Universal tracking control is investigated in the context of a class $\mathcal{S}$ of $M$-input, $M$-output dynamical systems modelled by functional differential equations. The class encompasses a wide variety of nonlinear and infinite-dimensional systems and contains - as a prototype subclass - all finite-dimensional linear single-input single-output minimum-phase systems with positive high-frequency gain. The control objective is to ensure that, for an arbitrary ${\mathbb{R}}^{M}$-valued reference signal $r$ of class ${W}^{1,\infty}$ (absolutely continuous and bounded with essentially bounded derivative) and every system of class $\mathcal{S}$, the tracking error $e$ between plant output and reference signal evolves within a prespecified performance envelope or funnel in the sense that $\varphi \left(t\right)\parallel e\left(t\right)\parallel <1$ for all $t\ge 0$, where $\varphi $ a prescribed real-valued function of class ${W}^{1,\infty}$ with the property that $\varphi \left(s\right)>0$ for all $s>0$ and ${lim\; inf}_{s\to \infty}\varphi \left(s\right)>0$. A simple (neither adaptive nor dynamic) error feedback control of the form $u\left(t\right)=-\alpha \left(\varphi \right(t)\parallel e(t)\parallel )e\left(t\right)$ is introduced which achieves the objective whilst maintaining boundedness of the control and of the scalar gain $\alpha \left(\varphi \right(\xb7)\parallel e(\xb7)\parallel )$.

Keywords: nonlinear systems, functional differential equations, feedback control, tracking, transient behaviour

@article{COCV_2002__7__471_0, author = {Ilchmann, Achim and Ryan, E. P. and Sangwin, C. J.}, title = {Tracking with prescribed transient behaviour}, journal = {ESAIM: Control, Optimisation and Calculus of Variations}, pages = {471--493}, publisher = {EDP-Sciences}, volume = {7}, year = {2002}, doi = {10.1051/cocv:2002064}, zbl = {1044.93022}, mrnumber = {1925038}, language = {en}, url = {http://www.numdam.org/articles/10.1051/cocv:2002064/} }

TY - JOUR AU - Ilchmann, Achim AU - Ryan, E. P. AU - Sangwin, C. J. TI - Tracking with prescribed transient behaviour JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2002 SP - 471 EP - 493 VL - 7 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/cocv:2002064/ UR - https://zbmath.org/?q=an%3A1044.93022 UR - https://www.ams.org/mathscinet-getitem?mr=1925038 UR - https://doi.org/10.1051/cocv:2002064 DO - 10.1051/cocv:2002064 LA - en ID - COCV_2002__7__471_0 ER -

%0 Journal Article %A Ilchmann, Achim %A Ryan, E. P. %A Sangwin, C. J. %T Tracking with prescribed transient behaviour %J ESAIM: Control, Optimisation and Calculus of Variations %D 2002 %P 471-493 %V 7 %I EDP-Sciences %U https://doi.org/10.1051/cocv:2002064 %R 10.1051/cocv:2002064 %G en %F COCV_2002__7__471_0

Ilchmann, Achim; Ryan, E. P.; Sangwin, C. J. Tracking with prescribed transient behaviour. ESAIM: Control, Optimisation and Calculus of Variations, Volume 7 (2002), pp. 471-493. doi : 10.1051/cocv:2002064. http://www.numdam.org/articles/10.1051/cocv:2002064/

[1] Adaptive stabilization of multivariable linear systems, in Proc. 23rd Conf. on Decision and Control. Las Vegas (1984) 1574-1577.

and ,[2] Systems of controlled functional differential equations and adaptive tracking. SIAM J. Control Optim. 40 (2002) 1746-1764. | MR | Zbl

, and ,[3] Low-gain integral control of infinite dimensional regular linear systems subject to input hysteresis, in Advances in Mathematical Systems Theory, edited by F. Colonius, U. Helmke, D. Prätzel-Wolters and F. Wirth. Birkhäuser Verlag, Boston, Basel, Berlin (2000) 255-293.

and ,[4] An adaptive controller which provides an arbitrarily good transient and steady-state response. IEEE Trans. Automat. Control 36 (1991) 68-81. | MR | Zbl

and ,[5] Controlled functional differential equations and adaptive stabilization. Int. J. Control 74 (2001) 77-90. | MR | Zbl

and ,[6] Smooth stabilization implies coprime factorization. IEEE Trans. Automat. Control 34 (1989) 435-443. | MR | Zbl

,[7] The Lorenz equations: Bifurcations, chaos and strange attractors. Springer-Verlag, New York (1982). | MR | Zbl

,[8] Transfer functions of regular linear systems, Part 1: Characterization of regularity. Trans. Amer. Math. Soc. 342 (1994) 827-854. | MR | Zbl

,*Cited by Sources: *