Time-delay Systems Of Neutral Type Research Articles

Multiplicity–Induced–Dominancy property for second–order neutral differential equations with application in oscillation damping

This paper addresses the exponential stability of linear time-delay systems of neutral type. In general, it is quite a challenge to establish conditions on the parameters of the system in order to guarantee such a stability. Recent works emphasized the link between maximal multiplicity and dominant roots. Indeed, conditions for a given multiple root to be necessarily dominant are investigated, this property is known as Multiplicity-Induced-Dominancy (MID). The aim of this paper is to explore the effect of multiple roots with admissible multiplicities exhibiting, under appropriate conditions, the validity of the MID property for second-order neutral time-delay differential equations with a single delay. The ensuing control methodology is summarized in a five-steps algorithm that can be exploited in the design of higher-order systems. The main ingredient of the proposed method is the dominancy proof for multiple spectral values based on frequency bounds established via integral equations. As an illustration, the stabilization of the classical oscillator benefits from the obtained results.

Asymptotic Analysis of Spectrum and Stability for One Class of Singularly Perturbed Neutral-Type Time-Delay Systems

In this study, a singularly perturbed linear time-delay system of neutral type is considered. It is assumed that the delay is small of order of a small positive parameter multiplying a part of the derivatives in the system. This system is decomposed asymptotically into two much simpler parameter-free subsystems, the slow and fast ones. Using this decomposition, an asymptotic analysis of the spectrum of the considered system is carried out. Based on this spectrum analysis, parameter-free conditions guaranteeing the exponential stability of the original system for all sufficiently small values of the parameter are derived. Illustrative examples are presented.

A Scalable Approach to Compute Delay Margin of a Class of Neutral-Type Time Delay Systems

An approach is presented to approximate the delay margin of a class of linear time-invariant neutral-type time delay systems. The approach is based on frequency sweeping tools and factorization of ...

Necessary and Sufficient Stability Condition by Finite Number of Mathematical Operations for Time-delay Systems of Neutral Type

We present a new necessary and sufficient condition of exponential stability of neutral-type linear time invariant (LTI) systems with one delay. The new stability criterion is given in terms of a block matrix, which exclusively depends on the delay Lyapunov matrix and is of finite dimension.

Development of an Effective Model for Computing Rightmost Eigenvalues of Power Systems With Inclusion of Time Delays

This paper proposes a new discretization model for computing the rightmost eigenvalues, i.e., the dominant modes, of the power system with inclusion of time delays. At first, a general model of the time-delay power system is introduced. This model can be adopted for analysis of both retarded and neutral-type time-delay systems. Then, based on the general model, a discretization method for computing the rightmost eigenvalues is developed. In the method, only the retarded variables are required to be discretized. As a result, the order of the proposed model is far less than the order of existing models for solving eigenvalues of time-delay power systems. Meanwhile, the sparsity of the original power system model is passed down to the discretized model. The proposed model is tested on 5 power systems, from 4 machines to 562 machines, to show its effectiveness and efficiency.

Stability of neutral type delay systems: A joint Lyapunov–Krasovskii and Razumikhin approach

In this paper, a new stability criterion for linear time-delay systems of neutral type is presented. By virtue of the well-known Krasovskii theorem, the system is exponentially stable, if and only if there exists a functional which (i) has a negative definite derivative along the solutions of the system, and (ii) admits a quadratic lower bound, of course under an assumption that the difference operator corresponding to the system is stable. We modify the second condition in the following manner: The functional is required to admit a quadratic lower bound only on the special set of initial functions in our stability result instead of the set of all appropriate functions in the Krasovskii theorem. This special set consists of the functions satisfying a Razumikhin-type inequality and a similar inequality on the derivative. Basing on such modification and staying within the framework of the functionals with a given derivative, we suggest a methodology for the stability analysis, which is described and tested on the examples in the case of a scalar equation. Unlike most of recent results, we use the functional with a derivative prescribed just as a negative definite quadratic form of the “current” state of a system, what does not impose restrictions on the approach.

A note on converse Lyapunov-Krasovskii theorems for nonlinear neutral systems in Sobolev spaces

The problem of existence of a Lyapunov-Krasovskii functional (LKF) for nonlinear neutral type time-delay systems is revisited considering the uniform stability analysis and the LKF in a Sobolev space of absolutely continuous functions with bounded derivatives.

