On the Characterization of Hankel and Toeplitz Operators Describing Switched Linear Dynamic Systems with Point Delays
This paper investigates the causality properties of a class of linear time-delay systems under constant point delays which possess a finite set of distinct linear time-invariant parameterizations (or configurations) which, together with some switching function, conform a linear time-varying switche...
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| Format: | Article |
| Language: | English |
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Wiley
2009-01-01
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| Series: | Abstract and Applied Analysis |
| Online Access: | http://dx.doi.org/10.1155/2009/670314 |
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| author | M. De la Sen |
| author_facet | M. De la Sen |
| author_sort | M. De la Sen |
| collection | DOAJ |
| description | This paper investigates the causality properties of a class of linear time-delay systems under constant point delays which possess a finite set of distinct linear time-invariant parameterizations (or configurations) which,
together with some switching function, conform a linear time-varying switched dynamic system. Explicit expressions
are given to define pointwisely the causal and anticausal Toeplitz and Hankel operators from the set of switching time instants generated from the switching function. The case of the auxiliary unforced system defined by the matrix of undelayed dynamics being dichotomic (i.e., it has no eigenvalue on the complex imaginary axis) is considered in detail. Stability conditions as well as dual instability ones are discussed for this case which guarantee that the whole system is either stable, or unstable but no configuration of the switched system has eigenvalues within some vertical strip including the imaginary axis. It is proved that if the system is causal and uniformly controllable and observable, then it is globally asymptotically Lyapunov stable independent of the delays, that is, for any possibly values of such delays, provided that a minimum residence time in-between consecutive switches is kept or if all the set of matrices describing the auxiliary unforced delay—free system parameterizations commute pairwise. |
| format | Article |
| id | doaj-art-309a1430dad84980a7489884d3004ba4 |
| institution | Kabale University |
| issn | 1085-3375 1687-0409 |
| language | English |
| publishDate | 2009-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Abstract and Applied Analysis |
| spelling | doaj-art-309a1430dad84980a7489884d3004ba42025-02-03T05:46:25ZengWileyAbstract and Applied Analysis1085-33751687-04092009-01-01200910.1155/2009/670314670314On the Characterization of Hankel and Toeplitz Operators Describing Switched Linear Dynamic Systems with Point DelaysM. De la Sen0IIDP, Faculty of Science and Technology, University of the Basque Country, Campus of Leioa (Bizkaia), Aptdo, 644-Bilbao, SpainThis paper investigates the causality properties of a class of linear time-delay systems under constant point delays which possess a finite set of distinct linear time-invariant parameterizations (or configurations) which, together with some switching function, conform a linear time-varying switched dynamic system. Explicit expressions are given to define pointwisely the causal and anticausal Toeplitz and Hankel operators from the set of switching time instants generated from the switching function. The case of the auxiliary unforced system defined by the matrix of undelayed dynamics being dichotomic (i.e., it has no eigenvalue on the complex imaginary axis) is considered in detail. Stability conditions as well as dual instability ones are discussed for this case which guarantee that the whole system is either stable, or unstable but no configuration of the switched system has eigenvalues within some vertical strip including the imaginary axis. It is proved that if the system is causal and uniformly controllable and observable, then it is globally asymptotically Lyapunov stable independent of the delays, that is, for any possibly values of such delays, provided that a minimum residence time in-between consecutive switches is kept or if all the set of matrices describing the auxiliary unforced delay—free system parameterizations commute pairwise.http://dx.doi.org/10.1155/2009/670314 |
| spellingShingle | M. De la Sen On the Characterization of Hankel and Toeplitz Operators Describing Switched Linear Dynamic Systems with Point Delays Abstract and Applied Analysis |
| title | On the Characterization of Hankel and Toeplitz Operators Describing Switched Linear Dynamic Systems with Point Delays |
| title_full | On the Characterization of Hankel and Toeplitz Operators Describing Switched Linear Dynamic Systems with Point Delays |
| title_fullStr | On the Characterization of Hankel and Toeplitz Operators Describing Switched Linear Dynamic Systems with Point Delays |
| title_full_unstemmed | On the Characterization of Hankel and Toeplitz Operators Describing Switched Linear Dynamic Systems with Point Delays |
| title_short | On the Characterization of Hankel and Toeplitz Operators Describing Switched Linear Dynamic Systems with Point Delays |
| title_sort | on the characterization of hankel and toeplitz operators describing switched linear dynamic systems with point delays |
| url | http://dx.doi.org/10.1155/2009/670314 |
| work_keys_str_mv | AT mdelasen onthecharacterizationofhankelandtoeplitzoperatorsdescribingswitchedlineardynamicsystemswithpointdelays |