The $ l_\infty $-induced norm of multivariable discrete-time linear systems: Upper and lower bounds with convergence rate analysis

Oe Ryung Kang; Jung Hoon Kim; Oe Ryung Kang; Jung Hoon Kim

doi:10.3934/math.20231492

AIMS Mathematics

2023, Volume 8, Issue 12: 29140-29157. doi: 10.3934/math.20231492

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The $ l_\infty $-induced norm of multivariable discrete-time linear systems: Upper and lower bounds with convergence rate analysis

Oe Ryung Kang ¹,
Jung Hoon Kim ^{1,2
,
,}

1.
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 Korea

Received: 26 July 2023 Revised: 19 September 2023 Accepted: 16 October 2023 Published: 26 October 2023
MSC : 47A30, 65Y20, 93-08, 93C05, 93C55

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.
- multivariable discrete-time linear systems,
- truncated fashion,
- $ l_\infty $-induced norm,
- generalized $ H_2 $ norm,
- convergence rate analysis
Citation: 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[J]. AIMS Mathematics, 2023, 8(12): 29140-29157. doi: 10.3934/math.20231492

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Abstract

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.

References

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