Lyapunov and non-Lyapunov stability of linear discrete time delay systems

This paper presents a series of new results in finite and infinite-memory modeling of discrete-time fractional differences. The introduced normalized finite fractional difference is shown to properly approximate its fractional difference original, in particular in terms of the steady-state properties. A stability analysis is also presented and a recursive computation algorithm is offered for finite fractional differences. A thorough analysis of computational and accuracy aspects is culminated with the introduction of a perfect finite fractional difference and, in particular, a powerful adaptive finite fractional difference, whose excellent performance is illustrated in simulation examples.

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“…According to the arguments presented by Debeljković et al (2002), a system described by the equation…”

Section: Appendix a Proof Of Theoremmentioning

confidence: 99%

Normalized finite fractional differences: Computational and accuracy breakthroughs

Stanisławski¹,

Latawiec²

2012

International Journal of Applied Mathematics and Computer Science

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“…Numerical stability of the different approaches considered to solve the moving-medium sound propagation equations is examined by using theory of discrete time-delay systems [8]. The spatial discretisation results in a number of local values for the acoustic pressures p and components of the particle velocities V i .…”

Section: Methodsmentioning

confidence: 99%

“…It can be proven that the discrete system is stable if and only if all eigenvalues (or poles) of the matrix A have a modulus smaller than one [8]. In other words, all eigenvalues must lie within a circle of radius 1 in the complex plane.…”

Section: Methodsmentioning

confidence: 99%

Prediction-step staggered-in-time FDTD: An efficient numerical scheme to solve the linearised equations of fluid dynamics in outdoor sound propagation

Renterghem

Botteldooren

2007

Applied Acoustics

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The finite-difference time-domain (FDTD) method to solve the linearised equations of fluid dynamics has shown to be very powerful and useful in outdoor sound propagation. Practical applications are however limited due to the large need for computational resources. The numerical discretisation influences computational efficiency to an important degree. In this paper, some possible ways to discretise temporal derivatives are studied. Two obvious ways of time-discretisation namely staggered-in-time (SIT) and a simple collocated-in-time (CIT) scheme are compared to the prediction-step staggered-in-time (PSIT) scheme. The latter is intended to be used for the calculation of sound propagation in the typical low wind speeds encountered in the outdoor environment at low heights above the earthÕs surface. It was shown that the PSIT scheme is more stable than the SIT scheme, so practical calculations are possible. Computational efficiency is increased to an important degree compared to the CIT scheme. The numerical accuracy (more precisely the amplitude error) of the PSIT scheme is an important improvement upon SIT. The CIT scheme on the other hand conserves amplitude better. The amplitude error becomes larger with increasing wind speed because of some simplifications during the numerical discretisation. In low wind speeds, the PSIT algorithm can serve as an interesting compromise between numerical accuracy and the required amount of computing power.

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“…It has a varying number of steps of time-delays, equal to k, i.e., increasing with time. Instead, the models addressed in [50][51][52] consider a finite constant number of steps of time-delays.…”

Section: The Discrete-time Fractional-order State-space Modelmentioning

confidence: 99%