# Controlled functional differential equations : approximate and exact asymptotic tracking with prescribed transient performance

Eugene P. Ryan; Chris J. Sangwin; Philip Townsend

ESAIM: Control, Optimisation and Calculus of Variations (2009)

- Volume: 15, Issue: 4, page 745-762
- ISSN: 1292-8119

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topRyan, Eugene P., Sangwin, Chris J., and Townsend, Philip. "Controlled functional differential equations : approximate and exact asymptotic tracking with prescribed transient performance." ESAIM: Control, Optimisation and Calculus of Variations 15.4 (2009): 745-762. <http://eudml.org/doc/245044>.

@article{Ryan2009,

abstract = {A tracking problem is considered in the context of a class $\mathcal \{S\}$ of multi-input, multi-output, nonlinear systems modelled by controlled functional differential equations. The class contains, as a prototype, all finite-dimensional, linear, $m$-input, $m$-output, minimum-phase systems with sign-definite “high-frequency gain”. The first control objective is tracking of reference signals $r$ by the output $y$ of any system in $\mathcal \{S\}$: given $\lambda \ge 0$, construct a feedback strategy which ensures that, for every $r$ (assumed bounded with essentially bounded derivative) and every system of class $\mathcal \{S\}$, the tracking error $e = y-r$ is such that, in the case $\lambda >0$, $\limsup _\{t\rightarrow \infty \}\Vert e(t)\Vert <\lambda $ or, in the case $\lambda =0$, $\lim _\{t\rightarrow \infty \}\Vert e(t)\Vert = 0$. The second objective is guaranteed output transient performance: the error is required to evolve within a prescribed performance funnel $\mathcal \{F\}_\varphi $ (determined by a function $\varphi $). For suitably chosen functions $\alpha $, $\nu $ and $\theta $, both objectives are achieved via a control structure of the form $u(t)=-\nu (k(t))\theta (e(t))$ with $k(t)=\alpha (\varphi (t)\Vert e(t)\Vert )$, whilst maintaining boundedness of the control and gain functions $u$ and $k$. In the case $\lambda =0$, the feedback strategy may be discontinuous: to accommodate this feature, a unifying framework of differential inclusions is adopted in the analysis of the general case $\lambda \ge 0$.},

author = {Ryan, Eugene P., Sangwin, Chris J., Townsend, Philip},

journal = {ESAIM: Control, Optimisation and Calculus of Variations},

keywords = {functional differential inclusions; transient behaviour; approximate tracking; asymptotic tracking},

language = {eng},

number = {4},

pages = {745-762},

publisher = {EDP-Sciences},

title = {Controlled functional differential equations : approximate and exact asymptotic tracking with prescribed transient performance},

url = {http://eudml.org/doc/245044},

volume = {15},

year = {2009},

}

TY - JOUR

AU - Ryan, Eugene P.

AU - Sangwin, Chris J.

AU - Townsend, Philip

TI - Controlled functional differential equations : approximate and exact asymptotic tracking with prescribed transient performance

JO - ESAIM: Control, Optimisation and Calculus of Variations

PY - 2009

PB - EDP-Sciences

VL - 15

IS - 4

SP - 745

EP - 762

AB - A tracking problem is considered in the context of a class $\mathcal {S}$ of multi-input, multi-output, nonlinear systems modelled by controlled functional differential equations. The class contains, as a prototype, all finite-dimensional, linear, $m$-input, $m$-output, minimum-phase systems with sign-definite “high-frequency gain”. The first control objective is tracking of reference signals $r$ by the output $y$ of any system in $\mathcal {S}$: given $\lambda \ge 0$, construct a feedback strategy which ensures that, for every $r$ (assumed bounded with essentially bounded derivative) and every system of class $\mathcal {S}$, the tracking error $e = y-r$ is such that, in the case $\lambda >0$, $\limsup _{t\rightarrow \infty }\Vert e(t)\Vert <\lambda $ or, in the case $\lambda =0$, $\lim _{t\rightarrow \infty }\Vert e(t)\Vert = 0$. The second objective is guaranteed output transient performance: the error is required to evolve within a prescribed performance funnel $\mathcal {F}_\varphi $ (determined by a function $\varphi $). For suitably chosen functions $\alpha $, $\nu $ and $\theta $, both objectives are achieved via a control structure of the form $u(t)=-\nu (k(t))\theta (e(t))$ with $k(t)=\alpha (\varphi (t)\Vert e(t)\Vert )$, whilst maintaining boundedness of the control and gain functions $u$ and $k$. In the case $\lambda =0$, the feedback strategy may be discontinuous: to accommodate this feature, a unifying framework of differential inclusions is adopted in the analysis of the general case $\lambda \ge 0$.

LA - eng

KW - functional differential inclusions; transient behaviour; approximate tracking; asymptotic tracking

UR - http://eudml.org/doc/245044

ER -

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