Stability Of Time-delay Systems Research Articles

Weak and strong stabilization for time-delay semi-linear systems governed by constrained feedback control

This paper is concerned with the issue of weak and strong stabilization for distributed semi-linear systems with time delay using a constrained feedback control. The results of the semi-linear systems without delay are generalized for strong and weak stabilization cases. Illustrating applications to hyperbolic and parabolic equations are considered.

Input-to-state stability theorem and strict Lyapunov functional constructions for time-delay systems on time scales

ABSTRACT In this paper, input-to-state stability of nonlinear time-delay systems on time scales is investigated. Due to the advantages of the strict Lyapunov functionals in uncertainty quantification and robustness analysis, one always prefers to construct the strict Lyapunov functionals to analyse stability of time-delay systems. However, it may be not an easy task to do this for some time-delay systems. This paper proposes an input-to-state stability theorem based on a time-scale uniformly asymptotically stable function. The advantage of this theorem is that it is dependent on the non-strict Lyapunov functional, whose time-scale derivative can be non-negative on some time intervals. Then, some approaches are established to construct the strict Lyapunov functionals based on the non-strict ones. It is shown that input-to-state stability theorems can be also formulated in terms of these strict Lyapunov functionals. Finally, to illustrate the effectiveness of the main results, an example is given.

An LMI Approach to Nonlinear State-Feedback Stability of Uncertain Time-Delay Systems in the Presence of Lipschitzian Nonlinearities

This article proposes a new nonlinear state-feedback stability controller utilizing linear matrix inequality (LMI) for time-delay nonlinear systems in the presence of Lipschitz nonlinearities and subject to parametric uncertainties. Following the Lyapunov–Krasovskii stabilization scheme, the asymptotic stability criterion resulted in the LMI form and the nonlinear state-feedback control technique was determined. Due to their significant contributions to the system stability, time delays and system uncertainties were taken into account while the suggested scheme was designed so that the system’s stabilization was satisfied in spite of time delays and system uncertainties. The benefit of the proposed method is that not only is the control scheme independent of the system order, but it is also fairly simple. Hence, there is no complexity in using the proposed technique. Finally, to justify the proficiency and performance of the suggested technique, a numerical system and a rotational inverted pendulum were studied. Numerical simulations and experimental achievements prove the efficiency of the suggested control technique.

Stability of time-delay systems with impulsive control involving stabilizing delays

This paper studies the stability of time-delay systems with impulsive control involving stabilizing delays. A Razumikhin-type inequality with delayed impulses is proposed, where the information of time delay existing in impulses is effectively fetched and then integrated to the stabilization of time-delay systems. Some sufficient conditions ensuring global exponential stability (GES) for a class of nonlinear time-delay systems are obtained. The proposed conditions do not impose any restriction on the upper bounds of two consecutive impulse signals and on the size relation between time delays in system and in impulses, which is improved in the existing results on this topic. Moreover, our proposed results show that the time delays in impulses can contribute to the stability of the time-delay systems. Two numerical examples are given to demonstrate the applicability of our proposed results.

Fast Generation of Stability Charts for Time-Delay Systems Using Continuation of Characteristic Roots

Abstract Many dynamic processes involve time delays, thus their dynamics are governed by delay differential equations (DDEs). Studying the stability of dynamic systems is critical, but analyzing the stability of time-delay systems is challenging because DDEs are infinite-dimensional. We propose a new approach to quickly generate stability charts for DDEs using continuation of characteristic roots (CCR). In our CCR method, the roots of the characteristic equation of a DDE are written as implicit functions of the parameters of interest, and the continuation equations are derived in the form of ordinary differential equations (ODEs). Numerical continuation is then employed to determine the characteristic roots at all points in a parametric space; the stability of the original DDE can then be easily determined. A key advantage of the proposed method is that a system of linearly independent ODEs is solved rather than the typical strategy of solving a large eigenvalue problem at each grid point in the domain. Thus, the CCR method can significantly reduce the computational effort required to determine the stability of DDEs. As we demonstrate with several examples, the CCR method generates highly accurate stability charts, and does so up to 10 times faster than the Galerkin approximation method.

