$ L_2/L_1 $ induced norm and Hankel norm analysis in sampled-data systems

Tomomichi Hagiwara; Masaki Sugiyama; Tomomichi Hagiwara; Masaki Sugiyama

doi:10.3934/math.2024149

AIMS Mathematics

2024, Volume 9, Issue 2: 3035-3075. doi: 10.3934/math.2024149

Previous Article Next Article

Research article Special Issues

$ L_2/L_1 $ induced norm and Hankel norm analysis in sampled-data systems

Tomomichi Hagiwara ^,,
Masaki Sugiyama

Department of Electrical Engineering, Kyoto University, Kyoto 615-8510, Japan

Received: 22 September 2023 Revised: 09 November 2023 Accepted: 21 December 2023 Published: 02 January 2024
MSC : 93C57, 93B28, 93B52, 47N70, 93C05

This paper is concerned with the $ L_2/L_1 $ induced and Hankel norms of sampled-data systems. In defining the Hankel norm, the $ h $-periodicity of the input-output relation of sampled-data systems is taken into account, where $ h $ denotes the sampling period; past and future are separated by the instant $ \Theta\in[0, h) $, and the norm of the operator describing the mapping from the past input in $ L_1 $ to the future output in $ L_2 $ is called the quasi $ L_2/L_1 $ Hankel norm at $ \Theta $. The $ L_2/L_1 $ Hankel norm is defined as the supremum over $ \Theta\in[0, h) $ of this norm, and if it is actually attained as the maximum, then a maximum-attaining $ \Theta $ is called a critical instant. This paper gives characterization for the $ L_2/L_1 $ induced norm, the quasi $ L_2/L_1 $ Hankel norm at $ \Theta $ and the $ L_2/L_1 $ Hankel norm, and it shows that the first and the third ones coincide with each other and a critical instant always exists. The matrix-valued function $ H(\varphi) $ on $ [0, h) $ plays a key role in the sense that the induced/Hankel norm can be obtained and a critical instant can be detected only through $ H(\varphi) $, even though $ \varphi $ is a variable that is totally irrelevant to $ \Theta $. The relevance of the induced/Hankel norm to the $ H_2 $ norm of sampled-data systems is also discussed.
- sampled-data systems,
- intersample behavior,
- induced norm,
- quasi Hankel norm,
- Hankel norm,
- $ H_2 $ norm,
- impulse response
Citation: Tomomichi Hagiwara, Masaki Sugiyama. $ L_2/L_1 $ induced norm and Hankel norm analysis in sampled-data systems[J]. AIMS Mathematics, 2024, 9(2): 3035-3075. doi: 10.3934/math.2024149

Related Papers:

Abstract

This paper is concerned with the $ L_2/L_1 $ induced and Hankel norms of sampled-data systems. In defining the Hankel norm, the $ h $-periodicity of the input-output relation of sampled-data systems is taken into account, where $ h $ denotes the sampling period; past and future are separated by the instant $ \Theta\in[0, h) $, and the norm of the operator describing the mapping from the past input in $ L_1 $ to the future output in $ L_2 $ is called the quasi $ L_2/L_1 $ Hankel norm at $ \Theta $. The $ L_2/L_1 $ Hankel norm is defined as the supremum over $ \Theta\in[0, h) $ of this norm, and if it is actually attained as the maximum, then a maximum-attaining $ \Theta $ is called a critical instant. This paper gives characterization for the $ L_2/L_1 $ induced norm, the quasi $ L_2/L_1 $ Hankel norm at $ \Theta $ and the $ L_2/L_1 $ Hankel norm, and it shows that the first and the third ones coincide with each other and a critical instant always exists. The matrix-valued function $ H(\varphi) $ on $ [0, h) $ plays a key role in the sense that the induced/Hankel norm can be obtained and a critical instant can be detected only through $ H(\varphi) $, even though $ \varphi $ is a variable that is totally irrelevant to $ \Theta $. The relevance of the induced/Hankel norm to the $ H_2 $ norm of sampled-data systems is also discussed.

