Stability Analysis Of Linear Systems Research Articles

Stability analysis of linear systems with multiple time-varying delays via a region partitioning approach and reciprocally convex combination lemmas

The delays-dependent stability analysis of linear systems with multiple time-varying delays is addressed in this study. To estimate the integral term that results from the differentiation of Lyapunov–Krasovskii functional (LKF), an improved region partitioning approach and relaxed lemmas are proposed. Based on all the delayed state information, the maximum delay interval [−h,0] is separated into 2N non-overlapping subintervals. Secondly, two novel generalized reciprocally convex combination lemmas (GRCCLs) are proposed, with Bessel–Legendre-based inequality, to estimate the integral terms generated by the region partitioning approach to obtain less conservative stability criteria. Finally, the obtained stability criteria is applied to simple linear systems and load frequency control (LFC) scheme of the two-area power system for stability analysis, and the effectiveness of proposed method is verified.

Parametrised auxiliary function-based integral inequality for time delay system

ABSTRACT This paper employs two scalar parameters to extend auxiliary function-based integral inequality into a new formulations, say a parametrised auxiliary function-based integral inequality. Using these formulations, the constituent signals are utilised more efficiently by exploiting the interaction between them to reduce conservatism. Numerical example for stability analysis of linear systems with time-varying delay shows the improved performance of the proposed new formulations in terms of maximum delay bounds and decision variables.

Stability analysis of linear delayed systems based on an allowable delay set partitioning approach

This note is concerned with the stability analysis of linear systems with a time-varying delay, where the delay is assumed to be a periodically varying bounded function with restricted derivatives. Combining the allowable delay set (ADS) definition and the region partitioning technique, an ADS partitioning approach is proposed, and then by with aid of this approach, a novel Lyapunov–Krasovskii functional (LKF), namely delay-set-partition-based (DSPB) LKF, is developed. Compared with the previous LKF method, the main advantage of the DSPB LKF lies in allowing the considered system to construct its own LKF in each sub-ADS. By utilizing the proposed LKF, less conservative stability criteria are derived. Finally, two simulation examples are provided to verify the effectiveness and superiority of the proposed method.

Stability analysis of linear systems with time-varying delay via some novel techniques

This paper concentrates on the stability analysis of linear systems with a time-varying delay. Firstly, a new Lyapunov–Krasovskii functional (LKF) is given, which involves more time delay cross-terms information. Secondly, an improved reciprocally convex inequality with higher estimation accuracy is derived. Then, on the basis of the constructed LKF and the improved reciprocally convex inequality, some new stability criteria with less conservative are proposed. Finally, the superiority of the obtained stability criteria is explained by two numerical examples.

The Stability Analysis of Linear Systems with Cauchy—Polynomial-Vandermonde Matrices

The numerical approximation of both eigenvalues and singular values corresponding to a class of totally positive Bernstein–Vandermonde matrices, Bernstein–Bezoutian structured matrices, Cauchy—polynomial-Vandermonde structured matrices, and quasi-rational Bernstein–Vandermonde structured matrices are well studied and investigated in the literature. We aim to present some new results for the numerical approximation of the largest singular values corresponding to Bernstein–Vandermonde, Bernstein–Bezoutian, Cauchy—polynomial-Vandermonde and quasi-rational Bernstein–Vandermonde structured matrices. The numerical approximation for the reciprocal of the largest singular values returns the structured singular values. The new results for the numerical approximation of bounds from below for structured singular values are accomplished by computing the largest singular values of totally positive Bernstein–Vandermonde structured matrices, Bernstein–Bezoutian structured matrices, Cauchy—polynomial-Vandermonde structured matrices, and quasi-rational Bernstein–Vandermonde structured matrices. Furthermore, we present the spectral properties of totally positive Bernstein–Vandermonde structured matrices, Bernstein–Bezoutian structured matrices, Cauchy—polynomial-Vandermonde structured matrices, and structured quasi-rational Bernstein–Vandermonde matrices by computing the eigenvalues, singular values, structured singular values and its lower and upper bounds and condition numbers.

A new stability criterion and its application to robust stability analysis for linear systems with distributed delays

Recently, the possibility to verify positive definiteness of Lyapunov–Krasovskii functionals only on a specific Razumikhin-type set of functions to conclude on the stability of linear time delay systems was shown. In this paper, for a class of systems with multiple concentrated and distributed delays, we extend the result and prove that this specific set may be applied while verifying the negative definiteness of the functionals derivatives along the solutions as well. The result is applied in the robustness analysis with respect to uncertainties both in the system matrices and in the delays. It provides a simple way to prove the robust stability conditions due to obviating the need to work with indefinite derivatives of functionals along the solutions of a perturbed system. As a by-product, the estimates for an unstable eigenvalue of the system are derived, and the functionals defined on complex-valued initial functions are presented.

