Exponential Input-to-state Stability Research Articles

Sliding-mode-based impulsive control for a class of time-delay systems with input disturbance

This paper addresses the impulsive stabilization problem for a class of time-delay systems in the presence of input disturbances. A novel impulsive control law is constructed, which is based on the discrete-time equivalent control design technique. The sliding-mode-based impulsive control law allows counteracting the disturbance effect using its previous step value. To adapt in real time to the variation in sampling periods, the sliding function is designed as a function of impulse interval. Piecewise Lyapunov functions based on the partition on the maximum impulse interval are introduced to deal with the time-varying structure of the sliding function. Sufficient conditions for exponential input-to-state stability (EISS) of the impulsively controlled systems are derived, in which the EISS gain characterizes the attenuation ability of the proposed impulsive control law on the disturbance estimation error. The disturbance attenuation performance of the proposed impulsive control strategy is demonstrated through the time-delay Chua’s circuit.

Building coercive Lyapunov–Krasovskii functionals based on Razumikhin and Halanay approaches

AbstractIn this article, we provide a systematic and constructive way to build a Lyapunov–Krasovskii functional for time‐delay systems whose stability can be established through the Razumikhin or the Halanay approaches. The constructed Lyapunov–Krasovskii functional turns out to be coercive, meaning sandwiched between functions of the state history norm, and to dissipate in terms of the whole history norm. We present these results in the framework of input‐to‐state stability (ISS) in order to further account for the influence of input disturbances. A special emphasis is also given on exponential stability and exponential ISS. We illustrate our findings though the study of a coupled ODE‐PDE model of a chemical reactor, and show that, unlike most results in that area, our approach happens to ensure ISS in terms of the supremum norm of the state.

Variable Gain Impulsive Synchronization for Discrete-Time Delayed Neural Networks and Its Application in Digital Secure Communication.

This article revisits the problems of impulsive stabilization and impulsive synchronization of discrete-time delayed neural networks (DDNNs) in the presence of disturbance in the input channel. A new Lyapunov approach based on double Lyapunov functionals is introduced for analyzing exponential input-to-state stability (EISS) of discrete impulsive delayed systems. In the framework of double Lyapunov functionals, a pair of timer-dependent Lyapunov functionals are constructed for impulsive DDNNs. The pair of Lyapunov functionals can introduce more degrees of freedom that not only can be exploited to reduce the conservatism of the previous methods, but also make it possible to design variable gain impulsive controllers. New design criteria for impulsive stabilization and impulsive synchronization are derived in terms of linear matrix inequalities. Numerical results show that compared with the constant gain design technique, the proposed variable gain design technique can accept larger impulse intervals and equip the impulsive controllers with a stronger disturbance attenuation ability. Applications to digital signal encryption and image encryption are provided which validate the effectiveness of the theoretical results.

Exponential input-to-state stability of delay Korteweg–de Vries–Burgers equations

The exponential input-to-state stability (EISS) is studied for delay Korteweg–de Vries–Burgers (DKdVB) equations. DKdVB equations are considered with distributed input and boundary input. Making use of the Lyapunov–Krasovskii functional method and inequality techniques, a sufficient condition is established to ensure the EISS for DKdVB equations. Besides the impact of the time delay on the EISS, this sufficient condition also shows the effect of the diffusion term on the EISS. In case of uncertainties of system’s coefficients, robust EISS of uncertain DKdVB equations is investigated. A criterion is obtained to guarantee the EISS for the uncertain DKdVB equation. Two numerical simulation examples are given to illustrate the validity of the derived results.

The exponential input-to-state stability property: characterisations and feedback connections

The exponential input-to-state stability (ISS) property is considered for systems of controlled nonlinear ordinary differential equations. A characterisation of this property is provided, including in terms of a so-called exponential ISS Lyapunov function and a natural concept of linear state/input-to-state L^2-gain. Further, the feedback connection of two exponentially ISS systems is shown to be exponentially ISS provided a suitable small-gain condition is satisfied.

