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We focus on the proportional-integral-derivative (PID) controller design for linear time-delay systems. All the controller gains ($k_P$ , $k_I$ , and $k_D$) and the delay ($\tau$) are treated as free parameters and no particular constraints are imposed on the controlled plants. Such a problem (involving totally four free parameters) is of theoretical as well as practical importance, but, to the best of the authors’ knowledge, it has not been fully explored. First, we will develop an algebraic algorithm to solve the complete stability problem w.r.t. $\tau$. Consequently, for any given PID controller vector $(k_P, k_I, k_D)$ , the distribution of ${NU}(\tau)$ (${NU}(\tau)$ denotes the number of characteristic roots in the right-half plane, as a function of $\tau$) can be accurately obtained and the exhaustive stability range of $\tau$ may be automatically calculated. Next, a global understanding of the distribution of ${NU}(\tau)$ over the whole $(k_P, k_I, k_D)$ -space may be achieved and all structural changes regarding the ${NU}(\tau)$ distribution can be analytically determined. To achieve such a goal, a complete positive real root classification (for some appropriate auxiliary characteristic equation) will be explicitly proposed. Finally, we will give a new methodology, a new parameter-space approach, for determining the stability set in the $(k_P, k_I, k_D, \tau)$ -space. [ABSTRACT FROM AUTHOR]

For time-delay systems, the asymptotic behavior analysis of the critical imaginary roots w.r.t. the infinitely many critical delays is an open problem. In order to find a general solution, we will exploit the link between the asymptotic behavior of critical imaginary roots and the asymptotic behavior of frequency-sweeping curves, from a new analytic curve perspective. As a consequence, we will establish a frequency-sweeping framework with three main results: (1) A finer (regularity-singularity) classification for time-delay systems will be obtained. (2) The general invariance property will be proved and hence the asymptotic behavior of the critical imaginary roots w.r.t. the infinitely many critical delays can be adequately studied. (3) The complete stability problem can be fully solved. Moreover, the frequency-sweeping framework is extended to cover a broader class of time-delay systems. Finally, the geometric insights of frequency-sweeping curves are investigated. Consequently, some deeper properties on the asymptotic behavior of time-delay systems and the link to frequency-sweeping curves are found. [ABSTRACT FROM AUTHOR]

A standard approach in analyzing dynamical systems consists in identifying and understanding the eigenvalues bifurcations when crossing the imaginary axis. Efficient methods for crossing imaginary roots identification exist. However, to the best of the author's knowledge, the multiplicity of such roots was not deeply investigated. In recent papers by the authors [1], [2], it is emphasized that the multiplicity of the zero spectral value can exceed the number of the coupled scalar delay-differential equations and a constructive approach Vandermonde-based allowing to an adaptive bound for such a multiplicity is provided. Namely, it is shown that the zero spectral value multiplicity depends on the system structure (number of delays and number of non zero coefficients of the associated quasipolynomial) rather than the degree of the associated quasipolynomial [3]. This technical note extends the constructive approach in investigating the multiplicity of crossing imaginary roots $j\omega$ where $\omega\neq 0$ and establishes a link with a new class of functional confluent Vandermonde matrices. A symbolic algorithm for computing the LU-factorization for such matrices is provided. As a byproduct of the proposed approach, a bound sharper than the Polya-Szegö generic bound arising from the principle argument is established. [ABSTRACT FROM AUTHOR]

Obtaining the Puiseux series of multiple imaginary (characteristic) roots (MIRs) is a fundamental issue in the stability analysis of time-delay systems. However, to the best of the authors' knowledge, this issue has not been fully investigated up to date. This note focuses on the Puiseux series expansion of MIRs of linear time-invariant systems including commensurate delays. For an MIR of any multiplicity, we propose an algorithm for defining the structure of the Puiseux series, as well as the explicit computation of the corresponding coefficients. By using the proposed method, we can find all the Puiseux series corresponding to all the root loci. [ABSTRACT FROM AUTHOR]

We address the problem of stabilizing systems belonging to a family of time-varying nonlinear systems with distributed input delay through state feedbacks without retarded term. The approach we adopt is based on a new technique that is inspired by the reduction model technique. The control laws we obtain are nonlinear and time-varying. They globally uniformly exponentially stabilize the origin of the considered system. We illustrate the construction with a networked control system. [ABSTRACT FROM AUTHOR]

