Dixon resultant theory for stability analysis of distributed delay systems and enhancement of delay robustness

Gao, Qingbin; Cai, Jiazhi; Firoozy, Peyman; Guo, Shenghui; Hong, Hanlin; Long, Zhili

doi:10.1016/j.jfranklin.2022.05.034

Cited by 18 publications

(10 citation statements)

References 25 publications

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“…Time delay phenomenon is widespread in practical engineering systems, such as the chemical reactors, 2 the pendulum systems, 3 and so on. 31,32 In this section, we extend the above result to nonlinear MASs with both state and input delay. In particular, the following nonlinear time-delay MASs are considered…”

Section: Extension To System (1) With Both State and Input Delaymentioning

confidence: 66%

Sampled‐data bipartite consensus control of nonlinear multi‐agent systems under denial‐of‐service attacks

Pan

Zhang

Fan

2022

Adaptive Control & Signal

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This article concentrates on the sampled-data secure bipartite consensus problem for a class of nonlinear multiagent systems under intermittent denial-of-service attacks. The network communication channels are destroyed when the attacks occur, which causes all the system states to be unavailable. A novel distributed sampled-data output feedback control protocol is developed by using the discontinuous sensor data. It is proved by the Lyapunov stability analysis that the considered nonlinear multiagent systems can achieve bipartite consensus exponentially in light of the designed control protocol. Then, the control algorithm is extended to systems with both state and input delay. Finally, a simulation example is presented to verify the effectiveness of the proposed control protocol.

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Section: Extension To System (1) With Both State and Input Delaymentioning

confidence: 66%

Sampled‐data bipartite consensus control of nonlinear multi‐agent systems under denial‐of‐service attacks

Pan

Zhang

Fan

2022

Adaptive Control & Signal

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“…are also used to convert quasi-polynomial (1) into an equivalent characteristic polynomial. In spite of the fact that the resultant-based approaches have been extended to dynamic systems with two or multiple time delays, including neutral systems [56], distributed delay systems [38,57], and fractional-order systems [58], an exact determination of crossing frequency set is still lack. Hence, it may waste time in sweeping frequencies that lie between the lower and upper bounds but not actually belong to the crossing frequency set.…”

Section:  Andmentioning

confidence: 99%

A Simple Procedure for the Creation of Stability Charts in Delay Parameter Space for a Class of LTI Systems With Two Delays

Cai

Hwang

2023

IEEE Access

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The central task of the stability robustness with respect to delay uncertainties lies in creating the whole stability chart (delay map) in the delay parameter space. In this paper, we present a very simple frequency-sweeping procedure for creating stability chart in the delay parameter plane for a class of linear time-invariant (LTI) two-delay systems with a delay crossing talk. Actually, the procedure based on using Rekasius pseudo-delay substitution and discriminant of quadratic polynomial to characterize the crossing frequency set. The exact and exhaustive determination of crossing frequency intervals involves only finding all positive real roots of a real-coefficient polynomial. With the availability of the entire crossing frequency set, the famous cluster treatment of characteristic roots (CTCR) paradigm is applied to construct complete stability chart in the delay plane. A by-product of the proposed procedure is the revelation of the non-zero finite frequencies corresponding to infinite pseudo-delays. These frequencies divide the crossing frequency intervals into subintervals over each of which the frequency-sweeping technique provides continuous stability crossing curves in the domain of pseudo delays. For illustration and validation, two examples are provided.INDEX TERMS Delay map, frequency sweeping, stability crossing curves, time-delay systems.

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“…It is also noteworthy that the resultant-based methods presented in the literature for determining the stability crossing frequency set still suffer from two major inadequacies. First, the existing methods provide only the lower and upper bounds [28,[ 44], [46] (or just the upper bound [18], [42], [43]) rather than the exact ranges of the stability crossing frequency set. This limitation can lead to an incorrect set of crossing frequencies, as demonstrated in Example 2.…”

Section: Introductionmentioning

confidence: 99%

On the Use of Resultant-Based Frequency-Sweeping Approaches to Construct Stability Charts for Systems With Two Delays

Hwang,

Cai,

et al. 2024

IEEE Access

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The resultant-based frequency-sweeping approach is appealing for creating a stability map to study stability robustness against delay uncertainties. It relies on converting the system's transcendental characteristic function into a polynomial form such that the crossing frequency set and the corresponding stability switching curves in the delay parameter plane can be exactly and exhaustively determined using only algebraic computations. In the existing literature, the equation conversion has been achieved by using Rekasius substitution, bilinear substitution, or half-angle tangent substitution, and variable eliminations are carried out by using Sylvester resultant or Dixon resultant. In this paper, we first point out that since the computation of Dixon resultant involves a polynomial division, it could be beneficial in a symbolic programming environment. Then, we provide numerical evidence to compare the computational complexity of using different equation conversion methods and the subsequent resultant elimination to obtain the critical crossing frequencies from which the exact crossing frequency set can be determined. Next, based on the resultant theory, we reveal the roles of the critical crossing frequencies that play in the characterization of the delay map. Finally, we provide an efficient procedure for constructing stability charts in the delay parameter plane for linear time-invariant systems with two delays in states. To demonstrate our contributions, explanatory examples are provided along with a complete illustrative example.

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Dixon resultant theory for stability analysis of distributed delay systems and enhancement of delay robustness

Cited by 18 publications

References 25 publications

Sampled‐data bipartite consensus control of nonlinear multi‐agent systems under denial‐of‐service attacks

Sampled‐data bipartite consensus control of nonlinear multi‐agent systems under denial‐of‐service attacks

A Simple Procedure for the Creation of Stability Charts in Delay Parameter Space for a Class of LTI Systems With Two Delays

On the Use of Resultant-Based Frequency-Sweeping Approaches to Construct Stability Charts for Systems With Two Delays

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