Chapter 1 Introduction

1-1 The Study of Systems

1-2 The Scope of the Book

Chapter 2 Linear Spaces and Linear Operators

2-1 Introduction

2-2 Linear Spaces over a Field

2-3 Linear Independence, Bases, and Representations

Change of Basis

2-4 Linear Operators and Their Representations

Matrix Representations of a Linear Operator

2-5 Systems of Linear Algebraic Equations

2-6 Eigenvectors, Generalized Eigenvectors, and Jordan-Form Representations of a Linear Operator

Derivation of a Jordan-Form Representation

2-7 Functions of a Square Matrix

Polynomials of a Square Matrix

Functions of a Square Matrix

Functions of a Matrix Defined by Means of Power Series

2-8 Norms and Inner Product

2-9 Concluding Remarks

Problems

Chapter 3 Mathematical Descriptlons of Systems

3-1 Introduction

3-2 The Input-Output Description

Linearity

Causality

Relaxedness

Time Invariance

Transfer-Function Matrix

3-3 The State-Variable Description

The Concept of State

Dynamical Equations

Linearity

Time Invariance

Transfer-Function Matrix

Analog and Digital Computer Simulations of Linear Dyna-mical Equations

3-4 Examples

Dynamical Equations for RLC Networks

3-5 Comparisons of the Input-Output Description and the State-Variable Description

3-6 Mathematical Descriptions of Composite Systems

Time-Varying Case

Time-Invariant Case

Well-Posedness Problem

3-7 Discrete-Time Systems

3-8 Concluding Remarks

Problems

Chapter 4 Linear Dynamical Equations and Impulse-Response Matrices

4-1 Introduction

4-2 Solutions of a Dynamical Equation

Time-Varying Case

Solutions of x = A（t）x

Solutions of the Dynamical Equation E

Time-Invariant Case

4-3 Equivalent Dynamical Equations

Time-Invariant Case

Time-Varying Case

Linear Time-Varying Dynamical Equation withPeriodic A（·）

4-4 Impulse-Response Matrices and Dynamical Equations

Time-Varying Case

Time-Invariant Case

4-5 Concluding Remarks

Problems

Chapter 5 Controllability and Observability of Linear Dynamical Equatlons

5-1 Introduction

5-2 Linear Independence of Time Functions

5-3 Controllability of Linear Dynamical Equations

Time-Varying Case

Differential Controllability, Instantaneous Controllabil-ity, and Uniform Controllability

Time-Invariant Case

Controllability Indices

5-4 Observability of Linear Dynamical Equations

Time-Varying Case

Differential Observability, Instantaneous Observabil-ity, and Uniform Observability

Linear Time-Invariant Dynamical Equations

Observability Indices

5-5 Canonical Decomposition of a Linear Time-Invariant Dyna-mical Equation

Irreducible Dynamical Equations

5-6 Controllability and Observability of Jordan-Form Dynamical Equations

5-7 Output Controllability and Output Function Controllability

5-8 Computational Problems

5-9 Concluding Remarks

Problems

Chapter 6 Irreducible Realizations, Strict System Equivalence, and Identification

