Contents

1
Mathematical Descriptions of Systems
1.1
Introduction
A. Physical Processes, Models, and Mathematical Descriptions /
B. Classification of Systems / C. Finite-Dimensional
Systems /D. Chapter Description /E. Guidelines for the
Reader
1.2
Preliminaries
A.Notation/B. Continuous Functions
1.3
Initial-Value Problems
A. Systems of First-Order Ordinary Differential Equations /
B. Classification of Systems of First-Order Ordinary
Differential Equations / C. nth-Order Ordinary Differential
Equations
1.4
Examples of Initial-Value Problems
1.5
More Mathematical Preliminaries
A. Sequences /B. Sequences of Functions / C. The Weierstrass
M-Test
1.6
Existence of Solutions of Initial-Value Problems
A. The Ascoli-Arzela Lemma / B. E-Approximate Solutions / C. The Cauchy-Peano Existence Theorem
1.7
Continuation of Solutions
A. Zorn's Lemma / B. Continuable Solutions / C. Continuation of Solutions to the Boundary of D
1.8
Uniqueness of Solutions
A. The Gronwall Inequality /B. Unique Solutions
1.9
Continuous Dependence of Solutions on Initial Conditions and Parameters
1.10
Systems of First-Order Ordinary Differential Equations
A. More Mathematical Preliminaries: Vector Spaces / B. Further Mathemafical Preliminaries: Normed Linear Spaces / C. Additional Mathematical Preliminaries: Convergence /D. Solutions of Systems of FirstOrder Ordinary Differenfial Equations: Existence, Continuation, Uniqueness, and Continuous Dependence on Initial Conditions
1.11
Systems of Linear First-Order Ordinary Differential Equations
A. Linearization/B. Examples
1.12
Linear Systems: Existence, Uniqueness, Continuation, and Continuity with Respect to Parameters of Solutions
1.13
Solutions of Linear State Equations
1.14
State-Space Description of Continuous-Time Systems
1.15
State-Space Description of Discrete-Time Systems
1.16
Input-Output Description of Systems
A. External Description of Systems: General Considerations / B. Linear Discrete-Time Systems / C. The Dirac Delta Distribution /D. Linear Continuous-Time Systems Summary Notes
1.17
Summary
1.18
Notes
1.19
References
1.20
Exercises
2
Response of Linear Systems
2.1
Introduction
A. Chapter Description / B. Guidelines for the Reader
2.2
Background Material
A. Linear Subspaces / B. Linear Independence / C. Bases /
D. Linear Transformations / E. Representafion of Linear
Transformations by Matrices / *F. Some Properties of
Matrices / *G. Determinants of Matrices / H. Solving Linear
Algebraic Equations /I. Equivalence and Similarity /
J. Eigenvalues and Eigenvectors / K. Direct Sums of
Linear Subspaces / L. Some Canonical Forms of Matrices /
M. Minimal Polynomials / N. Nilpotent Operators / O. The
Jordan Canonical Form
2.3
Linear Homogeneous and Nonhomogeneous Equations
A. The Fundamental Matrix / B. The State Transition
Matrix / C. Nonhomogeneous Equations / D. How to
Determine The State Transition Matrix
2.4
Linear Systems with Constant Coefficients
A. Some Properties of exp(At) /B. How to Determine exp(At) /
C. Modes and Asymptotic Behavior of Time-lavariant
Systems
2.5
Linear Periodic Systems
2.6
State Equation and Input-Output Description of Continuous-Time System
A. Response of Linear Continuous-Time Systems / B. Transfer
Functions / C. Equivalence of Internal Representations
2.7
State Equation and Input-Output Description of Discrete-Time Systems
A. Response of Linear Discrete-Time Systems /B. The
Transfer Function and the z-Transform / C. Equivalence
of Internal Representations / D. Sampled-Data Systems /
E. Modes and Asymptofic Behavior of Time-Invariant
Systems
2.8
An Important Comment on Notation
2.9
Summary
2.10
Notes
2.11
References
2.12
Exercises
3
Controllability, Observability, and Special Forms
3.1
Introduction
A. Brief Introduction to Reachability and Observability /
B. Chapter Description / C. Guidelines for the Reader
PART 1 Controllability and Observability
3.2
Reachability and Controllability
A. Continuous-Time Time-Varying Systems /B. Continuous-
Time Time-Invariant Systems / C. Discrete-Time Systems
3.3
Observability and Constructibility
A. Continuous-Time Time-Varying Systems /B. Continuous-
Time Time-Invariant Systems / C. Discrete-Time Systems
PART 2 Special Forms for Time-Invariant Systems
