Recursive formulation of the WKB solution for linear time-varying dynamic systems

Existence of periodic solutions of fractional order dynamic systems is an important and difficult issue in fractional order systems field. In this paper, the non existence of completely periodic solutions and existence of partly periodic solutions of fractional order linear time varying periodic systems and fractional order nonlinear time varying periodic systems are discussed. A new property of Laplace transform of periodic function is derived. The non existences of completely periodic solutions of fractional order linear time varying periodic systems and fractional order nonlinear time varying periodic fractional order systems are presented by Laplace transform method and contradiction approach. The existence of partly periodic solutions of fractional order dynamic systems are proved by constructing numerical examples and considering Laplace transform property approaches. The examples and state figures are given to illustrate the effectiveness of conclusion presented.

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Control of Dynamic Systems With Nonlinearities and Time Varying Parameters

14th Biennial Conference on Mechanical Vibration and Noise: Dynamics and Vibration of Time-Varying Systems and Structures ◽

10.1115/detc1993-0114 ◽

1993 ◽

Author(s):

Dirk Söffker ◽

Peter C. Müller

Keyword(s):

Dynamic Systems ◽

Disturbance Rejection ◽

Linear Time ◽

Time Varying ◽

Varying Parameters ◽

Linear Time Invariant ◽

Time Variance ◽

Disturbance Rejection Control ◽

Time Varying Parameters ◽

Time Invariant

Abstract The well-known theory of disturbance rejection control and the experience of using a generalized technique with universal fault model for building observers and regulators for the estimation and compensation of disturbances and unmodeled or uncertain effects as well, could be used for controlling dynamic systems with time varying parameters and nonlinearities. Based on a linear time-invariant model the effects of non-linearities and unmodeled dynamics are estimated by an extended observer scheme. Using this information these dynamic effects will be compensated by the developed compensation scheme. Here also different compensation techniques of disturbance rejection control are discussed, compared, and modified. The simulation example of an inverted flexible pendulum shows the efficiency of the method controlling an unstable mechanical system without exact knowledge of structure and parameters of nonlinearity and time-variance.

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SOME ANALYTIC CALCULATIONS OF THE CHARACTERISTIC EXPONENTS

International Journal of Bifurcation and Chaos ◽

10.1142/s021812740701955x ◽

2007 ◽

Vol 17 (10) ◽

pp. 3675-3678 ◽

Cited By ~ 2

Author(s):

P. VAN DER KLOET ◽

F. L. NEERHOFF ◽

N. H. WANING

Keyword(s):

Dynamic Systems ◽

Linear Time ◽

Equilibrium Points ◽

Analytic Solutions ◽

Nonlinear Dynamic Systems ◽

Time Varying ◽

Variational Equations ◽

Linear Time Invariant ◽

Characteristic Exponents ◽

Time Invariant

As is well known, the variational equations of nonlinear dynamic systems are linear time-varying (LTV) by nature. In the modal solutions for these LTV equations, the earlier introduced dynamic eigenvalues play a key role. They are closely related to the Lyapunov- and Floquet-exponents of the corresponding nonlinear systems. In this contribution, we present some simple examples for which analytic solutions exist. It is also demonstrated by example how the classical linear time-invariant (LTI) solutions are related to the equilibrium points of the general LTV solutions.

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Linear time-varying dynamic systems optimization via higher-order method using shifted Chevyshev's polynomials

10.2514/6.1998-4218 ◽

1998 ◽

Author(s):

Xiaochun Xu ◽

Sunil Agrawal

Keyword(s):

Dynamic Systems ◽

Linear Time ◽

Higher Order ◽

Time Varying ◽

Order Method ◽

Higher Order Method

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On Nonnegative Solutions of Fractionalq-Linear Time-Varying Dynamic Systems with Delayed Dynamics

Abstract and Applied Analysis ◽

10.1155/2014/247375 ◽

2014 ◽

Vol 2014 ◽

pp. 1-19 ◽

Cited By ~ 2

Author(s):

M. De la Sen

Keyword(s):

Dynamic Systems ◽

Fractional Derivatives ◽

Linear Time ◽

Asymptotic Properties ◽

Time Varying ◽

Caputo Fractional Derivatives ◽

Nonnegative Solutions ◽

Time Varying Systems ◽

Fractional Differential ◽

The Stability

This paper is devoted to the investigation of nonnegative solutions and the stability and asymptotic properties of the solutions of fractional differential dynamic linear time-varying systems involving delayed dynamics with delays. The dynamic systems are described based onq-calculus and Caputo fractional derivatives on any order.

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Global Stability of Polytopic Linear Time-Varying Dynamic Systems under Time-Varying Point Delays and Impulsive Controls

Mathematical Problems in Engineering ◽

10.1155/2010/693958 ◽

2010 ◽

Vol 2010 ◽

pp. 1-33 ◽

Cited By ~ 4

Author(s):

M. de la Sen

Keyword(s):

Dynamic System ◽

Linear Systems ◽

Dynamic Systems ◽

Linear Time ◽

Time Varying ◽

Linear Time Invariant ◽

Impulsive Controls ◽

The Stability ◽

Time Invariant ◽

Point Delays

This paper investigates the stability properties of a class of dynamic linear systems possessing several linear time-invariant parameterizations (or configurations) which conform a linear time-varying polytopic dynamic system with a finite number of time-varying time-differentiable point delays. The parameterizations may be timevarying and with bounded discontinuities and they can be subject to mixed regular plus impulsive controls within a sequence of time instants of zero measure. The polytopic parameterization for the dynamics associated with each delay is specific, so that(q+1)polytopic parameterizations are considered for a system withqdelays being also subject to delay-free dynamics. The considered general dynamic system includes, as particular cases, a wide class of switched linear systems whose individual parameterizations are timeinvariant which are governed by a switching rule. However, the dynamic system under consideration is viewed as much more general since it is time-varying with timevarying delays and the bounded discontinuous changes of active parameterizations are generated by impulsive controls in the dynamics and, at the same time, there is not a prescribed set of candidate potential parameterizations.

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