one example, a number of topics concerning nonlinear systems are treated in the book. Since nonlinear effects arise in all real systems, (e.g. at extremes of the operating ranges of system variables) and since there are important physical phenomena that require nonlinear models (e.g. Newton's law of gravitation, models for competition in population ecology, electronic device models, etc.), a knowledge of some fundamentals of nonlinear systems is quite important for applications of systems analysis and design. The uses of linearization and other approximations are covered, and other approaches arising from qualitative notions of nonlinear system behavior are discussed.
3.) Discretization, Discrete-Time Models, and Simulation. Since differential equations are rarely solvable by any kind of ``closed form'' analytical expression, numerical methods are indispensable for determining properties of solutions. Digital simulation methods require some kind of discretization process and provide information about the solutions on some grid of points rather than for all values of the independent variable. In applications such as audio signal processing and computerized control, it is necessary to interface a discrete-time system (the signal processor or the digital process controller) with the continuous-time variables of the ``real world'' (the output of a microphone or the measured values of temperature, pressure, etc.). The coverage of this topic is intended to provide insight and understanding by presenting and analyzing a variety of discretization techniques that are widely used.
4. Optimization. System design involves consideration of various performance objectives in view of physical and geometrical constraints and limitations on available resources. In short, a designer tries to achieve best possible performance within the limits posed by technology, economics, the time available for design, etc. By modeling performance criteria and constraints in mathematical form, analytical and numerical methods can be used to obtain optimum designs. Even in cases where system behavior is judged in a qualitative way by the designer or the end user of the system, numerical methods and graphical displays of system response characteristics may be used to guide the designer. This book covers the basic theory behind the principal kinds of optimization problems that have been used in systems applications, and it provides a selected overview of numerical optimization techniques.
review of basic operations, basis transformations, eigenvalues and eigenvectors, Cayley-Hamilton theorem
existence and uniqueness conditions, inverses and pseudoinverses, (triangular) factorization and solution of linear equations, eigenvector-eigenvalue factorization, QR factorization for least squares solutions and eigenvalue calculations, Singular Value Decomposition (SVD) and applications, splitting and iterative solution of linear equations, iterative methods for computing eigenvalues and eigenvectors.
differentiation and integration of matrix functions, Taylor series expansions
linear, time invariant state equations, homogeneous and forced solutions, general solution, matrix exponential for homogeneous solution, use of eigenvectors and eigenvalues, variation of parameters for forced solution
output equation, weighting pattern (impulse response) and convolution integral formula for output, high order differential equation for input-output description, (observer) canonical realization in state space, determination of initial conditions for (observable) state equations
types of stability, conditions on eigenvalues, connection with input-output stability
phasor response of input-output differential equation, transfer function, magnitude and phase description of transfer function, poles and zeros, Bode plots, bandwidth and damping ratio
step response, time constants, connections with frequency response
overview of same kinds of results in the case of discrete-time systems, special focus on state equations, input-output difference equations, stability, frequency response
sample-and-hold, forward difference (Euler) and backward difference methods, connections between approximation of derivatives and numerical integration of differential equations, trapezoidal method, implicit versus explicit discretization methods, discretization for nonlinear systems, Runge-Kutta methods
transfer function relations, stability and accuracy analysis of Euler, backward difference and trapezoidal methods, frequency warping
two design approaches: bilinear transformation and impulse invariance
partial differential equation models: telegraph equations, Poisson's equation, heat equation. Finite difference discretization, semi-discretization, finite element discretization.
derivatives as linear operators, examples, applications to Newton's method, local linear approximation
linearization of state equations, linearization of integral representation for solution of nonlinear differential equation and Picard iteration, application of Newton's method: waveform relaxation, Lyapunov's theorem on stability using linearization, Lyapunov functions
classification of equilibria, qualitative behavior of solutions: stability, instability, etc., phase plane, nonlinear phenomena: limit cycles and chaotic solutions
basic neural network model; analysis and design of associative memories, stability analysis, discrete-time neural network model
Volterra series representation of input-output systems, describing functions, quasilinearization and harmonic balancing, qualitative properties of forced response of nonlinear systems
global versus local approximation, piecewise linear analysis
necessary and sufficient conditions for unconstrained optimization problems, Lagrange multipliers for problems with equality constraints, neighboring optima, inequality constraints and Kuhn-Tucker conditions
numerical methods for minimization problems: gradient method, linear search, quasi-Newton methods, techniques for constrained problems
optimal control, principle of optimality, dynamic programming, Lagrange multiplier methods
Systems: Analysis, Design, and Computation
Bradley W. Dickinson
Prentice-Hall, 1991
ISBN: 0-13-338047-5