Linear state/signal systems:
"The theory presented in this book arose as a product of a continued collaboration between the two authors during the years 2003-2021. The basis for this collaboration was our common interest in passive linear time-invariant input/state/output systems theory. At the time when this project start...
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| Format: | Buch |
| Sprache: | English |
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Cambridge, United Kingdom
Cambridge University Press
2022
|
| Schriftenreihe: | Encyclopedia of mathematics and its applications
183 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Zusammenfassung: | "The theory presented in this book arose as a product of a continued collaboration between the two authors during the years 2003-2021. The basis for this collaboration was our common interest in passive linear time-invariant input/state/output systems theory. At the time when this project started O. Staffans was preparing a joint article (Ball and Staffans, 2006) with Prof. J. Ball which, in particular, explored the connections between conservative input/state/output systems theory on one hand and some results in the behavioral theory introduced by J.Willems in the late 80's on the other hand. After extensive discussions on the above approach, comparing it to the theory of passive electrical networks, we understood that this opens up a new direction in the study of passive linear time-invariant systems. We called the new class of systems that arose in this way passive state/signal systems. From the outset it was clear that the notion of passivity with an arbitrary supply rate fits more naturally into the state/signal setting than in the input/state/output setting, and that the standard "diagonal transformation" of Livšic, the Potapov-Ginzburg transformation, and the Redheffer and chain scattering transformations have natural interpretations as transformations between input/output resolvents of different input/state/output representations of a passive state/signal system. We also soon discovered that virtually all the standard control theory notions, such as controllability and observability, minimality, stability, stabilizability, detectability, and well-posedness have natural state/signal counterparts"-- |
| Beschreibung: | Includes bibliographical references and index |
| Beschreibung: | xxviii, 1019 Seiten |
| ISBN: | 9781316519677 |
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| 490 | 1 | |a Encyclopedia of mathematics and its applications |v 183 | |
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| 520 | 3 | |a "The theory presented in this book arose as a product of a continued collaboration between the two authors during the years 2003-2021. The basis for this collaboration was our common interest in passive linear time-invariant input/state/output systems theory. At the time when this project started O. Staffans was preparing a joint article (Ball and Staffans, 2006) with Prof. J. Ball which, in particular, explored the connections between conservative input/state/output systems theory on one hand and some results in the behavioral theory introduced by J.Willems in the late 80's on the other hand. After extensive discussions on the above approach, comparing it to the theory of passive electrical networks, we understood that this opens up a new direction in the study of passive linear time-invariant systems. We called the new class of systems that arose in this way passive state/signal systems. From the outset it was clear that the notion of passivity with an arbitrary supply rate fits more naturally into the state/signal setting than in the input/state/output setting, and that the standard "diagonal transformation" of Livšic, the Potapov-Ginzburg transformation, and the Redheffer and chain scattering transformations have natural interpretations as transformations between input/output resolvents of different input/state/output representations of a passive state/signal system. We also soon discovered that virtually all the standard control theory notions, such as controllability and observability, minimality, stability, stabilizability, detectability, and well-posedness have natural state/signal counterparts"-- | |
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Datensatz im Suchindex
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|---|---|
| adam_text | ENCYCLOPEDIA OF MATHEMATICS AND ITS APPLICATIONS
Linear State/Signal Systems
DAMIR Z AROV
South Ukrainian National Pedagogical University
OLOFJ STAFFANS
Abo Akademi University, Finland
Contents
Preface page xxi
List of Notations xx
1 Introduction and Overview 1
1 1 Linear Time-Invariant Dynamical Systems 1
111 State Systems 2
112 Systems That Interact with the Outside World 3
113 Input/State/Output Systems 3
114 ‘Input/Output Systems 4
115 Classical (Sub)networks 5
116_ Port-Hamiltonian Systems 5
117 Behavioral Systems 6
118 State/Signal Systems 7
119 State/Signal versus I/S/O Systems 8
1 1 10 Frequency Domain Systems 10
1 1 11 Boundary Triplets il
1 1 12 State/Signal versus Behavioral Systems 11
1 1 13 How to Read This Book 12
1 1 14 H-Spaces 12
1 1 15 Where to Go from Here? 13
1 2 An Overview of State/Signal and Input/State/Output Systems 14
121 Input/State/Output Systems 14
122 Well-Posed I/S/O Systems 17
1 23 State/Signal Systems 18
124 1/S/O Representations 19
125 Similarity of I/S/O and State/Signal Systems 20
126 Input/Output Invariant Properties of I/S/O Systems 23
127 Properties of I/S/O Systems in the State/Signal Sense 25
vi
Contents
128 Static Transformations of S/O and State/Signal
Systems
129_ Invariant Subspaces of I/S/O and State/Signal
Systems
1 2 10 Interconnections of 1/S/O and State/Signal Systems
1 2 11 External Characteristics of /S/O and State/Signal
Systems
1 2 12 Restrictions, Projections, and Compressions
1 2 13 YS/O and State/Signal Systems in Discrete Time
1 2 14 The Resolvent Matrix of an /S/O System
1 2 15 The Resolvent Set and the Characteristic Bundles of
State/Signal Systems
1 2 16 Well-Posed /S/O and State/Signal Systems
in the Frequency Domain
1 2 17 General Resolvable Frequency Domain I/S/O and
State/Signal Systems
1 2 18 Dual and Adjoint /S/O and State/Signal Systems
1 2 19 Passive /S/O and State/Signal Systems
1 2 20 Passive Finite-Dimensional Electrical n-Ports
1 2 21 Some Finite-Dimensional Passive 2-Ports
1 2 22 Some Distributed Parameter Passive Systems
Notes and Comments
State/Signal Systems: Trajectories, Transformations, and
Interconnections
State/Signal Nodes and State/Signal Systems
211 The State/Signal System and Its Trajectories
212 Regular and Semiregular State/Signal Nodes
213 Kernel and Image Representations of Closed
State/Signal Nodes
21 4 Bounded State/Signal Nodes and Systems
Some Basic Transformations of State/Signal Nodes
221 Similarity of Two State/Signal Nodes
222 Time Reflection of a State/Signal Node
