G. A. Leonov

Phase synchronization: Theory and applications

The state-of-the-art of the phase synchronization theory was reviewed. Consideration was given to its applications to the synchronous and induction electrical motors, phase locked loops, and autosynchronization of the unbalanced rotors. The Yakubovich-Kalman frequency theorem was widely used to study the phase synchronization systems for global stability.

The Brockett Problem for Linear Discrete Control Systems

The Brockett problem for linear discrete control systems is studied. A method for designing time-varying stabilizing feedback is developed. The Brockett problem for second-order systems with scalar feedback and nondegenerate transfer function is shown to have a positive solution.

On the method of harmonic linearization

The development of numerical methods, computational engineering, and the applied theory of bifurcations made it possible to consider the application and verification of the harmonic linearization procedure from new positions. The joint application of the harmonic linearization procedure, the classical method of a small parameter, and numerical methods enables calculating the periodic modes of automatic control systems by the multistage procedure, where at the first stage the method of harmonic linearization is used.

The Brocket Stabilization Problem

The Brocket stabilization problem for two-dimensional dynamic systems is solved.

On the Aizerman problem

The method of harmonic linearization in degenerate cases is stated and strictly mathematically justified. This method is applied to the search for periodic oscillations for systems satisfying the generalized Routh-Hurwitz condition.

Mathematical models of phase syncronization systems with quadrature and phase-quadrature units

Integro-differential and differential equations are derived for PLL systems with squarer and for the Costas loop. The equations are based on special asymptotic methods of high-frequency harmonic and pulse oscillations analysis. Methods to compute phase detector characteristics are proposed. New classes of such characteristics are introduced for highfrequency pulse clocks.

Pyragas Stabilizability of Unstable Equilibria by Nonstationary Time-Delayed Feedback

The problem of Pyragas stabilizability of unstable equilibria of nonlinear systems is considered. Stabilization algorithm by periodic time-delayed feedback is constructed. An analytical criterion for stabilization is obtained. The proposed approach is based on the method of nonstationary stabilization.

Analysis and synthesis of controlled delay lines

It is shown in the mathematically strict manner that RC-circuit is suitable as a controlled delay line for different problems of electronics if it is sequentially connected with a hysteretic relay. This relay is either artificially introduced or appears as non-idealness of logical elements. A possibility to apply phase locked loops to control the delay time is noted. The control here is made by changing the capacity in RC-circuit.

Circular Criteria for Linear Systems

Circular stability and unstability criteria for linear time-varying systems are formulated. They are applicable to verify the stability of linear systems for which the matrix in the right side have eigenvalues with positive real parts at certain specific time instants.