TY - JOUR

T1 - Coordinatewise estimates for vector outputs of multivariable phase control systems

AU - Smirnova, V. B.

AU - Utina, N. V.

AU - Shepelyavyi, A. I.

AU - Perkin, A. A.

PY - 2009/9/1

Y1 - 2009/9/1

N2 - Two classes of multidimensional phase control systems with differentiable vector periodic functions are considered, the class of continuous systems described by ordinary differential equations and the class of discrete systems described by difference equations. The number of cycle slips for angular coordinates in phase systems with differentiable nonlinearities is studied. The study is based on the direct Lyapunov method and uses periodic Lyapunov functions, extensions of the phase space of the system, and the Yakubovich-Kalman lemma. This lemma provides necessary and sufficient conditions for the existence of Lyapunov functions by using the transfer matrix of the linear part of the system. As a result, for phase systems possessing global asymptotics, frequency criteria making it possible to sharpen estimates for the deviation of angular coordinates from their initial values are obtained. These criteria contain multiparameter frequency inequalities with variable parameters satisfying certain algebraic inequalities.

AB - Two classes of multidimensional phase control systems with differentiable vector periodic functions are considered, the class of continuous systems described by ordinary differential equations and the class of discrete systems described by difference equations. The number of cycle slips for angular coordinates in phase systems with differentiable nonlinearities is studied. The study is based on the direct Lyapunov method and uses periodic Lyapunov functions, extensions of the phase space of the system, and the Yakubovich-Kalman lemma. This lemma provides necessary and sufficient conditions for the existence of Lyapunov functions by using the transfer matrix of the linear part of the system. As a result, for phase systems possessing global asymptotics, frequency criteria making it possible to sharpen estimates for the deviation of angular coordinates from their initial values are obtained. These criteria contain multiparameter frequency inequalities with variable parameters satisfying certain algebraic inequalities.

UR - http://www.scopus.com/inward/record.url?scp=84859720246&partnerID=8YFLogxK

U2 - 10.3103/S1063454109030091

DO - 10.3103/S1063454109030091

M3 - Article

AN - SCOPUS:84859720246

VL - 42

SP - 212

EP - 218

JO - Vestnik St. Petersburg University: Mathematics

JF - Vestnik St. Petersburg University: Mathematics

SN - 1063-4541

IS - 3

ER -