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Linear and Nonlinear Multivariable Feedback Control

Book Description

Automatic feedback control systems play crucial roles in many fields, including manufacturing industries, communications, naval and space systems. At its simplest, a control system represents a feedback loop in which the difference between the ideal (input) and actual (output) signals is used to modify the behaviour of the system. Control systems are in our homes, computers, cars and toys. Basic control principles can also be found in areas such as medicine, biology and economics, where feedback mechanisms are ever present.

Linear and Nonlinear Multivariable Feedback Control presents a highly original, unified control theory of both linear and nonlinear multivariable (also known as multi-input multi-output (MIMO)) feedback systems as a straightforward extension of classical control theory. It shows how the classical engineering methods look in the multidimensional case and how practising engineers or researchers can apply them to the analysis and design of linear and nonlinear MIMO systems.

This comprehensive book:

  • uses a fresh approach, bridging the gap between classical and modern, linear and nonlinear multivariable control theories;
  • includes vital nonlinear topics such as limit cycle prediction and forced oscillations analysis on the basis of the describing function method and  absolute stability analysis by means of the primary classical frequency-domain criteria (e.g. Popov, circle or parabolic criteria);
  • reinforces the main themes with practical worked examples solved by a special MATLAB-based graphical user interface, as well as with problems, questions and exercises on an accompanying website.  

The approaches presented in Linear and Nonlinear Multivariable Feedback Control form an invaluable resource for graduate and undergraduate students studying multivariable feedback control as well as those studying classical or modern control theories. The book also provides a useful reference for researchers, experts and practitioners working in industry

Table of Contents

  1. Cover Page
  2. Title Page
  3. Copyright Page
  4. Contents
  5. Preface
  6. Part I Linear Multivariable Control Systems
    1. Chapter 1: Canonical representations and stability analysis of linear MIMO systems
      1. 1.1 Introduction
      2. 1.2 General linear square MIMO systems
        1. 1.2.1 Transfer matrices of general MIMO systems
        2. 1.2.2 MIMO system zeros and poles
        3. 1.2.3 Spectral representation of transfer matrices: characteristic transfer functions and canonical basis
        4. 1.2.4 Stability analysis of general MIMO systems
        5. 1.2.5 Singular value decomposition of transfer matrices
      3. 1.3 Uniform MIMO systems
        1. 1.3.1 Characteristic transfer functions and canonical representations of uniform MIMO systems
        2. 1.3.2 Stability analysis of uniform MIMO systems
      4. 1.4 Normal MIMO systems
        1. 1.4.1 Canonical representations of normal MIMO systems
        2. 1.4.2 Circulant MIMO systems
        3. 1.4.3 Anticirculant MIMO systems
        4. 1.4.4 Characteristic transfer functions of complex circulant and anticirculant systems
      5. 1.5 Multivariable root loci
        1. 1.5.1 Root loci of general MIMO systems
        2. 1.5.2 Root loci of uniform systems
        3. 1.5.3 Root loci of circulant and anticirculant systems
    2. Chapter 2: Performance and design of linear MIMO systems
      1. 2.1 Introduction
      2. 2.2 Generalized frequency response characteristics and accuracy of linear MIMO systems under sinusoidal inputs
        1. 2.2.1 Frequency characteristics of general MIMO systems
        2. 2.2.2 Frequency characteristics and oscillation index of normal MIMO systems
        3. 2.2.3 Frequency characteristics and oscillation index of uniform MIMO systems
      3. 2.3 Dynamical accuracy of MIMO systems under slowly changing deterministic signals
        1. 2.3.1 Matrices of error coefficients of general MIMO systems
        2. 2.3.2 Dynamical accuracy of circulant, anticirculant and uniform MIMO systems
        3. 2.3.3 Accuracy of MIMO systems with rigid cross-connections
      4. 2.4 Statistical accuracy of linear MIMO systems
        1. 2.4.1 Accuracy of general MIMO systems under stationary stochastic signals
        2. 2.4.2 Statistical accuracy of normal MIMO systems
        3. 2.4.3 Statistical accuracy of uniform MIMO systems
        4. 2.4.4 Formulae for mean square outputs of characteristic systems
      5. 2.5 Design of linear MIMO systems
  7. Part II Nonlinear Multivariable Control Systems
    1. Chapter 3: Study of one-frequency self-oscillation in nonlinear harmonically linearized MIMO systems
      1. 3.1 Introduction
      2. 3.2 Mathematical foundations of the harmonic linearization method for one-frequency periodical processes in nonlinear MIMO systems
      3. 3.3 One-frequency limit cycles in general MIMO systems
        1. 3.3.1 Necessary conditions for the existence and investigation of the limit cycle in harmonically linearized MIMO systems
        2. 3.3.2 Stability of the limit cycle in MIMO systems
      4. 3.4 Limit cycles in uniform MIMO systems
        1. 3.4.1 Necessary conditions for the existence and investigation of limit cycles in uniform MIMO systems
        2. 3.4.2 Analysis of the stability of limit cycles in uniform systems
      5. 3.5 Limit cycles in circulant and anticirculant MIMO systems
        1. 3.5.1 Necessary conditions for the existence and investigation of limit cycles in circulant and anticirculant systems
        2. 3.5.2 Limit cycles in uniform circulant and anticirculant systems
    2. Chapter 4: Forced oscillation and generalized frequency response characteristics of nonlinear MIMO systems
      1. 4.1 Introduction
      2. 4.2 Nonlinear general MIMO systems
        1. 4.2.1 One-frequency forced oscillation and capturing in general MIMO systems
        2. 4.2.2 Generalized frequency response characteristics and oscillation index of stable nonlinear MIMO systems
        3. 4.2.3 Generalized frequency response characteristics of limit cycling MIMO systems
      3. 4.3 Nonlinear uniform MIMO systems
        1. 4.3.1 One-frequency forced oscillation and capturing in uniform systems
        2. 4.3.2 Generalized frequency response characteristics of stable nonlinear uniform systems
        3. 4.3.3 Generalized frequency response characteristics of limit cycling uniform systems
      4. 4.4 Forced oscillations and frequency response characteristics along the canonical basis axes of nonlinear circulant and anticirculant systems
      5. 4.5 Design of nonlinear MIMO systems
    3. Chapter 5: Absolute stability of nonlinear MIMO systems
      1. 5.1 Introduction
      2. 5.2 Absolute stability of general and uniform MIMO systems
        1. 5.2.1 Multidimensional Popov’s criterion
        2. 5.2.2 Application of the Bode diagrams and Nichols plots
        3. 5.2.3 Degree of stability of nonlinear MIMO systems
      3. 5.3 Absolute stability of normal MIMO systems
        1. 5.3.1 Generalized Aizerman’s hypothesis
      4. 5.4 Off-axis circle and parabolic criteria of the absolute stability of MIMO systems
        1. 5.4.1 Off-axis circle criterion
        2. 5.4.2 Logarithmic form of the off-axis criterion of absolute stability
        3. 5.4.3 Parabolic criterion of absolute stability
      5. 5.5 Multidimensional circle criteria of absolute stability
        1. 5.5.1 General and normal MIMO systems
        2. 5.5.2 Inverse form of the circle criterion for uniform systems
      6. 5.6 Multidimensional circle criteria of the absolute stability of forced motions
  8. Bibliography
  9. Index
  10. Footnotes