scholarly journals Explicit Commutativity and Stability for the Heun's Linear Time-Varying Differential Systems

2021 ◽  
Author(s):  
Salisu Ibrahim
Keyword(s):  
Differential System ◽  
Stability Problem ◽  
Significant Contribution ◽  
Initial Conditions ◽  
Linear Time ◽  
Time Varying ◽  
Differential Systems ◽  
System Sensitivity ◽  
System Robustness ◽  
The Stability

This paper studies the commutativity and stability for the Heun’s linear time-varying system (LTVS) with both zero and non-zero initial conditions(ICs). Given a LTVS A of order 2 , we find it’s commutative pair, that is a new LTVS B of order m ≤ n . Explicit commutative theories and conditions for second-order LTVSs are derived and solved to simplify and guarantee the equivalency between the connected input-output of systems A B and B A . The explicit results obtained are juxtaposed by simulation in order to investigate the commutativity of Heun’s differential system, sensitivity of Heun’s system, effects due to disturbance on Heun’s system, robustness on Heun’s system and problems regarding the stability of Heun’s system. This findings will help to fill the gap on stability problem, system behaviors, commutativity theory, and general theory for solutions of differential equations, which has significant contribution to science and unlimited application in engineering, our results are verify using Heun’s differential system as well as authenticated by Wolfrom Mathematica 1 1 and Matlab.

scholarly journals Commutativity of Linear Time-Varying Differential Systems with Nonzero Initial Conditions: A Review and Some New Extensions

2011 ◽  
Vol 2011 ◽  
pp. 1-25 ◽  
Author(s):  
Muhammet Koksal ◽  
Mehmet Emir Koksal
Keyword(s):  
Initial Conditions ◽  
Linear Time ◽  
Sufficient Conditions ◽  
Industrial Applications ◽  
Time Varying ◽  
Differential Systems ◽  
System Sensitivity ◽  
Time Varying Systems ◽  
A Chain ◽  
Fifth Order

Necessary and sufficient conditions for the commutativity of linear time-varying systems are derived in the case of nonzero initial conditions. It is shown that some commutative class of linear time-varying systems may not commute with arbitrary initial conditions. In this respect, commutativity of Euler differential systems is investigated. Explicit commutativity conditions for the fifth-order systems are solved. New results about the effects of commutativity on system sensitivity and disturbance properties are presented, which is very important for network design and industrial applications where many of the systems are composed of subsystems cooperating one after another in a chain. The results are supported by examples treated either analytically or numerically.

scholarly journals The VIT Transform Approach to Discrete-Time Signals and Linear Time-Varying Systems

Eng ◽  
2021 ◽  
Vol 2 (1) ◽  
pp. 99-125
Author(s):  
Edward W. Kamen
Keyword(s):  
Discrete Time ◽  
Initial Conditions ◽  
Linear Time ◽  
Order Term ◽  
Initial Time ◽  
Time Varying ◽  
Varying Coefficients ◽  
Time Signals ◽  
Time Varying Systems ◽  
Time Varying Coefficients

A transform approach based on a variable initial time (VIT) formulation is developed for discrete-time signals and linear time-varying discrete-time systems or digital filters. The VIT transform is a formal power series in z−1, which converts functions given by linear time-varying difference equations into left polynomial fractions with variable coefficients, and with initial conditions incorporated into the framework. It is shown that the transform satisfies a number of properties that are analogous to those of the ordinary z-transform, and that it is possible to do scaling of z−i by time functions, which results in left-fraction forms for the transform of a large class of functions including sinusoids with general time-varying amplitudes and frequencies. Using the extended right Euclidean algorithm in a skew polynomial ring with time-varying coefficients, it is shown that a sum of left polynomial fractions can be written as a single fraction, which results in linear time-varying recursions for the inverse transform of the combined fraction. The extraction of a first-order term from a given polynomial fraction is carried out in terms of the evaluation of zi at time functions. In the application to linear time-varying systems, it is proved that the VIT transform of the system output is equal to the product of the VIT transform of the input and the VIT transform of the unit-pulse response function. For systems given by a time-varying moving average or an autoregressive model, the transform framework is used to determine the steady-state output response resulting from various signal inputs such as the step and cosine functions.

