Stability Of Nonlinear Time-delay Systems Research Articles

Event-Triggered Impulsive Control for Input-to-State Stabilization of Nonlinear Time-Delay Systems.

This article investigates the event-triggered impulsive control (ETIC) problem for a class of nonlinear time-delay systems subject to exogenous disturbances. An original event-triggered mechanism (ETM) which utilizes the information of system state and external input is constructed based on Lyapunov function approach. To achieve the input-to-state stability (ISS) of the considered system, some sufficient conditions are presented, in which the underlying relationship among ETM, exogenous input, and impulse action is established. Furthermore, the possible Zeno behavior induced by the proposed ETM is excluded simultaneously. As an application, the design criterion of ETM and impulse gain is put forward for a class of impulsive control systems with delay according to the feasibility of some linear matrix inequalities (LMIs). Finally, two examples with numerical simulations are provided to confirm the effectiveness of the developed theoretical results, where the synchronization issue of delayed Chua's circuit is considered.

Stabilization of time-delay nonlinear systems using Takagi-Sugeno fuzzy models

We propose a controller design method for time-delay nonlinear systems with delays affecting both the states and the inputs, represented by Takagi-Sugeno fuzzy models with nonlinear consequents. To handle the nonlinearities in the consequents we assume that they are slope-bounded. Linear matrix inequality conditions are formulated to design the controller. The obtained results are compared to state-of-the-art approaches and illustrated on two examples.

Global rational stabilization of a class of nonlinear time-delay systems

The present paper is mainly aimed at introducing a novel notion of stability of nonlinear time-delay systems called Rational Stability. According to the Lyapunov-type, various sufficient conditions for rational stability are reached. Under delay dependent conditions, we suggest a nonlinear time-delay observer to estimate the system states, a state feedback controller and the observer-based controller rational stability is provided. Moreover, global rational stability using output feedback is given. Finally, the study presents simulation findings to show the feasibility of the suggested strategy.

On Sontag’s formula for the sampled-data observer-based stabilization of nonlinear time-delay systems

In this paper, a new methodology for the design of sampled-data dynamic output feedback (DOF) stabilizers for control-affine nonlinear time-delay systems is proposed. In particular, the methodology here developed is based on the Artstein’s theory of control Lyapunov functions and related Sontag’s formula, here extended, for the first time in the literature, to the case of DOF controllers. Firstly, the new notion of Steepest Descent Error Dynamics (SDED) is introduced and used to suitably re-formulate the well-known Sontag’s universal formula for the case of DOF controllers. Then, it is shown that, under suitably fast sampling, a proposed sampled-data DOF controller based on the new provided version of Sontag’s universal formula ensures the semiglobal practical stability of the related sampled-data closed-loop system. Time-varying sampling intervals, as well as the intersampling system behavior, are taken into account. In the theory here developed, the case of delay-free systems is addressed as a special case. Applications are presented for the validation of the proposed theoretical results.

A novel theorem for prescribed-time control of nonlinear uncertain time-delay systems

This paper considers the problem of prescribed-time control for a class of nonlinear uncertain time-delay systems. First, the definition of prescribed-time stability for nonlinear time-delay systems is given, and the necessary conditions to realize the finite-time (including fixed/prescribed-time) stabilization of the nonlinear time-delay system are illustrated. Then, the prescribed-time Lyapunov stability theorem is proposed by utilizing the prescribed-time adjustment function and Lyapunov–Krasovskii (L–K) functional method. The stability of the resulting closed-loop system is proved and the different situations for uncertain time-delay systems are discussed. For first-order fully actuated system, a continuous controller is designed. The effects of different prescribed-time adjustment functions on the controller are also studied. Finally, a simulation example is provided to show the effectiveness of the theoretical results.