223 Time Rescaling of a State/Signal Node
224 Exponentially Weighted State/Signal Nodes
Properties of Trajectories of State/Signal Systems
231 Classical, Generalized, and Mild Trajectories
232 Existence and Uniqueness of Trajectories
3]
2 4
Contents
233 Connections between Classical, Generalized, and
Mild Trajectories
Some Additional Transformations of State/Signal Nodes
241 The (P, Q)-Image of a State/Signal Node
2 42 Parts and Static Projections of a State/Signal Node
2 43 Adding Inputs and Output to a State/Signal Node
Interconnections of State/Signal Nodes
251 The Cross Product of Two State/Signal Nodes
252 (P, Q)-Interconnections of State/Signal Nodes
253A Short Circuit Connection of State/Signal Nodes
254| Examples of Interconnections of State/Signal Nodes
Examples of Infinite-Dimensional State/Signal Systems
State/Signal Systems: Dynamic and Frequency Domain Properties
Signal Behaviors and Their State/Signal Realizations
311 Future Signal Behaviors
312 External Equivalence of State/Signal Systems
Dynamic Properties of State/Signal Systems
321 Controllability and Observability of State/Signal
Systems
322 Intertwinements of State/Signal Systems
323 Compressions, Restrictions, and Projections of
State/Signal Systems
324 Examples of Minimal Compressions
325 State/Signal Systems with the Continuation Property
State Systems
331A State System and Its Trajectories
332 The Homogeneous Cauchy Problem
333 Bounded State Systems and Uniformly Continuous
Groups
334 Well-Posed State Systems and Strongly Continuous
Semigroups
335 Transformations and Interconnections of State Nodes
336 Invariant Subspaces of State Nodes
337 Intertwinement of State Nodes
Frequency Domain Characteristics of State/Signal Nodes
341 The Characteristic Node Bundle
342 The Characteristic Control Bundle
3 43 The Characteristic Observation Bundle
344 The Characteristic Signal Bundle
vii
vii
Contents
345 The Characteristic Bundles of Transformed State/Signal
Systems
346 The Resolvent of a Regular State Node
347 The Resolvent Set of a State/Signal Node
Invariance with Respect to Similarities
Dual and Adjoint State/Signal Nodes and Systems
361 The Dual of a State/Signal System
362 The Duals of Some Transformed State/Signal Nodes
363 The Characteristic Bundles of Dual State/Signal Nodes
364 The Adjoint State/Signal System
Notes to Chapters 2 and 3
Input/State/Output Representations
Input/State/Output Nodes and Systems
411 Regular /S/O Nodes
412 General I/S/O Nodes and Systems
413 Kemal and Image Representations of Closed 1/S/O
Nodes
414 State, Input/State and State/Output Nodes and Systems
415 Input/State and State/Output Representations of
State/Signal Systems
416 Free Inputs and Continuously Determined Outputs
417 Existence and Uniqueness of Trajectories
418| Bounded I/S/O Nodes and Systems
Input/State/Output Representations of State/Signal Nodes and
Systems
4 21 The State/Signal Node Induced by an /S/O Node
422 VO Representations of the Signal Space
423 General I/S/O Representations of a State/Signal Node
4 24 Semiregular /S/O Representations of a Semiregular
State/Signal Node
425 Regular VS/O Representations of a Regular State/Signal
Node
426 Parametrization of /S/O Representations
427 Bounded I/S/O Representations of Bounded State/Signal
Nodes
428 Parametrization of Bounded I/S/O Representations
State Feedback and Output Injection Representations
431 State Feedback Representations
432 Output Injection Representations
Contents
4 4 Basic Transformations of Input/State/Output Nodes
441 Similarity of I/S/O Nodes
44 2 Time Reflection of an I/S/O Node
443 Time Rescaling of an I/S/O Node
4 44 Exponentially Weighted /S/O Nodes
4 5 Properties of Trajectories of Input/State/Output Systems
451 Basic Properties of the Sets of Classical
and Generalized Trajectories
452 Solvability and the Uniqueness Property
453 Connections between Classical, Generalized,
and Mild Trajectories
4 6 Some Simple Input/State/Output Examples
Input/State/Output Systems: Dynamic and Frequency Domain
Properties
5 1 Additional Transformations of Input/State/Output Nodes
511 Adding a Feedthrough Term to an I/S/O Node
512 Modifying Inputs and Outputs of an /S/O Node
513 The (P, R, Q)-Image of an I/S/O Node
514 Parts and Static Projections of an I/S/O Node
515 Static Output Feedback
516| Adding Inputs and Outputs to an I/S/O Node
517A Second Look at State Feedbacks and Output
Injections
Interconnections of Input/State/Output Nodes
521 The Cross Product of Two I/S/O Nodes
522 (P, R, Q)-Interconnections of I/S/O Nodes
5 23 A Short Circuit Connection of I/S/O Nodes
524 T-Junctions, Sum Junctions, and Difference
Junctions
5 25 Parallel and Difference Connections
526 Cascade Connections
527 Dynamic Feedback
528 Examples of I/S/O Interconnections
Realizations of Input/Output Behaviors
531 Future I/O Behaviors
532 External Equivalence of YS/O Systems
Dynamic Properties of Input/State/Output Systems
541 Controllability and Observability of I/S/O Systems
542 Intertwinements of I/S/O Systems
Contents
5 43 Compressions, Restrictions, and Projections of US/O
Systems 326
5 44 _US/O Systems with the Continuation Property 330
5 5 Frequency Domain Characteristics of Input/State/Output Nodes 331
551 The Characteristic Node Bundle of an I/S/O Node 331
552 The I/S/O Resolvent Matrix of an I/S/O Node 334
553 Resolvability of Transformed I/S/O Nodes 338
554 Frequency Domain I/S/O-Admissible I/O
Representations 341
5 6 The Correspondence between State/Signal and Input/State/Output
Notions 343
561 VO Invariant Notions 343
562 Properties of I/S/O Systems in the State/Signal Sense 349
5 7 Adjoint and Dual Input/State/Output Nodes and Systems 351
571 The Adjoint and the Dual of an /S/O Node 351
572 Adjoint and Dual /S/O Representations 354
573 WS/O Lagrange Identities 355
574 Properties of Adjoint and Dual I/S/O Nodes and
Systems 359
575 The Adjoints and Duals of Some Transformed 1/S/O
Nodes 360
576 The Adjoints and Duals of Some Interconnected
VS/O Nodes 364
5 8 Notes to Chapters 4 and 5 366
Bounded Input/State/Output Systems in Continuous and Discrete
Time 370
6 1 Bounded State Operators and Nodes 370
611 The Spectral Radius of a Bounded State Operator 370
612 Invariant Subspaces of Bounded State Operators
and Uniformly Continuous Groups 372
613 Parts and Projections of Bounded State Operators 373
614 Parts and Projections of Uniformly Continuous
Groups 375
615 Intertwinements of Bounded State Operators and
Uniformly Continuous Groups 377
616| Compressions of Bounded State Operators and
Uniformly Continuous Groups 379
617 The General Structure of a Compression of a
Bounded State Operator 385
6 2
Contents
613 The Adjoints of Bounded State Operators and
Uniformly Continuous Groups
Static Properties of Bounded Input/State/Output Nodes
621 Transformations of Bounded I/S/O Nodes
622 Interconnections of Bounded I/S/O Nodes
6 23 The I/S/O Resolvent Matrix of a Bounded I/S/O Node
