On Computing Puiseux Series for Multiple Imaginary Characteristic Roots of LTI Systems With Commensurate Delays

Li, Xuguang; Niculescu, Silviu–Iulian; Çela, Arben; Wang, Honghai; Cai, Tiao-Yang

doi:10.1109/tac.2012.2226102

Cited by 42 publications

(36 citation statements)

References 18 publications

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“…The CIR λ = j is double at τ = π. Using the approach in [11], the asymptotic behavior of (j, π) corresponds to the Puiseux series Δλ = (0.0049 + 0.0248j)(Δτ ) By Theorem 6, we know that NU(τ ) → ∞ as τ → ∞, which is verified by the plot of NU(τ ), see Fig. 5.…”

Section: A Numerical Examplementioning

confidence: 65%

“…However, it was assumed in [15] and [17] that the neutral system under consideration has only simple CIRs. It was recently pointed out in [11] that a multiple CIR's asymptotic behavior is very complicated and the Puiseux series is Proceedings of the 33rd Chinese Control Conference July 28-30, 2014, Nanjing, China needed as the mathematical tool 2 . To the best of our knowledge, no research has been reported for neutral time-delay systems with multiple CIRs, so far.…”

Section: Introductionmentioning

confidence: 99%

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Complete stability for neutral time-delay systems: A unified frequency-sweeping approach

Niculescu

Çela

2014

Proceedings of the 33rd Chinese Control Conference

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This paper studies the complete stability of time-delay systems of neutral type (shortly, neutral systems), inspired by that the complete stability of time-delay systems of retarded type (shortly, retarded systems) was recently solved by establishing a new frequency-sweeping (mathematical) framework. It is not hard to see that the invariance property, the most important basis for building up the frequency-sweeping framework for retarded systems, also holds for neutral systems. We are hence motivated to apply the relevant results to neutral systems by considering the distinctions between two types of time-delay systems. It will be interesting to see that these distinctions can be effectively covered in the frequency-sweeping framework. More precisely, we will find: (1) the stability of the neutral operator (as a necessary stability condition additionally required by neutral systems) can be directly examined from the frequency-sweeping curves (FSCs); and (2) the ultimate stability problem for neutral systems can be fully studied from the FSCs. As a consequence, the frequency-sweeping framework is proved to be also a unified approach for the complete stability of neutral time-delay systems.

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Section: A Numerical Examplementioning

confidence: 65%

Section: Introductionmentioning

confidence: 99%

Complete stability for neutral time-delay systems: A unified frequency-sweeping approach

Niculescu

Çela

2014

Proceedings of the 33rd Chinese Control Conference

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“…If multiple critical eigenvalues appear, the problem will generally become more complicated and we may invoke the Puiseux series to treat such a case (see e.g., [14] for the analysis of multiple critical roots for time-delay systems).…”

Section: Remarkmentioning

confidence: 99%

“…In this chapter, we will only study the case with only simple critical characteristic roots (i.e., we suppose that no multiple critical characteristic roots appear). One may refer to [14] for a general method for asymptotic behavior analysis. The proposed procedure will be illustrated by a numerical example.…”

Section: Introductionmentioning

confidence: 99%

Stabilization of Networked Control Systems with Hyper-Sampling Periods

Li¹,

Çela²,

Niculescu³

2016

Delays and Networked Control Systems

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This chapter considers the stabilization of Networked Control Systems (NCSs) under the hyper-sampling mode. Such a sampling mode, recently proposed in the literature, appears naturally in the scheduling policies of real-time systems under constrained (calculation and communication) resources. Meanwhile, as expected, the stabilization problem under the hyper-sampling mode is much more complicated than in the case of single-sampling mode. In this chapter, we propose a procedure to design the feedback gain matrix such that we can obtain a stabilizable region as large as possible. In the first step, we determine the stabilizable region under the singlesampling period. This step can be easily obtained by solving some linear matrix inequalities (LMIs) and from the result we may obtain a stabilizable region for the hyper-sampling period. Then, in the second step, we further detect the stabilizable region, based on the one found in the first step, by adjusting the feedback gain matrices based on the asymptotic behavior analysis. By this step, a larger stabilizable region may be found and this step can be used in an iterative manner. The proposed procedure will be illustrated by a numerical example. We can see from the example that the stabilizable region under the hyper-sampling period may lead to a smaller average sampling frequency (ASF) guaranteeing the stability of the NCS than the single-sampling period, i.e., less system resources are required by the hyper-sampling period.

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“…Challenges due to non-differentiability arise when the imaginary roots are also multiple roots. Such problems have traditionally been solved using Puiseux series (Kato, 1980;Knopp, 1996), see, for example, Chen, Fu, Niculescu & Guan (2010a) and Li, Niculescu, Ç ela, Wang & Cai (2013) for systems with one parameter.…”

Section: Introductionmentioning

confidence: 99%

Some insights into the migration of double imaginary roots under small deviation of two parameters

Irofti

Boussaada

et al. 2018

Automatica

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This paper studies the migration of double imaginary roots of the systems' characteristic equation when two parameters are subjected to small deviations. The proposed approach covers a wide range of models. Under the least degeneracy assumptions, we found that the local stability crossing curve has a cusp at the point that corresponds to the double root, and it divides the neighborhood of this point into an S-sector and a G-sector. When the parameters move into the G-sector, one of the roots moves to the right half-plane, and the other moves to the left half-plane. When the parameters move into the S-sector, both roots move either to the left half-plane or the right half-plane depending on the sign of a quantity that depends on the characteristic function and its derivatives up to the third order.

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On Computing Puiseux Series for Multiple Imaginary Characteristic Roots of LTI Systems With Commensurate Delays

Cited by 42 publications

References 18 publications

Complete stability for neutral time-delay systems: A unified frequency-sweeping approach

Complete stability for neutral time-delay systems: A unified frequency-sweeping approach

Stabilization of Networked Control Systems with Hyper-Sampling Periods

Some insights into the migration of double imaginary roots under small deviation of two parameters

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