Delay Perturbations Research Articles

Delay-range-dependent stability for uncertain stochastic systems with interval time-varying delay and nonlinear perturbations

This paper considers the problem of robust stability for a class of uncertain stochastic systems with interval time-varying delay under nonlinear perturbations. A new delay-dependent method for robust stability of the systems is proposed. The innovation of the method includes employment of a tighter integral inequality and construction of an appropriate type of Lyapunov–Krasovskii functional. The restriction used to bound some trace term in the existing methods is also removed. The resulting criterion derived from this method has advantages over previous ones in that it has less conservatism and enlarges the scope of application. The reduction in conservatism of the proposed criterion is attributed to a method to estimate the upper bound on the stochastic differential of the Lyapunov–Krasovskii functional without neglecting any useful terms in the delay-dependent stability analysis. On the basis of the estimation and by utilizing free-weighting matrices, new delay-range-dependent stability criterion is established in terms of linear matrix inequality. Finally, numerical examples are provided to show the effectiveness and reduced conservatism of the proposed method.

Robust stability criterion for Markovian jump systems with nonlinear perturbations and mode-dependent time delays

In this paper, the problem of delay-dependent stability of a class of uncertain Markovian jump systems with mode-dependent interval time-varying delay and nonlinear perturbations is considered using Lyapunov–Krasovskii (LK) functional approach. In the proposed stability analysis, by exploiting a candidate LK functional and by imposing tighter bounding on the cross-terms using minimal number of slack matrix variables, a less conservative delay-dependent stability criterion is presented in linear matrix inequality framework. The effectiveness of the proposed stability criterion over some of the recently reported results is demonstrated through standard numerical examples.

Spectrum-based stability analysis and stabilisation of systems described by delay differential algebraic equations

An eigenvalue-based framework is developed for the stability analysis and stabilisation of coupled systems with time-delays, which are naturally described by delay differential algebraic equations. The spectral properties of these equations are analysed and a numerical method for computing characteristic roots and stability assessment is presented, thereby taking into account the effect of small delay perturbations on stability. Subsequently, the design of stabilising controllers with a prescribed structure or order is addressed, based on a direct optimisation approach. The effectiveness of the approach is illustrated with numerical examples. All algorithms have been implemented in publicly available software.

Stability analysis and output‐feedback stabilization of discrete‐time systems with a time‐varying state delay and nonlinear perturbation

AbstractThis paper aims to derive stability conditions and an output‐feedback stabilization method for discrete‐time systems with a time‐varying state delay and nonlinear perturbation. With a new way of handling the Lyapunov stability criterion, linear matrix inequality conditions are obtained for estimating bounds on delay to ensure the asymptotic stability. Based on the conditions, a synthesis procedure is developed for finding stabilizing output‐feedback gains, which are formulated as direct design variables. Three numerical examples are employed to demonstrate the effectiveness and advantages of the proposed method. Copyright © 2010 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society

Variation formulas of solution for a delay differential equation taking into account delay perturbation and the continuous initial condition

Abstract Variation formulas of solution are proved for a non-linear differential equation with constant delay. In this paper, the essential novelty is the effect of delay perturbation in the variation formulas. The continuity of the initial condition means that the values of the initial function and the trajectory always coincide at the initial moment.

Fixed-order strong H-infinity control of interconnected systems with time-delays

AbstractWe design fixed-order strong H-infinity controllers for general time-delay systems. The designer chooses the controller order and may introduce constant time-delays in the controller. We represent the closed-loop system of the plant and the controller as delay differential algebraic equations (DDAEs). This representation deals with any interconnection of systems with time-delays without any elimination techniques. We present a numerical algorithm to compute the strong H-infinity norm for DDAEs which is robust to arbitrarily small delay perturbations, unlike the standard H-infinity norm. We optimize the strong H-infinity norm of the closed-loop system based on non-smooth, non-convex optimization methods using this algorithm and the computation of the gradient of the strong H-infinity norm with respect to the controller parameters. We tune the controller parameters and design H-infinity controllers with a prescribed order or structure.