Dynamic Properties of Bounded Input/State/Output Systems
631 Strongly Invariant and Unobservably Invariant
Subspaces
632 External Equivalence of Bounded I/S/O Systems
633 Intertwinements of Bounded I/S/O Systems
6 34 Restrictions and Projections of Bounded I/S/O Systems
635 Compressions of Bounded I/S/O Systems
636 The General Structure of a Bounded I/S/O Compression
637 Compressions into Minimal Bounded I/S/O Systems
The Adjoint and the Dual of a Bounded Input/State/Output
Node
Discrete Time Input/State/Output Systems
651 Introduction to Discrete Time I/S/O Systems
652 Properties of Discrete Time US/O Systems
653 Time Reflection of Discrete Time I/S/O Systems
6 54 Power Weightings of Discrete Time I/S/O Systems
655 Frequency Domain Shifts of Discrete Time I/S/O
Systems
656 Stable Discrete Time I/S/O Systems
657 Connections between Continuous and Discrete Time
V/S/O Properties
658| Dynamic Notions for Bounded I/S/O Nodes
Bounded Input/State/Output Realizations
661 Analyticity at Infinity of the I/S/O Resolvent Matrix
662 Existence of a Bounded I/S/O Realization
Bounded State/Signal Systems in Continuous and Discrete Time
7A
Static Properties of Bounded State/Signal Nodes
711 The I/S/O-Bounded Resolvent Set of a Bounded
State/Signal Node
712 Transformations of Bounded State/Signal Nodes
713 Resolvability of Transformations of State/Signal
Nodes
Dynamic Properties of Bounded State/Signal Systems
xi
8
Contents
Strongly Invariant and Unobservably Invariant
Subspaces
External Equivalence of Bounded State/Signal Systems
Intertwinements of Bounded State/Signal Systems
Restrictions and Projections of Bounded State/Signal
Systems
Compressions of Bounded State/Signal Systems
The General Structure of a Bounded State/Signal
Compression
Compressions into Minimal Bounded State/Signal
Systems
Bounded State/Signal Realizations
7 3 The Dual and the Adjoint of a Bounded State/Signal Node
74 Discrete Time State/Signal Systems
Introduction to Discrete Time State/Signal Systems
Properties of Discrete Time State/Signal Systems
Time Reflection of Discrete Time State/Signal Systems
Power Weightings of Discrete Time State/Signal
Systems
Frequency Domain Shifts of Discrete Time State/Signal
Systems
Stable Discrete Time State/Signal Systems
Connections between Continuous and Discrete Time
State/Signal Properties
Dynamic Notions for Bounded State/Signal Nodes
7 5 Notes to Chapters 6 and 7
Semi-bounded Input/State/Output Systems
Co Semigroups and Well-Posed State Systems
On the Resolvents of Generators of Co
Semigroups
The Inhomogeneous Cauchy Problem
Invariant Subspaces of Cp Semigroups
Parts, Projections, and Restrictions of Single-Valued
Resolvable Main Operators
Parts and Projections of Co Semigroups
Intertwinements of Cy Semigroups
Compressions of Cp Semigroups
The General Structure of a Compression of a Co
Semigroup
81 9
Contents
The Adjoint of a Co Semigroup
8 2 Semi-bounded Input/State/Output Systems
Introduction to Semi-bounded I/S/O Systems
Transformations of Semi-bounded I/S/O Nodes
Interconnections of Semi-bounded I/S/O Nodes
The I/S/O Resolvent Matrix of a Semi-bounded 1/S/O
Node
8 3 Dynamic Properties of Semi-bounded Input/State/Output
Systems
Strongly Invariant and Unobservably Invariant
Subspaces
External Equivalence of Semi-bounded I/S/O Systems
Intertwinements of Semi-bounded I/S/O Systems
Restrictions and Projections of Semi-bounded I/S/O
Systems
Compressions of Semi-bounded US/O Systems
The General Structure of a Semi-bounded I/S/O
Compression
Compressions into Minimal Semi-bounded I/S/O
Systems
8 4 The Adjoint of a Semi-bounded Input/State/Output Node
Semi-bounded State/Signal Systems
9 1 Static Properties of Semi-bounded State/Signal Nodes
Introduction to Semi-bounded State/Signal Nodes
and Systems
The I/S/O Semi-bounded Resolvent Set of a
Semi-bounded State/Signal Node
Transformations and Interconnections
of Semi-bounded State/Signal Nodes
9 2 Dynamic Properties of Semi-bounded State/Signal Systems
Strongly Invariant and Unobservably Invariant
Subspaces
External Equivalence of Semi-bounded State/Signal
Systems
Intertwinements of Semi-bounded State/Signal Systems
Restrictions and Projections of Semi-bounded
State/Signal Systems
Compressions of Semi-bounded State/Signal Systems
xiii
xiv
Contents
The General Structure of a Semi-bounded State/Signal
Compression
Compressions into Minimal Semi-bounded State/Signal
Systems
9 3 The Adjoint of a Semi-bounded State/Signal Node
9 4 Notes to Chapters 8 and 9
Resolvable Input/State/Output and State/Signal Nodes
10 1 Resolvable State Nodes
Linear Operator-Valued Pencils
The Resolvent of a State Node
The Interpolation Space of a Semiregular State Node
The Extrapolation Space of a Regular Resolvable
State Node
The Duals of the Interpolation and Extrapolation
Spaces
The Interpolation and Extrapolation Spaces of a
Semigroup Generator
10 2 Resolvable Input/State/Output Nodes
Resolvability of an I/S/O Node
Kernel and Image Representations of the YS/O
Resolvent Matrix
The I/S/O Resolvent Identity
Representations of the System Operator
Semiregular and Regular Resolvable I/S/O Nodes
The Observation and Control Operators of a Regular
Resolvable I/S/O Node
Some Examples of Regular Resolvable I/S/O Nodes
Resolvability of Transformed 1/S/O Nodes
Resolvability of Interconnected I/S/O Nodes
The Resolvent Family of Bounded /S/O Nodes
A Finite-Dimensional Nonregular Resolvable /S/O
Node
The Adjoint and the Dual of a Resolvable 1/S/O Node
10 3 Resolvable State/Signal Nodes
On the Resolvent Set of a Closed State/Signal Node
Frequency Domain I/S/O-Admissible VO
Representations
Resolvability of Transformed State/Signal Nodes
The Resolvent Family of Bounded State/Signal Nodes
11
Contents
10 3 5 The Dual and the Adjoint of a Resolvable State/Signal
System
10 4 Notes and Comments
Frequency Domain Input/State/Output Systems
Frequency Domain Input/State/Output Systems
11 1 1 Introduction to Frequency Domain US/O Systems
11 1 2 Frequency Domain Controllability and Observability
11 1 3 Frequency Domain Invariance
11 1 4 The Frequency Domain Behavior and External
Equivalence
11 1 5 Frequency Domain Intertwinements
11 1 6 Frequency Domain Compressions, Restrictions,
and Projections
11 1 7 Resolvable Frequency Domain Compressions,
Restrictions, and Projections
11 1 8 The General Structure of a Resolvable Frequency
Domain Compression
11 1 9 Compressions into 2-Minimal /S/O Systems
11 1 10 Results for Connected Frequency Domains
The Adjoint and the Dual of a Frequency Domain
Input/State/Output System
11 2 1 Frequency Domain Lagrange Identities
11 2 2 Properties of Adjoint and Dual Frequency Domain
VS/O Systems
Frequency Domain Notions for 2-Resolvable Input/State/Output
Nodes
11 3 1 Dynamic Properties of the Resolvent Family of
Bounded J/S/O Nodes
Resolvable Frequency Domain State Systems
11 4 1 Frequency Domain Invariance
11 4 2 Frequency Domain Intertwinements and
Compressions
