Solution of Systems of Linear Delay Differential Equations via Laplace Transformation

Yi, Sun; Ulsoy, A. Galip

doi:10.1109/cdc.2006.377712

Cited by 27 publications

(32 citation statements)

References 8 publications

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“…where W k (·) is a k-th branch of the Lambert function, which is complex-valued and c k is a coefficient determined from the initial condition and history function of (2) [20]. Therefore, according to (5) u c is periodic and can be represented as a sum of Fourier series; however, it is limited to lowest frequency components [17].…”

Section: A Phase Description During Tdl R-osc Modementioning

confidence: 98%

Lightweight TRNG Based on Multiphase Timing of Bistables

Wieczorek

2016

IEEE Trans. Circuits Syst. I

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The paper presents a concept of a True Random Number Generator (TRNG) that utilizes phase noise of a pair of ring oscillators (ROs) to increase the variance of the initial condition of a bistable. For this purpose a special TRNG D-latch architecture (TDL) has been proposed, which can either operate in the oscillatory ring-oscillator mode or the nearly-metastable mode. The RO mode increases the probability of the nearly metastable operation of the TDL, which in turn increases the mean value and variance of the resolve time. Moreover, due to the oscillatory nature of the TDL metastability, the resolve time can be easily measured and used for further randomness harvesting in the TRNG. The proposed TRNG uses a pair of TDLs to reduce sensitivity to process or temperature variation, whereas TDLs' individual resolve time, their resolve time difference and logical state contribute to randomness. In the article, the impact of the devices' imbalance (tolerance) resulting from process variation on the resultant entropy is also explored. The proposed TRNG based on TDLs is scalable and has been implemented in modern CPLD and FPGA devices whereas the bit rate reached up to 1 Mbit. The article also discusses theoretical issues related to a transformation of phase and resolve time probability density functions and their influence on TRNG parameters.

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Section: A Phase Description During Tdl R-osc Modementioning

confidence: 98%

Lightweight TRNG Based on Multiphase Timing of Bistables

Wieczorek

2016

IEEE Trans. Circuits Syst. I

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“…The unavoidable matrix inversion in computing coefficients S k and C N k [30] makes such an evaluation too expensive in practice. Furthermore, it is common in VLSI circuit simulation that the original TDSs are of very high order; thus, this time-domain evaluation approach is infeasible due to its extremely high computational cost.…”

Section: Controllability and Observability Gramians Of Tdssmentioning

confidence: 99%

Gramian‐based model order reduction of parameterized time‐delay systems

Wang

Zhang

Wang

et al. 2012

Circuit Theory & Apps

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Time-delay systems (TDSs) frequently arise in circuit simulation especially in high-frequency applications. Model order reduction (MOR) techniques can be used to facilitate the simulation of TDSs. On the other hand, many kinds of variations, such as temperature and geometric uncertainties, can have significant impact on the transient responses of TDSs. Therefore, it is important to preserve parametric dependence during the MOR procedure. This paper presents a new parameterized MOR scheme for TDSs with parameter variations. We derive parameterized reduced-order models (ROMs) for TDSs using balanced truncation by approximating the Gramians in the multi-dimensional space of parameters. The resulting ROMs can preserve the parametric dependence, making it efficient for repeated simulations under different parameter settings. Numerical examples are presented to verify the accuracy and efficiency of our proposed algorithm.

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“…Apparently, time needed for the computation is inversely proportional to the delay value. On the contrary, the power of the Lambert W function in solving such equations has been recently proven and extended as much as possible (see [1,23,24] to name a few). As will be seen, the complexity of the calculation is delay-independent which makes analysis much more general for any initial function φ (t).…”

Section: Delay Differential Equationsmentioning

confidence: 99%

“…Then, oscillatory and asymptotic properties are examinated. Following [24], we can easily determine coefficient's expression for higher-order differential equations as same as for first-order ones. Taking the Laplace transform of (14) as Then, the inverse substitution of Q(λ k ) yields to the expression of C k as:…”

Section: Odd-order Casementioning

confidence: 99%

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Application of Lambert W Function in Oscillation Theory

Jadlovská¹

2014

AEI

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This paper is primarily concerned with analyzing some special properties of the characteristic equation of the linear delay differential equation of the first and odd order. The basic concepts and results keen to the oscillatory nature of solutions, as well as stability analysis are re-investigated using the properties of Lambert W function and compared with existing ones. A brief survey of the function origin and possible applications is presented together with a short behavior analysis.

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Solution of Systems of Linear Delay Differential Equations via Laplace Transformation

Cited by 27 publications

References 8 publications

Lightweight TRNG Based on Multiphase Timing of Bistables

Lightweight TRNG Based on Multiphase Timing of Bistables

Gramian‐based model order reduction of parameterized time‐delay systems

Application of Lambert W Function in Oscillation Theory

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