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A Generalized Framework of Linear Multivariable Control

Book Description

A Generalized Framework of Linear Multivariable Control proposes a number of generalized models by using the generalized inverse of matrix, while the usual linear multivariable control theory relies on some regular models.

The book supports that in H-infinity control, the linear fractional transformation formulation is relying on the inverse of the block matrix. If the block matrix is not regular, the H-infinity control does not apply any more in the normal framework. Therefore, it is very important to relax those restrictions to generalize the classical notions and models to include some non-regular cases.

This book is ideal for scholars, academics, professional engineer and students who are interested in control system theory.

  • Presents a comprehensive set of numerical procedures, algorithms, and examples on how to deal with irregular models
  • Provides a summary on generalized framework of linear multivariable control that focuses on generalizations of models and notions
  • Introduces a number of generalized models by using the generalized inverse of matrix

Table of Contents

  1. Cover image
  2. Title page
  3. Table of Contents
  4. Copyright
  5. 1: Introduction
    1. Abstract
  6. 2: Mathematical preliminaries
    1. Abstract
    2. 2.1 Vector algebra
    3. 2.2 Matrix algebra
    4. 2.3 Matrix inverse
    5. 2.4 Solving system of linear equation
    6. 2.5 Linear differential equation
    7. 2.6 Matrix differential equation
    8. 2.7 Laplace transform
  7. 3: Generalized inverse of matrix and solution of linear system equation
    1. Abstract
    2. 3.1 The generalized inverse of matrix
    3. 3.2 The full rank decomposition theorem
    4. 3.3 The least square solution to an algebraic matrix equation
    5. 3.4 The singular value decomposition
  8. 4: Polynomial fraction description
    1. Abstract
    2. 4.1 Introduction
    3. 4.2 Right polynomial fractions
    4. 4.3 Left polynomial fraction
    5. 4.4 Column and row degrees
    6. 4.5 Minimal realization
    7. 4.6 Poles and zeros
    8. 4.7 State feedback
  9. 5: Stability
    1. Abstract
    2. 5.1 Internal stability
    3. 5.2 Lyapunov stability
    4. 5.3 Input-output stability
  10. 6: Fundamental approaches to control system analysis
    1. Abstract
    2. 6.1 PMD theory of linear multivariable control systems
    3. 6.2 Behavioral approach in systems theory
    4. 6.3 Chain-scattering representations
    5. 6.4 Conclusions
  11. 7: Determination of finite and infinite frequency structure of a rational matrix
    1. Abstract
    2. 7.1 Introduction
    3. 7.2 The Toeplitz rank information
    4. 7.3 To determine the Smith form of a polynomial matrix
    5. 7.4 To determine the Smith-McMillan form at infinity of a rational matrix
    6. 7.5 To determine the Smith-McMillan form of a rational matrix
    7. 7.6 Conclusions
  12. 8: The solution of a regular PMD and the set of impulsive free initial conditions
    1. Abstract
    2. 8.1 Introduction
    3. 8.2 Preliminary results
    4. 8.3 A solution for the LNHMDEs
    5. 8.4 The smooth and impulsive solution components and impulsive free initial conditions: C∞ is of full row rank
    6. 8.5 The smooth and impulsive solution components and impulsive free initial conditions: C∞ is not of full row rank
    7. 8.6 Illustrative example
    8. 8.7 Conclusions
  13. 9: A refined resolvent decomposition of a regular polynomial matrix and application to the solution of regular PMDs
    1. Abstract
    2. 9.1 Introduction
    3. 9.2 Infinite Jordan pairs
    4. 9.3 The solution of regular PMDs
    5. 9.4 Algorithm and examples
    6. 9.5 Conclusions
  14. 10: Frequency structures of generalized companion form and application to the solution of regular PMDs
    1. Abstract
    2. 10.1 Introduction
    3. 10.2 The frequency structures of generalized companion form and a new resolvent decomposition
    4. 10.3 Application to the solution of regular PMDs
    5. 10.4 Conclusions
  15. 11: A generalized chain-scattering representation and its algebraic system properties
    1. Abstract
    2. 11.1 Introduction
    3. 11.2 Input-output consistency and GCSR
    4. 11.3 Algebraic system properties of GCSR and DGCSR
    5. 11.4 Realizations of GCSR and DGCSR
    6. 11.5 Conclusions
  16. 12: Realization of behavior
    1. Abstract
    2. 12.1 Introduction
    3. 12.2 Behavior realization
    4. 12.3 Realization of behavior for GCSRs and DGCSRs
    5. 12.4 Conclusions
  17. 13: Related extensions to system well-posedness and internal stability
    1. Abstract
    2. 13.1 Introduction
    3. 13.2 Input consistency, output uniqueness, fully internal well-posedness, and externally internal well-posedness
    4. 13.3 Further characterizations of externally internal well-posedness
    5. 13.4 Generalized linear fractional transformations, externally internal stability, and their characterizations
    6. 13.5 Conclusions
  18. 14: Nonstandard H∞ control problem: A generalized chain-scattering representation approach
    1. Abstract
    2. 14.1 Introduction
    3. 14.2 Reformulation of the nonstandard H∞ control problem via generalized chain-scattering representation
    4. 14.3 Solvability of nonstandard H∞ control problem
    5. 14.4 Conclusions
  19. 15: Internet congestion control: A linear multivariable control look
    1. Abstract
    2. 15.1 The basic model of Internet congestion control
    3. 15.2 Internet congestion control: A multivariable control look
    4. 15.3 Padé approximations to the system (15.7) and (15.8)
    5. 15.4 Analyses into system structure of congestion control of a simple network in frequency domain
    6. 15.5 Conclusions and further discussions
  20. 16: Conclusions and further research
    1. Abstract
  21. Bibliography
  22. Index