A Panoramic Sketch about the Robust Stability of Time-Delay Systems and Its Applications

This paper presents a brief review on the current applications and perspectives on the stability of complex dynamical systems, with an emphasis on three main classes of systems such as delay-free systems, time-delay systems, and systems with uncertainties in its parameters, which lead to some criteria with necessary and/or sufficient conditions to determine stability and/or stabilization in the domains of frequency and time. Besides, criteria on robust stability and stability of nonlinear time-delay systems are presented, including some numerical approaches.

An exploration of the Razumikhin stability theorem with applications in stabilization of delay systems

In this paper, we establish a new evolution property of Lyapunov functions that satisfy the Razumikhin Stability Theorem conditions. The derived property is shown to be a valid alternative criterion to prove the asymptotic stability of a delay system without calculating the derivative of the Lyapunov function, and it is especially useful in solving stabilization problems for time delay systems when the delay is unknown. As an example application of this derived property, we investigate the stabilization of continuous-time linear systems with an unknown time-varying delay, by proposing a switched low gain feedback control technique that utilizes the derived property. Existing literature on stabilization of continuous time delay systems all require some knowledge, such as an upper bound, of the delay. In this work, by proposing a switched low gain feedback control law utilizing the derived property, we are able to achieve asymptotic stabilization of the system without any knowledge of the delay. Furthermore, the proposed switched low gain feedback law also significantly improves the closed-loop system performance in terms of overshoot and convergence speed, in comparison with the traditional fixed gain low gain feedback design. Simulation study verifies the theoretical results.

Robust finite-time stability of neutral fractional time-delay systems

Robust finite-time stability of neutral fractional time-delay systems

Finite time stabilization of time-delay nonlinear systems with uncertainty and time-varying delay

Finite time stabilization of time-delay nonlinear systems with uncertainty and time-varying delay

Event-triggered Stabilization of Linear Time-delay Systems by Static Output Feedback Control

In this paper, we study the stabilization problem for a class of linear systems with a time-varying state delay. An event-triggered static output feedback controller is proposed, such that the sampling frequency and the update time of the controller are both lowered. First, we present a novel event-triggering mechanism depending not only on the output but also on an exponential term, with which less sampling is required and Zeno behavior can be excluded at the same time. Some sufficient conditions are then obtained, under which an exponential convergence can be achieved by means of the comparison principle approach. It is further shown that the parameters design can be easily given if the case reduces to the state feedback control. Moreover, two examples are presented to show the effectiveness of the results.

Integral input-to-state stability of nonlinear time-delay systems with delay-dependent impulse effects

This paper studies integral input-to-state stability (iISS) of nonlinear impulsive systems with time-delay in both the continuous dynamics and the impulses. Several iISS results are established by using the method of Lyapunov–Krasovskii functionals. For impulsive systems with iISS continuous dynamics and destabilizing impulses, we derive two iISS criteria that guarantee the uniform iISS of the whole system provided that the time period between two successive impulse moments is appropriately bounded from below. Then we provide an iISS result for systems with unstable continuous dynamics and stabilizing impulses. For this scenario, it is shown that the iISS properties are guaranteed if the impulses occur frequently enough. Last but not least, sufficient conditions are also obtained to guarantee the uniform iISS of the entire system over arbitrary impulse time sequences. As applications, iISS properties of a class of bilinear systems are studied in details with simulations to demonstrate the presented results.

Non-monotonic Lyapunov-Krasovskii functional approach to stability analysis and stabilization of discrete time-delay systems

In this paper, a novel non-monotonic Lyapunov-Krasovskii functional approach is proposed to deal with the stability analysis and stabilization problem of linear discrete time-delay systems. This technique is utilized to relax the monotonic requirement of the Lyapunov-Krasovskii theorem. In this regard, the Lyapunov-Krasovskii functional is allowed to increase in a few steps, while being forced to be overall decreasing. As a result, it relays on a larger class of Lyapunov-Krasovskii functionals to provide stability of a state-delay system. To this end, using the non-monotonic Lyapunov-Krasovskii theorem, new sufficient conditions are derived regarding linear matrix inequalities ( LMIs ) to study the global asymptotic stability of state-delay systems. Moreover, new stabilization conditions are also proposed for time-delay systems in this article. Both simulation and experimental results on a pH neutralizing process are provided to demonstrate the efficacy of the proposed method.