11 4 3 Results for Connected Frequency Domains
11 4 4 Frequency Domain Duality
Notes and Comments
Frequency Domain State/Signal Systems
12 1 Frequency Domain State/Signal Systems
12 1 1 Introduction to Frequency Domain State/Signal
Systems
KV
xvi
Contents
12 1 2 Separately and Jointly /S/O Admissible Frequency
Domains
12 1 3 Frequency Domain Controllability and Observability
12 1 4 Frequency Domain Invariance
12 1 5 The Frequency Domain Behavior and External
Equivalence
12 1 6 Frequency Domain Intertwinements
12 1 7 Frequency Domain Compressions, Restrictions,
and Projections
12 1 8 Resolvable Frequency Domain Compressions,
Restrictions, and Projections
12 1 9 The General Structure of a Resolvable Frequency
Domain Compression
12 1 10 Compressions into 2-Minimal State/Signal Systems
Local Frequency Domain Notions
12 2 1 Local Frequency Domain Notions for Q-Resolvable
State/Signal Systems
12 2 2 Connected Frequency Domains
The Dual and the Adjoint of a Frequency Domain State/Signal
System
12 3 1 Frequency Domain Lagrange Identities
12 3 2 Properties of Dual and Adjoint Frequency Domain
State/Signal Systems
Frequency Domain Notions for Q-Resolvable State/Signal
Nodes
12 4 1 Dynamic Properties of the Resolvent Family of
Bounded State/Signal Nodes
Notes and Comments
Internally Well-Posed Systems
Internally Well-Posed Input/State/Output Systems
13 1 1 Basic Definitions and Properties
13 1 2 Transformations and Interconnections
Frequency-Domain Internally Well-Posed Input/State/Output
Systems
13 2 1 Frequency Domain Invariance
13 2 2 Frequency Domain Intertwinements
13 2 3 Frequency-Domain Restrictions, Projections,
and Compressions
13 2 4 The General Structure of P4(%)-Compressions
14
Contents
13 3 Internally Well-Posed State/Signal Systems
13 3 1 Basic Definitions and Properties
13 3 2 Frequency-Domain Compressions of Internally
Well-Posed State/Signal Systems
13 4 Notes and Comments
Well-Posed Input/State/Output Systems
14 1 Basic Properties of Well-Posed Input/State/Output Systems
14 1 1 The Definition of a Well-Posed VS/O System
14 1 2 Alternative Conditions for Well-Posedness
14 1 3 The Fundamental I/S/O Solution of a Well-Posed
VS/O System
14 2 The Growth Bound of a Well-Posed Input/State/Output System
14 2 1 The Growth Bound of a Weill-Posed I/S/O System
14 2 2 Stable I/S/O Systems
14 3 Resolvability of Well-Posed Input/State/Output Systems
14 3 1 Well-Posed I/S/O Systems Are Resolvable
14 3 2 Growth Estimates for the /S/O Resolvent Matrix
14 4 Realizations of Shift-Invariant Causal Linear Operators
14 4 1 Shift Invariant Causal Linear Operators
14 4 2 Realizations of Shift Invariant Causal Linear Operators
14 4 3 Toeplitz and Hankel Operators
14 5 Transformations and Interconnections of Well-Posed
Input/State/Output Systems
14 5 1 Well-Posedness and Stability of Transformed I/S/O
Systems
14 5 2 Well-Posedness and Stability of Interconnected I/S/O
Systems
14 5 3 Stabilizable and Detectable I/S/O Systems
14 6 Dynamic Properties of Well-Posed Input/State/Output Systems
14 6 1 Strongly Invariant and Unobservably Invariant
Subspaces
14 6 2 Intertwinements of Well-Posed I/S/O Systems
14 6 3 Restrictions, Projections, and Compressions
14 6 4 The General Structure of a Well-Posed I/S/O
Compression
14 6 5 Compressions Into Minimal Well-Posed I/S/O Systems
14 7 Well-Posed Input/State/Output Systems in the Frequency
Domain
14 7 1 Time and Frequency Domain External Equivalence
14 7 2 Time and Frequency Domain Invariance
xvii
xviii
Contents
14 7 3 Time and Frequency Domain Compressions and
Intertwinements
14 7 4 Frequency Domain Stability
14 8 The Adjoint of a Well-Posed Input/State/Output System
14 9 Scattering Passive Input/State/Output Systems
14 9 1 Hilbert Space /S/O Nodes and Systems
14 9 2 Scattering Passive I/S/O Systems
14 9 3 The Internal I/S/O Cayley Transform
14 9 4 The Adjoint of a Passive Scattering System
14 10 Notes and Comments
Well-Posed State/Signal Systems
15 1 Basic Properties of Well-Posed State/Signal Systems
15 1 1 Basic Definitions
15 1 2 Well-Posedness and Stability of Transformed /S/O
Systems
15 1 3 The Behaviors Induced by a Well-Posed State/Signal
System
15 1 4 The Past/Present and Present/Future Maps of a
Well-Posed State/Signal System
15 2 Stable State/Signal Systems
15 2 1 Stable State/Signal Trajectories
15 2 2 Stable State/Signal Behaviors
15 2 3 Stabilizable and Detectable State/Signal Systems
15 3 Realizations of Well-Posed Behaviors
15 3 1 Well-Posed Future, Past, and Two-Sided Behaviors
15 3 2 State/Signal Realizations of Well-Posed Behaviors
15 3 3 The Past/Future Map of a Well-Posed Behavior
15 4 Dynamic Properties of Well-Posed State/Signal Systems
15 4 1 Strongly Invariant and Unobservably Invariant
Subspaces
15 4 2 Intertwinements of Well-Posed State/Signal Systems
15 4 3 Restrictions, Projections, and Compressions of Well-
Posed State/Signal Systems
15 4 4 The General Structure of a Compression
15 4 5 Compressions into Minimal Well-Posed State/Signal
Systems
15 5 Well-Posed State/Signal Systems in the Frequency Domain
15 6 The Adjoint of a Well-Posed State/Signal Node
15 7 Passive State/Signal Systems
15 7 1
Contents
Krein Spaces
The Krein Node Space of a Scattering Passive I/S/O
System
Passive State/Signal Systems
15 8 Notes and Comments
Appendix A —_ Operators and Analytic Vector Bundles in H-Spaces
A l H-Spaces
A 1 1
A 1 2
A 1 3
A 1 4
A 1 5
A 1 6
A 1 7
A 1 8
A 19
A 1 10
A111
A 1 12
Using More than One Norm in a Vector Space
Introduction to H-Spaces
Linear Operators in H-Spaces
Closed Linear Operators in H-Spaces
Complementary Projections and Coordinate
Respresentations of H-Spaces
Isomorphisms in H-Spaces
Partial Inverses of Bounded Linear Operators
Inversion of Block Matrix Operators
The Graph Norm and Graph Topology
Linear Multivalued Operators in H-Spaces
The Single-Valued and Injective Parts of a
Multivalued Operator
On the Resolvent of a Bounded Operator
A 2 Duality in H-Spaces
A21 The Dual of an H-Space
A22 The Adjoint of a Bounded Linear Operator
A23 Duals of Product Spaces
A24 The Duals of the Components of a Direct Sum
Decomposition
A25 The Adjoint of a Linear Operator with Dense Domain
A26 The Dual of a Continuous Dense Embedding
A27 The Adjoint of a Multivalued Operator
A 3 Analytic Vector Bundles and Analytic Operator-Valued
Functions
A31 The Dual Vector Bundle
References
Index
xix
|
| adam_txt |
ENCYCLOPEDIA OF MATHEMATICS AND ITS APPLICATIONS
Linear State/Signal Systems
DAMIR Z AROV
South Ukrainian National Pedagogical University
OLOFJ STAFFANS
Abo Akademi University, Finland
Contents
Preface page xxi
List of Notations xx
1 Introduction and Overview 1
1 1 Linear Time-Invariant Dynamical Systems 1
111 State Systems 2
112 Systems That Interact with the Outside World 3
113 Input/State/Output Systems 3
114 ‘Input/Output Systems 4
115 Classical (Sub)networks 5
116_ Port-Hamiltonian Systems 5
117 Behavioral Systems 6
118 State/Signal Systems 7
119 State/Signal versus I/S/O Systems 8
