Piecewise Linear Systems With Time Varying Coefficients

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.

BIMODAL PIECEWISE LINEAR DYNAMICAL SYSTEMS: REDUCED FORMS

2010 ◽  
Vol 20 (09) ◽  
pp. 2795-2808 ◽  
Author(s):  
JOSEP FERRER ◽  
M. DOLORS MAGRET ◽  
MARTA PEÑA
Keyword(s):  
Dynamical Systems ◽  
Linear Systems ◽  
Piecewise Linear ◽  
Equivalence Classes ◽  
State Variables ◽  
Linear Dynamical Systems ◽  
Piecewise Linear Systems ◽  
Wide Range ◽  
General Systems ◽  
Reduced Forms

Piecewise linear systems constitute a class of nonlinear systems which have recently attracted the interest of researchers because of their interesting properties and the wide range of applications from which they arise. Different authors have used reduced forms when studying these systems, mostly in the case where they are observable. In this work, we focus on bimodal continuous dynamical systems (those consisting of two linear systems on each side of a given hyperplane, having continuous dynamics along that hyperplane) depending on two or three state variables, which are the most common piecewise linear systems found in practice. Reduced forms are obtained for general systems, not necessarily observable. As an application, we calculate the dimension of the equivalence classes.

On Efficient Computation of the Steady-State Response of Linear Systems With Periodic Coefficients

1996 ◽  
Vol 118 (3) ◽  
pp. 522-526 ◽  
Author(s):  
T. J. Selstad ◽  
K. Farhang
Keyword(s):  
Steady State ◽  
Linear Systems ◽  
Steady State Solution ◽  
Time Varying ◽  
Steady State Response ◽  
Efficient Computation ◽  
Periodicity Condition ◽  
Varying Coefficients ◽  
State Response ◽  
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.

scholarly journals Model of time-varying linear systems and Kolmogorov equations

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.

Co-Movements of US and Asian Equity Markets: Evidence from Asymmetric and Time-Varying Coefficients

2014 ◽  
Vol 17 (04) ◽  
pp. 1450021 ◽  
Author(s):  
Bharat Kolluri ◽  
Susan Machuga ◽  
Mahmoud Wahab
Keyword(s):  
Piecewise Linear ◽  
Equity Markets ◽  
Time Varying ◽  
Conditional Mean ◽  
Varying Coefficients ◽  
The Us ◽  
Sharpe Ratios ◽  
Time Varying Coefficients ◽  
Movement Parameters ◽  
Performance Ranking

We examine co-movements of nine Asian equity markets with both the US and Japan with special interest in distinguishing co-movements during periods of positive returns from those during periods of negative returns. A discrete asymmetric piecewise linear conditional mean returns specification is adopted to generate asymmetric co-movement parameters for periods of positive and negative returns. Conditional heteroskedasticity is modeled using GARCH and EGARCH specifications. Predicted conditional volatilities are used to generate alternative estimates of asymmetric and time-varying co-movement parameters. Conditional mean returns from asymmetric and symmetric conditional mean return models along with GARCH and EGARCH volatilities are used to generate estimates of asymmetric and symmetric conditional (ex-ante) Sharpe ratios. Asian markets returns and volatilities show a clear tendency to move more with the US than with Japan; and their co-movements with negative US returns far exceed their co-movements with positive US returns, thereby suggesting that any diversification benefits into Asian equities are likely to manifest themselves more during periods of positive than negative US returns. Conditional asymmetric Sharpe ratios exceed conditional symmetric Sharpe ratios; however, and more importantly, performance-ranking differs depending on whether asymmetric versus symmetric conditional Sharpe ratios are used. Asymmetric conditional Sharpe ratios suggest that India (followed by Malaysia) offers the best return/risk tradeoff, with the least favorable market is South Korea (using GARCH) and Philippines (using EGARCH).

Stability of Linear Systems With Parametric Excitation

1970 ◽  
Vol 37 (1) ◽  
pp. 228-230 ◽  
Author(s):  
J. R. Dickerson
Keyword(s):  
Linear Systems ◽  
High Frequency ◽  
Mathieu Equation ◽  
Time Varying ◽  
Asymptotically Stable ◽  
System Matrix ◽  
Varying Coefficients ◽  
Stability Matrix ◽  
Time Varying Coefficients ◽  
Asymptotic Stability Conditions

A Lyapunov-type approach is used to develop sufficient asymptotic stability conditions for linear systems with time-varying coefficients. In particular, it is shown that parametric disturbances of high frequency cannot create instability in an already asymptotically stable system. Also it is shown that slowly varying parametric disturbances will not cause instability if the system matrix is a stability matrix for all values of time. The results are applied to the Mathieu equation to illustrate the character of the theorems. This example is chosen because of the availability of its exact stability boundaries.

Sign in / Sign up

Export Citation Format

Share Document