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Stability of composite systems with an asymptotically hyperstable subsystem. (English) Zbl 0606.93059

The system consists of two finite-dimensional linear subsystems described by state-space equations with scalar input u and outputs ${\sigma }_{1}$ and ${\sigma }_{2}:{\stackrel{˙}{x}}_{1}={A}_{1}{x}_{1}{b}_{1}u,\phantom{\rule{1.em}{0ex}}{\sigma }_{1}={c}_{1}^{T}{x}_{1},\phantom{\rule{1.em}{0ex}}{x}_{1}\left(0\right)={x}_{10}$

${\stackrel{˙}{x}}_{2}={A}_{2}{x}_{2}+{b}_{2}u,\phantom{\rule{1.em}{0ex}}{\sigma }_{2}={c}_{2}^{T}{x}_{2},\phantom{\rule{1.em}{0ex}}{x}_{2}\left(0\right)={x}_{20},$
$u\left(t\right)=-\phi \left({\sigma }_{1}\left(t\right)+{\sigma }_{2}\left(t\right)\right),$

where ${A}_{i}$, ${c}_{i}$, ${b}_{i}$ are ${n}_{i}×{n}_{i}$ real matrices and ${n}_{i}$-vectors, respectively $\left(i=1,2\right)$; ${x}_{i}$ $\left(i=1,2\right)$ are the state vectors and $\phi$ is a real-vaued continuous function. One of the two subsystems is assumed to be hyperstable. Under some additional assumptions stability properties of the composite system are studied.

Reviewer: L.Faibusovich
##### MSC:
 93D10 Popov-type stability of feedback systems 34D99 Stability theory of ODE 93A15 Large scale systems
##### Keywords:
hyperstability; scalar input; composite system