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E-grāmata: Linear Stochastic Systems: A Geometric Approach to Modeling, Estimation and Identification

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  • Formāts: PDF+DRM
  • Sērija : Series in Contemporary Mathematics 1
  • Izdošanas datums: 24-Apr-2015
  • Izdevniecība: Springer-Verlag Berlin and Heidelberg GmbH & Co. K
  • Valoda: eng
  • ISBN-13: 9783662457504
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Linear Stochastic Systems: A Geometric Approach to Modeling, Estimation and IdentificationPalielināt
  • Formāts: PDF+DRM
  • Sērija : Series in Contemporary Mathematics 1
  • Izdošanas datums: 24-Apr-2015
  • Izdevniecība: Springer-Verlag Berlin and Heidelberg GmbH & Co. K
  • Valoda: eng
  • ISBN-13: 9783662457504
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This book presents a treatise on the theory and modeling of second-order stationary processes, including an exposition on selected application areas that are important in the engineering and applied sciences. The foundational issues regarding stationary processes dealt with in the beginning of the book have a long history, starting in the 1940s with the work of Kolmogorov, Wiener, Cramér and his students, in particular Wold, and have since been refined and complemented by many others. Problems concerning the filtering and modeling of stationary random signals and systems have also been addressed and studied, fostered by the advent of modern digital computers, since the fundamental work of R.E. Kalman in the early 1960s. The book offers a unified and logically consistent view of the subject based on simple ideas from Hilbert space geometry and coordinate-free thinking. In this framework, the concepts of stochastic state space and state space modeling, based on the notion of the conditional independence of past and future flows of the relevant signals, are revealed to be fundamentally unifying ideas. The book, based on over 30 years of original research, represents a valuable contribution that will inform the fields of stochastic modeling, estimation, system identification, and time series analysis for decades to come. It also provides the mathematical tools needed to grasp and analyze the structures of algorithms in stochastic systems theory.

Recenzijas

The purpose of this book is to present the mathematical background necessary for understanding the linear state-space modeling of second-order random processes and its applications to estimation and identification theory. this monograph is an excellent reference for researchers interested in geometric theory of stochastic realization and its applications. (Viorica M. Ungureanu, Mathematical Reviews, January, 2016)

1 Introduction
1(24)
1.1 Geometric Theory of Stochastic Realization
2(8)
1.1.1 Markovian Splitting Subspaces
4(1)
1.1.2 Observability, Constructibility and Minimality
4(2)
1.1.3 Fundamental Representation Theorem
6(2)
1.1.4 Predictor Spaces and Partial Ordering
8(1)
1.1.5 The Frame Space
9(1)
1.1.6 Generalizations
10(1)
1.2 Spectral Factorization and Uniformly Chosen Bases
10(6)
1.2.1 The Linear Matrix Inequality and Hankel Factorization
11(2)
1.2.2 Minimality
13(1)
1.2.3 Rational Covariance Extension
13(1)
1.2.4 Uniform Choice of Bases
14(1)
1.2.5 The Matrix Riccati Equation
15(1)
1.3 Applications
16(5)
1.3.1 Smoothing
16(1)
1.3.2 Interpolation
17(1)
1.3.3 Subspace Identification
17(2)
1.3.4 Balanced Model Reduction
19(2)
1.4 An Brief Outline of the Book
21(2)
1.5 Bibliographical Notes
23(2)
2 Geometry of Second-Order Random Processes
25(40)
2.1 Hilbert Space of Second-Order Random Variables
