Matrix Polynomials, Similar Operators, and the Imaginary Axis Eigenvalues of a Matrix Delay Equation

Louisell, James

doi:10.1137/120886236

Cited by 12 publications

(5 citation statements)

References 15 publications

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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%

“…This “decoupling” idea is a common thread in several studies, including the work of Niculescu and Chen et al, see also the work of Gao and Liao for an extension of Chen's method to the case of linear fractional‐order retarded systems. 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.…”

Section: Introductionmentioning

confidence: 99%

“…As we are moving to the study of large‐scale problems affected by delays, especially those arising in network systems, neural networks, power networks, multi‐agent systems, and supply‐chain networks, it is becoming more and more often that we are in need of tools to analyze the stability of large‐size problems efficiently; the reader is referred to the works of Antoulas and Jarlebring for thorough studies on large‐scale systems and their treatment using Krylov‐based approximations and Arnoldi‐type projections methods. On the other hand, the major limitation in computing the DM based on the previously cited approaches is that they do not scale well with the size of the problem (see discussions in the works) and the references therein); and hence can be demonstrated on only small size problems. The main reason underlying this lack of progress is in the way the current approaches are constructed, as explained next.…”

Section: Introductionmentioning

confidence: 99%

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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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Section: Preliminaries and Problem Descriptionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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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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“…The condition in [13] needs to compute the characteristic equation of (1), which is not explicitly expressed as functions of the coefficients, and thus seems difficult to be used for robust stability analysis. For a single delay, strong stability can be checked by computing the generalized eigenvalues of a pair of matrices [16,17] as well as the matrix pencil based approach [19]. The method of cluster treatment of characteristic roots was used in [20] to derive the stability maps of (1) with three delays.…”

Section: Introduction and Literature Reviewmentioning

confidence: 99%

On strong stability and robust strong stability of linear difference equations with two delays

Zhou

2019

Automatica

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This paper provides a necessary and sufficient condition for guaranteeing exponential stability of the linear difference equationwhere a > 0, b > 0 are constants and A, B are n × n square matrices, in terms of a linear matrix inequality (LMI) of size (k + 1) n × (k + 1) n where k ≥ 1 is some integer. Different from an existing condition where the coefficients (A, B) appear as highly nonlinear functions, the proposed LMI condition involves matrices that are linear functions of (A, B) . Such a property is further used to deal with the robust stability problem in case of norm bounded uncertainty and polytopic uncertainty, and the state feedback stabilization problem. Solutions to these two problems are expressed by LMIs. A time domain interpretation of the proposed LMI condition in terms of Lyapunov-Krasovskii functional is given, which helps to reveal the relationships among the existing methods. Numerical example demonstrates the effectiveness of the proposed method.

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“…5,6,18 In the frequency domain, one determines explicitly the imaginary crossing of the poles. 19,20 Especially, matrix pencil approach 21,22 examines and calculates analytically the imaginary axis eigenvalues of the retarded matrix delay systems. In addition, application of Nyquist theorem 23 also lead to some conditions, which can be checked on Mikhailov diagrams.…”

mentioning

confidence: 99%

Frequency delay‐dependent stability criterion for time‐delay systems thanks to Fourier–Legendre remainders

Bajodek

Gouaisbaut

Seuret

2021

Intl J Robust & Nonlinear

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This paper investigates the stability of a linear finite-dimensional system interconnected to a single delay operator. From robust approaches, to derive delay-dependent frequency tests, a characterization of the delay behavior is required. Based on approximation methods, one describes the transported signal by lumped parameters. More precisely, by the use of the first Fourier-Legendre polynomials coefficients, we split the delay block into a finitedimensional part interconnected to a specific infinite-dimensional residual part. Two models are investigated with residuals related to two Fourier-Legendre remainders of the delayed transfer function. The main contribution is to highlight that the finite-dimensional models based on the first Legendre coefficients are proven to be related to Padé approximations and are recognized to be more and more accurate as the dimension increases. Interestingly, this modeling allows computing in an accurate manner the root locus of time-delay systems. Furthermore, as a by-product of this result, taking into account the infinite-dimensional remainders to keep track of the initial time-delay system, stability criteria are proposed by H∞ analysis. Considering both infinite-dimensional remainders as bounded delay-free uncertainties, the small-gain theorem provides a new sufficient condition of stability for retarded time-delay systems, which can be implemented as a delay-dependent frequency-sweeping test. Our results are illustrated on several academic examples.

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Matrix Polynomials, Similar Operators, and the Imaginary Axis Eigenvalues of a Matrix Delay Equation

Cited by 12 publications

References 15 publications

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

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

On strong stability and robust strong stability of linear difference equations with two delays

Frequency delay‐dependent stability criterion for time‐delay systems thanks to Fourier–Legendre remainders

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