Nonlinear control systems theory in Romania and the influence of the contributions of V. A. Yakubovich

Abstract

Theory of automatic control started in Romania as an essentially linear theory based on transfer functions (input/output representations). The introduction of the nonlinear theory started in the late 50ies of the past century by some research papers due to A. Halanay and V. M. Popov -with special reference to absolute stability. But only after the discovery by V.M. Popov of the famous frequency domain absolute stability inequality it became possible to connect his method to the classical ones which relied on Liapunov functions: this led to the assimilation of the results of V.A. Yakubovich on matrix inequalities which grew in the Yakubovich Kalman Popov lemma and the positiveness theory. The positiveness theory at its turn is strongly connected with the hyperstability theory and thus, with dissipativeness and passivity. The next decades witnessed applications to forced oscillations, generalization of the positiveness theory to the indefinite sign case, the start of the rigorous theory for adaptive systems -with some roots in Romania also -and the extension to discrete time and time delay systems. Consideration of the periodic discrete time systems witnessed a return to the book of Yakubovich and Starzhinskii on systems with periodic coefficients and its first extensions to the discrete time case, including some results on discrete time parametric resonance. Worth mentioning that the influence of V.A. Yakubovich generated the opening to the representatives of his school and, in general, to the Sankt Petesburg University school of differential equations and qualitative control theory.