Circle criterion and boundary control systems in factor form: input-output approach

Piotr Grabowski; Frank Callier

International Journal of Applied Mathematics and Computer Science (2001)

  • Volume: 11, Issue: 6, page 1387-1403
  • ISSN: 1641-876X

Abstract

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A circle criterion is obtained for a SISO Lur’e feedback control system consist- ing of a nonlinear static sector-type controller and a linear boundary control system in factor form on an infinite-dimensional Hilbert state space H previ- ously introduced by the authors (Grabowski and Callier, 1999). It is assumed for the latter that (a) the observation functional is infinite-time admissible, (b) the factor control vector satisfies a compatibility condition, and (c) the trans- fer function belongs to H∞ (Π+ ) and satisfies a frequency-domain inequality of the circle criterion type. We also require that the closed-loop system be well- posed, i.e. for any initial state x0 ∈ H the truncated input and output sig- nals uT , yT belong to L2 (0, T ) for any T > 0. The technique of the proof adapts Desoer-Vidyasagar’s circle criterion method (Desoer and Vidyasagar, 1975, Ch. 3, Secs. 1 and 2, pp. 37–43, Ch. 5, Sec. 2, pp. 139–142 and Ch. 6, Secs. 3 and 4, pp. 172–174), and uses the input-output map developed by the authors (Grabowski and Callier, 2001). The results are illustrated by two trans- mission line examples: (a) that of the loaded distortionless RLCG type, and (b) that of the unloaded RC type. The conclusion contains a discussion on improving the results by the loop-transformation technique.

How to cite

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Grabowski, Piotr, and Callier, Frank. "Circle criterion and boundary control systems in factor form: input-output approach." International Journal of Applied Mathematics and Computer Science 11.6 (2001): 1387-1403. <http://eudml.org/doc/207561>.

@article{Grabowski2001,
abstract = {A circle criterion is obtained for a SISO Lur’e feedback control system consist- ing of a nonlinear static sector-type controller and a linear boundary control system in factor form on an infinite-dimensional Hilbert state space H previ- ously introduced by the authors (Grabowski and Callier, 1999). It is assumed for the latter that (a) the observation functional is infinite-time admissible, (b) the factor control vector satisfies a compatibility condition, and (c) the trans- fer function belongs to H∞ (Π+ ) and satisfies a frequency-domain inequality of the circle criterion type. We also require that the closed-loop system be well- posed, i.e. for any initial state x0 ∈ H the truncated input and output sig- nals uT , yT belong to L2 (0, T ) for any T > 0. The technique of the proof adapts Desoer-Vidyasagar’s circle criterion method (Desoer and Vidyasagar, 1975, Ch. 3, Secs. 1 and 2, pp. 37–43, Ch. 5, Sec. 2, pp. 139–142 and Ch. 6, Secs. 3 and 4, pp. 172–174), and uses the input-output map developed by the authors (Grabowski and Callier, 2001). The results are illustrated by two trans- mission line examples: (a) that of the loaded distortionless RLCG type, and (b) that of the unloaded RC type. The conclusion contains a discussion on improving the results by the loop-transformation technique.},
author = {Grabowski, Piotr, Callier, Frank},
journal = {International Journal of Applied Mathematics and Computer Science},
keywords = {input-output relations; infinite-dimensional control systems; semigroups; boundary control; stabilization; well-posedness; transmission line; circle criterion; infinite-dimensional SISO systems},
language = {eng},
number = {6},
pages = {1387-1403},
title = {Circle criterion and boundary control systems in factor form: input-output approach},
url = {http://eudml.org/doc/207561},
volume = {11},
year = {2001},
}

TY - JOUR
AU - Grabowski, Piotr
AU - Callier, Frank
TI - Circle criterion and boundary control systems in factor form: input-output approach
JO - International Journal of Applied Mathematics and Computer Science
PY - 2001
VL - 11
IS - 6
SP - 1387
EP - 1403
AB - A circle criterion is obtained for a SISO Lur’e feedback control system consist- ing of a nonlinear static sector-type controller and a linear boundary control system in factor form on an infinite-dimensional Hilbert state space H previ- ously introduced by the authors (Grabowski and Callier, 1999). It is assumed for the latter that (a) the observation functional is infinite-time admissible, (b) the factor control vector satisfies a compatibility condition, and (c) the trans- fer function belongs to H∞ (Π+ ) and satisfies a frequency-domain inequality of the circle criterion type. We also require that the closed-loop system be well- posed, i.e. for any initial state x0 ∈ H the truncated input and output sig- nals uT , yT belong to L2 (0, T ) for any T > 0. The technique of the proof adapts Desoer-Vidyasagar’s circle criterion method (Desoer and Vidyasagar, 1975, Ch. 3, Secs. 1 and 2, pp. 37–43, Ch. 5, Sec. 2, pp. 139–142 and Ch. 6, Secs. 3 and 4, pp. 172–174), and uses the input-output map developed by the authors (Grabowski and Callier, 2001). The results are illustrated by two trans- mission line examples: (a) that of the loaded distortionless RLCG type, and (b) that of the unloaded RC type. The conclusion contains a discussion on improving the results by the loop-transformation technique.
LA - eng
KW - input-output relations; infinite-dimensional control systems; semigroups; boundary control; stabilization; well-posedness; transmission line; circle criterion; infinite-dimensional SISO systems
UR - http://eudml.org/doc/207561
ER -