Input-to-state stability of non-autonomous infinite-dimensional control systems

<p style='text-indent:20px;'>This paper addresses input-to-state stability (ISS) and integral input-to-state stability (iISS) for non-autonomous infinite-dimensional control systems. With the notion of uniformly exponential stability scalar function, ISS and iISS are considered based on indefinite Lyapunov functions. In addition, we obtain several necessary and sufficient characterizations of the iISS property, expressed in terms of dissipation inequalities. As a result, the iISS criteria of non-autonomous bilinear systems is also established. Furthermore, an illustrative example is given to show the applicability of the results.</p>

Input-to-state stability for time-varying delayed systems in Halanay-type inequality forms

This paper studies the input-to-state stability (ISS) for time-varying delayed systems (TVDS) in Halanay-type inequality forms. The time-delay in TVDS is allowed to be time-varying and unbounded. By introducing the notion of a uniform M-matrix, exponential ISS theorems are established respectively for continuous-time, discrete-time, and zero-order TVDS. The convergence rates of exponential ISS and ISS gains and their relation are subsequently estimated. These ISS theorems are less conservative and generalize the results of stability and ISS for Halanay-type inequalities in the literature. Moreover, necessary conditions of ISS are given for TVDS in Halanay-type equality forms. By specializing the ISS results to linear time-invariant delayed systems, the necessary and sufficient conditions of ISS are derived respectively. Three examples are given throughout the paper to illustrate the theoretical results.

On characterisations of the input to state stability properties for conformable fractional order bilinear systems

This paper proposes for the first time the theoretical requirements that a fractional-order bilinear system with conformable derivative has to fulfil in order to satisfy different input-to-state stability (ISS) properties. Variants of ISS, namely ISS itself, integral ISS, exponential integral ISS, small-gain ISS, and strong integral ISS for the general class of conformable fractional-order bilinear systems are investigated providing a set of necessary and sufficient conditions for their existence and then compared. Finally, the correctness of the obtained theoretical results is verified by numerical example.

Stabilization to Exponential Input-to-State Stability of a Class of Neural Networks with Delay by Observer-Based Aperiodic Intermittent Control

This study is devoted to investigating the stabilization to exponential input-to-state stability (ISS) of a class of neural networks with time delay and external disturbances under the observer-based aperiodic intermittent control (APIC). Compared with the general neural networks, the state of the neural network investigated is not yet fully available. Correspondingly, an observer-based APIC is constructed, and moreover, neither the observer nor the controller requires the information of time delay. Then, the stabilization to exponential ISS of the neural network is realized severally by the observer-based time-triggered APIC (T-APIC) and the observer-based event-triggered APIC (E-APIC), and corresponding criteria are given. Furthermore, the minimum activation time rate (MATR) of the observer-based T-APIC and the observer-based E-APIC is estimated, respectively. Finally, a numerical example is given, which not only verifies the effectiveness of our results but also shows that the observer-based E-APIC is superior to the observer-based T-APIC and the observer-based periodic intermittent control (PIC) in control times and the minimum activation time rate, and the function of the observer-based T-APIC is also better than the observer-based PIC.

Remote observer-based robust control for cyber-physical systems under asynchronous DoS attacks: an intelligent approach

In this paper, the input-to-state stability (ISS) control problem is studied for cyber-physical systems (CPSs) in the presence of asynchronous denial-of-service (DoS) attacks. To enhance the exponential ISS, an intelligent packet-based control method with buffering is proposed by introducing the acknowledgement signal (ACK), in which the trial transmission attempts of the control packet are intelligently determined by the situation (successful or failed) of its last sampling transmission instant after identifying the unnecessary transmission points and the necessary transmission points. Then, inspired by the packet-based control technology with buffering, the sufficient condition for attempting transmission of the control packet is given, in which the exponential ISS with maximum robustness index can be preserved. Compared with the existing method with a fixed transmission rate for the control packet, it is shown that the considered framework improves the exponential ISS performance with equal or less communication resource costs.