In this paper, we study the robust stability of uncertain time-delay systems. We consider uncertain quasi-polynomials whose coefficients may vary in a certain prescribed range. Our goal is to derive necessary and sufficient conditions for such uncertain quasi-polynomials to maintain stability independent of delay parameters. Our primary contributions are frequency-sweeping conditions for interval, diamond, and spherical quasi-polynomial families, which can be readily checked, requiring only the computation of two simple frequency-dependent functions. Additionally, we also obtain vertex- and edge-type results in the spirit of the Kharitonov approach known in robust stability analysis, showing that the stability of interval and diamond quasi-polynomials can be ascertained by checking the stability of certain special vertex and/or edge members in those families. Both type of results provide necessary and sufficient conditions for the quasi-polynomial families to be robustly stable independent of delay. [ABSTRACT FROM AUTHOR]

In this note, we study the stability properties of linear neutral delay systems. We consider systems described by both neutral differential-difference and state-space equations, and we seek to determine the delay margin of such systems, that is, the largest range of delay values for which a neutral delay system may preserve its stability. In both cases, we show that the delay margin can be found by computing the eigenvalues and generalized eigenvalues of certain constant matrices, which can be executed efficiently and with high precision. The results extend previously known work on retarded systems, and demonstrate that similar stability tests exist for neutral systems; in particular, the tests require essentially the same amount of computation required for retarded systems. [ABSTRACT FROM AUTHOR]

This note focuses on the static output feedback stabilization problem for a class of single-input--single-output systems when the control law includes multiple (distinct) delays. We are interested in giving necessary conditions for the existence of such stabilizing controllers, illustrative examples (second-order system, chain of integrators, or chain of oscillators) are presented, and, discussed. [ABSTRACT FROM AUTHOR]

This note addresses the output feedback stabilization problem of a chain of integrators using multiple delays. We shall prove that either n distinct delays or a proportional+delay compensator with n - 1 distinct delays are sufficient to stabilize a chain including n integrators. We present two different approaches. Both are constructive and rely on frequency-domain techniques: on a derivative feedback approximation idea, and a pole placement idea, respectively. An illustrative example (triple integrator) is presented. [ABSTRACT FROM AUTHOR]

This note focuses on the stability of some class of delay systems, including uncertainty in the delays. More precisely, we are interested in guaranteeing the stability of perturbed delay systems by assuming the stability of the nominal system. If the delay perturbation is constant, necessary and sufficient conditions are derived in terms of generalized eigenvalue distribution of some (finite-dimensional) constant matrix pencil. If the delay perturbation is time-varying, some sufficient stability conditions are derived using "exact" Lyapunov-Krasovskii functionals. Illustrative examples are also included. [ABSTRACT FROM AUTHOR]

Focuses on the analysis of conservatism introduced due to some model transformations used in control literature. Examination of the additional dynamics of the second order and parameterized model transformation; Additional constraints for stability introduced by the neutral model transformation.

Identifies the sufficient conditions for a linear system with point or distributed delay in the state to be passive. Clarification on passivity based on memoryless stabilization of linear delay systems; Lyapunov-Krasovskii functional construction.

Examines additional dynamics in transformed time-delay systems. Transformation of a system with single time delay to a system with distributed delay; Additional dynamics that can be characterized by appropriate additional eigenvalues; Stability criteria obtained.

Provides information on the robust exponential stability of a class of uncertain systems with time-varying delays. Description of these systems; Investigation of the conditions which guarantee exponential stability for all admissible uncertain systems; Simplicity of the proposed stability criteria.

Provides information on a ...memoryless control problem with supplementary ...-stability constraints for a class of linear systems with delayed state and disturbance inputs. Proposal of a linear matrix inequality (LMI) optimization for the considered problem; Conclusions reached.

Focuses on the problem of the control of a class of mechanical systems with a finite number of degrees-of-freedom. Details about the dynamic model; Stability analysis of some particular dynamics; Illustration of one-dof case and analization of some stability properties for various control strategies; Analysis of n-dof rigid manipulators; Conclusions.