3.4
Special Forms
A. Standard Forms for Uncontrollable and Unobservable
Systems /B. Eigenvalue/Eigenvector Tests for Controllability
and Observability / C. Relating State-Space and Input-
Output Descriptions / D. Controller and Observer Forms
3.5
Poles and Zeros
3.6
Summary
3.7
Notes
3.8
References
3.9
Exercises
4
State Feedback and State Observers
4.1
Introduction
A. A Brief Introduction to State-Feedback Controllers and
State Observers / B. Chapter Description / C. Guidelines for
the Reader
4.2
Linear State Feedback
A. Continuous-Time Systems / B. Eigenvalue Assignment /
C. The Linear Quadratic Regulator (LQR): Continuous-
Time Case/D. Input-OutputRelations/ E. Discrete-Time
Systems / F. The Linear Quadratic Regulator (LQR):
Discrete-Time Case
4.3
Linear State Observers
A. Full-Order Observers: Continuous-Time Systems /
B. Reduced-Order Observers: Continuous-Time Systems /
C. Optimal State Estimation: Continuous-Time Systems /
D. Full-Order Observers: Discrete-Time Systems /
E. Reduced-Order Observers: Discrete-Time Systems /
F: Optimal State Estimation: Discrete-Time Systems
4.4
Observer-Based Dynamic Controllers
A. State-Space Analysis / B. Transfer Function Analysis
4.5
Summary
4.6
Notes
4.7
References
4.8
Exercises
5
Realization Theory and Algorithms
5.1
Introduction
A. Chapter Description / B. Guidelines for the Reader
5.2
StateSpace Realizations of External Descriptions
A. Continuous-Time Systems / B. Discrete-Time Systems
5.3
Existence and Minimality of Realizations
A. Existence of Realizations / B. Minimality of Realizations /
C. The Order of Minimal Realizations /D. Minimality of
Realizations: Discrete-Time Systems
5.4
Realization Algorithms
A. Realizations Using Duality / B. Realizations in Controller/
Observer Form / C. Realizations with Matrix A Diagonal /
D. Realizations with Matrix A in Block Companion Form /
E. Realizations Using Singular Value Decomposition
5.5
Summary
5.6
Notes
5.7
References
5.8
Exercises
6
Stability
6.1
Introduction
A. Chapter Description / B. Guidelines for the Reader
6.2
Mathematical Background Material
A. Bilinear Functionals and Congruence / B. Euclidean
Vector Spaces / C. Linear Transformations on Euclidean
Vector Spaces
PART 1 Lyapunov Stability
6.3
The Concept of an Equilibrium
6.4
Qualitative Characterizations of an Equilibrium
6.5
Lyapunov Stability of Linear Systems
6.6
Some Geometric and Algebraic Stability Criteria
A. Some Graphical Criteria /B. Some Algebraic Criteria
6.7
The Matrix Lyapunov Equation
6.8
Linearization
PART 2 Input-Output Stability of Continuous-Time Systems
6.9
Input-Output Stability
PART 3 Stability of Discrete-Time Systems
6.10
Discrete-Time Systems
A. Preliminaries /B. Lyapunov Stability of an Equilibrium /
C. Linear Systems /D. The Schur-Cohn Criterion /E. The
Matrix Lyapunov Equation / F. Linearization / G. Input-
Output Stability
6.11
Summary
6.12
Notes
6.13
References
6.14
Exercises
7
Polynomial Matrix Descriptions and Matrix Fractional Descriptions of Systems
7.1
Introduction
A. A Brief Introduction to Polynomial and Fractional
Descriptions / B. Chapter Description / C. Guidelines for the
Reader
PART 1 Analysis of Systems
7.2
Background Material on Polynomial Matrices
A. Rank and Linear Independence / B. Unimodular and
Column (Row) Reduced Matrices / C. Hermite and Smith
Forms / D. Coprimeness and Common Divisors / E. The
Diophantine Equation
7.3
Systems Represented by Polynomial Matrix Descriptions
A. Equivalence of Representations / B. Controllability,
Observability, Stability, and Realizations / C. Interconnected
Systems
PART 2 Synthesis of Control Systems
7.4
Feedback Control Systems
A. Stabilizing Feedback Controllers / B. State Feedback
Control and State Estimation / C. Stabilizing Feedback
Controllers Using Proper and Stable MFDs / D. Two
Degrees of Freedom Feedback Controllers
7.5
Summary
7.6
Notes
7.7
References
7.8
Exercises
Appendix Numerical Considerations
A.1
Introduction
A.2
Solving Linear Algebraic Equations
A.3
Singular Values and Singular Value Decomposition
A.4
Solving Polynomial and Rational Matrix Equations Using Interpolation Methods
A.5
References

Index

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