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 of Linear Systems With Interval Time-Varying Delay

Stability Analysis of Linear Systems With Interval Time-Varying Delay

Advanced stability analysis for linear systems with time-varying delays via a generalized integral inequality

The improved stability conditions for linear systems with time-varying delays are investigated. The proposed integral inequality derived from the properties of simple matrices has been simply proved, and this is a generalized form of various integral inequalities. Thus, depending on N, various forms of integral inequality can be derived. Also, the stability results are obtained by extending the augmented terms of Lyapunov–Krasovskii functionals according to N. Finally, two well-known numerical examples show that the stability criteria derived through the proposed generalized integral inequality have less conservatism than those of the existing ones.

Stability Analysis for Linear Systems with a Differentiable Time-Varying Delay via Auxiliary Equation-Based Method

This paper concentrates on the stability problem for linear systems with a differentiable time-varying delay via an auxiliary equation-based method. By supposing that the second-order derivative of the system state is available, an auxiliary equation is obtained. On the basis of the system equation and the auxiliary equation, we define a suitable delay-product-type augmented Lyapunov-Krasovskii functional (LKF), under which more delay and system state information can be exploited. Based on the LKF, by utilizing some vital lemmas, adding zero terms, and the convex analysis method, we propose a new stability condition that is less conservative. Finally, to illustrate the merit of the obtained stability condition, two typical numerical examples are given.

Improved Stability Conditions for Time-Varying Delay Systems via Relaxed Lyapunov Functionals

In this paper, the stability analysis of linear systems with time-varying delays is studied. A novel Lyapunov method is presented, in which positive definiteness of the matrices in common Lyapunov functionals is relaxed by adding what is referred to as a zero-integral functional (ZIF). A general form of auxiliary polynomial-based functionals that contains such ZIF is given. Choosing polynomials of different order as well as exploring double-delay-product (DDP) terms, novel Lyapunov functionals are constructed, which contribute to a set of improved stability conditions expressed in terms of linear matrix inequalities. Finally, numerical examples are provided to corroborate the merits of the proposed method relative to a number of existing methods, and in particular, the effectiveness of the proposed ZIFs and DDP terms in reducing the conservatism of stability conditions.

A novel integral inequality and its application to stability analysis of linear system with multiple time delays

The integral inequality plays a key role in developing delay-dependent stability criteria for the linear system with multiple time delays. This paper investigates a novel integral inequality to bound the integral terms in the derivative of the Lyapunov–Krasovskii functional (LKF). Compared with the auxiliary function-based integral inequality, the presented integral inequality is more general. By removing the involved auxiliary function, the integral term in the bound is linearly represented with a sequence of integrals only related to the system state. After a newly augmented LKF is constructed with multiple integral terms concerning the relationships among multiple delays, an adjustable and relaxed stability criterion is established in terms of linear matrix inequalities using the proposed inequality. Two numerical examples are given to show the merits of the proposed method.

Improved stability criteria for linear systems with time-varying delays

This paper is concerned with the stability analysis of linear systems with time-varying delays. First, by introducing the quadratic terms of time-varying delays and some integral vectors, a more suitable Lyapunov-Krasovskii functional (LKF) is constructed. Second, two new delay-dependent estimation methods are developed in the stability analysis of linear system with time-varying delays, which include a reciprocally convex matrix inequality and an integral inequality. More information about time-varying delays and more free matrices are introduced into the two estimation approaches, which play a key role for obtaining an accurate upper bound of the integral terms in time derivative of LKFs. Third, based on the novel LKFs and new estimation approaches, some less conservative criteria are derived in the form of linear matrix inequality (LMI). Finally, three numerical examples are applied to verify the advantages and effectiveness of the newly proposed methods.

Stability Analysis of Systems With Time-Varying Delay via Improved Lyapunov–Krasovskii Functionals

This paper is concerned with the delay-dependent stability analysis of linear systems with a time-varying delay. Two types of improved Lyapunov-Krasovskii functionals (LKFs) are developed to derive less conservative stability criteria. First, a new delay-product-type LKF, including single integral terms with time-varying delays as coefficients is developed, and two stability criteria with less conservatism due to more delay information included are established for different allowable delay sets. Second, the delay-product-type LKF is further improved by introducing several negative definite quadratic terms based on the idea of matrix-refined-function-based LKF, and two stability criteria with more cross-term information and less conservatism for different allowable delay sets are also obtained. Finally, a numerical example is utilized to verify the effectiveness of the proposed methods.