1 1 10 Frequency Domain Systems 10
1 1 11 Boundary Triplets il
1 1 12 State/Signal versus Behavioral Systems 11
1 1 13 How to Read This Book 12
1 1 14 H-Spaces 12
1 1 15 Where to Go from Here? 13
1 2 An Overview of State/Signal and Input/State/Output Systems 14
121 Input/State/Output Systems 14
122 Well-Posed I/S/O Systems 17
1 23 State/Signal Systems 18
124 1/S/O Representations 19
125 Similarity of I/S/O and State/Signal Systems 20
126 Input/Output Invariant Properties of I/S/O Systems 23
127 Properties of I/S/O Systems in the State/Signal Sense 25
vi
Contents
128 Static Transformations of S/O and State/Signal
Systems
129_ Invariant Subspaces of I/S/O and State/Signal
Systems
1 2 10 Interconnections of 1/S/O and State/Signal Systems
1 2 11 External Characteristics of /S/O and State/Signal
Systems
1 2 12 Restrictions, Projections, and Compressions
1 2 13 YS/O and State/Signal Systems in Discrete Time
1 2 14 The Resolvent Matrix of an /S/O System
1 2 15 The Resolvent Set and the Characteristic Bundles of
State/Signal Systems
1 2 16 Well-Posed /S/O and State/Signal Systems
in the Frequency Domain
1 2 17 General Resolvable Frequency Domain I/S/O and
State/Signal Systems
1 2 18 Dual and Adjoint /S/O and State/Signal Systems
1 2 19 Passive /S/O and State/Signal Systems
1 2 20 Passive Finite-Dimensional Electrical n-Ports
1 2 21 Some Finite-Dimensional Passive 2-Ports
1 2 22 Some Distributed Parameter Passive Systems
Notes and Comments
State/Signal Systems: Trajectories, Transformations, and
Interconnections
State/Signal Nodes and State/Signal Systems
211 The State/Signal System and Its Trajectories
212 Regular and Semiregular State/Signal Nodes
213 Kernel and Image Representations of Closed
State/Signal Nodes
21 4 Bounded State/Signal Nodes and Systems
Some Basic Transformations of State/Signal Nodes
221 Similarity of Two State/Signal Nodes
222 Time Reflection of a State/Signal Node
223 Time Rescaling of a State/Signal Node
224 Exponentially Weighted State/Signal Nodes
Properties of Trajectories of State/Signal Systems
231 Classical, Generalized, and Mild Trajectories
232 Existence and Uniqueness of Trajectories
3]
2 4
Contents
233 Connections between Classical, Generalized, and
Mild Trajectories
Some Additional Transformations of State/Signal Nodes
241 The (P, Q)-Image of a State/Signal Node
2 42 Parts and Static Projections of a State/Signal Node
2 43 Adding Inputs and Output to a State/Signal Node
Interconnections of State/Signal Nodes
251 The Cross Product of Two State/Signal Nodes
252 (P, Q)-Interconnections of State/Signal Nodes
253A Short Circuit Connection of State/Signal Nodes
254| Examples of Interconnections of State/Signal Nodes
Examples of Infinite-Dimensional State/Signal Systems
State/Signal Systems: Dynamic and Frequency Domain Properties
Signal Behaviors and Their State/Signal Realizations
311 Future Signal Behaviors
312 External Equivalence of State/Signal Systems
Dynamic Properties of State/Signal Systems
321 Controllability and Observability of State/Signal
Systems
322 Intertwinements of State/Signal Systems
323 Compressions, Restrictions, and Projections of
State/Signal Systems
324 Examples of Minimal Compressions
325 State/Signal Systems with the Continuation Property
State Systems
331A State System and Its Trajectories
332 The Homogeneous Cauchy Problem
333 Bounded State Systems and Uniformly Continuous
Groups
334 Well-Posed State Systems and Strongly Continuous
Semigroups
335 Transformations and Interconnections of State Nodes
336 Invariant Subspaces of State Nodes
337 Intertwinement of State Nodes
Frequency Domain Characteristics of State/Signal Nodes
341 The Characteristic Node Bundle
342 The Characteristic Control Bundle
3 43 The Characteristic Observation Bundle
344 The Characteristic Signal Bundle
vii
vii
Contents
345 The Characteristic Bundles of Transformed State/Signal
Systems
346 The Resolvent of a Regular State Node
347 The Resolvent Set of a State/Signal Node
Invariance with Respect to Similarities
Dual and Adjoint State/Signal Nodes and Systems
361 The Dual of a State/Signal System
362 The Duals of Some Transformed State/Signal Nodes
363 The Characteristic Bundles of Dual State/Signal Nodes
364 The Adjoint State/Signal System
Notes to Chapters 2 and 3
Input/State/Output Representations
Input/State/Output Nodes and Systems
411 Regular /S/O Nodes
412 General I/S/O Nodes and Systems
413 Kemal and Image Representations of Closed 1/S/O
Nodes
414 State, Input/State and State/Output Nodes and Systems
415 Input/State and State/Output Representations of
State/Signal Systems
416 Free Inputs and Continuously Determined Outputs
417 Existence and Uniqueness of Trajectories
418| Bounded I/S/O Nodes and Systems
Input/State/Output Representations of State/Signal Nodes and
Systems
4 21 The State/Signal Node Induced by an /S/O Node
422 VO Representations of the Signal Space
423 General I/S/O Representations of a State/Signal Node
4 24 Semiregular /S/O Representations of a Semiregular
State/Signal Node
425 Regular VS/O Representations of a Regular State/Signal
Node
426 Parametrization of /S/O Representations
427 Bounded I/S/O Representations of Bounded State/Signal
Nodes
428 Parametrization of Bounded I/S/O Representations
State Feedback and Output Injection Representations
431 State Feedback Representations
432 Output Injection Representations
Contents
4 4 Basic Transformations of Input/State/Output Nodes
441 Similarity of I/S/O Nodes
44 2 Time Reflection of an I/S/O Node
443 Time Rescaling of an I/S/O Node
4 44 Exponentially Weighted /S/O Nodes
4 5 Properties of Trajectories of Input/State/Output Systems
451 Basic Properties of the Sets of Classical
and Generalized Trajectories
452 Solvability and the Uniqueness Property
453 Connections between Classical, Generalized,
and Mild Trajectories
4 6 Some Simple Input/State/Output Examples
Input/State/Output Systems: Dynamic and Frequency Domain
Properties
5 1 Additional Transformations of Input/State/Output Nodes
511 Adding a Feedthrough Term to an I/S/O Node
512 Modifying Inputs and Outputs of an /S/O Node
513 The (P, R, Q)-Image of an I/S/O Node
514 Parts and Static Projections of an I/S/O Node
515 Static Output Feedback
516| Adding Inputs and Outputs to an I/S/O Node
517A Second Look at State Feedbacks and Output
Injections
Interconnections of Input/State/Output Nodes
521 The Cross Product of Two I/S/O Nodes
522 (P, R, Q)-Interconnections of I/S/O Nodes
5 23 A Short Circuit Connection of I/S/O Nodes
524 T-Junctions, Sum Junctions, and Difference
Junctions
5 25 Parallel and Difference Connections
526 Cascade Connections
527 Dynamic Feedback
528 Examples of I/S/O Interconnections
Realizations of Input/Output Behaviors
531 Future I/O Behaviors
532 External Equivalence of YS/O Systems
Dynamic Properties of Input/State/Output Systems
541 Controllability and Observability of I/S/O Systems
542 Intertwinements of I/S/O Systems