References

[1]	M. A. Dahleh, J. B. Pearson, $L^1$-optimal compensators for continuous-time systems, IEEE T. Automat. Contr., 32 (1987), 889–895. https://doi.org/10.1109/TAC.1987.1104455 doi: 10.1109/TAC.1987.1104455
[2]	D. A. Wilson, Convolution and Hankel operator norms for linear systems, IEEE T. Automat. Contr., 34 (1989), 94–97. https://doi.org/10.1109/9.8655 doi: 10.1109/9.8655
[3]	W. W. Lu, G. J. Balas, A comparison between Hankel norms and induced system norms, IEEE T. Automat. Contr., 43 (1998), 1658–1662. https://doi.org/10.1109/9.728891 doi: 10.1109/9.728891
[4]	V. Chellaboina, W. M. Haddad, D. S. Bernstein, D. A. Wilson, Induced convolution operator norms of linear dynamical systems, Math. Control Signal., 13 (2000), 216–239. https://doi.org/10.1007/PL00009868 doi: 10.1007/PL00009868
[5]	K. Glover, All optimal Hankel-norm approximations of linear multivariable systems and their $L^\infty$-error bounds, Int. J. Control, 39 (1984), 1115–1193. https://doi.org/10.1080/00207178408933239 doi: 10.1080/00207178408933239
[6]	S. Y. Kung, D. W. Lin, Optimal Hankel-norm model reductions: Multivariable systems, IEEE T. Automat. Contr., 26 (1981), 832–852. https://doi.org/10.1109/TAC.1981.1102736 doi: 10.1109/TAC.1981.1102736
[7]	J. R. Partington, An introduction to Hankel operators, London Mathematical Society Student Texts 13, Cambridge University Press, Cambridge, 1988. https://doi.org/10.1017/CBO9780511623769
[8]	K. Zhou, J. C. Doyle, Essentials of robust control, Prentice Hall, 1998.
[9]	X. M. Zhang, Q. L. Han, X. Ge, B. Ning, B. L. Zhang, Sampled-data control systems with non-uniform sampling: A survey of methods and trends, Annu. Rev. Control, 55 (2023), 70–91. https://doi.org/10.1016/j.arcontrol.2023.03.004 doi: 10.1016/j.arcontrol.2023.03.004
[10]	T. Chen, B. A. Francis, $H_2$-optimal sampled-data control, IEEE T. Automat. Contr., 36 (1991), 387–397. https://doi.org/10.1109/9.75098 doi: 10.1109/9.75098
[11]	P. P. Khargonekar, N. Sivashankar, $H_2$ optimal control for sampled-data systems, Syst. Control Lett., 17 (1991), 425–436. https://doi.org/10.1016/0167-6911(91)90082-P doi: 10.1016/0167-6911(91)90082-P
[12]	B. Bamieh, J. B. Pearson, The $H^2$ problem for sampled-data systems, Syst. Control Lett., 19 (1992), 1–12. https://doi.org/10.1016/0167-6911(92)90033-O doi: 10.1016/0167-6911(92)90033-O
[13]	T. Hagiwara, M. Araki, FR-operator approach to the $H_2$ analysis and synthesis of sampled-data systems, IEEE T. Automat. Contr., 40 (1995), 1411–1421. https://doi.org/10.1109/9.402221 doi: 10.1109/9.402221
[14]	J. H. Kim, T. Hagiwara, Extensive theoretical/numerical comparative studies on $H_2$ and generalised $H_2$ norms in sampled-data systems, Int. J. Control, 90 (2017), 2538–2553. https://doi.org/10.1080/00207179.2016.1257158 doi: 10.1080/00207179.2016.1257158
[15]	T. Chen, B. A. Francis, On the $L_2$-induced norm of a sampled-data system, Syst. Control Lett., 15 (1990), 211–219. https://doi.org/10.1016/0167-6911(90)90114-A doi: 10.1016/0167-6911(90)90114-A
[16]	P. T. Kabamba, S. Hara, Worst-case analysis and design of sampled-data control systems, IEEE T. Automat. Contr., 38 (1993), 1337–1358. https://doi.org/10.1109/9.237646 doi: 10.1109/9.237646
[17]	B. Bamieh, J. B. Pearson, A general framework for linear periodic systems with application to $H^{\infty}$ sampled-data control, IEEE T. Automat. Contr., 37 (1992), 418–435. https://doi.org/10.1109/9.126576 doi: 10.1109/9.126576
[18]	H. T. Toivonen, Sampled-data control of continuous-time systems with an $H_\infty$ optimality criterion, Automatica, 28 (1992), 45–54. https://doi.org/10.1016/0005-1098(92)90006-2 doi: 10.1016/0005-1098(92)90006-2
[19]	Y. Hayakawa, S. Hara, Y. Yamamoto, $H_\infty$ type problem for sampled-data control systems—a solution via minimum energy characterization, IEEE T. Automat. Contr., 39 (1994), 2278–2284. https://doi.org/10.1109/9.333776 doi: 10.1109/9.333776
[20]	G. G. Zhu, R. E. Skelton, $L_2$ to $L_\infty$ gains for sampled-data systems, Int. J. Control, 61 (1995), 19–32. https://doi.org/10.1080/00207179508921890 doi: 10.1080/00207179508921890
