scholarly journals Positivity and Stability of the Solutions of Caputo Fractional Linear Time-Invariant Systems of Any Order with Internal Point Delays

2011 ◽  
Vol 2011 ◽  
pp. 1-25 ◽  
Author(s):  
M. De la Sen
Keyword(s):  
Dynamic Systems ◽  
Linear Time ◽  
Asymptotic Properties ◽  
Linear Time Invariant ◽  
Nonnegative Solutions ◽  
Fractional Differential ◽  
Linear Time Invariant Systems ◽  
The Stability ◽  
Time Invariant ◽  
Point Delays

This paper is devoted to the investigation of the nonnegative solutions and the stability and asymptotic properties of the solutions of fractional differential dynamic systems involving delayed dynamics with point delays. The obtained results are independent of the sizes of the delays.

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

2014 ◽  
Vol 2014 ◽  
pp. 1-19 ◽  
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.

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

2010 ◽  
Vol 2010 ◽  
pp. 1-33 ◽  
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.

scholarly journals Complete Controllability of Impulsive Fractional Linear Time-Invariant Systems with Delay

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Xian-Feng Zhou ◽  
Song Liu ◽  
Wei Jiang
Keyword(s):  
Dynamical Systems ◽  
Differential Equations ◽  
Fractional Differential Equations ◽  
Linear Time ◽  
Complete Controllability ◽  
Linear Time Invariant ◽  
Impulsive Fractional Differential Equations ◽  
Fractional Differential ◽  
Linear Time Invariant Systems ◽  
Time Invariant

Some flaws on impulsive fractional differential equations (systems) have been found. This paper is concerned with the complete controllability of impulsive fractional linear time-invariant dynamical systems with delay. The criteria on the controllability of the system, which is sufficient and necessary, are established by constructing suitable control inputs. Two examples are provided to illustrate the obtained results.

scholarly journals Torsion Discriminance for Stability of Linear Time-Invariant Systems

Mathematics ◽  
2020 ◽  
Vol 8 (3) ◽  
pp. 386
Author(s):  
Yuxin Wang ◽  
Huafei Sun ◽  
Yueqi Cao ◽  
Shiqiang Zhang
Keyword(s):  
Lebesgue Measure ◽  
Linear Time ◽  
Zero Solution ◽  
Positive Lebesgue Measure ◽  
Asymptotically Stable ◽  
Linear Time Invariant ◽  
Measurable Set ◽  
Linear Time Invariant Systems ◽  
The Stability ◽  
Time Invariant

This paper extends the former approaches to describe the stability of n-dimensional linear time-invariant systems via the torsion τ ( t ) of the state trajectory. For a system r ˙ ( t ) = A r ( t ) where A is invertible, we show that (1) if there exists a measurable set E 1 with positive Lebesgue measure, such that r ( 0 ) ∈ E 1 implies that lim t → + ∞ τ ( t ) ≠ 0 or lim t → + ∞ τ ( t ) does not exist, then the zero solution of the system is stable; (2) if there exists a measurable set E 2 with positive Lebesgue measure, such that r ( 0 ) ∈ E 2 implies that lim t → + ∞ τ ( t ) = + ∞ , then the zero solution of the system is asymptotically stable. Furthermore, we establish a relationship between the ith curvature ( i = 1 , 2 , ⋯ ) of the trajectory and the stability of the zero solution when A is similar to a real diagonal matrix.

Active Control of Linear Time-Varying Structures by Confinement of Vibrations

2005 ◽  
Vol 11 (1) ◽  
pp. 89-102 ◽  
Author(s):  
S. Choura ◽  
A. S. Yigit
Keyword(s):  
Control Strategy ◽  
Linear Time ◽  
Steady States ◽  
Time Varying ◽  
Linear Time Invariant ◽  
Slowing Down ◽  
Rapid Removal ◽  
Linear Time Invariant Systems ◽  
The Stability ◽  
Time Invariant

We propose a control strategy for the simultaneous suppression and confinement of vibrations in linear time-varying structures. The proposed controller has time-varying gains and can also be used for linear time-invariant systems. The key idea is to alter the original modes by appropriate feedback forces to allow parts of the structure reach their steady states at faster rates. It is demonstrated that the convergence of these parts to zero is improved at the expense of slowing down the settling of the remaining parts to their steady states. The proposed control strategy can be applied for the rapid removal of vibration energy in sensitive parts of a flexible structure for safety or performance reasons. The stability of the closed-loop system is proven through a Lyapunov approach. An illustrative example of a five-link manipulator with a periodic follower force is given to demonstrate the effectiveness of the method for time-varying as well as time-invariant systems.

Stability Analysis of LTI Systems With Three Independent Delays—A Computationally Efficient Procedure

2009 ◽  
Vol 131 (5) ◽  
Author(s):  
Rifat Sipahi ◽  
Hassan Fazelinia ◽  
Nejat Olgac
Keyword(s):  
Linear Time ◽  
Numerical Procedure ◽  
Computationally Efficient ◽  
Linear Time Invariant ◽  
Characteristic Roots ◽  
Stability Robustness ◽  
Linear Time Invariant Systems ◽  
The Stability ◽  
Operation Results ◽  
Time Invariant

A practical numerical procedure is introduced for determining the stability robustness map of a general class of higher order linear time invariant systems with three independent delays, against uncertainties in the delays. The procedure is based on an efficient and exhaustive frequency-sweeping technique within a single loop. This operation results in determination of the complete description of the kernel and the offspring hypersurfaces, which constitute exhaustively the potential stability switching loci in the space of the delays. The new numerical procedure corresponds to the first step in the overarching framework, called the cluster treatment of characteristic roots. The results of this treatment can also be represented in another domain (called the spectral delay space) within a finite dimensional cube called the building block, which is much simpler to view and analyze. The paper also offers several case studies to demonstrate the practicality of the new numerical methodology.

scholarly journals On the links between limit characteristic zeros and stability properties of linear time-invariant systems with point delays and their delay-free counterparts

2004 ◽  
Vol 2004 (4) ◽  
pp. 339-357
Author(s):  
M. De La Sen ◽  
J. Jugo
Keyword(s):  
Analytic Functions ◽  
Linear Time ◽  
System Stability ◽  
Stable Region ◽  
Finite Characteristic ◽  
Free System ◽  
Linear Time Invariant ◽  
Linear Time Invariant Systems ◽  
Time Invariant ◽  
Point Delays

We investigate the relationships between the infinitely many characteristic zeros (or modes) of linear systems subject to point delays and their delay-free counterparts based on algebraic results and theory of analytic functions. The cases when the delay tends to zero or to infinity are emphasized in the study. It is found that when the delay is arbitrarily small, infinitely many of those zeros are located in the stable region with arbitrarily large modulus, while their contribution to the system dynamics becomes irrelevant. The remaining finite characteristic zeros converge to those of the delay-free nominal system. When the delay tends to infinity, infinitely many zeros are close to the origin. Furthermore, there exist two auxiliary delay-free systems which describe the relevant dynamics in both cases for zero and infinite delays. The maintenance of the delay-free system stability in the presence of sufficiently small delayed dynamics is also discussed in light ofH∞-theory. The main mathematical arguments used to derive the results are based on the theory of analytic functions.

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