A Generalized Framework of Linear Multivariable Control

1: Introduction

Abstract

2: Mathematical preliminaries

Abstract

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

Abstract

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

Abstract

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

Abstract

5.1 Internal stability

5.2 Lyapunov stability

5.3 Input-output stability

6: Fundamental approaches to control system analysis

Abstract

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

Abstract

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

Abstract

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

Abstract

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

Abstract

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

Abstract

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

Abstract

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

Abstract

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

Abstract

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

Abstract

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