Qualitative Control Theory Research Articles

On Geometry of Vector Fields

It is well known that the study of the geometry and topology of the attainability set of a family of vector fields is one of the main tasks of the qualitative control theory, which is closely related to the geometry of orbits of vector fields. In this paper, we present the authors’ results on the geometry of the attainability set of a family of vector fields: results on the geometry of T -attainability sets and the geometry of orbits of Killing vector fields.

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

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.

Control of systems with aftereffect in scales of linear controllers with respect to the type of feedback

For control systems with deviating argument, we consider basic problems of qualitative control theory such as problems of stabilization and modal control in scales of linear controllers with respect to the type of feedback, from the simplest difference controllers to integral controllers of general form. We analyze the results obtained in this direction. Special attention is paid to the stabilization of two-dimensional systems by feedback in the form of difference controllers. For the case in which the construction of a difference controller is impossible or too difficult, an integral feedback is used. Unlike the well-known Krasovskii-Osipov method for the construction of integral feedback in delay systems, the suggested method is based on the Paley-Wiener theorem for entire functions of exponential type.

A Brief Review of the Development of Qualitative Control Theory in Belarus

The main stages of the development of the qualitative control and observation theory in Belarus are analyzed. Some fundamental results are presented. Unsolved problems are formulated.

Some Problems of Qualitative Control Theory for Time-delay Systems

The problem of modal control is extensively considered here for linear stationary time-delay systems of neutral type. To solve this problem a new type of regulators containing both time delays and state derivatives is introduced. Effective condition for the modal control by action of such regulators is given for the complete information case. In order to stUdy the modal control problem for systems with incomplete information an analog of the well-known dynamic regulator proposed by Pearson for systems with no delay is used.

On the Qualitative Control Theory in Dynamic Systems with Distributed Delay

The paper deals with the problems of controllability and observability of delay-differential systems which directly generalize the corresponding problems for ordinary linear dynamic systems without delay. We introduce a number of definitions which define more exactly a notion of the state for the hereditary systems as the minimal part of data needed for generating a unique trajectory of the system. We say that two initial data are s-equivalent if for every control function the corresponding solutions of the system coinside for t ≥ s (t is a time parameter). A set of classes of s-equivalent initial data is interpreted as a set of initial s-states of the system. Then problems of controllability and observability are formulated in a similar way as in Kaiman’s systems theory. We also introduce a notion of s-informator for the characterization of the s-states and generalize the well-known Lagrange’s identity to the systems with delay. This result is used for the investigation of some controllability problems of delay functional systems. Dual results and problem of computation of the minimum number of outputs needed for ∞-observability of the system is also presented.