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
Chapter
Chapter 2: Mathematical preliminaries
2.2.2 Basic matrix operations
2.4 Solving system of linear equation
2.4.2 A general scheme for solving system of linear equation
2.5 Linear differential equation
2.5.2 Homogeneous equations with constant coefficients
2.5.3 Nonhomogeneous equation with constant coefficients
2.5.4 Equation with variable coefficients
2.5.5 Systems of linear differential equations
2.6 Matrix differential equation
2.6.2 Stability and steady state of the matrix system
2.6.3 Solution in matrix form
2.6.4 Solving matrix ordinary differential equations
2.7.3 Region of convergence
2.7.4 Laplace transform pair table
2.7.5 Properties and theorems
2.7.6 Inverse Laplace transform
Chapter 3: Generalized inverse of matrix and solution of linear system equation
3.1 The generalized inverse of matrix
3.1.1 The left inverse and right inverse
3.1.2 Moore-Penrose inverse
3.1.3 The minimization approach to solve an algebraic matrix equation
3.2 The full rank decomposition theorem
3.3 The least square solution to an algebraic matrix equation
3.3.1 The solution to the compatible linear equations
3.3.2 The least square solution of incompatible equation
3.3.3 The minimum norm least squares solution for the equations
3.4 The singular value decomposition
Chapter 4: Polynomial fraction description
4.2 Right polynomial fractions
4.3 Left polynomial fraction
4.4 Column and row degrees
5.1.1 Uniform exponential stability
5.1.2 Uniform asymptotic stability
5.1.3 Lyapunov transformation
5.2.3 Uniform exponential stability
5.2.5 Time-invariant case
5.3 Input-output stability
5.3.1 Uniform bounded-input bounded-output stability
5.3.2 Relation to uniform exponential stability
5.3.3 Time-invariant case
Chapter 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
Chapter 7: Determination of finite and infinite frequency structure of a rational matrix
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
Chapter 8: The solution of a regular PMD and the set of impulsive free initial conditions
8.3 A solution for the LNHMDEs
8.4 The smooth and impulsive solution components and impulsive free initial conditions: C8 is of full row rank
8.5 The smooth and impulsive solution componentsand impulsive free initial conditions: C8 is not of full row rank
Chapter 9: A refined resolvent decomposition of a regular polynomial matrix and application to the solution of regular PMDs
9.2 Infinite Jordan pairs
9.3 The solution of regular PMDs
9.4 Algorithm and examples
Chapter 10: Frequency structures of generalized companion form and application to the solution of regular PMDs
10.2 The frequency structures of generalized companion form and a new resolvent decomposition
10.3 Application to the solution of regular PMDs
Chapter 11: A generalized chain-scattering representation and its algebraic system properties
11.2 Input-output consistency and GCSR
11.3 Algebraic system properties of GCSR and DGCSR
11.4 Realizations of GCSR and DGCSR
Chapter 12: Realization of behavior
12.2 Behavior realization
12.3 Realization of behavior for GCSRs and DGCSRs
Chapter 13: Related extensions to system well-posedness and internal stability
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
Chapter 14: Nonstandard H∞ control problem: A generalized chain-scattering representation approach
14.2 Reformulation of the nonstandard H∞ control problem via generalized chain-scattering representation
14.3 Solvability of nonstandard H∞ control problem
Chapter 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)
15.3.1 Padé approximation to the transfer functions of the time-delay system (15.7) and (15.8)
15.3.2 A proportional controller
15.4 Analyses into system structure of congestion control of a simple network in frequency domain
15.5 Conclusions and further discussions
Chapter 16: Conclusions and further research
A Generalized Framework of Linear Multivariable Control
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