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

Abstract

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A tracking problem is considered in the context of a class 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 𝒮 : given λ 0 , construct a feedback strategy which ensures that, for every r (assumed bounded with essentially bounded derivative) and every system of class 𝒮 , the tracking error e = y - r is such that, in the case λ > 0 , lim sup t e ( t ) < λ or, in the case λ = 0 , lim t e ( t ) = 0 . The second objective is guaranteed output transient performance: the error is required to evolve within a prescribed performance funnel ϕ (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 ) = - ν ( k ( t ) ) θ ( e ( t ) ) with k ( t ) = α ( ϕ ( t ) e ( t ) ) , whilst maintaining boundedness of the control and gain functions u and k. In the case λ = 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 λ 0 .

How to cite

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Ryan, 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 -

References

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