The $ l_\infty $-induced norm of multivariable discrete-time linear systems: Upper and lower bounds with convergence rate analysis
This paper develops a method for computing the $ l_{\infty} $-induced norm of a multivariable discrete-time linear system, for which an infinite-dimensional matrix should be intrinsically concerned with. To make such a computation feasible, we treat the infinite-dimensional matrix in a truncated fas...
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| Format: | Article |
| Language: | English |
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AIMS Press
2023-10-01
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| Series: | AIMS Mathematics |
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| Online Access: | https://www.aimspress.com/article/doi/10.3934/math.20231492?viewType=HTML |
| _version_ | 1797629023266799616 |
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| author | Oe Ryung Kang Jung Hoon Kim |
| author_facet | Oe Ryung Kang Jung Hoon Kim |
| author_sort | Oe Ryung Kang |
| collection | DOAJ |
| description | This paper develops a method for computing the $ l_{\infty} $-induced norm of a multivariable discrete-time linear system, for which an infinite-dimensional matrix should be intrinsically concerned with. To make such a computation feasible, we treat the infinite-dimensional matrix in a truncated fashion, and an upper bound and a lower bound on the $ l_\infty $-induced norm of the original multivariable discrete-time linear system are derived. More precisely, the matrix $ \infty $-norm of the (infinite-dimensional) tail part can be approximately computed by deriving its upper and lower bounds, while that of the (finite-dimensional) truncated part can be exactly obtained. With these values, an upper bound and a lower bound on the original $ l_\infty $-induced norm can be computed. Furthermore, these bounds are shown to converge to each other within an exponential order of $ N $, where $ N $ is the corresponding truncation parameter. Finally, some numerical examples are provided to demonstrate the theoretical validity and practical effectiveness of the developed computation method. |
| first_indexed | 2024-03-11T10:47:34Z |
| format | Article |
| id | doaj.art-e9b1942f353e4a0785d011041ff2d8e3 |
| institution | Directory Open Access Journal |
| issn | 2473-6988 |
| language | English |
| last_indexed | 2024-03-11T10:47:34Z |
| publishDate | 2023-10-01 |
| publisher | AIMS Press |
| record_format | Article |
| series | AIMS Mathematics |
| spelling | doaj.art-e9b1942f353e4a0785d011041ff2d8e32023-11-14T01:13:27ZengAIMS PressAIMS Mathematics2473-69882023-10-01812291402915710.3934/math.20231492The $ l_\infty $-induced norm of multivariable discrete-time linear systems: Upper and lower bounds with convergence rate analysisOe Ryung Kang0Jung Hoon Kim11. Department of Electrical Engineering, Pohang University of Science and Technology (POSTECH), Pohang 37673, Republic of Korea1. Department of Electrical Engineering, Pohang University of Science and Technology (POSTECH), Pohang 37673, Republic of Korea 2. Institute for Convergence Research and Education in Advanced Technology, Yonsei University, Incheon 21983, Republic of KoreaThis paper develops a method for computing the $ l_{\infty} $-induced norm of a multivariable discrete-time linear system, for which an infinite-dimensional matrix should be intrinsically concerned with. To make such a computation feasible, we treat the infinite-dimensional matrix in a truncated fashion, and an upper bound and a lower bound on the $ l_\infty $-induced norm of the original multivariable discrete-time linear system are derived. More precisely, the matrix $ \infty $-norm of the (infinite-dimensional) tail part can be approximately computed by deriving its upper and lower bounds, while that of the (finite-dimensional) truncated part can be exactly obtained. With these values, an upper bound and a lower bound on the original $ l_\infty $-induced norm can be computed. Furthermore, these bounds are shown to converge to each other within an exponential order of $ N $, where $ N $ is the corresponding truncation parameter. Finally, some numerical examples are provided to demonstrate the theoretical validity and practical effectiveness of the developed computation method.https://www.aimspress.com/article/doi/10.3934/math.20231492?viewType=HTMLmultivariable discrete-time linear systemstruncated fashion$ l_\infty $-induced normgeneralized $ h_2 $ normconvergence rate analysis |
| spellingShingle | Oe Ryung Kang Jung Hoon Kim The $ l_\infty $-induced norm of multivariable discrete-time linear systems: Upper and lower bounds with convergence rate analysis AIMS Mathematics multivariable discrete-time linear systems truncated fashion $ l_\infty $-induced norm generalized $ h_2 $ norm convergence rate analysis |
| title | The $ l_\infty $-induced norm of multivariable discrete-time linear systems: Upper and lower bounds with convergence rate analysis |
| title_full | The $ l_\infty $-induced norm of multivariable discrete-time linear systems: Upper and lower bounds with convergence rate analysis |
| title_fullStr | The $ l_\infty $-induced norm of multivariable discrete-time linear systems: Upper and lower bounds with convergence rate analysis |
| title_full_unstemmed | The $ l_\infty $-induced norm of multivariable discrete-time linear systems: Upper and lower bounds with convergence rate analysis |
| title_short | The $ l_\infty $-induced norm of multivariable discrete-time linear systems: Upper and lower bounds with convergence rate analysis |
| title_sort | l infty induced norm of multivariable discrete time linear systems upper and lower bounds with convergence rate analysis |
| topic | multivariable discrete-time linear systems truncated fashion $ l_\infty $-induced norm generalized $ h_2 $ norm convergence rate analysis |
| url | https://www.aimspress.com/article/doi/10.3934/math.20231492?viewType=HTML |
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