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Hybrid feedback laws for a class of cascade nonlinear control systems. (English) Zbl 0862.93048

The authors consider systems of the form

$\stackrel{˙}{\theta }=f\left(y,t\right),\phantom{\rule{1.em}{0ex}}\stackrel{˙}{x}=Ax+Bu,\phantom{\rule{1.em}{0ex}}y=Cx+Du·$

The main purpose is to construct an input function $u\left(t\right)$ which may depend on the fiber variable $\theta$ and such that the system becomes asymptotically stable at the origin when $u$ is replaced by $u\left(t\right)$.

The construction involves a family of functions $U\left(\alpha ,t\right)$ which are periodic of period $T$ with respect to $t$. The function $u\left(t\right)$ is defined according to the law

$u\left(t\right)=U\left({\alpha }_{k},t\right)\phantom{\rule{1.em}{0ex}}\text{for}\phantom{\rule{4.pt}{0ex}}t\in \left[kT,\left(k+1\right)T\right]$

for a suitable choice of the sequence $\left\{{\alpha }_{k}\right\}$.

The construction is valid under certain controllability-like assumptions on the $\theta$-subsystem.

##### MSC:
 93D15 Stabilization of systems by feedback 93A99 General systems theory