A matrix method for determining the imaginary axis eigenvalues of a delay system

Louisell, James

doi:10.1109/9.975510

Cited by 106 publications

(37 citation statements)

References 12 publications

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“…The fact that the spectrum of the delay free system of matrix equations (8) is symmetrical with respect of the imaginary axis follows from the idea that if for a value s there exist non trivial matrices X i i = −m, ..., 0, ...m − 1 that satisfy system (10). By tranposing and multiplying by (−1) system (10), it follows that…”

Section: Proposition 1 the Spectrum Of The Delay Free System (8) Is mentioning

confidence: 96%

“…In this section the key relationship between the spectrum of the delay free system (8) and the roots of the characteristic function of system (1) is established, [6], [10]. Theorem 1.…”

Section: Frequency Domain Stability Analysismentioning

confidence: 98%

“…This matrix has been shown to be the solution of a delay free system subject to boundary conditions [5], [10] and its existence and uniqueness were established in [7]. The extension of some of these results to the case of delays multiple of a basic one was recently addressed in [12].…”

Section: Introductionmentioning

confidence: 94%

See 2 more Smart Citations

Computation of Imaginary Axis Eigenvalues and Critical Parameters for Neutral Time Delay Systems

Ochoa

Mondié

Kharitonov

2012

Time Delay Systems: Methods, Applications and New Trends

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Abstract. In this paper, a procedure for the computation of the Lyapunov matrix of neutral type systems, with delays multiple of a basic one, is recalled: It consists in solving a boundary value problem for a delay free system of matrix equations. The important property that the elements of the spectrum of the delay system that are symmetric with respect to the imaginary axis belong to the spectrum of the delay free system is also established. This property is exploited for the determination of the critical values of the time delay system.

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Section: Proposition 1 the Spectrum Of The Delay Free System (8) Is mentioning

confidence: 96%

“…In this section the key relationship between the spectrum of the delay free system (8) and the roots of the characteristic function of system (1) is established, [6], [10]. Theorem 1.…”

Section: Frequency Domain Stability Analysismentioning

confidence: 98%

See 1 more Smart Citation

Computation of Imaginary Axis Eigenvalues and Critical Parameters for Neutral Time Delay Systems

Ochoa

Mondié

Kharitonov

2012

Time Delay Systems: Methods, Applications and New Trends

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“…The next method is a numerical approach to finding the imaginary root crossings that recasts the n × n transcendental determinant as a 2n 2 × 2n 2 eigenvalue problem in terms of the system matrices that can be solved numerically [1,7].…”

Section: B Kronecker Multiplicationmentioning

confidence: 99%

“…A numerical method is also formulated by introducing Kronecker multiplication to recast the n × n transcendental determinant as a 2n 2 × 2n 2 eigenvalue problem [1,7]. These eigenvalues indicate all the poles for which the stability changes and are used to find the associated time delays.…”

Section: Introductionmentioning

confidence: 99%

Efficient Numerical Analysis of Stability of High-Order Systems with a Time Delay

Armanious

Lind

2018

Journal of Guidance, Control, and Dynamics

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Time delays are a common perturbation in systems with many states, such as networked, distributed, or decentralized systems. Current methods analyzing the stability of large systems with time delay typically produce very conservative results. While more exact methods exist, these become inefficient for large systems. This paper provides a methodology for analyzing the stability of time-delayed systems that is derived from exact methods but is efficient for high-order systems. The computational and memory cost of this new technique is compared to the costs of existing techniques, and its efficiency is shown using a distributed system with over four hundred states I n = n × n identity matrix

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Predictive stability indicator: a novel approach to configuring a real‐time hybrid simulation

Maghareh

Dyke

RabieniaHaratbar

et al. 2016

Earthq Engng Struct Dyn

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Summary Real‐time hybrid simulation (RTHS) is an effective and versatile tool for the examination of complex structural systems with rate dependent behaviors. To meet the objectives of such a test, appropriate consideration must be given to the partitioning of the system into physical and computational portions (i.e., the configuration of the RTHS). Predictive stability and performance indicators (PSI and PPI) were initially established for use with only single degree‐of‐freedom systems. These indicators allow researchers to plan a RTHS, to quantitatively examine the impact of partitioning choices on stability and performance, and to assess the sensitivity of an RTHS configuration to de‐synchronization at the interface. In this study, PSI is extended to any linear multi‐degree‐of‐freedom (MDOF) system. The PSI is obtained analytically and it is independent of the transfer system and controller dynamics, providing a relatively easy and extremely useful method to examine many partitioning choices. A novel matrix method is adopted to convert a delay differential equation to a generalized eigenvalue problem using a set of vectorization mappings, and then to analytically solve the delay differential equations in a computationally efficient way. Through two illustrative examples, the PSI is demonstrated and validated. Validation of the MDOF PSI also includes comparisons to a MDOF dynamic model that includes realistic models of the hydraulic actuators and the control‐structure interaction effects. Results demonstrate that the proposed PSI can be used as an effective design tool for conducting successful RTHS. Copyright © 2016 John Wiley & Sons, Ltd

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A matrix method for determining the imaginary axis eigenvalues of a delay system

Cited by 106 publications

References 12 publications

Computation of Imaginary Axis Eigenvalues and Critical Parameters for Neutral Time Delay Systems

Computation of Imaginary Axis Eigenvalues and Critical Parameters for Neutral Time Delay Systems

Efficient Numerical Analysis of Stability of High-Order Systems with a Time Delay

Predictive stability indicator: a novel approach to configuring a real‐time hybrid simulation

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