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Positive-real structure and high-gain adaptive stabilization. (English) Zbl 0631.93058
This paper is a mathematical treatise presenting a proof of the stability conditions for a multivariable system with time-varying output feedback and subjected to linear or nonlinear perturbations of the state, output and input. It is assumed that the system is linear, controllable, observable and minimum-phase. The stability of the feedback system is proved with the Lyapunov equation and by relating ‘gain divergence’ $$(\lim_{t\to \infty}k(t)\to \infty$$, where k(t) is a feedback gain) to the system-theoretic criterion for positive-real matrices. No application is inserted and this paper is recommended for mathematicians with an interest in gain adaptation control rules.
Reviewer: H.H.van de Ven

##### MSC:
 93D15 Stabilization of systems by feedback 93C35 Multivariable systems, multidimensional control systems 34D10 Perturbations of ordinary differential equations 34D20 Stability of solutions to ordinary differential equations 93D05 Lyapunov and other classical stabilities (Lagrange, Poisson, $$L^p, l^p$$, etc.) in control theory 93C40 Adaptive control/observation systems
time-dependent
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