Tracking with prescribed transient behaviour

Ilchmann, Achim; Ryan, E. P.; Sangwin, C. J.

ESAIM: Control, Optimisation and Calculus of Variations, Volume 7 (2002), pp. 471-493.

Universal tracking control is investigated in the context of a class $𝒮$ 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 ${ℝ}^M$-valued reference signal $r$ of class $W^{1,\infty }$ (absolutely continuous and bounded with essentially bounded derivative) and every system of class $𝒮$, 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 $\alpha \left(\varphi \right(·)\parallel e(·)\parallel )$.

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