Improved Stability Criteria on Linear Systems with Distributed Interval Time-Varying Delays and Nonlinear Perturbations

The problem of delay-range-dependent stability analysis for linear systems with distributed time-varying delays and nonlinear perturbations is studied without using the model transformation and delay-decomposition approach. The less conservative stability criteria are obtained for the systems by constructing a new augmented Lyapunov–Krasovskii functional and various inequalities, which are presented in terms of linear matrix inequalities (LMIs). Four numerical examples are demonstrated for the results given to illustrate the effectiveness and improvement over other methods.

An Affine Integral Inequality of an Arbitrary Degree for Stability Analysis of Linear Systems With Time-Varying Delays

This paper is concerned with the stability analysis problems of linear systems with time-varying delays using integral inequalities. To reduce the conservatism of stability criteria obtained with Lyapunov-Kraosvksii approach, there has been a growing tendency to utilize various integral quadratic functions in the construction of Lyapunov-Krasovskii functionals. Consequently, integral inequalities also have played key roles to derive stability criteria guaranteeing the negativity of the Lyapunov-Krasovskii functional’s derivative. Recently, by utilizing first-degree or second-degree orthogonal polynomials, new free-matrix-based integral inequalities have been proposed for integral quadratic functions containing both a system state variable and its time derivative. This paper tries to generalize these inequalities and their stability criteria with a note on the relation among the existing integral inequalities and the proposed one. In this note, it is shown that increasing a degree of the proposed integral inequality only reduces or maintains the conservatism by deriving the hierarchical stability criteria. Four numerical examples including practical systems demonstrate the effectiveness of the proposed methods in terms of allowable upper delay bounds.

New Lyapunov-Krasovskii Functional for Stability Analysis of Linear Systems with Time-Varying Delay

This paper focuses on the problem of delay-dependent stability of linear systems with time-varying delay. A new delay-product-type augmented Lyapunov-Krasovskii functional (LKF) is constructed. Based on the LKF and by employing a generalized free-matrix-based integral inequality, less conservative delay-dependent stability criteria are obtained. Finally, two well-known numerical examples are used to confirm the effectiveness and the superiority of the presented stability criteria.

Robust stability analysis for linear systems with distributed delays: A time‐domain approach

SummaryThis work is devoted to the robust stability analysis of linear systems with distributed delays. The approach is based on recent results in the Lyapunov‐Krasovskii framework, where a Lyapunov‐Krasovskii functional with prescribed negative definite derivative is applied. The stability conditions obtained in this work, are simple inequalities which depend on the so‐called delay Lyapunov matrix. The cases of matrix parameters and delays uncertainties are addressed, accurately detecting exact stability bounds when applied in an iterative manner. Our method is used successfully to tackle the challenging problem of robust predictor‐based stabilization of systems with state and input delays.

Stability analysis of linear systems with time‐varying delay via a quadratic function negative‐definiteness determination method

This study aims to carry out the stability analysis of linear systems with a time-varying delay. It is known that the negative-definite condition of the derivative of a Lyapunov–Krasovskii functional (LKF) can be determined using the convex combination method if the convexity requirement is satisfied by the derivative of the LKF. However, this method is not feasible in cases where the LKF's derivative is a quadratic function. To address this problem, this study proposes a novel negative-definiteness determination lemma that encompasses the previous lemmas as its special cases and shows less conservatism. Then, this lemma is employed to derive a stability criterion, and its superiority is demonstrated using three examples.

Reducing conservatism of stability criteria for linear systems with time-varying delay using an improved triple-integral inequality

This paper focuses on stability analysis of linear systems with time-varying delay, where two cases of the time-varying delay are discussed, that is, the time-varying delay is either differentiable or just uniformly continuous. First, an improved triple-integral inequality is proposed to estimate triple-integrals tightly. Then, by introducing two novel Lyapunov–Krasovskii functionals catering for two cases of time-varying delay, two sufficient conditions on stability, respectively, for the system under two cases of time-varying delay are derived using the improved triple-integral inequality and a necessary and sufficient condition on quadratic matrix inequalities reported recently. Finally, three numerical examples show that the obtained results outperform some existing ones.