New results on exponential input-to-state stability analysis of memristor based complex-valued inertial neural networks with proportional and distributed delays

The present work accumulates the Exponential input-to-state stability (EISS) criteria of memristor based delayed complex-valued neural networks (DMCNN) associated with an inertial term and time-varying delays. Here two varieties of time-varying delays are provided, namely proportional and distributed delays. In this study, the delayed memristor neural networks (MNN) is constructed on the basis of second order complex-valued space. In addition, the sufficient conditions are proposed to ensure the EISS by using the combination of non-smooth analysis, set-valued maps, Lyapunov-Krasovskii functional having double integral terms and Kirchhoff’s matrix tree theorem, moreover we employ Cauchy-Schwarz inequality & some inequality techniques. At the end of this work, the hypothesis has been established with an illustrative example along with the simulations.

Comparison principles and input-to-state stability for stochastic impulsive systems with delays

ABSTRACT In this paper, the input-to-state stability (ISS), the pth moment exponential ISS and the stochastic input-to-state stability issues are investigated for the stochastic impulsive systems with time delays via the comparison principle method. Firstly, several general comparison principles in vector-version are proposed guaranteeing the existence, uniqueness and magnitude for solutions of the addressed system. Then based on these established comparison principles, the ISS-related properties are investigated for the stochastic impulsive delayed model. Finally, one example is given to illustrate effectiveness of the obtained results.

Exponential input-to-state stability of stochastic delay reaction–diffusion neural networks

This paper considers the mean square exponential input-to-state stability (EISS) for stochastic delay reaction–diffusion neural networks (SDRDNNS). SDRDNNS with distributed input and boundary input are investigated. In addition, constant delay and time-varying delay are considered. With the help of Lyapunov–Krasovskii functional method, Itô formula and Wirtinger-type inequality, delay-dependent sufficient conditions on mean square EISS of SDRDNNS are presented. These sufficient conditions show the effects of time-delay and diffusion term on mean-square EISS. Moreover, by means of numerical simulation, the effectiveness of our theoretical results is illustrated.

Input-to-State Stability of a Clamped-Free Damped String in the Presence of Distributed and Boundary Disturbances

This note establishes the Exponential Input-to-State Stability (EISS) property for a clamped-free damped string with respect to distributed and boundary disturbances. While efficient methods for establishing ISS properties for distributed parameter systems with respect to distributed disturbances have been developed during the last decades, establishing ISS properties with respect to boundary disturbances remains challenging. One of the well-known methods for well-posedness analysis of systems with boundary inputs is the use of a lifting operator for transferring the boundary disturbance to a distributed one. However, the resulting distributed disturbance involves time derivatives of the boundary perturbation. Thus, the subsequent ISS estimate depends on its amplitude, and may not be expressed in the strict form of ISS properties. To solve this problem, we show for a clamped-free damped string equation that the projection of the original system trajectories in an adequate Riesz basis can be used to establish the desired EISS property.

PI Regulation of a Reaction–Diffusion Equation With Delayed Boundary Control

The general context of this work is the feedback control of an infinite-dimensional system so that the closed-loop system satisfies a fading-memory property and achieves the setpoint tracking of a given reference signal. More specifically, this paper is concerned with the Proportional Integral (PI) regulation control of the left Neumann trace of a one-dimensional reaction-diffusion equation with a delayed right Dirichlet boundary control. In this setting, the studied reaction-diffusion equation might be either open-loop stable or unstable. The proposed control strategy goes as follows. First, a finite-dimensional truncated model that captures the unstable dynamics of the original infinite-dimensional system is obtained via spectral decomposition. The truncated model is then augmented by an integral component on the tracking error of the left Neumann trace. After resorting to the Artstein transformation to handle the control input delay, the PI controller is designed by pole shifting. Stability of the resulting closed-loop infinite-dimensional system, consisting of the original reaction-diffusion equation with the PI controller, is then established thanks to an adequate Lyapunov function. In the case of a time-varying reference input and a time-varying distributed disturbance, our stability result takes the form of an exponential Input-to-State Stability (ISS) estimate with fading memory. Finally, another exponential ISS estimate with fading memory is established for the tracking performance of the reference signal by the system output. In particular, these results assess the setpoint regulation of the left Neumann trace in the presence of distributed perturbations that converge to a steady-state value and with a time-derivative that converges to zero. Numerical simulations are carried out to illustrate the efficiency of our control strategy.