25(2)
2.1.1 Notations and Conventions
26(1)
2.2 Orthogonal Projections
27(6)
2.2.1 Linear Estimation and Orthogonal Projections
28(3)
2.2.2 Facts About Orthogonal Projections
31(2)
2.3 Angles and Singular Values
33(5)
2.3.1 Canonical Correlation Analysis
36(2)
2.4 Conditional Orthogonality
38(2)
2.5 Second-Order Processes and the Shift Operator
40(4)
2.5.1 Stationarity
42(2)
2.6 Conditional Orthogonality and Modeling
44(14)
2.6.1 The Markov Property
44(3)
2.6.2 Stochastic Dynamical Systems
47(2)
2.6.3 Factor Analysis
49(6)
2.6.4 Conditional Orthogonality and Covariance Selection
55(2)
2.6.5 Causality and Feedback-Free Processes
57(1)
2.7 Oblique Projections
58(4)
2.7.1 Computing Oblique Projections in the Finite-Dimensional Case
61(1)
2.8 Stationary Increments Processes in Continuous Time
62(1)
2.9 Bibliographical Notes
63(2)
3 Spectral Representation of Stationary Processes
65(38)
3.1 Orthogonal-Increments Processes and the Wiener Integral
65(5)
3.2 Harmonic Analysis of Stationary Processes
70(3)
3.3 The Spectral Representation Theorem
73(6)
3.3.1 Connections to the Classical Definition of Stochastic Fourier Transform
75(2)
3.3.2 Continuous-Time Spectral Representation
77(1)
3.3.3 Remark on Discrete-Time White Noise
78(1)
3.3.4 Real Processes
78(1)
3.4 Vector-Valued Processes
79(3)
3.5 Functionals of White Noise
82(6)
3.5.1 The Fourier Transform
85(3)
3.6 Spectral Representation of Stationary Increment Processes
88(3)
3.7 Multiplicity and the Module Structure of H(y)
91(9)
3.7.1 Definition of Multiplicity and the Module Structure of H(y)
92(3)
3.7.2 Bases and Spectral Factorization
95(4)
3.7.3 Processes with an Absolutely Continuous Distribution Matrix
99(1)
3.8 Bibliographical Notes
100(3)
4 Innovations, Wold Decomposition, and Spectral Factorization
103(50)
4.1 The Wiener-Kolmogorov Theory of Filtering and Prediction
103(7)
4.1.1 The Role of the Fourier Transform and Spectral Representation
104(1)
4.1.2 Acausal and Causal Wiener Filters
105(3)
4.1.3 Causal Wiener Filtering
108(2)
4.2 Orthonormalizable Processes and Spectral Factorization
110(5)
4.3 Hardy Spaces
115(3)
4.4 Analytic Spectral Factorization
118(1)
4.5 The Wold Decomposition
119(10)
4.5.1 Reversibility
127(2)
4.6 The Outer Spectral Factor
129(9)
4.6.1 Invariant Subspaces and the Factorization Theorem
131(4)
4.6.2 Inner Functions
135(1)
4.6.3 Zeros of Outer Functions
136(2)
4.7 Toeplitz Matrices and the Szego Formula
138(12)
4.7.1 Algebraic Properties of Toeplitz Matrices
146(4)
4.8 Bibliographical Notes
150(3)
5 Spectral Factorization in Continuous Time
153(22)
5.1 The Continuous-Time Wold Decomposition
153(1)
5.2 Hardy Spaces of the Half-Plane
154(5)
5.3 Analytic Spectral Factorization in Continuous Time
159(4)
5.3.1 Outer Spectral Factors in W2
160(3)
5.4 Wide Sense Semimartingales
163(5)
5.4.1 Stationary Increments Semimartingales
166(2)
5.5 Stationary Increments Semimartingales in the Spectral Domain
168(6)
5.5.1 Proof of Theorem 5.4.4
171(1)
5.5.2 Degenerate Stationary Increments Processes
172(2)
5.6 Bibliographical Notes
174(1)
6 Linear Finite-Dimensional Stochastic Systems
175(40)
6.1 Stochastic State Space Models
175(4)
6.2 Anticausal State Space Models
179(4)
6.3 Generating Processes and the Structural Function
183(3)
6.4 The Idea of State Space and Coordinate-Free Representation
186(2)
6.5 Observability, Constructibility and Minimality
188(3)
6.6 The Forward and the Backward Predictor Spaces
191(5)
6.7 The Spectral Density and Analytic Spectral Factors
196(8)
6.7.1 The Converse Problem
198(6)
6.8 Regularity
204(3)
6.9 The Riccati Inequality and Kalman Filtering
207(6)
6.10 Bibliographic Notes
213(2)