References

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  1. Bucci F. (1999): Stability of holomorphic semigroup systems under boundary perturbations, In: Optimal Control of Partial Differential Equations (K.-H. Hoffmann, G. Leugering and F. Tröltzsch, Eds.). — Proc. IFIP WG 7.2 Int. Conf., Chemnitz, Germany, 20–25 April, 1998, ISNM Series, Vol.133, Basel: Birkhäuser, pp.63–76. Zbl0951.47041
  2. Bucci F. (2000): Frequency domain stability of nonlinear feedback systems with unbounded input operator. — Dyn. Cont. Discr. Impuls. Syst., Vol.7, No.3, pp.351–368. Zbl0966.93089
  3. Desoer C.A. and Vidyasagar M. (1975): Feedback Systems: Input-Output Properties. — New York: Academic Press. Zbl0327.93009
  4. Duren P. (1970): Theory of Hp Spaces. — New York: Academic Press. Zbl0215.20203
  5. Górecki H., Fuksa S., Grabowski P. and Korytowski A. (1989): Analysis and Synthesis of Time-Delay Systems. — Chichester: Wiley. Zbl0695.93002
  6. Grabowski P. (1990): On the spectral – Lyapunov approach to parametric optimization of distributed parameter systems. — IMA J. Math. Contr. Inf., Vol.7, No.4, pp.317–338. Zbl0721.49006
  7. Grabowski P. (1994): The LQ controller problem: An example. — IMA J. Math. Contr. Inf., Vol.11, No.4, pp.355–368. Zbl0825.93201
  8. Grabowski P. and Callier F.M. (1999): Admissible observation operators. Duality of observation and control using factorizations. — Dyn. Cont., Discr. Impuls. Syst., Vol.6, pp.87– 119. Zbl0932.93009
  9. Grabowski P. and Callier F.M. (2000): On the circle criterion for boundary control systems in factor form: Lyapunov approach — Facultés Universitaires Notre–Dame de la Paix a Namur, Publications du Département de Mathématique, Research Report 00–07, Namur, Belgium: FUNDP. Submitted to Int. Eqns. Oper. Theory. 
  10. Grabowski P. and Callier F.M. (2001): Boundary control systems in factor form: Transfer functions and input-output maps — Int. Eqns. Oper. Theory, Vol.41, pp.1–37. Zbl1009.93041
  11. Logemann H. (1991): Circle criterion, small-gain conditions and internal stability for infinite-dimensional systems. — Automatica, Vol.27, No.4, pp.677–690. Zbl0749.93073
  12. Logemann H. and Curtain R.F. (2000): Absolute stability results for well-posed infinite- dimensional systems with low-gain integral control. — ESAIM: Contr. Optim. Calc. Var., Vol.5, pp.395–424. Zbl0964.93048
  13. Pazy A. (1993): Semigroups of Linear Operators and Applications to PDEs. — Berlin: Springer. 
  14. Vidyasagar M. (1993): Nonlinear Systems Analysis, 2nd Ed. — Englewood Cliffs NJ: Prentice-Hall. Zbl0900.93132

Citations in EuDML Documents

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  1. Piotr Grabowski, Frank M. Callier, On the circle criterion for boundary control systems in factor form: Lyapunov stability and Lur'e equations
  2. Piotr Grabowski, Frank M. Callier, On the circle criterion for boundary control systems in factor form : Lyapunov stability and Lur’e equations
  3. Krzysztof Bartecki, A general transfer function representation for a class of hyperbolic distributed parameter systems
  4. Łukasz Bartczuk, Andrzej Przybył, Krzysztof Cpałka, A new approach to nonlinear modelling of dynamic systems based on fuzzy rules

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