Stability analysis of systems with stochastically varying delays

Stability analysis of linear systems with two additive time-varying delays via delay-product-type Lyapunov functional

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.

The main objective of this paper is to present the time domain, frequency domain and stability analysis of linear systems represented by differential equations with complex-order derivatives. The impulse and step response of three different complex-order systems have been presented numerically with the help of MATLAB. For frequency domain analysis, Bode-plots of the same three complex-order systems have been sketched. Complex-order systems have infinite numbers of complex-conjugate poles. The stability analysis of the complex-order systems has been done in two ways. Firstly, for systems to be stable, the complex-conjugate poles in the principle Riemann sheet must be in the left half plane. Secondly, the complex-order q = u + iv of the complex-order systems must be interior to an open disk in the u-v plane, for systems to be stable.

This paper develops a quadratic function convex approximation approach to deal with the negative definite problem of the quadratic function induced by stability analysis of linear systems with time-varying delays. By introducing two adjustable parameters and two free variables, a novel convex function greater than or equal to the quadratic function is constructed, regardless of the sign of the coefficient in the quadratic term. The developed lemma can also be degenerated into the existing quadratic function negative-determination (QFND) lemma and relaxed QFND lemma respectively, by setting two adjustable parameters and two free variables as some particular values. Moreover, for a linear system with time-varying delays, a relaxed stability criterion is established via our developed lemma, together with the quivalent reciprocal combination technique and the Bessel-Legendre inequality. As a result, the conservatism can be reduced via the proposed approach in the context of constructing Lyapunov-Krasovskii functionals for the stability analysis of linear time-varying delay systems. Finally, the superiority of our results is illustrated through three numerical examples.

Multiple-integral inequalities to stability analysis of linear time-delay systems

Allowable delay sets for the stability analysis of linear time-varying delay systems using a delay-dependent reciprocally convex lemma

This paper addresses the stability analysis of linear continuous systems under interval uncertainties. An interval generalization of the known Raus criterion is suggested to estimate the stability of the system considered. It is based on obtaining the interval extensions of the elements of the Raus matrix which are nonlinear functions of independent system parameters. The case when these elements are independent intervals is considered. The interval extensions are also determined by using modified affine arithmetic. Two sufficient conditions on stability and instability of the linear system considered are obtained. A numerical example illustrating the applicability of the method suggested is solved at the end of the paper.

We consider linear systems with distributed delays where delay kernels are assumed to be finite duration impulse responses of finite dimensional systems. We show that stability analysis for this class of systems can be reduced to stability analysis of linear systems with discrete delays, for which many algorithms are available in the literature. The results are illustrated on a mathematical model of hematopoietic cell maturation dynamics.

Stability analysis of systems with time-varying delay via novel augmented Lyapunov–Krasovskii functionals and an improved integral inequality

This paper addresses the problem of stability analysis for linear systems with interval timevarying delay. A general form of the delay-fractioning approach is proposed, which not only takes advantage of all possible information on the delay's lower bound, but also exploits further information between the delay's upper and lower bounds. A new Lyapunov-Krasovskii functional (LKF) is constructed and delay-dependent stability criteria are derived in terms of linear matrix inequalities (LMIs) by using the piecewise analysis method. The convexity of the matrix function is used to avoid the conservatism caused by enlarging the time-varying delay to its upper bound in each subinterval. Numerical examples are given finally to verify the effectiveness of the proposed criteria.

Stability analysis of linear systems subject to regenerative switchings

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

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

Stability analysis of linear systems with time- varying delay is investigated. In order to highlight the relations between the variation of the delay and the states, redundant equations are introduced to construct a new modeling of the delay system. New types of Lyapunov Krasovskii functionals are then proposed allowing to reduce the conservatism of the stability criterion. Delay dependent stability conditions are then formulated in terms of linear matrix inequalities (LMI). Finally, an example shows the effectiveness of the proposed methodology.

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.