Discrete H¿ Optimization is concerned with the study of
H¿ optimization for digital signal processing and discrete-time control systems. The first three chapters present the basic theory and standard methods in digital filtering and systems from the frequency-domain approach, followed by a discussion of the general theory of approximation in Hardy spaces. AAK theory is introduced, first for finite-rank operators and then more generally, before being extended to the multi-input/multi-output setting. This mathematically rigorous book is self-contained and suitable for self-study. The advanced mathematical results derived here are applicable to digital control systems and digital filtering.|This book discusses in detail
H¿ optimization for digital signal processing and discrete-time control systems. The rigorous derivations of the results, many examples and the discussion of research problems, together with the fact that it is self-contained, make it suitable for self-study. It provides readers with sufficient background in the subject to enable them to begin research in this field.
1. Digital Signals and Digital Filters.- 1.1 Analog and Digital Signals.- 1.1.1 Band-Limited Analog Signals.- 1.1.2 Digital Signals and the Sampling Theorem.- 1.2 Time and Frequency Domains.- 1.2.1 Fourier Transforms and Convolutions on Three Basic Groups.- 1.2.2 Frequency Spectra of Digital Signals.- 1.3 z-Transforms.- 1.3.1 Properties of the z-Transform.- 1.3.2 Causal Digital Signals.- 1.3.3 Initial Value Problems.- 1.3.4 Singular and Analytic Discrete Fourier Transforms.- 1.4 Digital Filters.- 1.4.1 Basic Properties of Digital Filters.- 1.4.2 Transfer Functions and IIR Digital Filters.- 1.5 Optimal Digital Filter Design Criteria.- 1.5.1 An Interpolation Method.- 1.5.2 Ideal Filter Characteristics.- 1.5.3 Optimal IIR Filter Design Criteria.- Problems.- 2. Linear Systems.- 2.1 State-Space Deillegalscriptions.- 2.1.1 An Example of Flying Objects.- 2.1.2 Properties of Linear Time-Invariant Systems.- 2.1.3 Properties of State-Space Deillegalscriptions.- 2.2 Transfer Matrices and Minimal Realization.- 2.2.1 Transfer Matrices of Linear Time-Invariant Systems.- 2.2.2 Minimal Realization of Linear Systems.- 2.3 SISO Linear Systems.- 2.3.1 Kronecker´s Theorem.- 2.3.2 Minimal Realization of SISO Linear Systems.- 2.3.3 System Reduction.- 2.4 Sensitivity and Feedback Systems.- 2.4.1 Plant Sensitivity.- 2.4.2 Feedback Systems and Output Sensitivity.- 2.4.3 Sensitivity Minimization.- Problems.- 3. Approximation in Hardy Spaces.- 3.1 Hardy Space Preliminaries.- 3.1.1 Definition of Hardy Space Norms.- 3.1.2 Inner and Outer Functions.- 3.1.3 The Hausdorff-Young Inequalities.- 3.2 Least-Squares Approximation.- 3.2.1 Beurling´s Approximation Theorem.- 3.2.2 An All-Pole Filter Design Method.- 3.2.3 A Pole-Zero Filter Design Method.- 3.2.4 A Stabilization Procedure.- 3.3 Minimum-Norm Interpolation.- 3.3.1 Statement of the Problem.- 3.3.2 Extremal Kernels and Generalized Extremal Functions.- 3.3.3 An Application to Minimum-Norm Interpolation.- 3.3.4 Suggestions for Computation of Solutions.- 3.4 Nevanlinna-Pick Interpolation.- 3.4.1 An Interpolation Theorem.- 3.4.2 Nevanlinna-Pick´s Theorem and Pick´s Algorithm.- 3.4.3 Verification of Pick´s Algorithm.- Problems.- 4. Optimal Hankel-Norm Approximation and H?-Minimization.- 4.1 The Nehari Theorem and Related Results.- 4.1.1 Nehari´s Theorem.- 4.1.2 The AAK Theorem and Optimal Hankel-Norm Approximations.- 4.2 s-Numbers and Schmidt Pairs.- 4.2.1 Adjoint and Normal Operators.- 4.2.2 Singular Values of Hankel Matrices.- 4.2.3 Schmidt Series Representation of Compact Operators.- 4.2.4 Approximation of Compact Hankel Operators.- 4.3 System Reduction.- 4.3.1 Statement of the AAK´s Theorem.- 4.3.2 Proof of the AAK Theorem for Finite-Rank Hankel Matrices.- 4.3.3 Reformulation of AAK´s Result.- 4.4 H?-Minimization.- 4.4.1 Statement of the Problem.- 4.4.2 An Example of H?-Minimization.- 4.4.3 Existence, Uniqueness, and Construction of Optimal Solutions.- Problems.- 5. General Theory of Optimal Hankel-Norm Approximation.- 5.1 Existence and Preliminary Results.- 5.1.1 Solvability of the Best Approximation Problem.- 5.1.2 Characterization of the Bounded Operators that Commute with the Shift Operator.- 5.1.3 Beurling´s Theorem.- 5.1.4 Operator Norms of Hankel Matrices in Terms of Inner and Outer Factors.- 5.1.5 Properties of the Norm of Hankel Matrices.- 5.2 Uniqueness of Schmidt Pairs.- 5.2.1 Uniqueness of Ratios of Schmidt Pairs.- 5.2.2 Hankel Operators Generated by Schmidt Pairs.- 5.3 The Greatest Common Divisor: The Inner Function ?I0(z).- 5.3.1 Basic Properties of the Inner Function ?I0(z).- 5.3.2 Relations Between Dimensions and Degrees.- 5.3.3 Relations Between ?I0(z) and s-Numbers.- 5.4 AAK´s Main Theorem on Best Hankel-Norm Approximation.- 5.4.1 Proof of AAK´s Main Theorem: Case 1.- 5.4.2 Proof of AAK´s Main Theorem: Case 2.- 5.4.3 Proof of AAK´s Main Theorem: Case 3.- Problems.- 6. H?-Optimization and System Reduction for MIMO Systems.- 6.1 Balanced Realization of MIMO Linear Systems.- 6.1.1 Lyapunov´s Equations.- 6.1.2 Balanced Realizations.- 6.2 Matrix-Valued All-Pass Transfer Functions.- 6.3 Optimal Hankel-Norm Approximation for MIMO Systems.- 6.3.1 Preliminary Results.- 6.3.2 Matrix-Valued Extensions of the Nehari and AAK Theorems.- 6.3.3 Derivation of Results.- Problems.- References.- Further Reading.- List of Symbols.