7 The Geometry of Splitting Subspaces
215(36)
7.1 Deterministic Realization Theory Revisited: The Abstract Idea of State Space Construction
215(2)
7.2 Perpendicular Intersection
217(3)
7.3 Splitting Subspaces
220(5)
7.4 Markovian Splitting Subspaces
225(7)
7.5 The Markov Semigroup
232(2)
7.6 Minimality and Dimension
234(4)
7.7 Partial Ordering of Minimal Splitting Subspaces
238(12)
7.7.1 Uniform Choices of Bases
240(3)
7.7.2 Ordering and Scattering Pairs
243(3)
7.7.3 The Tightest Internal Bounds
246(4)
7.8 Bibliographic Notes
250(1)
8 Markovian Representations
251(62)
8.1 The Fundamental Representation Theorems
252(5)
8.2 Normality, Properness and the Markov Semigroup
257(5)
8.3 The Forward and Backward Systems (The Finite-Dimensional Case)
262(4)
8.4 Reachability, Controllability and the Deterministic Subspace
266(10)
8.5 Markovian Representation of Purely Deterministic Processes
276(5)
8.6 Minimality and Nonminimality of Finite-Dimensional Models
281(3)
8.7 Parameterization of Finite-Dimensional Minimal Markovian Representations
284(6)
8.8 Regularity of Markovian Representations
290(4)
8.9 Models Without Observation Noise
294(3)
8.10 The Forward and Backward Systems (The General Case)
297(14)
8.10.1 State-Space Isomorphisms and the Infinite-Dimensional Positive-Real-Lemma Equations
306(3)
8.10.2 More About Regularity
309(1)
8.10.3 Models Without Observation Noise
310(1)
8.11 Bibliographical Notes
311(2)
9 Proper Markovian Representations in Hardy Space
313(42)
9.1 Functional Representations of Markovian Representations
313(13)
9.1.1 Spectral Factors and the Structural Function
315(2)
9.1.2 The Inner Triplet of a Markovian Representation
317(2)
9.1.3 State Space Construction
319(4)
9.1.4 The Restricted Shift
323(3)
9.2 Minimality of Markovian Representations
326(13)
9.2.1 Spectral Representation of the Hankel Operators
328(3)
9.2.2 Strictly Noncyclic Processes and Properness
331(2)
9.2.3 The Structural Functions of Minimal Markovian Representations
333(4)
9.2.4 A Geometric Conditions for Minimality
337(2)
9.3 Degeneracy
339(9)
9.3.1 Regularity, Singularity, and Degeneracy of the Error Spaces
340(2)
9.3.2 Degenerate Processes
342(4)
9.3.3 Some Examples
346(2)
9.4 Coercivity Revisited
348(2)
9.5 Models Without Observation Noise
350(3)
9.6 Bibliographical Notes
353(2)
10 Stochastic Realization Theory in Continuous Time
355(58)
10.1 Continuous-Time Stochastic Models
355(6)
10.1.1 Minimality and Nonminimality of Models
356(2)
10.1.2 The Idea of State Space and Markovian Representations
358(2)
10.1.3 Modeling Stationary Processes
360(1)
10.2 Markovian Representations
361(15)
10.2.1 State Space Construction
363(6)
10.2.2 Spectral Factors and the Structural Function
369(4)
10.2.3 From Spectral Factors to Markovian Representations
373(3)
10.3 Forward and Backward Realizations for Finite-Dimensional Markovian Representations
376(10)
10.4 Spectral Factorization and Kalman Filtering
386(11)
10.4.1 Uniform Choice of Bases
387(1)
10.4.2 Spectral Factorization, the Linear Matrix Inequality and set T
388(5)
10.4.3 The Algebraic Riccati Inequality
393(1)
10.4.4 Kalman Filtering
394(3)
10.5 Forward and Backward Stochastic Realizations (The General Case)
397(14)
10.5.1 Forward State Representation
398(5)
10.5.2 Backward State Representation
403(3)
10.5.3 Stochastic Realizations of a Stationary Process
406(3)
10.5.4 Stochastic Realizations of a Stationary-Increment Process
409(2)
10.6 Bibliographical Notes
411(2)
11 Stochastic Balancing and Model Reduction
413(50)
11.1 Canonical Correlation Analysis and Stochastic Balancing
414(11)
11.1.1 Observability and Constructibility Gramians
416(4)