An Archaeological Approach to the Study of Cultural Stability

1955 ◽  
Vol 11 ◽  
pp. 32-57 ◽  
Author(s):  
Emil W. Haury ◽  
Robert L. Rands ◽  
Albert C. Spaulding ◽  
Walter W. Taylor ◽  
Raymond H. Thompson ◽  
...  
Keyword(s):  
Stability Problem ◽  
Significant Contribution ◽  
Time Perspective ◽  
Problem Area ◽  
Cultural Development ◽  
Instability Problem ◽  
Long Span ◽  
The Stability ◽  
Definition Of

The study of cultural stability requires a knowledge of cultural development over a reasonably long span of time. The definition of this time perspective is one of the major contributions of archaeology to the study of culture. The archaeologist therefore should be in a position to make a significant contribution to the appraisal of the stability problem itself. However, the lack of a commonly accepted anthropological definition of the concept of cultural stability imposes semantic difficulties which hinder the determination of practical limits for the stability- instability problem area. Moreover, the nature of the data available to the archaeologist conditions the kind of contribution he can make.

scholarly journals 1.17. Bulge and bar: a possible way of their formation

1998 ◽  
Vol 184 ◽  
pp. 49-50
Author(s):  
S.N. Nuritdinov ◽  
E.R. Gaynullia ◽  
K.T. Mirtodjieva
Keyword(s):  
Observational Data ◽  
Stability Problem ◽  
Initial Conditions ◽  
Present Report ◽  
Spherical Collapse ◽  
The Subject ◽  
The Law ◽  
The Stability ◽  
Gravitating Systems

Some observational data indicate that galaxy subsystems, including their central areas, first of all are the result of their global nonstationary evolution. That is why we earlier built (Nuritdinov 1992) the exact non-linearly pulsing rotating models of disklike and spherical self-gravitating systems. Unlike other authors we want to research the stability problem of nonlinear nonstationary models. In the present report we want to give only those results of the instability studied, which have a direct attitude to the subject under discussion. We put a certain question: what initial conditions have to exist, for instance, for the value of the virial parameter (2T/|U|)0 and the parameter of anisotropy < Tr > / < T⊥ >, that the collapse of a disk should result in a bar, and the spherical collapse will result in a thick ellipsoidal bulge. To answer the question it is very important to study stability of the solvable nonlinear unequilibrium models. All models discussed below pulsate under the law R = II(ψ)R0, where (Nuritdinov 1985)

scholarly journals Robust Time-Varying Output Formation Control for Swarm Systems with Nonlinear Uncertainties

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-13
Author(s):  
Xiaofeng Chai ◽  
Qing Wang ◽  
Yao Yu ◽  
Changyin Sun
Keyword(s):  
Stability Problem ◽  
Formation Control ◽  
Directed Network ◽  
Control Problems ◽  
Time Varying ◽  
Robust Controller ◽  
Nonlinear Uncertainties ◽  
The Stability ◽  
High Order Time ◽  
Time Invariant

Time-varying output formation control problems for high-order time-invariant swarm systems are studied with nonlinear uncertainties and directed network topology in this paper. A robust controller which consists of a nominal controller and a robust compensator is applied to achieve formation control. The nominal controller based on the output feedback is designed to achieve desired time-varying formation properties for the nominal system. And the robust compensator based on the robust signal compensator technology is constructed to restrain nonlinear uncertainties. The time-varying formation problem is transformed into the stability problem. And the formation errors can be arbitrarily small with expected convergence rate. Numerical examples are provided to illustrate the effectiveness of the proposed strategy.

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