Contents
5 43 Compressions, Restrictions, and Projections of US/O
Systems 326
5 44 _US/O Systems with the Continuation Property 330
5 5 Frequency Domain Characteristics of Input/State/Output Nodes 331
551 The Characteristic Node Bundle of an I/S/O Node 331
552 The I/S/O Resolvent Matrix of an I/S/O Node 334
553 Resolvability of Transformed I/S/O Nodes 338
554 Frequency Domain I/S/O-Admissible I/O
Representations 341
5 6 The Correspondence between State/Signal and Input/State/Output
Notions 343
561 VO Invariant Notions 343
562 Properties of I/S/O Systems in the State/Signal Sense 349
5 7 Adjoint and Dual Input/State/Output Nodes and Systems 351
571 The Adjoint and the Dual of an /S/O Node 351
572 Adjoint and Dual /S/O Representations 354
573 WS/O Lagrange Identities 355
574 Properties of Adjoint and Dual I/S/O Nodes and
Systems 359
575 The Adjoints and Duals of Some Transformed 1/S/O
Nodes 360
576 The Adjoints and Duals of Some Interconnected
VS/O Nodes 364
5 8 Notes to Chapters 4 and 5 366
Bounded Input/State/Output Systems in Continuous and Discrete
Time 370
6 1 Bounded State Operators and Nodes 370
611 The Spectral Radius of a Bounded State Operator 370
612 Invariant Subspaces of Bounded State Operators
and Uniformly Continuous Groups 372
613 Parts and Projections of Bounded State Operators 373
614 Parts and Projections of Uniformly Continuous
Groups 375
615 Intertwinements of Bounded State Operators and
Uniformly Continuous Groups 377
616| Compressions of Bounded State Operators and
Uniformly Continuous Groups 379
617 The General Structure of a Compression of a
Bounded State Operator 385
6 2
Contents
613 The Adjoints of Bounded State Operators and
Uniformly Continuous Groups
Static Properties of Bounded Input/State/Output Nodes
621 Transformations of Bounded I/S/O Nodes
622 Interconnections of Bounded I/S/O Nodes
6 23 The I/S/O Resolvent Matrix of a Bounded I/S/O Node
Dynamic Properties of Bounded Input/State/Output Systems
631 Strongly Invariant and Unobservably Invariant
Subspaces
632 External Equivalence of Bounded I/S/O Systems
633 Intertwinements of Bounded I/S/O Systems
6 34 Restrictions and Projections of Bounded I/S/O Systems
635 Compressions of Bounded I/S/O Systems
636 The General Structure of a Bounded I/S/O Compression
637 Compressions into Minimal Bounded I/S/O Systems
The Adjoint and the Dual of a Bounded Input/State/Output
Node
Discrete Time Input/State/Output Systems
651 Introduction to Discrete Time I/S/O Systems
652 Properties of Discrete Time US/O Systems
653 Time Reflection of Discrete Time I/S/O Systems
6 54 Power Weightings of Discrete Time I/S/O Systems
655 Frequency Domain Shifts of Discrete Time I/S/O
Systems
656 Stable Discrete Time I/S/O Systems
657 Connections between Continuous and Discrete Time
V/S/O Properties
658| Dynamic Notions for Bounded I/S/O Nodes
Bounded Input/State/Output Realizations
661 Analyticity at Infinity of the I/S/O Resolvent Matrix
662 Existence of a Bounded I/S/O Realization
Bounded State/Signal Systems in Continuous and Discrete Time
7A
Static Properties of Bounded State/Signal Nodes
711 The I/S/O-Bounded Resolvent Set of a Bounded
State/Signal Node
712 Transformations of Bounded State/Signal Nodes
713 Resolvability of Transformations of State/Signal
Nodes
Dynamic Properties of Bounded State/Signal Systems
xi
8
Contents
Strongly Invariant and Unobservably Invariant
Subspaces
External Equivalence of Bounded State/Signal Systems
Intertwinements of Bounded State/Signal Systems
Restrictions and Projections of Bounded State/Signal
Systems
Compressions of Bounded State/Signal Systems
The General Structure of a Bounded State/Signal
Compression
Compressions into Minimal Bounded State/Signal
Systems
Bounded State/Signal Realizations
7 3 The Dual and the Adjoint of a Bounded State/Signal Node
74 Discrete Time State/Signal Systems
Introduction to Discrete Time State/Signal Systems
Properties of Discrete Time State/Signal Systems
Time Reflection of Discrete Time State/Signal Systems
Power Weightings of Discrete Time State/Signal
Systems
Frequency Domain Shifts of Discrete Time State/Signal
Systems
Stable Discrete Time State/Signal Systems
Connections between Continuous and Discrete Time
State/Signal Properties
Dynamic Notions for Bounded State/Signal Nodes
7 5 Notes to Chapters 6 and 7
Semi-bounded Input/State/Output Systems
Co Semigroups and Well-Posed State Systems
On the Resolvents of Generators of Co
Semigroups
The Inhomogeneous Cauchy Problem
Invariant Subspaces of Cp Semigroups
Parts, Projections, and Restrictions of Single-Valued
Resolvable Main Operators
Parts and Projections of Co Semigroups
Intertwinements of Cy Semigroups
Compressions of Cp Semigroups
The General Structure of a Compression of a Co
Semigroup
81 9
Contents
The Adjoint of a Co Semigroup
8 2 Semi-bounded Input/State/Output Systems
Introduction to Semi-bounded I/S/O Systems
Transformations of Semi-bounded I/S/O Nodes
Interconnections of Semi-bounded I/S/O Nodes
The I/S/O Resolvent Matrix of a Semi-bounded 1/S/O
Node
8 3 Dynamic Properties of Semi-bounded Input/State/Output
Systems
Strongly Invariant and Unobservably Invariant
Subspaces
External Equivalence of Semi-bounded I/S/O Systems
Intertwinements of Semi-bounded I/S/O Systems
Restrictions and Projections of Semi-bounded I/S/O
Systems
Compressions of Semi-bounded US/O Systems
The General Structure of a Semi-bounded I/S/O
Compression
Compressions into Minimal Semi-bounded I/S/O
Systems
8 4 The Adjoint of a Semi-bounded Input/State/Output Node
Semi-bounded State/Signal Systems
9 1 Static Properties of Semi-bounded State/Signal Nodes
Introduction to Semi-bounded State/Signal Nodes
and Systems
The I/S/O Semi-bounded Resolvent Set of a
Semi-bounded State/Signal Node
Transformations and Interconnections
of Semi-bounded State/Signal Nodes
9 2 Dynamic Properties of Semi-bounded State/Signal Systems
Strongly Invariant and Unobservably Invariant
Subspaces
External Equivalence of Semi-bounded State/Signal
Systems
Intertwinements of Semi-bounded State/Signal Systems
Restrictions and Projections of Semi-bounded
State/Signal Systems
Compressions of Semi-bounded State/Signal Systems
xiii
xiv
Contents
The General Structure of a Semi-bounded State/Signal
Compression
Compressions into Minimal Semi-bounded State/Signal
Systems
9 3 The Adjoint of a Semi-bounded State/Signal Node
9 4 Notes to Chapters 8 and 9
Resolvable Input/State/Output and State/Signal Nodes
10 1 Resolvable State Nodes
Linear Operator-Valued Pencils
The Resolvent of a State Node
The Interpolation Space of a Semiregular State Node
The Extrapolation Space of a Regular Resolvable
State Node
The Duals of the Interpolation and Extrapolation
Spaces
The Interpolation and Extrapolation Spaces of a
Semigroup Generator
10 2 Resolvable Input/State/Output Nodes
Resolvability of an I/S/O Node
Kernel and Image Representations of the YS/O
Resolvent Matrix
The I/S/O Resolvent Identity
