Book contents
- Frontmatter
- Contents
- Preface
- Notations
- 1 Introduction and Overview
- 2 State/Signal Systems: Trajectories, Transformations, and Interconnections
- 3 State/Signal Systems: Dynamic and Frequency Domain Properties
- 4 Input/State/Output Representations
- 5 Input/State/Output Systems: Dynamic and Frequency Domain Properties
- 6 Bounded Input/State/Output Systems in Continuous and Discrete Time
- 7 Bounded Input/State/Output Systems in Continuous and Discrete Time
- 8 Semi-bounded Input/State/Output Systems
- 9 Semi-bounded State/Signal Systems
- 10 Resolvable Input/State/Output and State/Signal Nodes
- 11 Frequency Domain Input/State/Output Systems
- 12 Frequency Domain State/Signal Systems
- 13 Internally Well-Posed Systems
- 14 Well-Posed Input/State/Output Systems
- 15 Well-Posed State/Signal Systems
- Appendix A Operators and Analytic Vector Bundles in H-Spaces
- References
- Index
12 - Frequency Domain State/Signal Systems
Published online by Cambridge University Press: 05 May 2022
- Frontmatter
- Contents
- Preface
- Notations
- 1 Introduction and Overview
- 2 State/Signal Systems: Trajectories, Transformations, and Interconnections
- 3 State/Signal Systems: Dynamic and Frequency Domain Properties
- 4 Input/State/Output Representations
- 5 Input/State/Output Systems: Dynamic and Frequency Domain Properties
- 6 Bounded Input/State/Output Systems in Continuous and Discrete Time
- 7 Bounded Input/State/Output Systems in Continuous and Discrete Time
- 8 Semi-bounded Input/State/Output Systems
- 9 Semi-bounded State/Signal Systems
- 10 Resolvable Input/State/Output and State/Signal Nodes
- 11 Frequency Domain Input/State/Output Systems
- 12 Frequency Domain State/Signal Systems
- 13 Internally Well-Posed Systems
- 14 Well-Posed Input/State/Output Systems
- 15 Well-Posed State/Signal Systems
- Appendix A Operators and Analytic Vector Bundles in H-Spaces
- References
- Index
Summary
In this chapter we extend the frequency domain results for i/s/o systems presented in Chapter 11 into analogous frequency domain results for s/s systems. The main definitions are given for general s/s systems, but most of the more specific results require the s/s system to be resolvable (i.e., to have a nonempty resolvent set). We introduce s/s versions of Ω-resolvability, Ω-controllability, Ω-observability, Ω-minimality, Ω-intertwinements, Ω-restrictions, Ω-projections, and Ω-compressions for s/s systems, where Ω is a given nonempty open subset of the complex plane. In this setting the usual (time domain) regularity conditions are less significant. Some of the s/s results are proved directly in the s/s setting, but most of them are reduced to the corresponding i/s/o results by means of i/s/o representations of the given s/s systems.
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- Chapter
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- Linear State/Signal Systems , pp. 743 - 801Publisher: Cambridge University PressPrint publication year: 2022