Discrete H¿ Optimization is concerned with the study of
H¿ optimization for digital signal processing and discrete-time control systems. The first three chapters present the basic theory and standard methods in digital filtering and systems from the frequency-domain approach, followed by a discussion of the general theory of approximation in Hardy spaces. AAK theory is introduced, first for finite-rank operators and then more generally, before being extended to the multi-input/multi-output setting. This mathematically rigorous book is self-contained and suitable for self-study. The advanced mathermatical results derived here are applicabel to digital control systems and digital filtering.
1. Digital Signals and Digital Filters.- 1.1 Analog and Digital Signals.- 1.2 Time and Frequency Domains.- 1.3 z-Transforms.- 1.4 Digital Filters.- 1.5 Optimal Digital Filter Design Criteria.- Problems.- 2. Linear Systems.- 2.1 State-Space Deillegalscriptions.- 2.2 Transfer Matrices and Minimal Realization.- 2.3 SISO Linear Systems.- 2.4 Sensitivity and Feedback Systems.- Problems.- 3. Approximation in Hardy Spaces.- 3.1 Hardy Space Preliminaries.- 3.2 Least-Squares Approximation.- 3.3 Minimum-Norm Interpolation.- 3.4 Nevanlinna-Pick Interpolation.- Problems.- 4. Optimal Hankel-Norm Approximation and H?-Minimization.- 4.1 The Nehari Theorem and Related Results.- 4.2 s-Numbers and Schmidt Pairs.- 4.3 System Reduction.- 4.4 H?-Minimization.- Problems.- 5. General Theory of Optimal Hankel-Norm Approximation.- 5.1 Existence and Preliminary Results.- 5.2 Uniqueness of Schmidt Pairs.- 5.3 The Greatest Common Divisor: The Inner Function ?I0(z).- 5.4 AAK's Main Theorem on Best Hankel-Norm Approximation.- Problems.- 6. H?-Optimization and System Reduction for MIMO Systems.- 6.1 Balanced Realization of MIMO Linear Systems.- 6.2 Matrix-Valued All-Pass Transfer Functions.- 6.3 Optimal Hankel-Norm Approximation for MIMO Systems.- Problems.- References.- Further Reading.- List of Symbols.
