Nonlinear Delay Systems Research Articles

Stability of nonlinear distributed delay system with parameter uncertainties: Integral‐based event‐triggered impulsive control strategy

AbstractThis article investigates the stability of nonlinear uncertain distributed delay system via integral‐based event‐triggered impulsive control (IETIC) strategy. First, a IETIC mechanism is presented to reduce the redundant data transmission over the system, in which the integral‐based event‐triggered mechanism uses the integration of system states over a time period in the past. Second, a new lemma is proposed to eliminate the Zeno behavior of the established model through the IETIC mechanism. Third, a novel Lyapunov–Krasovskii functional (LKF) method related to probability density function is constructed to guarantee the stability of the established model based on LMI conditions, where a probability density function is introduced as a distributed delay kernel. Compared with existing methods, the constructed novel LKF method is less conservative or requiring less number of decision variables. Numerical examples are further provided to confirm the effectiveness and advantages of the proposed approach.

Controllability of higher order stochastic fractional control delay systems involving damping behavior

This article focuses on the problem of controllability of both cases linear and nonlinear higher order stochastic fractional control delay systems with damping behavior, which involving Caputo fractional derivative (CFD). The proposed approach utilizes the ideas of controllability Grammian matrix involving Mittag-Leffler function (MLF) and Burkholder-Davis-Gundy’s inequality. By employing Banach fixed point theorem, we establish the exact method to design a stochastic perturbation to control the considered nonlinear higher order fractional differential systems. As a final point, the derived design is illustrated with two numerical examples.

Lyapunov stability analysis for nonlinear delay systems under random effects and stochastic perturbations with applications in finance and ecology

This manuscript is involved in the study of stability of the solutions of functional differential equations (FDEs) with random coefficients and/or stochastic terms. We focus on the study of different types of stability of random/stochastic functional systems, specifically, stochastic delay differential equations (SDDEs). Introducing appropriate Lyapunov functionals enables us to investigate the necessary conditions for stochastic stability, asymptotic stochastic stability, asymptotic mean square stability, mean square exponential stability, global exponential mean square stability, and practical uniform exponential stability. Some examples with numerical simulations are presented to strengthen the theoretical results. Using our theoretical study, important aspects of epidemiological and ecological mathematical models can be revealed. In ecology, the dynamics of Nicholson’s blowflies equation is studied. Conditions of stochastic stability and stochastic global exponential stability of the equilibrium point at which the blowflies become extinct are investigated. In finance, the dynamics of the Black–Scholes market model driven by a Brownian motion with random variable coefficients and time delay is also studied.

Dependent delay stability characterization for a polynomial T-S Carbon Dioxide model

<p style='text-indent:20px;'>By extending some linear time delay systems stability techniques, this paper, focuses on continuous time delay nonlinear systems (TDNS) dependent delay stability conditions. First, by using the Takagi Sugeno Fuzzy Modeling, a novel relaxed dependent delay stability conditions involving uncommon free matrices, are addressed in Linear Matrix Inequalities (LMI). Then, as application a Nonlinear Carbon Dioxide Model is used and rewritten by a change of coordinate to the interior equilibrium point. Next, by using the non-linearity sector method the model is transformed to a corresponding Fuzzy Takagi Sugeno (TS) multi-model. Also, the maximum delay margin to which the model is stable, is identified. Finally, to prove the analytic results a numerical simulation is also performed and compared to other methods.</p>

Universal adaptive control for a class of nonlinear time-varying delay systems with unknown output function

This paper considers the stabilization problem for a class of nonlinear time-varying delay systems with unknown homogeneous growth rates and unknown output function. By introducing the dynamic gain into the system, an adaptive observer is designed. Then, combined with the Lyapunov–Krasovskii functional, an adaptive controller is proposed to guarantee the global stabilization of the closed-loop system. Two examples are provided to verify the effectiveness of the designed controller.

Adaptive Finite-Time Tracking Control for Nonlinear Time-Varying Delay Systems With Full State Constraints and Input Saturation

This paper investigates the adaptive finite-time tracking control problem for a class of nonlinear time-varying delay systems subject to full state constraints and input saturation. The nonlinear state-dependent functions (NSDFs) are introduced to handle the asymmetric time-varying full state constraints without feasibility conditions. To cope with the unknown time-varying delays, the radial basis function neural networks (RBF NNs) and the finite covering lemma are utilized. Avoiding the use of the Lyapunov-Krasovskii functionals (LKFs), no restriction on the derivative of the time-varying delays is needed. Meanwhile, the effect of the input saturation is eliminated by the augmented function with an auxiliary control signal. The adaptive finite-time tracking controller with only one adaptive parameter is constructed by applying the command filter approach and backstepping technique. It is proved that the proposed controller can ensure that all signals in the closed-loop system are bounded and the tracking error converges to a small neighborhood of the origin in a finite time. Finally, the effectiveness of the proposed scheme is demonstrated by two simulation examples.

