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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 $\mathbb{R}^M$-valued reference signal
r of class W1,∞ (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 ${\varphi}(t)\| e(t)\| < 1$ for all t ≥ 0, where φ a prescribed real-valued function of class
W1,∞ with the property that φ(s) > 0 for all s >
0 and $\liminf_{s\rightarrow\infty}{\varphi}(s)>0$. A simple (neither
adaptive nor dynamic) error feedback control of the form $u(t)=-
\alpha ({\varphi}(t)\|e(t)\|)e(t)$ is introduced which achieves the
objective whilst maintaining boundedness of the control and of the
scalar gain
$\alpha ({\varphi}(\cdot )\|e(\cdot )\|)$.
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