Inhaltsverzeichnis
1. Digital Signals and Digital Filters.- 1.1 Analog and Digital Signals.- 1.1.1 Band-Limited Analog Signals.- 1.1.2 Digital Signals and the Sampling Theorem.- 1.2 Time and Frequency Domains.- 1.2.1 Fourier Transforms and Convolutions on Three Basic Groups.- 1.2.2 Frequency Spectra of Digital Signals.- 1.3 z-Transforms.- 1.3.1 Properties of the z-Transform.- 1.3.2 Causal Digital Signals.- 1.3.3 Initial Value Problems.- 1.3.4 Singular and Analytic Discrete Fourier Transforms.- 1.4 Digital Filters.- 1.4.1 Basic Properties of Digital Filters.- 1.4.2 Transfer Functions and IIR Digital Filters.- 1.5 Optimal Digital Filter Design Criteria.- 1.5.1 An Interpolation Method.- 1.5.2 Ideal Filter Characteristics.- 1.5.3 Optimal IIR Filter Design Criteria.- Problems.- 2. Linear Systems.- 2.1 State-Space Deillegalscriptions.- 2.1.1 An Example of Flying Objects.- 2.1.2 Properties of Linear Time-Invariant Systems.- 2.1.3 Properties of State-Space Deillegalscriptions.- 2.2 Transfer Matrices and Minimal Realization.- 2.2.1 Transfer Matrices of Linear Time-Invariant Systems.- 2.2.2 Minimal Realization of Linear Systems.- 2.3 SISO Linear Systems.- 2.3.1 Kronecker's Theorem.- 2.3.2 Minimal Realization of SISO Linear Systems.- 2.3.3 System Reduction.- 2.4 Sensitivity and Feedback Systems.- 2.4.1 Plant Sensitivity.- 2.4.2 Feedback Systems and Output Sensitivity.- 2.4.3 Sensitivity Minimization.- Problems.- 3. Approximation in Hardy Spaces.- 3.1 Hardy Space Preliminaries.- 3.1.1 Definition of Hardy Space Norms.- 3.1.2 Inner and Outer Functions.- 3.1.3 The Hausdorff-Young Inequalities.- 3.2 Least-Squares Approximation.- 3.2.1 Beurling's Approximation Theorem.- 3.2.2 An All-Pole Filter Design Method.- 3.2.3 A Pole-Zero Filter Design Method.- 3.2.4 A Stabilization Procedure.- 3.3 Minimum-Norm Interpolation.- 3.3.1 Statement of the Problem.- 3.3.2 Extremal Kernels and Generalized Extremal Functions.- 3.3.3 An Application to Minimum-Norm Interpolation.- 3.3.4 Suggestions for Computation of Solutions.- 3.4 Nevanlinna-Pick Interpolation.- 3.4.1 An Interpolation Theorem.- 3.4.2 Nevanlinna-Pick's Theorem and Pick's Algorithm.- 3.4.3 Verification of Pick's Algorithm.- Problems.- 4. Optimal Hankel-Norm Approximation and H?-Minimization.- 4.1 The Nehari Theorem and Related Results.- 4.1.1 Nehari's Theorem.- 4.1.2 The AAK Theorem and Optimal Hankel-Norm Approximations.- 4.2 s-Numbers and Schmidt Pairs.- 4.2.1 Adjoint and Normal Operators.- 4.2.2 Singular Values of Hankel Matrices.- 4.2.3 Schmidt Series Representation of Compact Operators.- 4.2.4 Approximation of Compact Hankel Operators.- 4.3 System Reduction.- 4.3.1 Statement of the AAK's Theorem.- 4.3.2 Proof of the AAK Theorem for Finite-Rank Hankel Matrices.- 4.3.3 Reformulation of AAK's Result.- 4.4 H?-Minimization.- 4.4.1 Statement of the Problem.- 4.4.2 An Example of H?-Minimization.- 4.4.3 Existence, Uniqueness, and Construction of Optimal Solutions.- Problems.- 5. General Theory of Optimal Hankel-Norm Approximation.- 5.1 Existence and Preliminary Results.- 5.1.1 Solvability of the Best Approximation Problem.- 5.1.2 Characterization of the Bounded Operators that Commute with the Shift Operator.- 5.1.3 Beurling's Theorem.- 5.1.4 Operator Norms of Hankel Matrices in Terms of Inner and Outer Factors.- 5.1.5 Properties of the Norm of Hankel Matrices.- 5.2 Uniqueness of Schmidt Pairs.- 5.2.1 Uniqueness of Ratios of Schmidt Pairs.- 5.2.2 Hankel Operators Generated by Schmidt Pairs.- 5.3 The Greatest Common Divisor: The Inner Function ?I0(z).- 5.3.1 Basic Properties of the Inner Function ?I0(z).- 5.3.2 Relations Between Dimensions and Degrees.- 5.3.3 Relations Between ?I0(z) and s-Numbers.- 5.4 AAK's Main Theorem on Best Hankel-Norm Approximation.- 5.4.1 Proof of AAK's Main Theorem: Case 1.- 5.4.2 Proof of AAK's Main Theorem: Case 2.- 5.4.3 Proof of AAK's Main Theorem: Case 3.- Problems.- 6. H?-Optimization and System Reduction for MIMO Systems.- 6.1 Balanced Realization of MIMO Linear Systems.- 6.1.1 Lyapunov's Equations.- 6.1.2 Balanced Realizations.- 6.2 Matrix-Valued All-Pass Transfer Functions.- 6.3 Optimal Hankel-Norm Approximation for MIMO Systems.- 6.3.1 Preliminary Results.- 6.3.2 Matrix-Valued Extensions of the Nehari and AAK Theorems.- 6.3.3 Derivation of Results.- Problems.- References.- Further Reading.- List of Symbols.
Klappentext
Discrete H¿ Optimization is concerned with the study of H¿ optimization for digital signal processing and discrete-time control systems. The first three chapters present the basic theory and standard methods in digital filtering and systems from the frequency-domain approach, followed by a discussion of the general theory of approximation in Hardy spaces. AAK theory is introduced, first for finite-rank operators and then more generally, before being extended to the multi-input/multi-output setting. This mathematically rigorous book is self-contained and suitable for self-study. The advanced mathematical results derived here are applicable to digital control systems and digital filtering.
This book discusses in detail H¿ optimization for digital signal processing and discrete-time control systems. The rigorous derivations of the results, many examples and the discussion of research problems, together with the fact that it is self-contained, make it suitable for self-study. It provides readers with sufficient background in the subject to enable them to begin research in this field.