11.1.2 Stochastic Balancing
420(1)
11.1.3 Balanced Stochastic Realizations
421(4)
11.2 Stochastically Balanced Realizations from the Hankel Matrix
425(5)
11.3 Basic Principles of Stochastic Model Reduction
430(9)
11.3.1 Stochastic Model Approximation
432(5)
11.3.2 Relations to the Maximum Likelihood Criterion
437(2)
11.4 Prediction-Error Approximation in Restricted Model Classes
439(1)
11.5 Relative Error Minimization in H∞
440(12)
11.5.1 A Short Review of Hankel Norm Approximation
441(6)
11.5.2 Relative Error Minimization
447(5)
11.6 Stochastically Balanced Truncation
452(9)
11.6.1 The Continuous-Time Case
453(2)
11.6.2 The Discrete-Time Case
455(4)
11.6.3 Balanced Discrete-Time Model Reduction
459(2)
11.7 Bibliographical Notes
461(2)
12 Finite-Interval and Partial Stochastic Realization Theory
463(44)
12.1 Markovian Representations on a Finite Interval
464(4)
12.2 Kalman Filtering
468(8)
12.2.1 The Invariant Form of the Kalman Filter
470(1)
12.2.2 A Fast Kalman Filtering Algorithm
471(5)
12.3 Realizations of the Finite-Interval Predictor Spaces
476(4)
12.4 Partial Realization Theory
480(11)
12.4.1 Partial Realization of Covariance Sequences
480(2)
12.4.2 Hankel Factorization of Finite Covariance Sequences
482(4)
12.4.3 Coherent Bases in the Finite-Interval Predictor Spaces
486(2)
12.4.4 Finite-Interval Realization by Canonical Correlation Analysis
488(3)
12.5 The Rational Covariance Extension Problem
491(14)
12.5.1 The Maximum-Entropy Solution
492(4)
12.5.2 The General Case
496(5)
12.5.3 Determining P from Logarithmic Moments
501(4)
12.6 Bibliographical Notes
505(2)
13 Subspace Identification of Time Series
507(36)
13.1 The Hilbert Space of a Second-Order Stationary Time Series
508(5)
13.2 The Geometric Framework with Finite Data
513(2)
13.3 Principles of Subspace Identification
515(11)
13.3.1 Coherent Factorizations of Sample Hankel Matrices
516(3)
13.3.2 Approximate Partial Realization
519(1)
13.3.3 Approximate Finite-Interval Stochastic Realization
520(2)
13.3.4 Estimating B and D (The Purely Nondeterministic Case)
522(2)
13.3.5 Estimating B and D (The General Case)
524(2)
13.3.6 LQ Factorization in Subspace Identification
526(1)
13.4 Consistency of Subspace Identification Algorithms
526(15)
13.4.1 The Data Generating System
527(3)
13.4.2 The Main Consistency Result
530(1)
13.4.3 Convergence of the Sample Covariances
531(4)
13.4.4 The Convergence of (AN, CN, CN, AN(0))
535(3)
13.4.5 The Ergodic Case
538(1)
13.4.6 Concluding the Proof of Theorem 13.4.6
539(1)
13.4.7 On Order Estimation
540(1)
13.5 Bibliographical Notes
541(2)
14 Zero Dynamics and the Geometry of the Riccati Inequality
543(48)
14.1 The Zero Structure of Minimal Spectral Factors (The Regular Case)
543(23)
14.1.1 The Discrete-Time Regular Case
548(9)
14.1.2 The Continuous-Time Case
557(8)
14.1.3 Zero Dynamics and Geometric Control Theory
565(1)
14.2 Zero Dynamics in the General Discrete-Time Case
566(13)
14.2.1 Output-Induced Subspaces
567(7)
14.2.2 Invariant Directions
574(5)
14.3 The Local Frame Space
579(4)
14.3.1 The Geometric Problem
579(2)
14.3.2 The Tightest Local Frame
581(2)
14.4 Invariant Subspaces and the Algebraic Riccati Inequality
583(7)
14.5 Bibliographical Notes
590(1)
15 Smoothing and Interpolation
591(46)
15.1 Smoothing in Discrete Time
592(5)
15.1.1 The Frame Space
594(1)
15.1.2 Two-Filter Formulas
595(1)
15.1.3 Order Reduction in the Nonregular Case
596(1)
15.2 Finite-Interval Realization Theory for Continuous-Time Systems
597(10)
15.2.1 Time-Reversal of the State Equations
598(4)
15.2.2 Forward and Backward Stochastic Realizations
602(5)
15.3 Smoothing in Continuous Time (The General Case)
607(5)