Stabilization of nonlinear time-delay systems: Flexible delayed impulsive control

This paper studies the stabilization of nonlinear time-delay systems under flexible delayed impulsive control. Some sufficient conditions are provided for establishing stability property in terms of exponential Lyapunov-Razumikhin functions. It is shown that the size of delay in continuous dynamics can be flexible. Specially, it can be smaller or larger than the impulsive intervals, and there is no magnitude relationship between the delay in continuous flow and impulsive delay. In most existing results, from the impulsive control point of view, the Lyapunov functions were based on the assumption that there was a common threshold at every impulse point. In this study, utilizing the proposed method of average impulsive estimation (AIE), the rate coefficients are flexible, and the impulsive delay can be integrated to guarantee the effect of stabilization of impulses. As an application, the theoretical results are applied to the synchronization of a chaotic neural network, and the impulsive control input is formalized in terms of linear matrix inequalities (LMIs). The efficiency of the derived results is illustrated by two numerical examples.

Positive Solutions and Exponential Stability of Nonlinear Time-Delay Systems in the Model of BAM-Cohen-Grossberg Neural Networks

Positive Solutions and Exponential Stability of Nonlinear Time-Delay Systems in the Model of BAM-Cohen-Grossberg Neural Networks

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.

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.

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

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.

Stabilization of nonlinear time-delay systems: Distributed-delay dependent impulsive control

This technical note studies impulsive stabilization of general nonlinear systems with time-delay. Distributed time-delay is considered in the proposed nonlinear impulsive controller. Using Lyapunov–Razumikhin method, an exponential stability criterion is constructed, which is then applied to investigate stabilization of a linear time-delay system under linear distributed-delay dependent impulsive control. Sufficient conditions on the system parameters, impulsive control gains, impulsive instants and distributed delays are obtained in the form of an inequality for global exponential stability. In these results, it is shown that an unstable time-delay system can be successfully stabilized by distributed-delay dependent impulses. It is worth noting that the proposed impulsive controllers are independent of the system states at each impulsive instant, and the states with distributed delays play the key role in the stabilization process. A numerical example is provided to demonstrate the efficiency of the main results.

Global Output-feedback Stabilization for Nonlinear Time-delay Systems with Unknown Control Coefficients

This paper investigates the problem of global stabilization by output-feedback for a class of planar nonlinear systems with unknown time-delay and control coefficients. Compared with the closely related works, the highlight of the paper is that it is not required that the upper and lower bounds of unknown control coefficients are known a prior, and meanwhile, the derivative of unknown time-varying time-delay is also not required to have known upper bound. This makes the problem essentially different, and much more difficult to solve. Motivated by our recent works, by combining the method of universal control with backstepping technique, and skillfully constructing a new Lyapunov-Krasoviskii functional, we design a new adaptive stabilizing controller with the suitable design parameters which guarantees that the resulting closed-loop system state is globally bounded while the state of the original system converges to the origin after a finite time. A numerical example is given to illustrate the effectiveness of the theoretical results.

Membership-Function-Dependent Stability and Stabilization Conditions for T–S Fuzzy Time-Delay Systems

ABSTRACTIn this paper, the stability analysis and stabilization of time-delay nonlinear systems represented by a Takagi–Sugeno fuzzy model are investigated. Combining with a recently developed integral inequality, relaxed delay-dependent stability conditions are derived by means of linear matrix inequalities. To obtain the further relaxed stability criteria, the membership-function-dependent method is applied via the piecewise membership functions (PMFs) and some slack variables. The PMFs are selected to approximate the membership functions (MFs), and then the characteristics of the MFs are taken into consideration adequately. Under the imperfect premise matching design scheme, this paper also proposes a flexible fuzzy controller design strategy that the fuzzy controllers do not employ the identical MFs and the number of rules as the fuzzy models. Finally, this proposed approach is tested successfully and its effectiveness is validated by the simulation examples.

Global output feedback stabilisation of nonlinear time-delay systems with unknown output function

ABSTRACTThis paper studies the problem of global output feedback stabilisation for a class of nonlinear time-delay systems with the unknown output function. By constructing the appropriate Lyapunov–Krasovskii functional and observer, skillfully combining generalised adding a power integrator technique, sign function and homogeneous domination approach, an output feedback controller is designed to guarantee globally uniformly asymptotical stability of nonlinear time-delay systems with the unknown output function.

