Stability Analysis Of Time-delay Systems Research Articles

An investigation into the stability of helicopter control system under the influence of time delay using an improved LKF

This paper presents an improved Lyapunov–Krasovskii functional (LKF)-based stability analysis of time-delay systems (TDS) and its application to helicopter control systems. A novel LKF is proposed for the stability analysis of linear TDS after thoroughly examining the effects of each integral associated with a general class of LKF. The suggested stability results in the form of linear matrix inequalities (LMIs) are obtained using the Wirtinger-based integral inequality (WII), reciprocally convex combination lemma (RCCL), and extended reciprocally convex matrix inequality (ERCMI). Numerical examples are utilized to evaluate the impact of the presented theorems on stability. The proposed theorems are then used to examine the stability of helicopter control systems, and the results of these stability analyses are verified using a laboratory model of a helicopter control system.

On the necessity of sufficient LMI conditions for time-delay systems arising from Legendre approximation

This work is dedicated to the stability analysis of time-delay systems with a single constant delay using the Lyapunov–Krasovskii theorem. This approach has been widely used in the literature and numerous sufficient conditions of stability have been proposed and expressed as linear matrix inequalities (LMI). The main criticism of the method that is often pointed out is that these LMI conditions are only sufficient, and there is a lack of information regarding the reduction of the conservatism. Recently, scalable methods have been investigated using Bessel–Legendre inequality or orthogonal polynomial-based inequalities. The interest of these methods relies on their hierarchical structure with a guarantee of reduction of the level of conservatism. However, the convergence is still an open question that will be answered for the first time in this paper. The objective is to prove that the stability of a time-delay system implies the feasibility of these scalable LMI, at a sufficiently large order of the Legendre polynomials. Moreover, the proposed contribution is even able to provide an analytic estimation of this order, giving rise to a necessary and sufficient LMI for the stability of time-delay systems.

Stability analysis of time-delay systems in the parametric space

This paper presents a novel method for stability analysis of a wide class of linear, time-delay systems (TDS), including retarded, incommensurate and distributed delays. The proposed method is based on frequency domain analysis and application of Rouché’s theorem. Given a parametrized TDS and an arbitrary parametric point, the proposed method is capable of identifying the surrounding region in the parametric space for which the number of unstable poles remains invariant. First, a procedure for investigating stability along a line is developed. Then, the results are extended by application of Hölder’s inequality to investigate stability within a region. The proposed method is uniformly applicable to parameters of different types (simple delays, distributed delay limits, time constants, etc.), as illustrated by examples.

Criteria on Exponential Incremental Stability of Dynamical Systems with Time Delay

Incremental stability analysis for time-delay systems has attracted more and more attention for its contemporary applications in transportation processes, population dynamics, economics, satellite positions, etc. This paper researches the criteria for exponential incremental stability for time-delay systems with continuous or discontinuous right-hand sides. Firstly, the sufficient conditions for exponential incremental stability for time-delay systems with continuous right-hand sides are studied, and several corollaries for specific cases are provided. As for time-delay systems with discontinuous right-hand sides, after expounding the relevant conditions for the existence and uniqueness of the Filippov solution, by using approximation methods, sufficient conditions for exponential incremental stability are obtained. The conclusions are applied to linear switched time-delay systems and Hopfield neural network systems with composite right-hand sides.

Interval approximation method for stability analysis of time-delay systems

This paper studies the stability analysis of linear systems with time-varying delay, which is supposed to be the trigonometric form. By utilizing the characteristics between time-varying delay and its derivative, a novel interval approximation method is proposed, which provides the new allowable delay sets. Then making use of Wirtinger inequality, reciprocally convex inequality and the looped Lyapunov–Krasovskii functionals, the stability criteria with less conservatism are obtained. Finally, two examples are used to show the effectiveness and efficiency of the stability criteria.

Stability Analysis for Time-Delay Systems via a New Negativity Condition on Quadratic Functions

This article studies the stability problem of linear systems with time-varying delays. First, a new negative condition is established for a class of quadratic functions whose variable is within a closed set. Then, based on this new condition, a couple of stability criteria for the system under study are derived by constructing an appropriate Lyapunov–Krasovskii functional. Finally, it is demonstrated through two numerical examples that the proposed stability criteria are efficient and outperform some existing methods.

Stability Analysis of Time-Delay Systems via a Delay-Derivative-Partitioning Approach

This paper is devoted to the study of delay-dependent stability of time-varying delay systems. A delay-derivative-partitioning approach is proposed. By constructing an augmented Lyapunov functional that contains two delay-product-dependent terms and using the delay-derivative-partitioning approach, an improved stability condition is established. The derived condition is described as a set of linear matrix inequalities, which can be readily implemented. Finally, four examples are carried out so as to attest the effect and merit of the presented approach.

