Discrete H [∞] optimization : with applications in signal processing and control systems
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Bibliographic Information
Discrete H [∞] optimization : with applications in signal processing and control systems
（Springer series in information sciences, 26）
Springer, 1997
2nd ed
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Note
Enl. ed. of: Signal processing and systems theory, 1992
Includes bibliographical references and index
Description and Table of Contents
Description
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.
Table of Contents
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 Descriptions.- 2.1.1 An Example of Flying Objects.- 2.1.2 Properties of Linear Time-Invariant Systems.- 2.1.3 Properties of State-Space Descriptions.- 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.
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