Critical parameters of integral delay systems

Ochoa, Gilberto; Melchor-Aguilar, Daniel; Mondié, Sabine

doi:10.1002/rnc.3132

Cited by 12 publications

(7 citation statements)

References 23 publications

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“…The maximum delay value for exponential stability of Equation (34 ) obtained from the norm condition (5) is h max = 0.1850 that it is significantly better than h max = 0.1420 obtained from the linear matrix inequalities conditions in Corollary 4.2. For comparison, the critical delay computed from the approach in Ochoa et al (2014) is h * ≈ 41.537. Example 4.4: Now, let us consider the perturbed scalar integral delay system:…”

Section: Remark 4: Corollary 42 Shows That the Assumption (3) Imposementioning

confidence: 99%

“…Based on the general expressions of Lyapunov functionals introduced in Melchor- , some particular functionals were constructed for the case of constant kernels in Melchor-Aguilar (2010) and for a class of analytic kernels in Mondié and MelchorAguilar (2012) to obtain stability conditions formulated directly in terms of the coefficients of integral delay systems. The forthcoming paper of Ochoa, Melchor-Aguilar, and Mondié (2014) gives a methodology for computing the exact critical delays of the class of integral delay systems with analytic kernels via an auxiliary delay free system. The three works, Melchor-Aguilar (2010), Mondié and Melchor-Aguilar (2012) and Ochoa et al (2014), deal only with the unperturbed case of integral delay systems.…”

Section: Introductionmentioning

confidence: 99%

“…The forthcoming paper of Ochoa, Melchor-Aguilar, and Mondié (2014) gives a methodology for computing the exact critical delays of the class of integral delay systems with analytic kernels via an auxiliary delay free system. The three works, Melchor-Aguilar (2010), Mondié and Melchor-Aguilar (2012) and Ochoa et al (2014), deal only with the unperturbed case of integral delay systems.…”

Section: Introductionmentioning

confidence: 99%

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New results on robust exponential stability of integral delay systems

Melchor-Aguilar

2014

International Journal of Systems Science

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The robust exponential stability of integral delay systems with exponential kernels is investigated. Sufficient delay-dependent robust conditions expressed in terms of linear matrix inequalities and matrix norms are derived by using the LyapunovKrasovskii functional approach. The results are combined with a new result on quadratic stabilisability of the state-feedback synthesis problem in order to derive a new linear matrix inequality methodology of designing a robust non-fragile controller for the finite spectrum assignment of input delay systems that guarantees simultaneously a numerically safe implementation and also the robustness to uncertainty in the system matrices and to perturbation in the feedback gain.

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Section: Remark 4: Corollary 42 Shows That the Assumption (3) Imposementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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New results on robust exponential stability of integral delay systems

Melchor-Aguilar

2014

International Journal of Systems Science

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“…While the previous problem has already been studied from both time‐domain and frequency‐domain techniques, implementation of these techniques is prohibitive for large dimensional problems, due to the arising of large‐dimensional matrices or high‐order characteristic equations. Moreover, existing techniques do not relate the DM to the finite number of eigenvalues of system matrices.…”

Section: Preliminaries and Problem Descriptionmentioning

confidence: 99%

“…Along the same lines, we cite the works of Louisell, where the author presents an elegant technique based on Kronecker sum operations to eliminate the variable z to then compute the admissible imaginary poles at s = ∓ ω j to next solve the variable z and ultimately, the delay τ . Following Louisell's work, Ochoa et al establish a bridge between frequency‐ and time‐domain stability analysis by relating the spectrum of the LTI system and that of the so‐called delay‐free system, which is used to compute the Lyapunov matrices associated with the original system. These efforts are in line with recent studies in the frequency domain, where one aims to establish the understanding between the stability of open‐loop systems and the largest DM that these systems can achieve in closed‐loop settings …”

Section: Introductionmentioning

confidence: 99%

An approach to compute and design the delay margin of a large‐scale matrix delay equation

Ramírez

Koh

Sipahi

2018

Intl J Robust & Nonlinear

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A methodology to compute and design the delay margin (DM) of large-scale linear time-invariant systems is presented. Different from existing work, this methodology is scalable; does not impose restrictions on the system to be able to invoke simultaneous triangularization or simultaneous diagonalization; and sheds light on how the finite number of delay-free system eigenvalues can be effectively utilized to compute or design the DM. KEYWORDSdelay margin, delay margin design, time-delay systems INTRODUCTIONThe presence of time delays in feedback control systems has attracted tremendous interest in engineering, mathematics, and physics communities for over six decades. [1][2][3][4][5][6] The main reason for this is that delays can dramatically affect the dynamics of control systems. Notably, they can cause poor performance and even instability; and this must be properly dealt with either through controller design or through system design. Virtually, in all control applications, delays exist because it takes time to transmit information, sense information, formulate a proper decision, and execute such decisions. Widely studied problems include machine tool chatter, 7,8 teleoperation, 9,10 human reaction times in driving 11,12 and when operating an aircraft, 13,14 power networks, 15,16 population dynamics, 17,18 supply chain networks, 19,20 and gene regulatory networks. 21,22 In the pursuit of addressing the fundamental stability analysis and control design needs, one faces numerous challenges and opportunities. For example, the presence of delays makes the system infinite dimensional. In linear time-invariant (LTI) systems, this means that one has to deal with infinitely many poles, and thus, standard tools available for finite-dimensional problems are immediately ruled out as prospects to analyze stability and/or design controllers. Moreover, the presence of delay is not necessarily detrimental to the dynamics. An appropriate amount of delay can render improved disturbance rejection capabilities, 23 can increase stability margins, 24 can be deliberately introduced as part of a controller to enhance the performance of a closed-loop system, 25-28 or can provide stability. 29,30 One of the widely studied problems in the context of time-delay systems is the problem of computing the delay margin * (DM) of a system. An LTI system is stable for all delays from zero up to the DM ∈ [0, * ), which implies that when the delay value is equal to the DM = * , the system has at least one pole on the imaginary axis of the complex plane. 1 Many techniques have been developed to calculate the DM of LTI systems. Efforts in this direction possibly go back to 31 where the authors developed an approach to create "stability maps" of LTI systems with delay. These maps represent, on the plane of the delay and a parameter of interest, the boundaries along which the system can have imaginary poles ∓ j and which side of the boundaries indeed correspond to the stable operation of the system. Approaches to obtain similar maps have also been deve...

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Critical parameters for a class of integral difference equations

Ochoa-Ortega

Gomez

2021

Intl J Robust & Nonlinear

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This contribution addresses the problem of calculating the critical parameters of a class of integral delay systems. The starting point of the analysis is a neutral-type system with distributed delay that results from the application of a differential operator, and whose spectrum contains the spectrum of the integral system. Based on recent results concerning the delay Lyapunov matrix for the class of obtained neutral-type systems, it is shown that the set of critical frequencies of the integral delay system is contained in the set of critical frequencies of a delay-free system of matrix equations. K E Y W O R D Scritical frequencies, delay-free system, integral difference equations, Lyapunov matrix, neutral systemswhere 𝛼 k,l ∈ R and k = 1, … , m.

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Critical parameters of integral delay systems

Abstract: Published online in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/rnc.313

Cited by 12 publications

References 23 publications

New results on robust exponential stability of integral delay systems

New results on robust exponential stability of integral delay systems

An approach to compute and design the delay margin of a large‐scale matrix delay equation

Critical parameters for a class of integral difference equations

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