Controlled functional differential equations : approximate and exact asymptotic tracking with prescribed transient performance

A tracking problem is considered in the context of a class $𝒮$ 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 $𝒮$: 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 $𝒮$, 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 ${ℱ}_{\varphi }$ (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 $\lambda \ge 0$.