Representations of the System Operator
Semiregular and Regular Resolvable I/S/O Nodes
The Observation and Control Operators of a Regular
Resolvable I/S/O Node
Some Examples of Regular Resolvable I/S/O Nodes
Resolvability of Transformed 1/S/O Nodes
Resolvability of Interconnected I/S/O Nodes
The Resolvent Family of Bounded /S/O Nodes
A Finite-Dimensional Nonregular Resolvable /S/O
Node
The Adjoint and the Dual of a Resolvable 1/S/O Node
10 3 Resolvable State/Signal Nodes
On the Resolvent Set of a Closed State/Signal Node
Frequency Domain I/S/O-Admissible VO
Representations
Resolvability of Transformed State/Signal Nodes
The Resolvent Family of Bounded State/Signal Nodes
11
Contents
10 3 5 The Dual and the Adjoint of a Resolvable State/Signal
System
10 4 Notes and Comments
Frequency Domain Input/State/Output Systems
Frequency Domain Input/State/Output Systems
11 1 1 Introduction to Frequency Domain US/O Systems
11 1 2 Frequency Domain Controllability and Observability
11 1 3 Frequency Domain Invariance
11 1 4 The Frequency Domain Behavior and External
Equivalence
11 1 5 Frequency Domain Intertwinements
11 1 6 Frequency Domain Compressions, Restrictions,
and Projections
11 1 7 Resolvable Frequency Domain Compressions,
Restrictions, and Projections
11 1 8 The General Structure of a Resolvable Frequency
Domain Compression
11 1 9 Compressions into 2-Minimal /S/O Systems
11 1 10 Results for Connected Frequency Domains
The Adjoint and the Dual of a Frequency Domain
Input/State/Output System
11 2 1 Frequency Domain Lagrange Identities
11 2 2 Properties of Adjoint and Dual Frequency Domain
VS/O Systems
Frequency Domain Notions for 2-Resolvable Input/State/Output
Nodes
11 3 1 Dynamic Properties of the Resolvent Family of
Bounded J/S/O Nodes
Resolvable Frequency Domain State Systems
11 4 1 Frequency Domain Invariance
11 4 2 Frequency Domain Intertwinements and
Compressions
11 4 3 Results for Connected Frequency Domains
11 4 4 Frequency Domain Duality
Notes and Comments
Frequency Domain State/Signal Systems
12 1 Frequency Domain State/Signal Systems
12 1 1 Introduction to Frequency Domain State/Signal
Systems
KV
xvi
Contents
12 1 2 Separately and Jointly /S/O Admissible Frequency
Domains
12 1 3 Frequency Domain Controllability and Observability
12 1 4 Frequency Domain Invariance
12 1 5 The Frequency Domain Behavior and External
Equivalence
12 1 6 Frequency Domain Intertwinements
12 1 7 Frequency Domain Compressions, Restrictions,
and Projections
12 1 8 Resolvable Frequency Domain Compressions,
Restrictions, and Projections
12 1 9 The General Structure of a Resolvable Frequency
Domain Compression
12 1 10 Compressions into 2-Minimal State/Signal Systems
Local Frequency Domain Notions
12 2 1 Local Frequency Domain Notions for Q-Resolvable
State/Signal Systems
12 2 2 Connected Frequency Domains
The Dual and the Adjoint of a Frequency Domain State/Signal
System
12 3 1 Frequency Domain Lagrange Identities
12 3 2 Properties of Dual and Adjoint Frequency Domain
State/Signal Systems
Frequency Domain Notions for Q-Resolvable State/Signal
Nodes
12 4 1 Dynamic Properties of the Resolvent Family of
Bounded State/Signal Nodes
Notes and Comments
Internally Well-Posed Systems
Internally Well-Posed Input/State/Output Systems
13 1 1 Basic Definitions and Properties
13 1 2 Transformations and Interconnections
Frequency-Domain Internally Well-Posed Input/State/Output
Systems
13 2 1 Frequency Domain Invariance
13 2 2 Frequency Domain Intertwinements
13 2 3 Frequency-Domain Restrictions, Projections,
and Compressions
13 2 4 The General Structure of P4(%)-Compressions
14
Contents
13 3 Internally Well-Posed State/Signal Systems
13 3 1 Basic Definitions and Properties
13 3 2 Frequency-Domain Compressions of Internally
Well-Posed State/Signal Systems
13 4 Notes and Comments
Well-Posed Input/State/Output Systems
14 1 Basic Properties of Well-Posed Input/State/Output Systems
14 1 1 The Definition of a Well-Posed VS/O System
14 1 2 Alternative Conditions for Well-Posedness
14 1 3 The Fundamental I/S/O Solution of a Well-Posed
VS/O System
14 2 The Growth Bound of a Well-Posed Input/State/Output System
14 2 1 The Growth Bound of a Weill-Posed I/S/O System
14 2 2 Stable I/S/O Systems
14 3 Resolvability of Well-Posed Input/State/Output Systems
14 3 1 Well-Posed I/S/O Systems Are Resolvable
14 3 2 Growth Estimates for the /S/O Resolvent Matrix
14 4 Realizations of Shift-Invariant Causal Linear Operators
14 4 1 Shift Invariant Causal Linear Operators
14 4 2 Realizations of Shift Invariant Causal Linear Operators
14 4 3 Toeplitz and Hankel Operators
14 5 Transformations and Interconnections of Well-Posed
Input/State/Output Systems
14 5 1 Well-Posedness and Stability of Transformed I/S/O
Systems
14 5 2 Well-Posedness and Stability of Interconnected I/S/O
Systems
14 5 3 Stabilizable and Detectable I/S/O Systems
14 6 Dynamic Properties of Well-Posed Input/State/Output Systems
14 6 1 Strongly Invariant and Unobservably Invariant
Subspaces
14 6 2 Intertwinements of Well-Posed I/S/O Systems
14 6 3 Restrictions, Projections, and Compressions
14 6 4 The General Structure of a Well-Posed I/S/O
Compression
14 6 5 Compressions Into Minimal Well-Posed I/S/O Systems
14 7 Well-Posed Input/State/Output Systems in the Frequency
Domain
14 7 1 Time and Frequency Domain External Equivalence
14 7 2 Time and Frequency Domain Invariance
xvii
xviii
Contents
14 7 3 Time and Frequency Domain Compressions and
Intertwinements
14 7 4 Frequency Domain Stability
14 8 The Adjoint of a Well-Posed Input/State/Output System
14 9 Scattering Passive Input/State/Output Systems
14 9 1 Hilbert Space /S/O Nodes and Systems
14 9 2 Scattering Passive I/S/O Systems
14 9 3 The Internal I/S/O Cayley Transform
14 9 4 The Adjoint of a Passive Scattering System
14 10 Notes and Comments
Well-Posed State/Signal Systems
15 1 Basic Properties of Well-Posed State/Signal Systems
15 1 1 Basic Definitions
15 1 2 Well-Posedness and Stability of Transformed /S/O
Systems
15 1 3 The Behaviors Induced by a Well-Posed State/Signal
System
15 1 4 The Past/Present and Present/Future Maps of a
Well-Posed State/Signal System
15 2 Stable State/Signal Systems
15 2 1 Stable State/Signal Trajectories
15 2 2 Stable State/Signal Behaviors
15 2 3 Stabilizable and Detectable State/Signal Systems
15 3 Realizations of Well-Posed Behaviors
15 3 1 Well-Posed Future, Past, and Two-Sided Behaviors
15 3 2 State/Signal Realizations of Well-Posed Behaviors
15 3 3 The Past/Future Map of a Well-Posed Behavior
15 4 Dynamic Properties of Well-Posed State/Signal Systems
15 4 1 Strongly Invariant and Unobservably Invariant
Subspaces
15 4 2 Intertwinements of Well-Posed State/Signal Systems
15 4 3 Restrictions, Projections, and Compressions of Well-
Posed State/Signal Systems
15 4 4 The General Structure of a Compression
15 4 5 Compressions into Minimal Well-Posed State/Signal
Systems
15 5 Well-Posed State/Signal Systems in the Frequency Domain
