Funnel Control for Systems with Relative Degree Two

Tracking of reference signals $y_{\mathrm{ref}}(\cdot)$ by the output $y(\cdot)$ of linear (as well as a considerably large class of nonlinear) single-input, single-output systems is considered. The system is assumed to have strict relative degree two with (weakly) stable zero dynamics. The control objective is tracking of the error $e=y-y_{\mathrm{ref}}$ and its derivative $\dot{e}$ within two prespecified performance funnels, respectively. This is achieved by the so-called funnel controller $u(t)=-k_0(t)^2e(t)-k_1(t)\dot{e}(t)$, where the simple proportional error feedback has gain functions $k_0$ and $k_1$ designed in such a way to preclude contact of $e$ and $\dot{e}$ with the funnel boundaries, respectively. The funnel controller also ensures boundedness of all signals. We also show that the same funnel controller (i) is applicable to relative degree one systems, (ii) allows for input constraints provided a feasibility condition (formulated in terms of the system data, the saturation bounds, the funn...