Stability Criteria For Time-delay Systems Research Articles

Relaxed Stability Criteria for Time-Delay Systems: A Novel Quadratic Function Convex Approximation Approach

Relaxed Stability Criteria for Time-Delay Systems: A Novel Quadratic Function Convex Approximation Approach

General single/multiple integral inequalities and their applications to stability of time-delay systems

This paper is concerned with the stability of time delay systems. A new type of multiple integral inner product and a new class of multiple integral inner product space are firstly introduced. Based on them, general single/multiple integral inequalities are developed, and some previous integral inequalities including single and multiple integral inequalities can be regarded as their special cases. Secondly, suitable Lyapunov–Krasovskii functionals (LKF) are constructed. On the basis of the proposed integral inequalities, the derivative of the LKF can be gauged more accurate than those in the literature. As a result, some improved stability criteria of time-delay systems are formulated in terms of linear matrix inequalities. Through two numerical examples, it is shown that these criteria can provide larger delay upper bounds than some existing ones, showing the effectiveness and superiority of the proposed method.

A Hierarchical Stability Criterion for Time-Delay Systems based on the Affine Bessel-Legendre Inequality

A Hierarchical Stability Criterion for Time-Delay Systems based on the Affine Bessel-Legendre Inequality

Stability Criteria for Time-Delay Systems From an Insightful Perspective on the Characteristic Equation

This article provides a delay-dependent and a necessary and sufficient delay-independent stability criterion for linear autonomous continuous-time systems with a discrete delay. We take a simple perspective on the two-variable formulation of the characteristic equation, which leads to the following advantageous but not widespread delay-independent criterion: The sum of the coefficient matrix of the delay-free term and the elementwise unitarily rotated matrix of the delay term must remain Hurwitz for all rotation angles. A graphical test for the latter is shown to require no more than three lines of code. Concerning delay-independent stability, our main contribution is to extend this sufficient criterion to a necessary and sufficient one. Concerning delay-dependent stability, the focus is on the critical delay that bounds the initial delay interval of stability. We formulate a constrained minimization problem that gives the exact value of this critical delay. The taken perspective is especially insightful in terms of how the coefficient matrices may look like, and, for scalar systems, the delay-dependent stability chart becomes obvious at a first glance. The presented criteria are complementary to the well-known frequency-sweeping test, which results from another of three possible perspectives on the two-variable criterion. Besides of a unified treatment of these different perspectives, the article also discusses corollaries of the one taken, which include Mori’s famous criterion.

Input-to-State Stability for Time-Delay Systems With Large Delays

In this article, we consider the input-to-state stability (ISS) problem for a class of time-delay systems with intermittent large delays, which may cause the invalidation of traditional delay-dependent stability criteria. The topic of this article features that it proposes a novel kind of stability criterion for time-delay systems, which is delay dependent if the time delay is smaller than a prescribed allowable size. While if the time delay is larger than the allowable size, the ISS can be preserved as well provided that the large-delay periods satisfy the kind of duration condition. Different from existing results on similar topics, we present the main result based on a unified Lyapunov-Krasovskii function (LKF). In this way, the frequency restriction can be removed and the analysis complexity can be simplified. A numerical example is provided to verify the proposed results.

A new design method to global asymptotic stabilization of strict-feedforward stochastic nonlinear time delay systems

The stabilization problems of stochastic nonlinear systems remain challenging because many new obstacles will be encountered and need to be cautiously handled. This paper discusses the global asymptotic stabilization problem of stochastic nonlinear systems which possess strict-feedforward structure and multiple time delays. In the first place, we are trying to transform the considered system into an equivalent one by constructing the novel parameter-dependent state feedback controller and the suitable coordinate transformation. And then, through using the stochastic time delay system stability criterion and introducing the proper Lyapunov–Krasovskii functional, the global asymptotic stability of stochastic closed-loop system is proved. Finally, simulation results show the effectiveness of the parameter-dependent state feedback controller.

