Controlled functional differential equations : approximate and exact asymptotic tracking with prescribed transient performance
ESAIM: Control, Optimisation and Calculus of Variations, Tome 15 (2009) no. 4, pp. 745-762.

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 $\lambda \ge 0$, 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 $\lambda >0$, ${lim sup}_{t\to \infty }\parallel e\left(t\right)\parallel <\lambda$ or, in the case $\lambda =0$, ${lim}_{t\to \infty }\parallel e\left(t\right)\parallel =0$. The second objective is guaranteed output transient performance: the error is required to evolve within a prescribed performance funnel ${ℱ}_{\varphi }$ (determined by a function $\varphi$). For suitably chosen functions $\alpha$, $\nu$ and $\theta$, both objectives are achieved via a control structure of the form $u\left(t\right)=-\nu \left(k\left(t\right)\right)\theta \left(e\left(t\right)\right)$ with $k\left(t\right)=\alpha \left(\varphi \left(t\right)\parallel e\left(t\right)\parallel \right)$, 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 $\lambda \ge 0$.

DOI : https://doi.org/10.1051/cocv:2008045
Classification : 93D15,  93C30,  34K20,  34A60
Mots clés : functional differential inclusions, transient behaviour, approximate tracking, asymptotic tracking
@article{COCV_2009__15_4_745_0,
author = {Ryan, Eugene P. and Sangwin, Chris J. and Townsend, Philip},
title = {Controlled functional differential equations : approximate and exact asymptotic tracking with prescribed transient performance},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {745--762},
publisher = {EDP-Sciences},
volume = {15},
number = {4},
year = {2009},
doi = {10.1051/cocv:2008045},
zbl = {1175.93188},
mrnumber = {2567243},
language = {en},
url = {http://www.numdam.org/item/COCV_2009__15_4_745_0/}
}
Ryan, Eugene P.; Sangwin, Chris J.; Townsend, Philip. Controlled functional differential equations : approximate and exact asymptotic tracking with prescribed transient performance. ESAIM: Control, Optimisation and Calculus of Variations, Tome 15 (2009) no. 4, pp. 745-762. doi : 10.1051/cocv:2008045. http://www.numdam.org/item/COCV_2009__15_4_745_0/

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