15.3.1 Basic Representation Formulas
607(3)
15.3.2 Mayne-Fraser Two-Filter Formula
610(1)
15.3.3 The Smoothing Formula of Bryson and Frazier
610(1)
15.3.4 The Smoothing Formula of Rauch, Tung and Striebel
611(1)
15.4 Steady-State Smoothers in Continuous-Time
612(8)
15.4.1 The Two-Filter Formula
613(2)
15.4.2 Reduced-Order Smoothing
615(5)
15.5 Steady-State Smoothers in Discrete-Time
620(8)
15.5.1 The Two-Filter Formula
622(2)
15.5.2 Reduced Order Smoothing
624(4)
15.6 Interpolation
628(8)
15.6.1 State Interpolation
628(5)
15.6.2 Output Interpolation
633(3)
15.7 Bibliographical Notes
636(1)
16 Acausal Linear Stochastic Models and Spectral Factorization
637(38)
16.1 Acausal Stochastic Systems
637(2)
16.2 Rational Spectral Factorization
639(5)
16.3 Duality and Rational All-Pass Functions
644(8)
16.3.1 Rational All-Pass Functions
647(2)
16.3.2 Generalizing the Concept of Structural Function
649(3)
16.4 Equivalent Representations of Markovian Splitting Subspaces
652(5)
16.4.1 Invariance with Respect to Duality
652(1)
16.4.2 Invariance with Respect to Pole Structure
653(3)
16.4.3 Invariance with Respect to Zero Structure
656(1)
16.5 The Riccati Inequality and the Algebraic Riccati Equation
657(10)
16.5.1 Zeros of the Spectral Density
659(2)
16.5.2 Zero Flipping by Feedback in Minimal Stochastic Realizations
661(1)
16.5.3 Partial Ordering of the Set P
662(2)
16.5.4 The Solution Set P0 of the Algebraic Riccati Equation
664(3)
16.5.5 Zeros on the Unit Circle Only
667(1)
16.6 Equivalent Representations of Stochastic Realizations, Continued
667(5)
16.6.1 The Structure of Rational All-Pass Functions
670(2)
16.7 Bibliographic Notes
672(3)
17 Stochastic Systems with Inputs
675(50)
17.1 Causality and Feedback
676(4)
17.2 Oblique Markovian Splitting Subspaces
680(4)
17.2.1 Coordinate-Free Representation of Stochastic Systems with Inputs
680(4)
17.3 State Space Construction from Basic Geometric Principles
684(11)
17.3.1 One-Step-Ahead Oblique Markovian Splitting Subspaces
688(3)
17.3.2 The Oblique Predictor Space
691(4)
17.4 Geometric Theory in the Absence of Feedback
695(15)
17.4.1 Feedback-Free Oblique Splitting Subspaces
697(1)
17.4.2 Observability, Constructibility and Minimality
698(4)
17.4.3 The Feedback-Free Oblique Predictor Space
702(1)
17.4.4 Extended Scattering Pairs
703(4)
17.4.5 Stochastic and Deterministic Minimality
707(3)
17.5 Applications to Subspace Identification
710(14)
17.5.1 The Basic Idea of Subspace Identification
711(2)
17.5.2 Finite-Interval Identification
713(4)
17.5.3 The N4SID Algorithm
717(4)
17.5.4 Conditioning in Subspace Identification
721(1)
17.5.5 Subspace Identification with Feedback
721(3)
17.6 Bibliographical Notes
724(1)
A Basic Principles of Deterministic Realization Theory
725(12)
A.1 Realization Theory
725(8)
A.1.1 The Hankel Factorization
727(1)
A.1.2 Solving the Realization Problem
728(5)
A.2 Balancing
733(2)
A.3 Bibliographical Notes
735(2)
B Some Topics in Linear Algebra and Hilbert Space Theory
737(24)
B.1 Some Facts from Linear Algebra and Matrix Theory
737(13)
B.1.1 Inner Product Spaces and Matrix Norms
737(3)
B.1.2 Cholesky Factorization
740(1)
B.1.3 Sylvester's Inequality
741(1)
B.1.4 The Moore-Penrose Pseudo-inverse
741(3)
B.1.5 Connections to Least-Squares Problems
744(2)
B.1.6 Matrix Inversion Lemma
746(1)
B.1.7 Logarithm of a Matrix
747(1)
B.1.8 Lyapunov Equations
747(2)
B.1.9 Inertia Theorems
749(1)
B.2 Hilbert Spaces
750(6)
B.2.1 Operators and Their Adjoints
753(3)
B.3 Subspace Algebra
756(3)
B.3.1 The Shift Acting on Subspaces
758(1)
B.4 Bibliographical Notes
759(2)
Bibliography 761(14)
Index 775
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