Abstract Most of the reported Lotka–Volterra examples have at most one stability interval for the delay parameters. Furthermore, the existing methods fall short in treating more general case studies. Inspired by some recent results for analyzing the stability of time-delay systems, this paper focuses on a deeper characterization of the stability of Lotka–Volterra systems w.r.t. the delay parameters. More precisely, we will introduce the recently-proposed frequency-sweeping approach to study the complete stability problem for a broad class of linearized Lotka–Volterra systems. As a result, the whole stability delay-set can be analytically determined. Moreover, as a significant byproduct of the proposed approach, some Lotka–Volterra examples are found to have multiple stability delay-intervals. To the best of the authors' knowledge, such a characterization represents a novelty for having some insights in the population dynamics: in some situations, a longer maturation period of species is helpful for the stability of a population system. [ABSTRACT FROM AUTHOR]

We study stabilization problems for time-varying linear systems with constant input delays. Our reduction method ensures input-to-state stability with respect to additive uncertainties, under arbitrarily long delays. It applies to rapidly time-varying systems, and gives a lower bound on the admissible rapidness parameters. We also cover slowly time-varying systems, including upper bounds on the allowable slowness parameters. We illustrate our work using a pendulum model. [ABSTRACT FROM AUTHOR]

We study the stability of a linear system with a point- wise, time-varying delay. We assume that the delay varies around a nominal value in a deterministic way and investigate the influence of this variation on stability. More precisely we are interested in characterizing situations where the time-varying delay system is stable, whereas the system with constant delay is unstable. Our approach consists of relating the stability properties of a system with a fast varying point-wise delay with these of a time-invariant system with a distributed delay. Then we can use frequency domain methods to analyze the problem and to derive stability criteria. The results are first illustrated with two theoretical examples. Then, we study a model of a variable speed rotating cutting tool. Based on the developed theory, we thereby provide both a theoretical explanation and a quantitative analysis tool for the beneficial effect of a variation of the machine speed on enhancing stability properties, which was reported in the literature. [ABSTRACT FROM AUTHOR]

The problem of the global uniform asymptotic stabilization by bounded feedback of a chain of integrators with a delay in the input is solved. No limitation on the size of the delay is imposed. To validate the approach, a third-order example is presented. [ABSTRACT FROM AUTHOR]

This paper deals with a new approach to develop an envelope that engulfs all poles of a time-delay system. Through linear matrix inequalities, we are able to determine envelopes for retarded and neutral time-delay systems. The envelopes proposed are not only tighter than the ones in the literature, but, with our procedure, they can also be applied to verify the stability of the system and design state-feedback controllers, which cope with design requirements regarding $\alpha$ -stability. [ABSTRACT FROM AUTHOR]

This paper presents a systematic method to analyze the stability of systems with single delay in which the coefficient polynomials of the characteristic equation depend on the delay. Such systems often arise in, for example, life science and engineering systems. A method to analyze such systems was presented by Beretta and Kuang in a 2002 paper, but with some very restrictive assumptions. This paper extends their results to the general case with the exception of some degenerate cases. It is found that a much richer behavior is possible when the restrictive assumptions are removed. The interval of interest for the delay is partitioned into subintervals so that the magnitude condition generates a fixed number of frequencies as functions of the delay within each subinterval. The crossing conditions are expressed in a general form, and a simplified derivation for the first-order derivative criterion is obtained. Illustrative examples are also presented in this paper. [ABSTRACT FROM AUTHOR]

In this technical note we present a stability analysis approach for polynomially-dependent one-parameter systems. The approach, which appears to be conceptually appealing and computationally efficient and is referred to as an eigenvalue perturbation approach, seeks to characterize the analytical and asymptotic properties of eigenvalues of matrix-valued functions or operators. The essential problem dwells on the asymptotic behavior of the critical eigenvalues on the imaginary axis, that is, on how the imaginary eigenvalues may vary with respect to the varying parameter. This behavior determines whether the imaginary eigenvalues cross from one half plane into another, and hence plays a critical role in determining the stability of such systems. Our results reveal that the eigenvalue asymptotic behavior can be characterized by solving a simple generalized eigenvalue problem, leading to numerically efficient stability conditions. [ABSTRACT FROM AUTHOR]