New stability criterion for time-delay systems via an augmented Lyapunov–Krasovskii functional

This paper focuses on the stability of linear systems with time-delay. An augmented Lyapunov–Krasovskii functional (LKF) is first constructed, and then a generalized delay-dependent stability criterion is obtained by combining the newly constructed LKF and the generalized vectors-based multiple integral inequality (GVMII). By deriving one type of the concrete orthogonal function in the criterion, an easy-to-use criterion has been derived, which is less conservative than before. Finally, one numerical example is given to show the superiority of our method.

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.

Stability Analysis for Systems With Time Delays via New Integral Inequalities

This correspondence is concerned with the stability for interval time-delay systems. Triple integral inequalities without slack variables are proposed, which show more and more evident improvements as the degrees of intermediate polynomials increase. Driven by the proposed inequalities, the Lyapunov-Krasovskii functional with quadruple integrals and augmented vector consisting of triple integrals is constructed to derive less conservative stability criteria for time-delay systems. The numerical examples are provided to demonstrate the effectiveness of the proposed approaches.

Stability Analysis for Time-delay Systems with Nonlinear Disturbances via New Generalized Integral Inequalities

This paper represents a novel less conservative stability criterion for time-delay systems with nonlinear disturbances. The main purpose is to obtain larger upper bound of the time-varying delay. A suitable Lyapunov- Krasovskii functional (LKF) with triple integral terms is constructed. Then, two new generalized double integral (GDI) inequalities are proposed which encompass Wirtinger-based double inequality as a special case. A simple case of the proposed GDI inequality is utilized to estimate double integral terms in the time derivative of the constructed LKF. Further, an improved delay-dependent stability criterion is derived in the form of linear matrix inequalities (LMIs). Finally, some numerical examples are given to illustrate the improvement of the proposed criteria.

Improved exponential stability criteria of time-delay systems via weighted integral inequalities

In this paper, we establish new weighted integral inequalities by considering polynomials which are orthogonal in a weighted sense. These inequalities generalize some results established recently. They are applied to study exponential stability of some time-delay systems under the framework of linear matrix inequalities. Numerical tests are given to demonstrate the efficiency of the derived stability conditions.

Hierarchy of stability criterion for time-delay systems based on multiple integral approach

Taking a class of time-delay systems as research object, this brief aims at developing a theoretical support on the hierarchy of stability criterion which is derived by the multiple integral approach and free-weighting matrix technique. The hierarchy implies that the conservatism of stability criterion can be reduced by increasing the ply of integral terms in Lyapunov–Krasovskii functional (LKF). Together with three numerical experiments, the hierarchy of stability criterion is further shown.

Further results on output-feedback stabilization of stochastic upper-triangular systems with state and input time-delays

Based on stochastic time-delay system stability criterion and a homogeneous domination approach, the output-feedback stabilization problem for a class of more general stochastic upper-triangular systems with state and input time-delays has been solved in this paper. Firstly, the initial system is changed into an equivalent one with a designed scalar by introducing a set of coordinate transformations. After that, by designing an implementable homogeneous reduced-order observer, and tactfully selecting a suitable Lyapunov–Krasoviskii functional and a low gain scale, a delay-independent output-feedback controller is explicitly constructed. Finally, the globally asymptotically stability in probability of the closed-loop system is ensured by rigorous proof. The simulation results demonstrate the efficiency of the proposed design scheme.

Improved Results on Asymptotic Stabilization for Stochastic Nonlinear Time-Delay Systems With Application to a Chemical Reactor System

The global asymptotic stabilization problem is investigated for a class of stochastic nonlinear time-varying delay systems under the weaker condition on nonlinear functions. The new parameter-dependent state and output feedback controllers are, respectively, proposed. Based on the stochastic time-delay system stability criterion, by tactfully introducing a suitable Lyapunov-Krasovskii functional, the globally asymptotically stable in probability of the closed-loop system is guaranteed by rigorous proof. As a practical application, the stochastic model of a two-stage chemical reactor system is established by reasonably introducing the Gaussian white noise. The developed approach is applied to the control design for this practical system. The simulation results demonstrate the efficiency of the proposed design approach.

