Integral conditions on the uniform asymptotic stability for two-dimensional linear systems with time-varying coefficients

An efficient method for obtaining the steady-state response of linear systems with periodically time varying coefficients is developed. The steady-state solution is obtained by dividing the fundamental period into a number of intervals and establishing, based on a fourth-order Rung-Kutta formulation, the relation between the response at the start and end of the period. Imposition of periodicity condition upon the response facilitates computation of the initial condition that yields the steady-state values in a single pass; i.e., integration over only one period. Through a practical example, the method is shown to be more accurate and computationally more efficient than other known methods for computing the steady-state response.

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Asymptotic behaviour and boundedness of linear systems with time varying coefficients

Acta Astronautica ◽

10.1016/0094-5765(89)90015-5 ◽

1989 ◽

Vol 19 (10) ◽

pp. 787-795 ◽

Cited By ~ 1

Author(s):

S. Pradeep ◽

S.K. Shrivastava

Keyword(s):

Asymptotic Behaviour ◽

Linear Systems ◽

Time Varying ◽

Varying Coefficients ◽

Time Varying Coefficients

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Piecewise Linear Systems With Time Varying Coefficients

Volume 1D: 16th Biennial Conference on Mechanical Vibration and Noise ◽

10.1115/detc97/vib-3926 ◽

1997 ◽

Author(s):

S. Natsiavas ◽

S. Theodossiades

Keyword(s):

Linear Systems ◽

Piecewise Linear ◽

Duffing Oscillator ◽

Constant Load ◽

Time Varying ◽

Gear Pair ◽

Linear Dynamical Systems ◽

Varying Coefficients ◽

Piecewise Linear Systems ◽

Time Varying Coefficients

Abstract A new method is presented for determining periodic steady state response of piecewise linear dynamical systems with time varying coefficients. As an example mechanical model, a gear-pair system with backlash is examined, under the action of a constant torque. Originally, some useful insight is gained on the type of motions expected by investigating the response of a weakly nonlinear Mathieu-Duffing oscillator, subjected to a constant external load. The information obtained is then used in seeking the appropriate form of approximate periodic solutions of the piecewise linear system. Finally, these solutions are determined by developing a new analytical method. This method combines elements from approaches applied for piecewise linear systems with constant coefficients as well as classical perturbation techniques applied for systems with time varying coefficients. The validity and accuracy of the approach is verified by numerical results. In addition, response diagrams are presented, illustrating the effect of the constant load and the damping on the gear-pair response.

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Model of time-varying linear systems and Kolmogorov equations

Archives of Control Sciences ◽

10.1515/acsc-2015-0013 ◽

2015 ◽

Vol 25 (2) ◽

pp. 201-214

Author(s):

Assen V. Krumov

Keyword(s):

Analytical Solution ◽

Linear Systems ◽

Sufficient Conditions ◽

Original System ◽

Approximate Model ◽

Time Varying ◽

Kolmogorov Equations ◽

Varying Coefficients ◽

Method Of Solution ◽

Time Varying Coefficients

Abstract In the paper an approximate model of time-varying linear systems using a sequence of time-invariant systems is suggested. The conditions for validity of the approximation are proven with a theorem. Examples comparing the numerical solution of the original system and the analytical solution of the model are given. For the system under the consideration a new criterion giving sufficient conditions for robust Lagrange stability is suggested. The criterion is proven with a theorem. Examples are given showing stable and non stable solutions of a time-varying system and the results are compared with the numerical Runge-Kutta solution of the system. In the paper an important application of the described method of solution of linear systems with time-varying coefficients, namely analytical solution of the Kolmogorov equations is shown.

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