On the circle criterion for boundary control systems in factor form : Lyapunov stability and Lur’e equations

Piotr Grabowski; Frank M. Callier

ESAIM: Control, Optimisation and Calculus of Variations (2006)

  • Volume: 12, Issue: 1, page 169-197
  • ISSN: 1292-8119

Abstract

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A Lur’e feedback control system consisting of a linear, infinite-dimensional system of boundary control in factor form and a nonlinear static sector type controller is considered. A criterion of absolute strong asymptotic stability of the null equilibrium is obtained using a quadratic form Lyapunov functional. The construction of such a functional is reduced to solving a Lur’e system of equations. A sufficient strict circle criterion of solvability of the latter is found, which is based on results by Oostveen and Curtain [Automatica 34 (1998) 953–967]. All the results are illustrated in detail by an electrical transmission line example of the distortionless loaded ℜ𝔏ℭ𝔊 -type. The paper uses extensively the philosophy of reciprocal systems with bounded generating operators as recently studied and used by Curtain in (2003) [Syst. Control Lett. 49 (2003) 81–89; SIAM J. Control Optim. 42 (2003) 1671–1702].

How to cite

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Grabowski, Piotr, and Callier, Frank M.. "On the circle criterion for boundary control systems in factor form : Lyapunov stability and Lur’e equations." ESAIM: Control, Optimisation and Calculus of Variations 12.1 (2006): 169-197. <http://eudml.org/doc/244909>.

@article{Grabowski2006,
abstract = {A Lur’e feedback control system consisting of a linear, infinite-dimensional system of boundary control in factor form and a nonlinear static sector type controller is considered. A criterion of absolute strong asymptotic stability of the null equilibrium is obtained using a quadratic form Lyapunov functional. The construction of such a functional is reduced to solving a Lur’e system of equations. A sufficient strict circle criterion of solvability of the latter is found, which is based on results by Oostveen and Curtain [Automatica 34 (1998) 953–967]. All the results are illustrated in detail by an electrical transmission line example of the distortionless loaded $\mathfrak \{RLCG\}$-type. The paper uses extensively the philosophy of reciprocal systems with bounded generating operators as recently studied and used by Curtain in (2003) [Syst. Control Lett. 49 (2003) 81–89; SIAM J. Control Optim. 42 (2003) 1671–1702].},
author = {Grabowski, Piotr, Callier, Frank M.},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {infinite-dimensional control systems; semigroups; Lyapunov functionals; circle criterion},
language = {eng},
number = {1},
pages = {169-197},
publisher = {EDP-Sciences},
title = {On the circle criterion for boundary control systems in factor form : Lyapunov stability and Lur’e equations},
url = {http://eudml.org/doc/244909},
volume = {12},
year = {2006},
}

TY - JOUR
AU - Grabowski, Piotr
AU - Callier, Frank M.
TI - On the circle criterion for boundary control systems in factor form : Lyapunov stability and Lur’e equations
JO - ESAIM: Control, Optimisation and Calculus of Variations
PY - 2006
PB - EDP-Sciences
VL - 12
IS - 1
SP - 169
EP - 197
AB - A Lur’e feedback control system consisting of a linear, infinite-dimensional system of boundary control in factor form and a nonlinear static sector type controller is considered. A criterion of absolute strong asymptotic stability of the null equilibrium is obtained using a quadratic form Lyapunov functional. The construction of such a functional is reduced to solving a Lur’e system of equations. A sufficient strict circle criterion of solvability of the latter is found, which is based on results by Oostveen and Curtain [Automatica 34 (1998) 953–967]. All the results are illustrated in detail by an electrical transmission line example of the distortionless loaded $\mathfrak {RLCG}$-type. The paper uses extensively the philosophy of reciprocal systems with bounded generating operators as recently studied and used by Curtain in (2003) [Syst. Control Lett. 49 (2003) 81–89; SIAM J. Control Optim. 42 (2003) 1671–1702].
LA - eng
KW - infinite-dimensional control systems; semigroups; Lyapunov functionals; circle criterion
UR - http://eudml.org/doc/244909
ER -

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