Order of Legendre-LMI conditions to assess stability of time-delay systems

This paper investigates the stability analysis of time-delay systems through Lya-punov arguments. Using the existence of a complete Lyapunov-Krasovskii functional and relying on the polynomial approximation theory, our main goal is to approximate the complete Lyapunov functional and to take profit of the supergeometric convergence rate of the truncated error part. Necessary and sufficient conditions in the linear matrix inequality (LMI) framework for sufficiently large approximated orders are consequently proposed. Moreover, an estimation of the necessary order is provided analytically with respect to system parameters.

On the time-varying Halanay inequality with applications to stability analysis of time-delay systems

The main results of the paper are improvements on the stability analysis of Halanay inequalities with time-varying coefficients in both continuous-time and discrete-time setting. Three classes of improved conditions are established to ensure that the solution to the Halanay inequality is uniformly exponentially stable. The merit of the proposed new conditions is that the coefficients of the Halanay inequality can be unbounded and sign indefinite. This is achieved by using the notion and properties of uniformly asymptotic stable (UAS) functions. Based on the improved stability conditions for the Halanay inequality and the Lyapunov Razumikhin approach, three classes of sufficient conditions are established for testing the stability of time-varying time-delay systems. Finally, the advantages of the proposed methods are illustrated by some numerical examples with some of them borrowed from the literature.

Stability Analysis for Time-delay Systems via a Novel Negative Condition of the Quadratic Polynomial function

Stability Analysis for Time-delay Systems via a Novel Negative Condition of the Quadratic Polynomial function

Composite slack-matrix-based integral inequality and its application to stability analysis of time-delay systems

This paper focuses on the delay-dependent stability problem of time-varying delay linear systems. A composite slack-matrix-based integral inequality (CSMBII) is presented in delay-product types. It overcomes the limitation of reciprocal convexity in Bessel–Legendre integral inequality such that CSMBII is more convenient to cope with time-varying delay systems. And it also overcomes the shortcoming that the slack matrices in previous works are independent of time-varying delay. Based on CSMBII, a new delay-dependent stability criterion is derived for time delay systems. A numerical example is given to illustrate the effectiveness of the stability criterion.

Extended Integral Inequality for Stability Analysis of Time-delay Systems

In this paper, we propose an extension of the generalized free-weighting-matrix based integral inequality, which is an enriched special case of the conventional inequality. This improved inequality requires less conservative criteria, since the vectors can be chosen independently and freely. By applying the proposed inequality combined with a double integral inequality to evaluate the derivative of Lyapunov-Krasovskii functional, a new stability criterion is formulated for a system with a time delay. To illustrate the effectiveness of the new stability criteria, we take three commonly used numerical examples. Simulation results confirm the effectiveness of the proposed approach.

Active vibration control and stability analysis of a time-delay system subjected to friction-induced vibration

The use of active vibration control may induce a delay leading to detrimental degradation of the performance of active vibration control. This is particularly true in the case of mechanical systems subjected to friction-induced vibration and noise for which such time-delays can lead to the appearance of undesirable instability. Furthermore, conducting a stability analysis of time-delay systems and estimation of the critical time delay are challenging, due to the infinite nature of the characteristic (quasi) polynomial of the associated closed-loop system, having an infinite number of roots.The objective of this paper is to discuss a strategy for the estimation of the critical time delay for the problem of Friction-Induced Vibration and noisE (FIVE). To achieve such an objective, the prediction of the stability analysis of time delay systems and the estimation of the associated critical time delay are first performed by applying the frequency sweep test and the eigenvalue problem approximation using the Taylor series expansion of the delayed term. In a second time, a mixed approach is proposed to predict effectively the real critical time delay of autonomous controlled systems subjected to friction-induced vibration. The efficiency of the proposed approach is illustrated by numerical examples for the prediction of self-sustaining vibrations of a phenomenological model with two degrees of freedom for which it is possible to provide a clear understanding and illustration of the phenomena involved and observed.

Observer Design for Cyber-Physical Systems With State Delay and Sparse Sensor Attacks

This paper studies the problem of the secure state estimation of cyber-physical systems. The system model we studied is a class of discrete-time systems with state delay and sparse sensor attacks. In the design of the observer, a Projection Operator is utilized to determine the correct attack set, and the gain matrix will be switched accordingly. In this way, the excessive estimation error could be avoided effectively. In addition, based on the generalization of the classical Krasovskii Stability Theorem for stability analysis of time-delay systems, a Lyapunov-Krasovskii functional is chosen and a sufficient condition for the existence of the observer together with its design method is determined. The designed observer overcomes the difficulties caused by state delay and sensor attacks, and has good performance in secure estimation. Finally, two examples are given to verify that the observer can restore the desired estimate in a short time when a sudden attack occurs.

