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The poles method for second-order linear time-varying systems

Siwczyński M., Jaraczewski M.

Czasopismo Techniczne. Mechanika

2015

Y. 112, iss. 4-M

123--130

In dynamic linear systems described by differential equations with constant parameters, the poles of the rational function (transfer function of the system) play an important role. This article attempts to expand the poles concept in a situation where the system is described by the second-order linear system with timevarying parameters. It then introduces the concept of characteristic equations and time-dependent poles.

W opisie liniowych systemów dynamicznych opisanych przez równania różniczkowe o stałych parametrach ważną rolę odgrywają bieguny funkcji wymiernej (transmitancji systemu). Ten artykuł rozszerza koncepcję biegunów na przypadek, gdy system jest opisany równaniem liniowym drugiego rzędu o zmiennych w czasie parametrach. Pojawiają się tu pojęcia zależnego od czasu równania charakterystycznego i zależnych od czasu biegunów transmitancji.

Positive time-varying continuous-time linear systems and electrical circuits

Kaczorek T.

Bulletin of the Polish Academy of Sciences. Technical Sciences

2015

Vol. 63, nr 4

837--842

The positivity of time-varying continuous-time linear systems and electrical circuits are addressed. Necessary and sufficient conditions for the positivity of the systems and electrical circuits are established. It is shown that there exists a large class of positive electrical circuits with time-varying parameters. Examples of positive electrical circuits are presented.

Analysis and optimal control of linear time varying systems via the linear legendre mother wavelets

Khellat F., Vasegh N.

Systems Science

2008

Vol. 34, no 2

5-10

A method for finding the optimal control of a linear time varying system with quadratic performance index is discussed. This method is based on using the linear Legendre mother wavelets. The properties of the linear Legendre mother wavelets are presented. The operational matrices of integral and product are utilized to reduce the solution of optimal control to the explicit solution of algebraic equations. An illustrative example is included to demonstrate the validity and applicability of the technique.

An extension of the Cayley-Hamilton theorem for nonlinear time-varying systems

Kaczorek T.

International Journal of Applied Mathematics and Computer Science

2006

Vol. 16, no 1

141-145

The classical Cayley-Hamilton theorem is extended to nonlinear time-varying systems with square and rectangular system matrices. It is shown that in both cases system matrices satisfy many equations with coefficients being the coefficients of characteristic polynomials of suitable square matrices. The proposed theorems are illustrated with numerical examples.