# 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 (2008)

- 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 (2008): 745-762. <http://eudml.org/doc/90935>.

@article{Ryan2008,

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 $\lambda \geq 0$, 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
$\lambda >0$, $\limsup_\{t\rightarrow\infty\}\|e(t)\|<\lambda$ or, in
the case $\lambda=0$, $\lim_\{t\rightarrow\infty\}\|e(t)\| = 0$. 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 φ). For
suitably chosen functions α, ν and θ, both
objectives are achieved via a control structure of the form
$u(t)=-\nu (k(t))\theta (e(t))$ with $k(t)=\alpha (\varphi
(t)\|e(t)\|)$, 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 $\lambda \geq 0$.
},

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; functional differential inclusions; asymptotic tracking},

language = {eng},

month = {7},

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/90935},

volume = {15},

year = {2008},

}

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

DA - 2008/7//

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 $\lambda \geq 0$, 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
$\lambda >0$, $\limsup_{t\rightarrow\infty}\|e(t)\|<\lambda$ or, in
the case $\lambda=0$, $\lim_{t\rightarrow\infty}\|e(t)\| = 0$. 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 φ). For
suitably chosen functions α, ν and θ, both
objectives are achieved via a control structure of the form
$u(t)=-\nu (k(t))\theta (e(t))$ with $k(t)=\alpha (\varphi
(t)\|e(t)\|)$, 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 $\lambda \geq 0$.

LA - eng

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

UR - http://eudml.org/doc/90935

ER -

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