Exponential input-to-state stability of quaternion-valued neural networks with time delay

This paper debated the exponential input-to-state stability (EITSS) of the solution for a kind of quaternion-valued neural networks (QVNNs) with time delay. It fills the blank of QVNN in the aspect of input-to-state stability (ITSS). In virtue of the quaternion multiplication is not suitable for commutative law, QVNN is ordinarily resolved into four real-valued neural networks (RVNNs). Making use of a novel Lyapunov–Krasovskii function and some inequalities, we obtain a little sufficient conditions to assure the considered system is EITSS. Finally, by means of two examples, it is certified that the calculation results in this paper are fine.

Pth moment exponential input‐to‐state stability of nonlinear discrete‐time impulsive stochastic delay systems

SummaryThis paper investigates the pth moment exponential input‐to‐state stability (ISS) of nonlinear discrete‐time impulsive stochastic delay systems. By employing Lyapunov functionals, some pth moment exponential ISS criteria are provided. The obtained results show that a discrete‐time stochastic delay system can become pth moment exponential input‐to‐state stable by impulsive controls even if it may be not input‐to‐state stable itself. On the other hand, the original system without impulses can retain its ISS property with appropriate destabilizing impulses. As an application, the theoretical results are used to test the ISS for a class of recurrent neural networks under stochastic perturbations. Finally, a numerical example is presented to illustrate the effectiveness of the results.

Moment exponential input‐to‐state stability of non‐linear switched stochastic systems with Lévy noise

The moment exponential input-to-state stability (ISS) problem for a class of non-linear switched stochastic systems is studied. A continuously differentiable Lyapunov function with indefinite derivative is introduced, which generalises classic Lyapunov function method. Two situations are considered: (i) synchronous switching, i.e. candidate controllers coincide with system modes; (ii) asynchronous switching, i.e. the candidate controllers have a lag to the switching of the system modes. By employing indefinite derivative Lyapunov function method and average dwell-time approach, sufficient conditions for moment exponential ISS of the systems are derived. Finally, an example is given to illustrate the effectiveness of the authors' results.

Input‐to‐state exponents and related ISS for delayed discrete‐time systems with application to impulsive effects

SummaryThis paper aims to investigate the input‐to‐state exponents (IS‐e) and the related input‐to‐state stability (ISS) for delayed discrete‐time systems (DDSs). By using the method of variation of parameters and introducing notions of uniform and weak uniform M‐matrix, the estimates for 3 kinds of IS‐e are derived for time‐varying DDSs. The exponential ISS conditions with parts suitable for infinite delays are thus established, by which the difference from the time‐invariant case is shown. The exponential stability of a time‐varying DDS with zero external input cannot guarantee its ISS. Moreover, based on the IS‐e estimates for DDSs, the exponential ISS under events criteria for DDSs with impulsive effects are obtained. The results are then applied in 1 example to test synchronization in the sense of ISS for a delayed discrete‐time network, where the impulsive control is designed to stabilize such an asynchronous network to the synchronization.

Relaxed ISS Small-Gain Theorems for Discrete-Time Systems

In this paper ISS small-gain theorems for discrete-time systems are stated, which do not require input-to-state stability (ISS) of each subsystem. This approach weakens conservatism in ISS small-gain theory, and for the class of exponential ISS systems we are able to prove that the proposed relaxed small-gain theorems are nonconservative in a sense to be made precise. The proofs of the small-gain theorems rely on the construction of a dissipative finite-step ISS Lyapunov function which is introduced in this work. Furthermore, dissipative finite-step ISS Lyapunov functions, as relaxations of ISS Lyapunov functions, are shown to be sufficient and necessary to conclude ISS of the overall system.