Adaptive Observer for Systems with Distributed Output Delay

Abstract We are considering the problem of sampled-data observer design for nonlinear time-varying delay systems that are state- and parameter-affine. The novelty is that the system is subject to both distributed delay and parameter uncertainty. A Kalman-like observer is developed to deal with both state and parameter estimation. Its main components are: (i) a time-varying-gain state-estimator involving both output and parameter rate injections; (ii) a distributed-nature adaptive output-predictor that compensates for delay and output sampling delay; (iii) an optimized parameter-estimator that copes with parameter uncertainty, optimization is intended in the sense that we make use of all available information, unlike similar existing estimators. The resulting observer is shown to be exponentially convergent, for small delays and sampling intervals, provided the input signal is sufficiently exciting. The analysis is performed using a Lyapunov-Krasovskii functional, Halanay’s lemma, Wirtinger’s inequality and other tools.

A fractional nonlinear evolutionary delay system driven by a hemi-variational inequality in Banach spaces

This article deals with a new fractional nonlinear delay evolution system driven by a hemi-variational inequality in a Banach space. Utilizing the KKM theorem, a result concerned with the upper semicontinuity and measurability of the solution set of a hemivariational inequality is established. By using a fixed point theorem for a condensing set-valued map, the nonemptiness and compactness of the set of mild solutions are also obtained for such a system under mild conditions. Finally, an example is presented to illustrate our main results.

Design of disturbance observer based on adaptive‐neural control for large‐scale time‐delay systems in the presence of actuator fault and unknown dead zone

SummaryThis article presents an adaptive neural compensation scheme for a class of large‐scale time delay nonlinear systems in the presence of unknown dead zone, external disturbances, and actuator faults. In this article, the quadratic Lyapunov–Krasovskii functionals are introduced to tackle the system delays. The unknown functions of the system are estimated by using radial basis function neural networks. Furthermore, a disturbance observer is developed to approximate the external disturbances. The proposed adaptive neural compensation control method is constructed by utilizing a backstepping technique. The boundedness of all the closed‐loop signals is guaranteed via Lyapunov analysis and the tracking errors are proved to converge to a small neighborhood of the origin. Simulation results are provided to illustrate the effectiveness of the proposed control approach.

Finite-Time Tracking Control for Nonstrict-Feedback State-Delayed Nonlinear Systems with Full-State Constraints and Unmodeled Dynamics

The problem of finite-time tracking control is discussed for a class of uncertain nonstrict-feedback time-varying state delay nonlinear systems with full-state constraints and unmodeled dynamics. Different from traditional finite-control methods, a C 1 smooth finite-time adaptive control framework is introduced by employing a smooth switch between the fractional and cubic form state feedback, so that the desired fast finite-time control performance can be guaranteed. By constructing appropriate Lyapunov-Krasovskii functionals, the uncertain terms produced by time-varying state delays are compensated for and unmodeled dynamics is coped with by introducing a dynamical signal. In order to avoid the inherent problem of “complexity of explosion” in the backstepping-design process, the DSC technology with a novel nonlinear filter is introduced to simplify the structure of the controller. Furthermore, the results show that all the internal error signals are driven to converge into small regions in a finite time, and the full-state constraints are not violated. Simulation results verify the effectiveness of the proposed method.

M-matrix based stability analysis switched nonlinear time-varying delay systems

This paper addresses the stability analysis problem for a class of switched nonlinear time-varying delays systems. In particular, the considered time-varying delays depend on the subsystem number. Based on a common Lyapunov function, the M-matrix proprieties, and through the $$ l_{1,\infty } $$ induced norms notion, new algebraic stability criteria under arbitrary switching are derived. Then, by employing the aggregation techniques and the Borne–Gentina criterion, the obtained results are extended to a class of switched nonlinear systems modeled by difference equations. By some comparisons, it is shown that the proposed approach allows us to avoid searching a common Lyapunov function, considered a hard task even for simple cases. Lastly, numerical simulations are included to illustrate the effectiveness of the obtained results.

An improved -gain analysis method of nonlinear delay systems and its application to delayed neural networks

The Lyapunov–Krasovskii functional (LKF) method is widely used to derive -gain analysis criteria of delay systems. Nevertheless, how to improve the method for obtaining less conservative -gain analysis criteria is still an open problem. In this paper, an improved -gain analysis method called a delay-mode-based Lyapunov–Krasovskii functional (DMBLKF) method is proposed to replace the LKF method. This is achieved by weakening the condition that a LKF is required for a system with a delay varying in a delay interval. With the help of the DMBLKF method, new -gain analysis criteria of delayed neural networks are obtained and a numerical example is given to show that the DMBLKF method is more effective than the LKF method.

Robust stability analysis of uncertain fractional order neutral-type delay nonlinear systems with actuator saturation

In this paper, we study the robust stability of uncertain fractional order (FO) nonlinear systems having neutral-type delay and input saturation. From the Lyapunov–Krasovskii functional, sufficient criteria on asymptotic robust stability of such FO systems with the help of linear matrix inequalities are specified to compute the gain of state-feedback controller. An optimization is also derived using the cone complementarity linearization method for finding the controller gains subject to maximizing the domain of attraction. The main results are confirmed via numerical examples.

