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 $\lambda >0$, ${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 ${\mathcal{F}}_{\varphi}$ (determined by a function $\varphi $). For suitably chosen functions $\alpha $, $\nu $ and $\theta $, both objectives are achieved via a control structure of the form $u\left(t\right)=-\nu \left(k\right(t\left)\right)\theta \left(e\right(t\left)\right)$ with $k\left(t\right)=\alpha \left(\varphi \right(t)\parallel e(t)\parallel )$, 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 $\lambda \ge 0$.

Keywords: 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}, mrnumber = {2567243}, zbl = {1175.93188}, language = {en}, url = {http://archive.numdam.org/articles/10.1051/cocv:2008045/} }

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 SP - 745 EP - 762 VL - 15 IS - 4 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/cocv:2008045/ DO - 10.1051/cocv:2008045 LA - en ID - COCV_2009__15_4_745_0 ER -

%0 Journal Article %A Ryan, Eugene P. %A Sangwin, Chris J. %A Townsend, Philip %T Controlled functional differential equations : approximate and exact asymptotic tracking with prescribed transient performance %J ESAIM: Control, Optimisation and Calculus of Variations %D 2009 %P 745-762 %V 15 %N 4 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/cocv:2008045/ %R 10.1051/cocv:2008045 %G en %F 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, Volume 15 (2009) no. 4, pp. 745-762. doi : 10.1051/cocv:2008045. http://archive.numdam.org/articles/10.1051/cocv:2008045/

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