Design of robust decentralised controllers for MIMO plants with delays through network structure exploitation

ABSTRACT A methodology is proposed for the design of robust controllers for retarded and neutral-type time-delay systems, focusing on decentralised and overlapping fixed-order controllers for Multiple Input Multiple Output (MIMO) systems. The methodology is grounded in a direct optimisation approach and relies on the minimisation of spectral abscissa and cost functions, as a function of the controller or design parameters. First, an approach applicable to generic MIMO systems is presented, which imposes a suitable sparsity pattern with the possibility of fixing elements in the controller parameterisation. Second, if the system to be controlled has the structure of a network of coupled identical subsystems, then it is shown that this structure can be exploited by an improved algorithm for the design of decentralised controllers, thereby improving computational efficiency and scalability with the number of subsystems. Several numerical examples illustrate the effectiveness of the methodology, and its extension towards consensus type problems.

Stability analysis of linear stochastic neutral-type time-delay systems with two delays

This paper is concerned with stability analysis of stochastic neutral-type time-delay systems with two different delays in both the state variables and the retarded derivatives of state variables. Sufficient conditions in terms of linear matrix inequalities (LMIs) are obtained by applying the Lyapunov–Krasovskii functional (LKF) based approach. A novel spectral radius based condition is imposed to guarantee the stability of the associated difference operator and a novel method is also proposed to analyze the stability of the overall system. Moreover, by carefully analyzing the relationship between these two different delays, a method is provided to construct all possible yet the minimal number of simple quadratic integral functions in the LKF, which helps to reduce both the conservatism and complexity of the resulting stability conditions. The effectiveness of the proposed approaches is illustrated by two numerical examples.

Necessary conditions of exponential stability for a class of linear neutral-type time-delay systems

ABSTRACTFor a class of linear neutral-type time-delay systems (NTTDSs), this paper will present necessary exponential stability conditions by employing the Lyapunov--Krasovskii functional approach. Since these conditions are represented by the Lyapunov matrix and the neutral coefficient matrix, they not only offer a novel tool for analysing stability of linear NTTDSs by characterising instability domains, but also extend the existing results of the neutral-delay-free systems. As a medium step, the relations between the Lyapunov matrix and the fundamental matrix are characterised. The validation of the obtained results is explained by numerical examples and comparison with some existing results.

Necessary Stability Conditions for Neutral Type Systems With a Single Delay

Necessary conditions for the exponential stability of time-delay systems of neutral type, and a new stability criterion for the scalar case are presented in this contribution. They are obtained in the framework of Lyapunov-Krasovskii functionals of complete type. The particularity of these conditions is that they depend exclusively on the delay Lyapunov matrix.

New robustness bounds for neutral type delay systems via functionals with prescribed derivative

The paper is devoted to the robust stability analysis of linear neutral type time delay systems with a constant delay and norm-bounded uncertainties. The method is based on the Lyapunov–Krasovskii functional with a derivative prescribed as a negative definite quadratic form of the “current” system state, which is considered to be not suitable for the robustness analysis due to the fact that it does not admit a quadratic lower bound. Unlike existing results, our approach does not require the derivative of the functional along the solutions of the perturbed system to be negative definite. Instead, we need just an essential part of the integral of the derivative to be negative. The resulting stability condition is presented in the form of a simple inequality depending on the so-called Lyapunov matrix, under an assumption that the difference operator of the perturbed system is stable. The result is applicable to all exponentially stable systems.

Stability Analysis of Neutral Type Time-Delay Positive Systems with Commensurate Delays

Abstract Recently, the author derived a necessary and sufficient condition for the asymptotic stability of neutral type time-delay positive systems (TDPSs), where the neutral type TDPS of interest is given by a feedback connection between a finite-dimensional LTI positive system and the pure delay The goal of this paper is to extend this result to neutral type TDPSs with multiple commensurate delays. To this end, we first represent a neutral type TDPS with multiple commensurate delays as a neutral type TDPS with a single delay, by augmenting input and output signals in the feedback connection. By this conversion, we can readily apply the latest result already established for single delay neutral type TDPSs. As the main result, we show that a neutral type TDPS with commensurate delays is stable if and only if its delay-free counterpart is stable and an additional admissibility condition is satisfied. This result in particular implies that the stability is irrelevant of the length of delays.