Fixed-Order H-Infinity Control for Interconnected Systems Using Delay Differential Algebraic Equations

We analyze and design H-infinity controllers for general time-delay systems with time-delays in systems' state, inputs, and outputs. We allow the designer to choose the order of the controller and to introduce constant time-delays into the controller. The closed-loop system of the plant and the controller is modeled by a system of delay differential algebraic equations (DDAEs). The advantage of the DDAE modeling framework is that any interconnection of systems and controllers prone to various types of delays can be dealt with in a systematic way, without using any elimination technique. We present a predictor-corrector algorithm for the H-infinity norm computation of systems described by DDAEs. Instrumental to this we analyze the properties of the H-infinity norm. In particular, we illustrate that it may be sensitive with respect to arbitrarily small delay perturbations. Due to this sensitivity, we introduce the strong H-infinity norm, which explicitly takes into account small delay perturbations, inevitable in any practical control application. We present a numerical algorithm to compute the strong H-infinity norm for DDAEs. Using this algorithm and the computation of the gradient of the strong H-infinity norm with respect to the controller parameters, we minimize the strong H-infinity norm of the closed-loop system based on nonsmooth, nonconvex optimization methods. By this approach, we tune the controller parameters and design H-infinity controllers with a prescribed order or structure.

On the sensitivity of the H∞ norm of systems described by delay differential algebraic equations

AbstractWe consider delay differential algebraic equations (DDAEs) to model interconnected systems with time-delays. The DDAE framework does not require any elimination techniques and can directly deal with any interconnection of systems and controllers with time-delays. In this framework, we analyze the properties of the H∞ norm of systems described by delay differential algebraic equations. We show that the standard H∞ norm may be sensitive to arbitrarily small delay perturbations. We introduce the strong H∞ norm which is insensitive to small delay perturbations and describe its properties. We conclude that the strong H∞ norm is more appropriate in any practical control application compared to the standard H∞ norm for systems with time-delays whenever there are high-frequency paths in control loops.

Preservation of exponential stability for linear non-autonomous functional differential systems

We consider preservation of exponential stability for a system of linear equations with a distributed delay under the addition of new terms and a delay perturbation. As particular cases, the system includes models with concentrated delays and systems of integrodifferential equations. Our method is based on Bohl–Perron type theorems.

Time-dependent vehicle routing subject to time delay perturbations

This paper extends the existing vehicle routing models by incorporating unanticipated delays at the customers' docking stations in time-dependent environments, i.e., where speeds are not constant throughout the planning horizon. A model is presented that is capable of optimizing the relevant costs taking into account these unplanned delays. The theoretical implications on the composition of the routing schedules are examined in detail. Experiments using a Tabu Search heuristic on a large number of datasets are provided. Based on these experiments, the generated routing schedules are analyzed and the cost-benefit trade-off between routing schedules that are protected against delays and routing schedules that are not protected against these delays are examined. Structural properties of the different solutions that have higher endurance capabilities with regards to the disruptions are highlighted.

OPTIMAL ADAPTIVE PREDICTION FOR SISO SYSTEMS

An adaptive predictor that is optimal in the minimum mean-square error (MMSE) sense at each step is obtained for single-input, single-output (SISO) linear time-invariant discrete-time systems having general delay and white noise perturbation. The adaptive d-step-ahead predictor includes the uncertainty associated with the parameter and state estimates whereas conventional adaptive predictors, that are asymptotically optimal, ignore the uncertainty in the parameter estimates.

A NEW CHANNEL FOR DETECTING DARK MATTER SUBSTRUCTURE IN GALAXIES: GRAVITATIONAL LENS TIME DELAYS

We show that dark matter substructure in galaxy-scale halos perturbs the time delays between images in strong gravitational lens systems. The variance of the effect depends on the subhalo mass function, scaling as the product of the substructure mass fraction and a characteristic mass of subhalos (namely <m^2>/<m>). Time delay perturbations therefore complement gravitational lens flux ratio anomalies and astrometric perturbations by measuring a different moment of the subhalo mass function. Unlike flux ratio anomalies, "time delay millilensing" is unaffected by dust extinction or stellar microlensing in the lens galaxy. Furthermore, we show that time delay ratios are immune to the radial profile degeneracy that usually plagues lens modeling. We lay out a mathematical theory of time delay perturbations and find it to be tractable and attractive. We predict that in "cusp" lenses with close triplets of images, substructure may change the arrival-time order of the images (compared with smooth models). We discuss the possibility that this effect has already been observed in RX J1131-1231.