Global decentralized output-feedback stabilization of large-scale stochastic high-order nonlinear systems with multi-delays

Based on the stochastic nonlinear time-delay system stability criterion and homogeneous domination approach, this paper solves the global decentralized output-feedback stabilization problem for a class of large-scale stochastic high-order nonlinear systems with multi-delays. By skillfully choosing an appropriate Lyapunov–Krasoviskii functional and successfully overcoming some troublesome obstacles in the design and analysis procedure, the decentralized output-feedback controller and homogeneous observer are constructed for each subsystem to guarantee that the closed-loop system is globally asymptotically stable (GAS) in probability. The simulation example is given to verify the effectiveness of the proposed design scheme.

Asymptotic stabilization of stochastic high-order upper-triangular nonlinear systems with input time-varying delay

Based on the homogeneous domination approach and stochastic nonlinear time-delay system stability criterion, this paper investigates the global state-feedback stabilization problem for a class of stochastic high-order upper-triangular nonlinear systems with input time-varying delay. By skilfully choosing an appropriate Lyapunov–Krasoviskii functional and successfully solving several troublesome obstacles in the design and analysis procedure, a delay-independent state-feedback controller is designed to render the closed-loop system globally asymptotically stable in probability. The simulation example is given to verify the effectiveness of the proposed design scheme.

An enhanced stability criterion for time-delay systems via a new bounding technique

In the past few years, a great deal of effort has been devoted to improve delay-dependent conditions for stability of linear systems with an interval time-varying delay. To reduce conservatism of stability conditions, in the framework of the Lyapunov–Krasovskii functional (LKF) method, the bounding technique plays a key role. In this paper, a new bounding technique based on a new integral inequality is proposed. By employing the newly bounding technique proposed in this paper, an enhanced stability criterion for a class of linear systems with an interval time-varying delay is derived. The effectiveness and a significant improvement of the obtained results are shown by numerical examples.

Further Stability Criteria for Time-Delay Systems with Interval Time-Varying Delays

AbstractIn this paper, a novel stability criterion is developed for time-delay systems having time-varying delays that belong to a given range. Based on a choice of different type of Lyapunov-Krasovskii functional in an extensively augmented form, some new delay-range dependent stability criteria are proposed. This developed stability result has advantages over some previous ones. First, the method is based on the selection of a new, extensively augmented Lyapunov-Krasovskii functional which does not only take the delay range into account but also the weighted version of it. Second it estimates the upper bound of the derivative of the Lyapunov functional without ignoring some useful integral terms. Finally, we have introduced several free slack variables in relation with Newton-Leibniz formula to provide some kind of relaxation for the proposed stability criteria. In addition to this, we have also employed the method of completing to squares which has enabled to provide further relaxation with some additional decision variables.

Equivalence of stability criteria for time-delay systems

During the past few decades, there have been a number of new delay-dependent stability criteria based on linear matrix inequalities published in the literature. These criteria, to some extent, can reduce the conservativeness. In fact, some criteria are equivalence. In this paper, we aim to theoretically establish equivalence of two stability criteria. One was obtained by Sun et al. [Sun et al. In: Proceedings of IEEE Power & Energy Society General Meeting. 2009, 1–7] and the other was given by Xu et al. [Xu et al. IEEE Transactions on Automatic Control, 2005, 50(3): 384–387]. Also, we theoretically establish equivalence of the robust stability criterion presented by Jia et al. [Jia et al. Automation of Electric Power Systems, 2010, 34(3): 6–11] and the one given by Xu et al. (2005).

Improved robust stability and feedback stabilization criteria for time-delay systems

This paper develops novel robust stability and feedback stabilization criteria with guaranteed performance for a class of linear continuous time-delay systems with polytopic uncertainties. The time-varying delay function is unknown and differentiable within bounded interval and the input delay is constant. The criteria is derived based on the constructive use of a new Lyapunov–Krasovskii functional together with the integral inequality. The developed stability condition is expressed in terms of linear matrix inequality that manipulates fewer decision variables and requires reduced computational load. Through a comparison with other existing stability methods, it is established that the developed method retains some useful terms that are frequently dropped out and does not employ any free-weighting matrices to avoid redundancy. A state-feedback stabilizing controller is designed to ensure that the closed loop is robustly stable with guaranteed performance. Representative examples are simulated to illustrate the developed results.