15 6 The Adjoint of a Well-Posed State/Signal Node
15 7 Passive State/Signal Systems
15 7 1
Contents
Krein Spaces
The Krein Node Space of a Scattering Passive I/S/O
System
Passive State/Signal Systems
15 8 Notes and Comments
Appendix A —_ Operators and Analytic Vector Bundles in H-Spaces
A l H-Spaces
A 1 1
A 1 2
A 1 3
A 1 4
A 1 5
A 1 6
A 1 7
A 1 8
A 19
A 1 10
A111
A 1 12
Using More than One Norm in a Vector Space
Introduction to H-Spaces
Linear Operators in H-Spaces
Closed Linear Operators in H-Spaces
Complementary Projections and Coordinate
Respresentations of H-Spaces
Isomorphisms in H-Spaces
Partial Inverses of Bounded Linear Operators
Inversion of Block Matrix Operators
The Graph Norm and Graph Topology
Linear Multivalued Operators in H-Spaces
The Single-Valued and Injective Parts of a
Multivalued Operator
On the Resolvent of a Bounded Operator
A 2 Duality in H-Spaces
A21 The Dual of an H-Space
A22 The Adjoint of a Bounded Linear Operator
A23 Duals of Product Spaces
A24 The Duals of the Components of a Direct Sum
Decomposition
A25 The Adjoint of a Linear Operator with Dense Domain
A26 The Dual of a Continuous Dense Embedding
A27 The Adjoint of a Multivalued Operator
A 3 Analytic Vector Bundles and Analytic Operator-Valued
Functions
A31 The Dual Vector Bundle
References
Index
xix |
| any_adam_object | 1 |
| any_adam_object_boolean | 1 |
| author | Arov, Damir Z. 1934- Staffans, Olof 1947- |
| author_GND | (DE-588)172974356 (DE-588)142388831 |
| author_facet | Arov, Damir Z. 1934- Staffans, Olof 1947- |
| author_role | aut aut |
| author_sort | Arov, Damir Z. 1934- |
| author_variant | d z a dz dza o s os |
| building | Verbundindex |
| bvnumber | BV048240032 |
| callnumber-first | Q - Science |
| callnumber-label | QA402 |
| callnumber-raw | QA402 |
| callnumber-search | QA402 |
| callnumber-sort | QA 3402 |
| callnumber-subject | QA - Mathematics |
| classification_rvk | SK 880 |
| ctrlnum | (OCoLC)1322044017 (DE-599)KXP1788052560 |
| dewey-full | 003/.74 |
| dewey-hundreds | 000 - Computer science, information, general works |
| dewey-ones | 003 - Systems |
| dewey-raw | 003/.74 |
| dewey-search | 003/.74 |
| dewey-sort | 13 274 |
| dewey-tens | 000 - Computer science, information, general works |
| discipline | Informatik Mathematik |
| discipline_str_mv | Informatik Mathematik |
| format | Book |
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| id | DE-604.BV048240032 |
| illustrated | Not Illustrated |
| index_date | 2024-07-03T19:53:52Z |
| indexdate | 2024-07-10T09:32:50Z |
| institution | BVB |
| isbn | 9781316519677 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-033620546 |
| oclc_num | 1322044017 |
| open_access_boolean | |
| owner | DE-384 |
| owner_facet | DE-384 |
| physical | xxviii, 1019 Seiten |
| publishDate | 2022 |
| publishDateSearch | 2022 |
| publishDateSort | 2022 |
| publisher | Cambridge University Press |
| record_format | marc |
| series | Encyclopedia of mathematics and its applications |
| series2 | Encyclopedia of mathematics and its applications |
| spelling | Arov, Damir Z. 1934- Verfasser (DE-588)172974356 aut Linear state/signal systems Damir Z. Arov, South Ukrainian National Pedagogical University, Olof J. Staffans, Åbo Akademi University, Finland Cambridge, United Kingdom Cambridge University Press 2022 xxviii, 1019 Seiten txt rdacontent n rdamedia nc rdacarrier Encyclopedia of mathematics and its applications 183 Includes bibliographical references and index "The theory presented in this book arose as a product of a continued collaboration between the two authors during the years 2003-2021. The basis for this collaboration was our common interest in passive linear time-invariant input/state/output systems theory. At the time when this project started O. Staffans was preparing a joint article (Ball and Staffans, 2006) with Prof. J. Ball which, in particular, explored the connections between conservative input/state/output systems theory on one hand and some results in the behavioral theory introduced by J.Willems in the late 80's on the other hand. After extensive discussions on the above approach, comparing it to the theory of passive electrical networks, we understood that this opens up a new direction in the study of passive linear time-invariant systems. We called the new class of systems that arose in this way passive state/signal systems. From the outset it was clear that the notion of passivity with an arbitrary supply rate fits more naturally into the state/signal setting than in the input/state/output setting, and that the standard "diagonal transformation" of Livšic, the Potapov-Ginzburg transformation, and the Redheffer and chain scattering transformations have natural interpretations as transformations between input/output resolvents of different input/state/output representations of a passive state/signal system. We also soon discovered that virtually all the standard control theory notions, such as controllability and observability, minimality, stability, stabilizability, detectability, and well-posedness have natural state/signal counterparts"-- Lineares Regelungssystem (DE-588)4167730-4 gnd rswk-swf Linear systems Operator theory Linear control systems MATHEMATICS / General Lineares Regelungssystem (DE-588)4167730-4 s DE-604 Staffans, Olof 1947- Verfasser (DE-588)142388831 aut Erscheint auch als Online-Ausgabe, ebook 9781009024921 Encyclopedia of mathematics and its applications 183 (DE-604)BV000903719 183 HEBIS Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=033620546&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Arov, Damir Z. 1934- Staffans, Olof 1947- Linear state/signal systems Encyclopedia of mathematics and its applications Lineares Regelungssystem (DE-588)4167730-4 gnd |
| subject_GND | (DE-588)4167730-4 |
| title | Linear state/signal systems |
| title_auth | Linear state/signal systems |
| title_exact_search | Linear state/signal systems |
| title_exact_search_txtP | Linear state/signal systems |
| title_full | Linear state/signal systems Damir Z. Arov, South Ukrainian National Pedagogical University, Olof J. Staffans, Åbo Akademi University, Finland |
| title_fullStr | Linear state/signal systems Damir Z. Arov, South Ukrainian National Pedagogical University, Olof J. Staffans, Åbo Akademi University, Finland |
| title_full_unstemmed | Linear state/signal systems Damir Z. Arov, South Ukrainian National Pedagogical University, Olof J. Staffans, Åbo Akademi University, Finland |
| title_short | Linear state/signal systems |
| title_sort | linear state signal systems |
| topic | Lineares Regelungssystem (DE-588)4167730-4 gnd |
| topic_facet | Lineares Regelungssystem |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=033620546&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000903719 |
| work_keys_str_mv | AT arovdamirz linearstatesignalsystems AT staffansolof linearstatesignalsystems |