Input delay compensation for neutral type time-delay systems

In this contribution we investigate the input delay compensation for neutral type time-delay systems with both state and input delays. A nested predictor is established to predict the future states such that the input delay that can be arbitrarily large yet bounded are compensated completely. It is shown that the compensated closed-loop system in the presence of input delay possesses the same characteristic equation as the closed-loop system in the absence of input delay. An implementation scheme by adding input filters is also proposed. Under an additional assumption, explicit nested predictor feedback controllers involving only 1-fold integrals are established. A numerical example is carried out to support the obtained theoretical results.

On Exact Controllability of Time-Delay Systems of Neutral Type

Данная работа посвящена решению задачи точной управляемости для достаточно широкого класса систем с запаздыванием нейтрального и смешанного типов. Рассматривая эквивалентную операторную модель в гильбертовом пространстве, мы формулируем условия управляемости в виде некоторой векторной проблемы моментов. Вид данной проблемы моментов существенно упрощается при наличии базиса пространства из собственных векторов оператора системы с запаздыванием. Замена управления позволяет преобразовать структуру системы, и гарантировать существование базиса из собственных векторов соответствующего оператора. Мы доказываем критерий точной управляемости и устанавливаем точное время управления.

Predicting the secondary dynamic mode interference phenomenon in thermoacoustic instability control

This paper brings a novel mathematical perspective in assessing the rise of the secondary dynamic modes to prominence during the suppression of thermoacoustic instability. This phenomenon is observed by many earlier investigators; however, without a complete analytical reasoning. We consider a Rijke tube with both a passive Helmholtz resonator and an active feedback control to suppress instabilities. The core dynamics is represented as a linear time-invariant multiple time-delay system of neutral type. Parametric stability of the resulting infinite-dimensional dynamics is investigated using a recent analytical tool: cluster treatment of characteristic roots paradigm. This tool reveals the stability outlook of such systems exhaustively and non-conservatively in the parameter space of the system. First, we examine the stability with and without the Helmholtz resonator. We then select an unstable operation for the resonator-mounted Rijke tube, impose a time-delayed integral feedback control over it and reveal the stabilizing controller parameters using the cluster treatment of characteristic roots methodology. When high control gains are inappropriately selected, the new analytical procedure declares how the secondary dynamic modes of the system exhibit instability although the initially unstable mode is now stabilized. All of these stability assessments are cross-validated using experimental results from a laboratory-scale Rijke tube set-up.

Obtention of the functional of complete type for neutral type delay systems via a new Cauchy formula

In this paper, we present a new manner for computing the so-called new form of the functional of complete type for neutral type time-delay systems without assumptions of continuity and differentiability on the initial function. It is obtained by using a new Cauchy formula. We present briefly the necessary stability conditions depending on the delay Lyapunov matrix that this result makes possible to prove.

Estimates for solutions to a class of time-delay systems of neutral type with periodic coefficients and several delays

Estimates for solutions to a class of time-delay systems of neutral type with periodic coefficients and several delays

A neutral system approach to H∞ PD/PI controller design of processes with uncertain input delay

This paper presents a neutral system approach to the design of an H∞ controller for input delay systems in presence of uncertain time-invariant delay. It is shown that when proportional derivative (PD) controller is applied to a time-delay system, the resulting closed loop system is generally a time-delay system of neutral type with delay term coefficients depending on the controller parameters. A descriptor model transformation is used to derive an advantageous bounded real lemma representation for the system. Furthermore, new delay-dependent sufficient conditions for the existence of an H∞ PD and PI controller in presence of uncertain delay are derived in terms of matrix inequalities. Some case studies and numerical examples are given in order to illustrate the advantages of the proposed method.

Practical stability and stabilization of a class of nonlinear neutral type time delay systems with multiple delays: BMI’s approaches

In this paper, the practical stability and stabilization of a class of nonlinear neutral type time delay systems with multiple delays and bounded perturbations is discussed. Sufficient conditions based on Lyapunov-Krasovskii functionals are derived. They are formulated in terms of the feasibility of a set of Bilinear Matrix Inequalities (BMI’s). A practical exponential estimate of the system response is also obtained. This approach is shown to be useful in the solution of an engineering problem: the elimination of the stick-slip phenomenon in the drilling process.