DYNAMIC COMPLEXITIES OF HOLLING TYPE II FUNCTIONAL RESPONSE PREDATOR–PREY SYSTEM WITH DIGEST DELAY AND IMPULSIVE HARVESTING ON THE PREY

In this paper, we introduce and study Holling type II functional response predator–prey system with digest delay and impulsive harvesting on the prey, which contains with periodically pulsed on the prey and time delay on the predator. We investigate the existence and global attractivity of the predator-extinction periodic solutions of the system. By using the theory on delay functional and impulsive differential equation, we obtain the sufficient condition with time delay and impulsive perturbations for the permanence of the system.

ROBUST EXPONENTIAL STABILITY OF STOCHASTIC TIME-DELAY SYSTEMS WITH UNCERTAINTIES AND NONLINEAR PERTURBATIONS

In this paper, the delay-dependent robust exponential mean-square stability analysis problem is considered for a class of uncertain stochastic systems with time-varying delay and nonlinear perturbations. Some sufficient conditions on delay-dependent robust exponential stability in the mean square are established in terms of linear matrix inequalities (LMIs) by exploiting a novel Lyapunov–Krasovskii functional and by making use of zero equations methods. These developed results indicate less conservatism than the existing ones due to the introduction of some free weighting matrices which can be selected properly. The new delay-dependent stability criteria are expressed as a set of LMIs, which can be readily solved by using standard numerical software. Numerical examples are provided to demonstrate the effectiveness and the applicability of the proposed criteria.

A Predator-Prey Gompertz Model with Time Delay and Impulsive Perturbations on the Prey

We introduce and study a Gompertz model with time delay and impulsive perturbations on the prey. By using the discrete dynamical system determined by the stroboscopic map, we obtain the sufficient conditions for the existence and global attractivity of the “predator-extinction” periodic solution. With the theory on the delay functional and impulsive differential equation, we obtain the appropriate condition for the permanence of the system.

Strong Stability of Neutral Equations with an Arbitrary Delay Dependency Structure

The stability theory for linear neutral equations subjected to delay perturbations is addressed. It is assumed that the delays cannot necessarily vary independently of each other, but depend on a possibly smaller number of independent parameters. As a main result, necessary and sufficient conditions for strong stability are derived along with bounds on the spectrum, which take into account the precise dependency structure of the delays. In the derivation of the stability theory, results from realization theory and determinantal representations of multivariable polynomials play an important role. The observations and results obtained in the paper are first illustrated and validated with a numerical example. Next, the effects of small feedback delays on the stability of a boundary controlled hyperbolic partial differential equation and of a control system involving state derivative feedback are analyzed.

Stability impact of small delays in proportional–derivative state feedback

This paper points out specific stability problems of the state derivative and proportional–derivative state feedback which may result from the presence of otherwise negligible delays in the control loops. It is shown that the stability problems arise from the neutral character of the closed-loop system with the state derivative action under small delay perturbations. As the main result, a stability criterion is derived for a safe implementation of the state derivative feedback. Furthermore, a scheme including filtered state derivative feedback is proposed to overcome the destabilizing effects of small feedback delays. Two application examples are presented. From a theoretical point of view, the main contribution of the paper lies in the stability theory of linear neutral equations with dependent delays, subjected to delay perturbations.

Necessity and sufficiency conditions for the absolute null controllability for Linear delay perturbations

We are inspired by the works of Chukwu [1], Eke [2], Schinterdorf and Barmish [4] to unveil necessary and sufficient conditions for the absolute null controllability of a linear delay perturbed system with zero in the domain of null controllability. Journal of the Nigerian Association of Mathematical Physics Vol. 10 2006: pp. 549-552

On loop phase margins of multivariable control systems

This paper has two contributions. One is introduction of loop phase margins for multivariable control systems, which extends the concept of phase margin in SISO systems to MIMO systems. The other one is development of an algorithm for computing the loop phase margins. The algorithm is composed of two steps. The first is to find the stabilizing ranges of loop time delay perturbations of the MIMO system using an LMI-based stability criterion derived here. The second is to convert these stabilizing ranges of loop time delays into the stabilizing ranges of loop phases. Two numerical examples are given to illustrate the effectiveness of the proposed approach.

Delay-dependent criteria for robust stability of linear neutral systems with time-varying delay and nonlinear perturbations

This article investigates the robust stability of linear neutral systems with time-varying delay and nonlinear perturbations. Using a new Lyapunov–Krasovskii functional and employing some free weighting matrices, less conservative delay-dependent robust stability conditions for such systems in terms of linear matrix inequalities are derived. Numerical examples are given to indicate significant improvements over some existing results.