Stability of nonlinear time-delay systems satisfying a quadratic constraint

The stability problem of nonlinear time-delay systems is addressed. A quadratic constraint is employed to exploit the structure of nonlinearity in dynamical systems via a set of multiplier matrices. This yields less conservative results concerning stability analysis. By employing a Wirtinger-based inequality, a delay-dependent stability criterion is derived in terms of linear matrix inequalities for the nominal and uncertain systems. A numerical example is used to demonstrate the effectiveness of the proposed stability conditions in dealing with some larger class of nonlinearities.

Delay-range-dependent control of nonlinear time-delay systems under input saturation

Summary This paper describes a delay-range-dependent local state feedback controller synthesis approach providing estimation of the region of stability for nonlinear time-delay systems under input saturation. By employing a Lyapunov–Krasovskii functional, properties of nonlinear functions, local sector condition and Jensen's inequality, a sufficient condition is derived for stabilization of nonlinear systems with interval delays varying within a range. Novel solutions to the delay-range-dependent and delay-dependent stabilization problems for linear and nonlinear time-delay systems, respectively, subject to input saturation are derived as specific scenarios of the proposed control strategy. Also, a delay-rate-independent condition for control of nonlinear systems in the presence of input saturation with unknown delay-derivative bound information is established. And further, a robust state feedback controller synthesis scheme ensuring L2 gain reduction from disturbance to output is devised to address the problem of the stabilization of input-constrained nonlinear time-delay systems with varying interval lags. The proposed design conditions can be solved using linear matrix inequality tools in connection with conventional cone complementary linearization algorithms. Simulation results for an unstable nonlinear time-delay network and a large-scale chemical reactor under input saturation and varying interval time-delays are analyzed to demonstrate the effectiveness of the proposed methodology. Copyright © 2015 John Wiley & Sons, Ltd.

Event-based Stabilization of Nonlinear Time-Delay Systems

In this paper, a universal formula is proposed for event-based stabilization of nonlinear time-delay systems affine in the control. The feedback is derived from the seminal law proposed by E. Sontag (1989) and then extended to event-based control of nonlinear undelayed systems. Under the assumption of the existence of a control Lyapunov-Krasovsky functional (CLKF), it enables smooth (except at the origin) asymptotic stabilization while ensuring that the sampling intervals do not contract to zero. Global asymptotic stability is obtained under the small control property assumption. Moreover, the control can be proved to be smooth anywhere under certain conditions. Simulation results highlight the ability of the proposals.

Globally asymptotic stabilization for nonlinear time-delay systems with ISS inverse dynamics

The output feedback stabilization is considered for a class of nonlinear time-delay systems with inverse dynamics in this paper. An appropriate state observer is constructed for the unmeasurable system states in order to realize the control objective. By adopting the backstepping and Lyapunov-Krasovskii functional methods, a systematic design procedure for a memoryless output feedback control law is presented. It is shown that the designed controller can make the closed-loop system globally asymptotically stable while keeping all signals bounded. An illustrative example is discussed to show the effectiveness of the proposed control strategy.

Constrained receding horizon controls for nonlinear time-delay systems

This paper proposes a constrained receding horizon control (RHC) for a nonlinear time-delay system with input and state delays. The control law is obtained by minimizing a receding horizon cost function with weighting functions of inputs and states on the end portion of the horizon. For stability, a general condition on the weighting functions is presented and its feasibility is illustrated via a certain type of nonlinear time-delay systems. In order to deal with input and state constraints, an invariant set is obtained, where the trajectories of the inputs and the states satisfy given constraints and stay forever under some conditions. It is shown in a numerical example that the proposed RHC guarantees the closed-loop stability for nonlinear time-delay systems while meeting the constraints.