Polynomial-Type Lyapunov–Krasovskii Functional and Jacobi–Bessel Inequality: Further Results on Stability Analysis of Time-Delay Systems

To derive a less conservative stability criterion via Lyapunov-Krasovskii functional (LKF) method, in previous literature, multiple integral terms are usually introduced into the construction of LKFs. This article generalizes the results of previous literature by proposing a polynomial-type LKF, which contains the LKFs with multiple integral terms as special cases. In addition, a Jacobi-Bessel inequality is presented to bound the derivative of such LKF. As a result, an improved stability criterion of time-delay systems is established. Finally, two numerical examples are given to illustrate the effectiveness, and advantages of our method.

Novel stability criteria for linear time-delay systems using Lyapunov-Krasovskii functionals with a cubic polynomial on time-varying delay

One of challenging issues on stability analysis of time-delay systems is how to obtain a stability criterion from a matrix-valued polynomial on a time-varying delay. The first contribution of this paper is to establish a necessary and sufficient condition on a matrix-valued polynomial inequality over a certain closed interval. The degree of such a matrix-valued polynomial can be an arbitrary finite positive integer. The second contribution of this paper is to introduce a novel Lyapunov-Krasovskii functional, which includes a cubic polynomial on a time-varying delay, in stability analysis of time-delay systems. Based on the novel Lyapunov-Krasovskii functional and the necessary and sufficient condition on matrix-valued polynomial inequalities, two stability criteria are derived for two cases of the time-varying delay. A well-studied numerical example is given to show that the proposed stability criteria are of less conservativeness than some existing ones.

A new multiple integral inequality and its application to stability analysis of time-delay systems

This paper focuses on the stability problem of linear time-delay systems. Firstly, a generalized vectors-based multiple integral inequality (GVMII) is presented, which can treat some existing results as its special cases. Secondly, based on the multiple integral inequality, a new delay-dependent stability criterion is derived for time-delay systems. Finally, a numerical example is given to illustrate the effectiveness of the stability criterion.

Finite-interval quadratic polynomial inequalities and their application to time-delay systems

This paper proposes two inequality lemmas for the stability analysis of time-delay systems to obtain the stability criteria in terms of linear matrix inequalities from ones in terms of the convex of concave finite-interval quadratic polynomials. The first lemma improves the existing work for the concave finite-interval quadratic polynomial by utilizing the property of a cross point between two tangent lines at the boundaries of the finite interval. The second lemma introduces an inequality for the convex or concave finite-interval quadratic polynomials by decomposing the finite-interval quadratic polynomials into aTMa, where M is a 2 × 2 block matrix and a is a vector consisting of a constant and a variable, and by exploiting information on the finite interval. To illustrate the potential of the proposed lemmas, this paper presents feasible regions for the finite-interval quadratic polynomials and maximum time-delay upper bounds for time-delay systems, comparing to those in the literature.

Insight into stability analysis of time-delay systems using Legendre polynomials

In this paper, a numerical analysis to assess stability of time-delay systems is investigated. The proposed approach is based on the design of a finite-dimensional approximation of the infinite-dimensional space of solutions of the system. Indeed, based on the dynamical coefficients on the sequence made of the first Legendre polynomials, the original time-delay system is modelled by a finite-dimensional model interconnected to a modelling error.Putting aside the interconnection, the resulting finite-dimensional system turns out to be a nice approximation of the time-delay system. Using Padé arguments, the eigenvalues of this finite-dimensional system are proven to converge towards a set of characteristic roots of the original time-delay system. Furthermore, considering now the whole interconnected system and having a deeper look at the interconnection, an enriched Lyapunov-Krasovskii functional is proposed to develop a sufficient condition expressed in terms of linear matrix inequalities for the stability of the time-delay system. Both results are illustrated on a toy example.

A generalized multiple-integral inequality and its application on stability analysis for time-varying delay systems

This paper is concerned with the stability analysis of time-delay systems. Lyapunov–Krasovskii functional method is utilized to obtain stability criteria in the form of linear matrix inequalities. The main purpose is to obtain less conservative stability criteria by reducing the estimation gap of the time derivative of the constructed Lyapunov–Krasovskii functional. First, a generalized multiple-integral inequality is put forward based on Schur complement lemma. Then, some special cases of the proposed generalized multiple-integral inequality are given to estimate single and double integral terms in the derivative of Lyapunov–Krasovskii functional. Furthermore, less conservative stability criteria are derived. Finally, three examples are given to illustrate the improvement of the proposed criteria.