State feedback stabilization of stochastic nonlinear time-delay systems: a dynamic gain method

State feedback stabilization of stochastic nonlinear time-delay systems: a dynamic gain method

Finite-time stability of neutral fractional order time delay systems with Lipschitz nonlinearities

This paper is concerned with finite-time stability of neutral fractional order time delay systems with Lipschitz nonlinearities. By use of the method of steps and the generalized Gronwall inequality, a new criterion on finite-time stability is obtained. Two numerical examples are given to illustrate the effectiveness of our main results.

New versions of Bessel–Legendre inequality and their applications to systems with time-varying delay

This paper is concerned with the stability of linear time-delay systems by applying the Lyapunov–Krasovskii (L–K) functional method. Two versions of Bessel–Legendre (B-L) inequality are developed that are more suitable to deal with the stability problem of systems with a time-varying delay. Meanwhile, in order to take full advantage of the interest of the new versions of B-L inequality, a novel L–K functional is properly tailored by integrating the integral information of the state into quadratic and integral terms. As a result, more relaxed stability conditions are obtained, whose effectiveness is illustrated by two commonly-used numerical examples.

Impulsive Stabilization of Nonlinear Time-Delay System With Input Saturation via Delay-Dependent Polytopic Approach

The impulsive stabilization of nonlinear time-delay system with input saturation via delay-dependent polytopic approach is studied in this article. Different from polytopic representation technique, delay-dependent polytopic technique is able to estimate a larger domain of attraction. Based on this approach, the actuator saturation term is first introduced into the design of impulsive controller, which is expressed as a convex combination of the product of delay-dependent state vectors and auxiliary matrices. By applying delay-dependent polytopic technique and delay-dependent Lyapunov–Krasovskii functional (LKF) approach, a new series of less conservative linear matrix inequality (LMI) criteria are obtained to ensure the stability of the established model. Finally, two examples are presented to claim the effectiveness of theoretical analysis results.

Stability and stabilization for delayed fuzzy systems via reciprocally convex matrix inequality

This paper focuses on the problems of stability and stabilization for time-delayed fuzzy systems. By establishing a novel Lyapunov-Krasovskii functional (LKF) and using an extended reciprocally convex matrix inequality with auxiliary function integral inequality, an improved stability condition is proposed. To further improve the results, a delay-product-type term is introduced into the LKF. Then, a delay-dependent stabilization condition is developed based on parallel distributed compensation (PDC) scheme by some linearization techniques. Finally, the efficiency and benefits of the stability criterion and controller design method are illustrated by several classical numerical examples.

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.

Analysis of Integral Input-to-State Stable time-delay systems in cascade

We consider the cascade interconnection of two nonlinear time-delay systems, each of which being integral input-to-state stable (iISS). We provide an explicit growth rate condition on the dissipation rate of the driving system and the input rate of the driven system under which the overall cascade is globally asymptotically stable in the absence of inputs (0-GAS), and its solutions are bounded in response to any input with a suitable bounded-energy assumption.

Delay-dependent stability of pitch control system in a large wind turbine plant

The use of hydraulic pressure drive unit as actuator element in the pitch control scheme of a large wind turbine plant introduces inevitable time-delay in the feedback loop. This delay has an adverse impact on the performance and stability of the closed-loop system; nevertheless, if the delay magnitude exceeds a critical margin, the closed-loop system is driven to instability. In a large wind turbine plant, owing to its massive tower structure and enormous blade span, unstable, or for that matter, even marginally stable operation is highly detrimental. If the unboundedly evolving or sustained oscillations in pitch angle, signifying unstable or marginally stable operation respectively, are not curtailed using efficient braking systems, it might lead to the loss of whole structure. Hence, in this paper, the problem of delay-dependent stability of time-delayed pitch control system of a large wind turbine plant has been addressed using the classical Lyapunov-Krasovskii functional approach combined with Wirtinger inequality. The derived stability criterion which is in LMI framework, is tested on a benchmark wind turbine plant. In the sequel, simulation results are also provided to validate the effectiveness of the presented analytical results.