Further Results on the Achievable Delay Margin Using LTI Control

Ju, Peijun; Zhang, Huanshui

doi:10.1109/tac.2015.2502399

Cited by 28 publications

(28 citation statements)

References 15 publications

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“…Moreover, the analysis in [22] was carried out largely case by case, and for this reason, its technique does not appear readily generalizable. The same can be said of the improvement in [13].…”

Section: Introductionmentioning

confidence: 89%

“…which gives rise to (13). The remaining results then follow by solving the polynomials in question explicitly.…”

Section: B Special Casesmentioning

confidence: 99%

“…Furthermore, we compare τ 6 to the improved upper bound in [13] (Theorem 1, Theorem 2). The results are shown in Figure 5.…”

Section: Examplesmentioning

confidence: 99%

“…154), the delay margin was determined for the first-order system achievable by static feedback, while in [31], it was also found for the first-order system when PID controllers are used instead. Other related results concerning stabilizability via delayed feedback can be found in, e.g., [16,26] In the recent work [13,22], upper bounds on the delay margin were obtained for general SISO systems subject to an uncertain constant delay. These bounds serve to provide a limit beyond which no single LTI output feedback controller may exist to robustly stabilize the delay plant family within the margin.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Fundamental Limits on Uncertain Delays: When Is a Delay System Stabilizable by LTI Controllers?

Zhu

Chen

2017

IEEE Trans. Automat. Contr.

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This paper concerns the stabilization of linear timeinvariant (LTI) systems subject to uncertain, possibly timevarying delays. The fundamental issue under investigation, referred to as the delay margin problem, addresses the question: What is the largest range of delay such that there exists a single LTI feedback controller capable of stabilizing all the plants for delays within that range? Drawing upon analytic interpolation and rational approximation techniques, we derive fundamental bounds on the delay margin, within which the delay plant is guaranteed to be stabilizable by a certain LTI output feedback controller. Our contribution is threefold. First, for single-input single-output (SISO) systems with an arbitrary number of plant unstable poles and nonminimum phase zeros, we provide an explicit, computationally efficient bound on the delay margin, which requires computing only the largest real eigenvalue of a constant matrix. Second, for multi-input multi-output (MIMO) systems, we show that estimates on the variation ranges of multiple delays can be obtained by solving LMI problems, and further, by finding bounds on the radius of delay variations. Third, we show that these bounds and estimates can be extended to systems subject to time-varying delays. When specialized to more specific cases, e.g., to plants with one unstable pole but possibly multiple nonminimum phase zeros, our results give rise to analytical expressions exhibiting explicit dependence of the bounds and estimates on the pole and zeros, thus demonstrating how fundamentally unstable poles and nonminimum phase zeros may limit the range of delays over which a plant can be stabilized by a LTI controller.

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Section: Introductionmentioning

confidence: 89%

“…which gives rise to (13). The remaining results then follow by solving the polynomials in question explicitly.…”

Section: B Special Casesmentioning

confidence: 99%

“…Furthermore, we compare τ 6 to the improved upper bound in [13] (Theorem 1, Theorem 2). The results are shown in Figure 5.…”

Section: Examplesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Fundamental Limits on Uncertain Delays: When Is a Delay System Stabilizable by LTI Controllers?

Zhu

Chen

2017

IEEE Trans. Automat. Contr.

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show abstract

“…In [2] (p. 154), the delay margin was examined for the first-order system with a constant delay stabilized by static feedback, while in [15], the stabilization was achieved by using PID controllers for first-order systems. Furthermore, for single-input single-output (SISO) systems with constant delays, the upper bound was determined in [16,17] for general LTI systems with an arbitrary number of unstable poles. These bounds consequently provide a limit beyond which no single LTI output feedback controller may exist to robustly stabilize a delay plant family within the delay margin.…”

Section: Introductionmentioning

confidence: 99%

On the Achievable Stabilization Delay Margin for Linear Plants with Time-Varying Delays

Zhu

2017

Mathematics

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Abstract:The paper contributes to stabilization problems of linear systems subject to time-varying delays. Drawing upon small gain criteria and robust analysis techniques, upper and lower bounds on the largest allowable time-varying delay are developed by using bilinear transformation and rational approximates. The results achieved are not only computationally efficient but also conceptually appealing. Furthermore, analytical expressions of the upper and lower bounds are derived for specific situations that demonstrate the dependence of those bounds on the unstable poles and nonminumum phase zeros of systems.

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Delay effects on the mean‐square stabilization of stochastic systems with input delay

Sun

Liu

et al. 2022

Intl J Robust & Nonlinear

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This paper studies delay effects on the mean-square stabilization of stochastic systems with a single input delay. Both continuous-and discrete-time systems are investigated. A continuous-time system is described by a stochastic differential equation with high-dimensional Brownian motion. Two conclusions are proved: (1) Once a system is unstabilizable for some value of delay, it will be unstabilizable for any larger delay; (2) If a system is stabilizable for some value of delay, it will be stabilizable in a small neighborhood of that delay. Hence, by excluding the trivial cases that the system is unstabilizable with zero delay and is stabilizable for any delay, there always exists a finite and positive number CM such that the system is stabilizable if and only if the delay is less than CM. Upper bounds of CM are presented. In a special case, an upper bound is given analytically in terms of eigenvalues of the matrix in the drift coefficient before the state. In general cases, upper bounds are given via LMI optimization problems. Two cases in which CM can be computed are studied. In the case that the system has no state-dependent noise, CM is presented via an LMI optimization problem. In the case that both the state and the control of the system are scalars, CM is expressed analytically. All the above results have counterparts in discrete-time system.

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Further Results on the Achievable Delay Margin Using LTI Control

Cited by 28 publications

References 15 publications

Fundamental Limits on Uncertain Delays: When Is a Delay System Stabilizable by LTI Controllers?

Fundamental Limits on Uncertain Delays: When Is a Delay System Stabilizable by LTI Controllers?

On the Achievable Stabilization Delay Margin for Linear Plants with Time-Varying Delays

Delay effects on the mean‐square stabilization of stochastic systems with input delay

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