On approximation of stability radius for an infinite-dimensional feedback control system

Hideki Sano

Kybernetika (2016)

  • Volume: 52, Issue: 5, page 824-835
  • ISSN: 0023-5954

Abstract

top
In this paper, we discuss the problem of approximating stability radius appearing in the design procedure of finite-dimensional stabilizing controllers for an infinite-dimensional dynamical system. The calculation of stability radius needs the value of H ∞ -norm of a transfer function whose realization is described by infinite-dimensional operators in a Hilbert space. From the computational point of view, we need to prepare a family of approximate finite-dimensional operators and then to calculate the H ∞ -norm of their transfer functions. However, it is not assured that they converge to the value of H ∞ -norm of the original transfer function. The purpose of this study is to justify the convergence. In a numerical example, we treat parabolic distributed parameter systems with distributed control and distributed/boundary observation.

How to cite

top

Sano, Hideki. "On approximation of stability radius for an infinite-dimensional feedback control system." Kybernetika 52.5 (2016): 824-835. <http://eudml.org/doc/287547>.

@article{Sano2016,
abstract = {In this paper, we discuss the problem of approximating stability radius appearing in the design procedure of finite-dimensional stabilizing controllers for an infinite-dimensional dynamical system. The calculation of stability radius needs the value of $H_\infty $-norm of a transfer function whose realization is described by infinite-dimensional operators in a Hilbert space. From the computational point of view, we need to prepare a family of approximate finite-dimensional operators and then to calculate the $H_\infty $-norm of their transfer functions. However, it is not assured that they converge to the value of $H_\infty $-norm of the original transfer function. The purpose of this study is to justify the convergence. In a numerical example, we treat parabolic distributed parameter systems with distributed control and distributed/boundary observation.},
author = {Sano, Hideki},
journal = {Kybernetika},
keywords = {distributed parameter system; finite-dimensional controller; stability radius; transfer function; semigroup; distributed parameter system; finite-dimensional controller; stability radius; transfer function; semigroup},
language = {eng},
number = {5},
pages = {824-835},
publisher = {Institute of Information Theory and Automation AS CR},
title = {On approximation of stability radius for an infinite-dimensional feedback control system},
url = {http://eudml.org/doc/287547},
volume = {52},
year = {2016},
}

TY - JOUR
AU - Sano, Hideki
TI - On approximation of stability radius for an infinite-dimensional feedback control system
JO - Kybernetika
PY - 2016
PB - Institute of Information Theory and Automation AS CR
VL - 52
IS - 5
SP - 824
EP - 835
AB - In this paper, we discuss the problem of approximating stability radius appearing in the design procedure of finite-dimensional stabilizing controllers for an infinite-dimensional dynamical system. The calculation of stability radius needs the value of $H_\infty $-norm of a transfer function whose realization is described by infinite-dimensional operators in a Hilbert space. From the computational point of view, we need to prepare a family of approximate finite-dimensional operators and then to calculate the $H_\infty $-norm of their transfer functions. However, it is not assured that they converge to the value of $H_\infty $-norm of the original transfer function. The purpose of this study is to justify the convergence. In a numerical example, we treat parabolic distributed parameter systems with distributed control and distributed/boundary observation.
LA - eng
KW - distributed parameter system; finite-dimensional controller; stability radius; transfer function; semigroup; distributed parameter system; finite-dimensional controller; stability radius; transfer function; semigroup
UR - http://eudml.org/doc/287547
ER -

References

top
  1. Balas, M. J., 10.1016/0022-247x(88)90401-5, J. Math. Anal. Appl. 133 (1988), 283-296. Zbl0647.93036MR0954706DOI10.1016/0022-247x(88)90401-5
  2. Chicone, C., Latushkin, Y., 10.1090/surv/070, Mathematical Surveys and Monographs, vol. 70. American Mathematical Society, 1999. Zbl0970.47027MR1707332DOI10.1090/surv/070
  3. Curtain, R. F., 10.1137/0322018, SIAM J. Control Optim. 22 (1984), 255-276. Zbl0542.93056MR0732427DOI10.1137/0322018
  4. Glowinski, R., Lions, J.-L., He, J., Exact and Approximate Controllability for Distributed Parameter Systems: A Numerical Approach., In: Encyclopedia of Mathematics and Its Applications, vol. 117. Cambridge University Press, Cambridge 2008. Zbl1142.93002MR2404764
  5. Jai, A. El, Pritchard, A. J., Sensors and Controls in the Analysis of Distributed Systems. English Language Edition., Ellis Horwood Limited, Chichester 1988. MR0933327
  6. Guglielmi, N., Manetta, M., 10.1093/imanum/dru038, IMA J. Numerical Analysis 35 (2015), 1402-1425. Zbl1328.65168MR3407266DOI10.1093/imanum/dru038
  7. Lasiecka, I., Triggiani, R., 10.1017/cbo9780511574801, In: Encyclopedia of Mathematics and Its Applications, vol. 74. Cambridge University Press, Cambridge 2000. MR1745475DOI10.1017/cbo9780511574801
  8. Nambu, T., On stabilization of partial differential equations of parabolic type: boundary observation and feedback., Funkcialaj Ekvacioj, Serio Internacia 28 (1985), 267-298. Zbl0614.93049MR0852115
  9. Pazy, A., 10.1007/978-1-4612-5561-1, In: Applied Mathematical Sciences, vol. 44. Springer-Verlag, New York 1983. Zbl0516.47023MR0710486DOI10.1007/978-1-4612-5561-1
  10. Pritchard, A. J., Townley, S., 10.1016/0167-6911(88)90108-9, Systems Control Lett. 11 (1988), 33-37. Zbl0648.93047MR0949887DOI10.1016/0167-6911(88)90108-9
  11. Pritchard, A. J., Townley, S., 10.1016/0022-0396(89)90144-7, J. Differential Equations 77 (1989), 254-286. Zbl0674.34061MR0983295DOI10.1016/0022-0396(89)90144-7
  12. Sakawa, Y., 10.1137/0321040, SIAM J. Control Optim. 21 (1983), 667-676. Zbl0527.93050MR0710993DOI10.1137/0321040
  13. Sano, H., Kunimatsu, N., 10.1016/0893-9659(94)90065-5, Appl. Math. Lett. 7 (1994), 5, 17-22. MR1350603DOI10.1016/0893-9659(94)90065-5
  14. Sano, H., 10.1080/002071799220119, Int. J. Control 72 (1999), 16, 1466-1479. MR1723338DOI10.1080/002071799220119
  15. Sano, H., 10.1016/j.jmaa.2011.11.011, J. Math. Anal. Appl. 388 (2012), 1194-1204. Zbl1243.35166MR2869818DOI10.1016/j.jmaa.2011.11.011
  16. Schumacher, J. M., 10.1137/0321050, SIAM J. Control Optim. 21 (1983), 823-836. Zbl0524.93054MR0719515DOI10.1137/0321050
  17. Zhou, K., Doyle, J., Glover, K., Robust and Optimal Control. (In Japanese, translated from the English by K. Z. Liu and Z. H. Luo.), Corona, Tokyo 1997. 

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.