A Generalized Framework of Linear Multivariable Control - 1st Edition - ISBN: 9780081019467, 9780081019474

A Generalized Framework of Linear Multivariable Control

1st Edition

Authors: Liansheng Tan
eBook ISBN: 9780081019474
Paperback ISBN: 9780081019467
Imprint: Butterworth-Heinemann
Published Date: 11th January 2017
Page Count: 322
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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.

Key Features

  • 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


Academics and industrial researchers, postgraduate students in the fields of control systems and optimizations

Table of Contents

1: Introduction


2: Mathematical preliminaries


2.1 Vector algebra

2.2 Matrix algebra

2.3 Matrix inverse

2.4 Solving system of linear equation

2.5 Linear differential equation

2.6 Matrix differential equation

2.7 Laplace transform

3: Generalized inverse of matrix and solution of linear system equation


3.1 The generalized inverse of matrix

3.2 The full rank decomposition theorem

3.3 The least square solution to an algebraic matrix equation

3.4 The singular value decomposition

4: Polynomial fraction description


4.1 Introduction

4.2 Right polynomial fractions

4.3 Left polynomial fraction

4.4 Column and row degrees

4.5 Minimal realization

4.6 Poles and zeros

4.7 State feedback

5: Stability


5.1 Internal stability

5.2 Lyapunov stability

5.3 Input-output stability

6: Fundamental approaches to control system analysis


6.1 PMD theory of linear multivariable control systems

6.2 Behavioral approach in systems theory

6.3 Chain-scattering representations

6.4 Conclusions

7: Determination of finite and infinite frequency structure of a rational matrix


7.1 Introduction

7.2 The Toeplitz rank information

7.3 To determine the Smith form of a polynomial matrix

7.4 To determine the Smith-McMillan form at infinity of a rational matrix

7.5 To determine the Smith-McMillan form of a rational matrix

7.6 Conclusions

8: The solution of a regular PMD and the set of impulsive free initial conditions


8.1 Introduction

8.2 Preliminary results

8.3 A solution for the LNHMDEs

8.4 The smooth and impulsive solution components and impulsive free initial conditions: C∞ is of full row rank

8.5 The smooth and impulsive solution components and impulsive free initial conditions: C∞ is not of full row rank

8.6 Illustrative example

8.7 Conclusions

9: A refined resolvent decomposition of a regular polynomial matrix and application to the solution of regular PMDs


9.1 Introduction

9.2 Infinite Jordan pairs

9.3 The solution of regular PMDs

9.4 Algorithm and examples

9.5 Conclusions

10: Frequency structures of generalized companion form and application to the solution of regular PMDs


10.1 Introduction

10.2 The frequency structures of generalized companion form and a new resolvent decomposition

10.3 Application to the solution of regular PMDs

10.4 Conclusions

11: A generalized chain-scattering representation and its algebraic system properties


11.1 Introduction

11.2 Input-output consistency and GCSR

11.3 Algebraic system properties of GCSR and DGCSR

11.4 Realizations of GCSR and DGCSR

11.5 Conclusions

12: Realization of behavior


12.1 Introduction

12.2 Behavior realization

12.3 Realization of behavior for GCSRs and DGCSRs

12.4 Conclusions

13: Related extensions to system well-posedness and internal stability


13.1 Introduction

13.2 Input consistency, output uniqueness, fully internal well-posedness, and externally internal well-posedness

13.3 Further characterizations of externally internal well-posedness

13.4 Generalized linear fractional transformations, externally internal stability, and their characterizations

13.5 Conclusions

14: Nonstandard H∞ control problem: A generalized chain-scattering representation approach


14.1 Introduction

14.2 Reformulation of the nonstandard H∞ control problem via generalized chain-scattering representation

14.3 Solvability of nonstandard H∞ control problem

14.4 Conclusions

15: Internet congestion control: A linear multivariable control look


15.1 The basic model of Internet congestion control

15.2 Internet congestion control: A multivariable control look

15.3 Padé approximations to the system (15.7) and (15.8)

15.4 Analyses into system structure of congestion control of a simple network in frequency domain

15.5 Conclusions and further discussions

16: Conclusions and further research


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About the Author

Liansheng Tan

He received his PhD degree from Loughborough University in the UK in 1999. He was a research fellow in Research School of Information Sciences and Engineering, The Australian National University, Australia from 2006 til 2009, and a postdoctoral research fellow in 2001 in School of Information Technology and Engineering at University of Ottawa, Canada. He has also held a number of visiting research positions at Loughborough University, University of Tsukuba, City University of Hong Kong and University of Melbourne. He is currently Editor-in-Chief of Journal of Computers, an Editor of International Journal of Computer Networks and Communications. He was an Editor of Dynamics of Continuous, Discrete and Impulsive Systems (Series B: Applications and Algorithms) (2006-2008), and an Editor of International Journal of Communication Systems. He has published over 120 referred papers widely in international journals and conferences.

He has been intensively worked on the subject of linear multivariable control theory since his PhD study at Loughborough University in the UK. His main contributions on this subject have been to generalize, extend and widen the scope of the linear multivariable control theory. Therefore, he has accumulated focuses on this very important and fundamental subject.

Affiliations and Expertise

Professor, Department of Computer Science, Central China Normal University