[21]	G. E. Dullerud, B. A. Francis, $L_1$ analysis and design of sampled-data systems, IEEE T. Automat. Contr., 37 (1992), 436–446. https://doi.org/10.1109/9.126577 doi: 10.1109/9.126577
[22]	N. Sivashankar, P. P. Khargonekar, Induced norms for sampled-data systems, Automatica, 28 (1992), 1267–1272. https://doi.org/10.1016/0005-1098(92)90072-N doi: 10.1016/0005-1098(92)90072-N
[23]	B. A. Bamieh, M. A. Dahleh, J. B. Pearson, Minimization of the $L^\infty$-induced norm for sampled-data systems, IEEE T. Automat. Contr., 38 (1993), 717–732. https://doi.org/10.1109/9.277236 doi: 10.1109/9.277236
[24]	J. H. Kim, T. Hagiwara, $L_1$ discretization for sampled-data controller synthesis via piecewise linear approximation, IEEE T. Automat. Contr., 61 (2016), 1143–1157. https://doi.org/10.1109/TAC.2015.2452815 doi: 10.1109/TAC.2015.2452815
[25]	T. Hagiwara, A. Inai, J. H. Kim, The $L_\infty/L_2$ Hankel operator/norm of sampled-data systems, SIAM J. Control Optim., 56 (2018), 3685–3707. https://doi.org/10.1137/17M1123146 doi: 10.1137/17M1123146
[26]	T. Hagiwara, A. Inai, J. H. Kim, On well-definablity of the $L_\infty/L_2$ Hankel operator and detection of all the critical instants in sampled-data systems, IET Control Theory A., 15 (2021), 668–682. https://doi.org/10.1049/cth2.12069 doi: 10.1049/cth2.12069
[27]	T. Hagiwara, H. Hara, Quasi $L_2/L_2$ Hankel norms and $L_2/L_2$ Hankel norm/operator of sampled-data systems, IEEE T. Automat. Contr., 68 (2023), 4428–4434. https://doi.org/10.1109/TAC.2022.3205270 doi: 10.1109/TAC.2022.3205270
[28]	R. Shiga, T. Hagiwara, Y. Ebihara, External positivity of sampled-data systems and their $L_q/L_\infty$ Hankel norm analysis, Trans. Soc. Instrum. Control Eng., 59 (2023), 92–102. https://doi.org/10.9746/sicetr.59.92 doi: 10.9746/sicetr.59.92
[29]	K. Chongsrid, S. Hara, Hankel norm of sampled-data systems, IEEE T. Automat. Contr., 40 (1995), 1939–1942. https://doi.org/10.1109/9.471220 doi: 10.1109/9.471220
[30]	Y. Yamamoto, A function space approach to sampled data control systems and tracking problems, IEEE T. Automat. Contr., 39 (1994), 703–713. https://doi.org/10.1109/9.286247 doi: 10.1109/9.286247
[31]	T. Hagiwara, H. Umeda, Modified fast-sample/fast-hold approximation for sampled-data system analysis, Eur. J. Control, 14 (2008), 286–296. https://doi.org/10.3166/ejc.14.286-296 doi: 10.3166/ejc.14.286-296
[32]	C. F. V. Loan, Computing integrals involving the matrix exponential, IEEE T. Automat. Contr., 23 (1978), 395–404. https://doi.org/10.1109/TAC.1978.1101743 doi: 10.1109/TAC.1978.1101743
[33]	B. Bamieh, J. N. Pearson, B. A. Francis, A. Tannenbaum, A lifting technique for linear periodic systems with applications to sampled-data control, Syst. Control Lett., 17 (1991), 79–88. https://doi.org/10.1016/0167-6911(91)90033-B doi: 10.1016/0167-6911(91)90033-B
[34]	G. C. Goodwin, M. Salgado, Frequency domain sensitivity functions for continuous time systems under sampled data control, Automatica, 30 (1996), 1263–1270. https://doi.org/10.1016/0005-1098(94)90107-4 doi: 10.1016/0005-1098(94)90107-4
[35]	M. Araki, Y. Ito, T. Hagiwara, Frequency response of sampled-data systems, Automatica, 32 (1996), 483–497. https://doi.org/10.1016/0005-1098(95)00162-X doi: 10.1016/0005-1098(95)00162-X
[36]	J. H. Kim, T. Hagiwara, A discretization approach to the analysis of yet another $H_2$ norm of LTI sampled-data systems, In: Proc. 56th IEEE Conference on Decision and Control, IEEE, Australia, 2017, 3594–3599. https://doi.org/10.1109/CDC.2017.8264187
[37]	J. H. Kim, T. Hagiwara, A study on the optimal controller synthesis for minimizing the $L_2$ norm of the response to the worst-timing impulse disturbance in LTI sampled-data systems, In: Proc. 57th IEEE Conference on Decision and Control, IEEE, USA, 2018, 6626–6631. https://doi.org/10.1109/CDC.2018.8619492

Reader Comments

Your name:*

Email:*
© 2024 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)