Stability of Discrete-Time Systems with Time-Varying Delay: Delay Decomposition Approach

Stojanović, Sreten B.; Debeljković, Dragutin Lj.; Dimitrijevic, Nebojsa

doi:10.15837/ijccc.2012.4.1375

Cited by 8 publications

(14 citation statements)

References 28 publications

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“…We apply these results to a number of well-studied systems to demonstrate that the notion of intrinsic stability is both computationally inexpensive, relative to other methods, and can be used to improve on some of the best known results. (See for instance Example 4.4, compared to results found in [14,15,16,17,18].) We also show that the asymptotic state of intrinsically stable switched systems is independent of the system's initial conditions (see Main Result 1).…”

Section: Introductionsupporting

confidence: 55%

“…which satisfies ρ( A ) = 1. Hence this system is not intrinsically stable, even though [14] showed that it is stable for all 0 ≤ L ≤ 9.61 × 10 8 .…”

Section: The Transition Matrix Of the Lifted Representation Ismentioning

confidence: 96%

“…Then this system is intrinsically stable, so is in fact stable for an arbitrarily large delay bound L > 0. In the following table, we compare this delay-independent result with the delay-dependent results of [14,15,16,17,18]: Method Max Upper Bound L Theorem 3.1 of [15] 10 Theorem 1, Theorem 2 of [16] 13 Theorem 3.2 of [17] 12 Theorem 1 of [18] 15 Theorem 2 of [14] 10 • 10 21 Main Result 2 of this paper ∞…”

Section: Application: Linear Systems With Distinct Delayed and Undela...mentioning

confidence: 99%

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Intrinsic stability: stability of dynamical networks and switched systems with any type of time-delays

Reber

Webb

2020

Nonlinearity

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In real-world networks the interactions between network elements are inherently time-delayed. These timedelays can not only slow the network but can have a destabilizing effect on the network's dynamics leading to poor performance. The same is true in computational networks used for machine learning etc. where time-delays increase the network's memory but can degrade the network's ability to be trained. However, not all networks can be destabilized by time-delays. Previously, it has been shown that if a network or high-dimensional dynamical system is intrinsically stabile, which is a stronger form of the standard notion of global stability, then it maintains its stability when constant time-delays are introduced into the system. Here we show that intrinsically stable systems, including intrinsically stable networks and a broad class of switched systems, i.e. systems whose mapping is time-dependent, remain stable in the presence of any type of time-varying time-delays whether these delays are periodic, stochastic, or otherwise. We apply these results to a number of well-studied systems to demonstrate that the notion of intrinsic stability is both computationally inexpensive, relative to other methods, and can be used to improve on some of the best known stability results. We also show that the asymptotic state of an intrinsically stable switched system is exponentially independent of the system's initial conditions.Here we refer to the emergent behavior of these interacting elements, which is the changing state of these network elements, as the network's dynamics. The network's dynamics can be periodic, as is found in many biological networks [2], synchronizing, which is the desired condition for transmitting power over large distance in power grids [3], and stable or multistable dynamics such as is found in gene-regulatory networks [4], etc.In real-world networks the interactions between network elements are inherently time-delayed. This comes from the fact that network elements are spatially separated, that information and other quantities can only be processed and transmitted at finite speeds, and that these quantities can be slowed by network traffic [5]. These time-delays not only slow the network, leading to poor performance, but can have a destabilizing effect on the network's dynamics, which can lead to network failure [6,7]. The same is true in computational networks used for machine learning etc. where time-delays can be used to increase the network's ability to detect long term temporal dependencies [8] but at the cost of potentially degrading the ability to train the network.Not all networks can be destabilized by time-delays. In [9] the notion of intrinsic stability was introduced, which is a stronger form of the standard notion of stability, i.e. a system in which there is a globally attracting fixed point (see [10] for more details). If a network is intrinsically stable the authors showed that constant-type time-delays, which are delays that do not vary in time, have no effect on the network's stabi...

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Section: Introductionsupporting

confidence: 55%

“…which satisfies ρ( A ) = 1. Hence this system is not intrinsically stable, even though [14] showed that it is stable for all 0 ≤ L ≤ 9.61 × 10 8 .…”

Section: The Transition Matrix Of the Lifted Representation Ismentioning

confidence: 96%

Section: Application: Linear Systems With Distinct Delayed and Undela...mentioning

confidence: 99%

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Intrinsic stability: stability of dynamical networks and switched systems with any type of time-delays

Reber

Webb

2020

Nonlinearity

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“…For delayed systems, the most common approaches for characterizing stability are Lyapunov methods, Linear Matrix Inequalities, and Semi-Definite Programming (see, for example, [15,16]. In [10,11] the notion of intrinsic stability was developed as an alternative to these approaches for studying delayed systems.…”

Section: Introductionmentioning

confidence: 99%

Stability of Stochastically Switched and Stochastically Time-Delayed Systems

Carter,

Murri,

Reber

et al. 2020

Preprint

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In this paper we introduce the notion of a patient first-mean stable system. Such systems are switched systems that are first-mean stable meaning that they converge to a globally attracting fixed point on average. They are also patient so that they do not lose their first-mean stability when time-delays are introduced into the system. As time-delays are, in general, a source of instability and poor performance patient first-mean stability is a much stronger condition than first-mean stability. This notion of patient stability allows one to design systems that cannot be destabilized via time-delays. It also significantly reduces the difficulty of modeling such systems since in patient systems time-delays can, to a large extent, be safely ignored. The paper's main focus is on giving a sufficient criteria under which a system is patient first-mean stable and we give a number of examples that demonstrate the simplicity of this criteria.

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“…Moreover, taking an automatic computerbased control system or a process control with networked transmission for example, it is necessary and more reasonable to simultaneously consider the possible transmission delay for the corresponding control law and the device delay due to computing or processing. Recently, much attention has been paid to the subjects of stability, stabilization and control of the time delay systems [1] - [5]. Fractional order models would be more accurate than integer order models.…”

Section: Introductionmentioning

confidence: 99%

Direct Method for Stability Analysis of Fractional Delay Systems

Pakzad

Nekui

2013

INT J COMPUT COMMUN, Int. J. Comput. Commun. Control

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In this paper, a direct method is presented to analyze the stability of fractional order systems with single and multiple commensurate time delays. Using the approach presented in this study, first, without using any approximation, the transcendental characteristic equation is converted to an algebraic one with some specific crossing points. Then, an expression in terms of system parameters and imaginary root of the characteristic equation is derived for computing the delay margin. Finally, the concept of stability is expressed as a function of delay. An illustrative example is presented to confirm the proposed method results.

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Stability of Discrete-Time Systems with Time-Varying Delay: Delay Decomposition Approach

Cited by 8 publications

References 28 publications

Intrinsic stability: stability of dynamical networks and switched systems with any type of time-delays

Intrinsic stability: stability of dynamical networks and switched systems with any type of time-delays

Stability of Stochastically Switched and Stochastically Time-Delayed Systems

Direct Method for Stability Analysis of Fractional Delay Systems

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