6-1 Introduction

6-2 The Characteristic Polynomial and the Degree of a Proper Rational Matrix

6-3 Irreducible Realizations of Proper Rational Functions

Irreducible Realization of β3/D（s）

Irreducible Realizations of g（s） = N（s）/D（s）

Observable Canonical-Form Realization

Controllable Canonical-Form Realization

Realization from the Hankel Matrix

Jordan-Canonical-Form Realization

Realization of Linear Time-Varying Differential Equations

6-4 Realizations of Vector Proper Rational Transfer Functions

Realization from the Hankel Matrix

6-5 Irreducible Realizations of Proper Rational Matrices: Hankel Methods

Method Ⅰ.Singular Value Decomposition

Method Ⅱ.Row Searching Method

6-6 Irreducible Realizations of （s）: Coprime Fraction Method

Controllable-Form Realization

Realization of N（s）D-1（s）, Where D（s） and N（s） Are NotRight Coprime

Column Degrees and Controllability Indices

Observable-Form Realization

6-7 Polynomial Matrix Description

6-8 Strict System Equivalence

6-9 Identification of Discrete-Time Systems from Noise-Free Data

Persistently Exciting Input Sequences

Nonzero Initial Conditions

6-10 Concluding Remarks

Problems

Chapter 7 State Feedback and State Estimators

7-1 Introduction

7-2 Canonical-Form Dynamical Equations

Single-Variable Case

Multivariable Case

7-3 State Feedback

Single-Variable Case

Stabilization

Effect on the Numerator of g（s）

Asymptotic Tracking Problem—Nonzero SetPoint

Multivariable Case

Method Ⅰ

Method Ⅱ

Method Ⅲ

Nonuniqueness of Feedback Gain Matrix

Assignment of Eigenvalues and Eigenvectors

Effect on the Numerator Matrix of G（s）

Computational Problems

7-4 State Estimators

Full-Dimensional State Estimator

Method Ⅰ

Method Ⅱ

Reduced-Dimensional State Estimator

Method Ⅰ

Method Ⅱ

7-5 Connection of State Feedback and State Estimator

Functional Estimators

7-6 Decoupling by State Feedback

7-7 Concluding Remarks

Problems

Chapter 8 Stablllty of Llnear Systems

8-1 Introduction

8-2 Stability Criteria in Terms of the Input-Output Description

Tine-Varying Case

Time-Invariant Case

8-3 Routh-Hurwitz Criterion

8-4 Stability of Linear Dynamical Equations

Time-Varying Case

Time-Invariant Case

8-5 Lyapunov Theorem

A Proof of the Routh-Hurwitz Criterion

8-6 Discrete-Time Systems

8-7 Concluding Remarks

Problems

Chapter 9 Llnear Tlme-Invarlant Composlte Systems: Characterlza-tlon, Stablllty, and Deslgns

9-1 Introduction

9-2 Complete Characterization of Single-Variable Composite Systems

9-3 Controllability and Observability of Composite Systems

Parallel Connection

Tandem Connection

Feedback Connection

9-4 Stability of Feedback Systems

Single-Variable Feedback System

Multivariable Feedback System

9-5 Design of Compensators: Unity Feedback Systems

Single-Variable Case

Single-Input or Single-Output Case

Multivariable Case—Arbitrary Pole Assignment

Multivariable Case—Arbitrary Denominator-Matrix Assignment

Decoupling

9-6 Asymptotic Tracking and Disturbance Rejection

Single-Variable Case

Multivariable Case

Static Decoupling—Robust and NonrobustDesigns

State-Variable Approach

9-7 Design of Compensators: Input-Output FeedbackSytems

Single-Variable Case

Multivariable Case

Implementations of Open-Loop Compensators

Implementation Ⅰ

Implementation Ⅱ

Applications

Decoupling

Asymptotic Tracking, Disturbance Rejection, andDecoupling

9-8 Concluding Remarks

Problems

Appendix A Elementary Transformations

A-1 Gaussian Elimination

A-2 Householder Transformation

A-3 Row Searching Algorithm

A-4 Hessenberg Form

Problems

Appendix B Analytic Functions of a Real Variable

Appendix C Minimum Energy Control

Appendix D Controllability after the Introduction of Sampling

Problems

Appendix E Hermitian Forms and Singular Value Decomposition

Problems

Appendix F On the Matrix Equation AM + MB = N

Problems

Appendix G Polynomials and Polynomial Matrices

G-1 Coprimeness of Polynomials

G-2 Reduction of Reducible Rational Functions

G-3 Polynomial Matrices

G-4 Coprimeness of Polynomial Matrices

G-5 Column- and Row-Reduced Polynomial Matrices

G-6 Coprime Fractions of Proper Rational Matrices

Problems

Appendix H Poles and Zeros

Problems

References

Index