An LMI based approach to stabilize a type of nonlinear uncertain neutral-type delay systems

This paper proposes a linear matrix inequality (LMI) based approach for stability analysis of a type of nonlinear uncertain neutral-type delay systems. The delays in states, state derivatives and inputs are known and constant. By constructing a Lyapunov–Krasovskii functional (LKF), a stability criterion in the form of LMI is derived. In this paper, we treat these nonlinear terms to be norm-bounded and construct a Lyapunov matrix inequality considering system states, delayed states, delayed state derivatives, and delayed inputs. The stability criterion of this nonlinear neutral-type delay system is then obtained. To this end, a numerical example is given to demonstrate the feasibility of the proposed approach. The developed method for stability analysis can potentially be applied to critical real-world applications such as the down-hole drilling system.

Uniform bounded input bounded output stability of fractional‐order delay nonlinear systems with input

SummaryThe bounded input bounded output (BIBO) stability for a nonlinear Caputo fractional system with time‐varying bounded delay and nonlinear output is studied. Utilizing the Razumikhin method, Lyapunov functions and appropriate fractional derivatives of Lyapunov functions some new bounded input bounded output stability criteria are derived. Also, explicit and independent on the initial time bounds of the output are provided. Uniform BIBO stability and uniform BIBO stability with input threshold are studied. A numerical simulation is carried out to show the system's dynamic response, and demonstrate the effectiveness of our theoretical results.

Adaptive asymptotic tracking control of uncertain nonlinear time-delay systems depended on delay estimation information

This paper investigates the issue of adaptive asymptotic tracking control for high-order strict-feedback systems with mismatched unstructured nonlinearities and unknown time delays. Under the proposed control strategy, unknown time delays are estimated with the help of adaptation mechanism and projection operator, which abandons the use of the conventional Lyapunov-Krasovskii functional method and avoids the discussion of controller singularity problem. It is mentioned that most of current reported results on adaptive control of nonlinear delay systems under fuzzy approximation are semi-globally bounded. To achieve the asymptotic control objective, the upper bounds of approximation errors and other residual errors are properly estimated by establishing appropriate adaptive laws. Based on the estimation information of time delays and errors bounds, a novel adaptive sliding tracking control scheme is developed such that the tracking error will be forced to keep sliding along the origin once the upper bounds are reached. Therefore, the asymptotic tracking control objective is achieved. Simulation example verifies the effectiveness of the proposed scheme.

Exponential stability analysis and control design for nonlinear system with time-varying delay

This paper investigates the problem of exponential stability analysis and control design for time delay nonlinear systems with unknown control coefficient. Nussbaum gain function is utilized to solve the problem of unknown control directions at every step. By designing a new Lyapunov-Krasovskii functional, the problem of unknown time-varying delay is solved. Under the frame of adaptive backstepping recursive design, an exponential stabilization control algorithm is developed, which demonstrates that all solutions of controlled system are ultimately uniformly bounded (UUB) and exponential converge to zero. Finally, simulation results are displayed to explain the superiority and effectiveness of the developed control method.

A General Framework for Optimal Control of Fractional Nonlinear Delay Systems by Wavelets

An iterative procedure to find the optimal solutions of general fractional nonlinear delay systems with quadraticperformance indices is introduced. The derivatives of state equations are understood in the Caputo sense. By presenting and applying a general framework, we use the Chebyshev wavelet method developed for fractional linear optimal control to convert fractional nonlinear optimal control problems as a sequence of quadratic programming ones. The concepts and computational procedure that are used for fractional linear optimal control are applied on fractional nonlinear optimal control. Different types of nonlinear optimal control problems with fractional or integer order can be solved. To see this, some numerical examples are solved. Another operational property of Chebyshev wavelets is presented.

Almost sure exponential stability of semi-Euler numerical scheme for nonlinear stochastic functional differential equation

It is generally known that explicit solution can rarely be obtained for stochastic differential equations (SDEs), not to mention the nonlinear stochastic functional differential equation (SFDEs). Therefore, this paper proposes a numerical scheme called semi-Euler numerical scheme for general SFDEs, which adapts better to the connotation of the Itô differential equations. Under a generalized polynomial growth condition, the scheme admits high nonlinearity of the system, the almost sure exponential stability of the continuous model and the numerical scheme is investigated by contrast. It is confirmed that the numerical scheme preserves the stability property of the continuous model with no restriction to the step size. Compared to the literature, the polynomial growth condition in [L. Liu, F. Deng, and T. Hou, Almost sure exponential stability of implicit numerical solution for stochastic functional differential equation with extended polynomial growth condition, Applied Mathematics and Computation. 330 (2018), pp. 201–212.], [W. Fei, L. Hu, X. Mao, and M. Shen, Delay dependent stability of highly nonlinear hybrid stochastic systems, Automatica, 82 (2017), pp. 165–170.], [M. Shen, W. Fei, X. Mao, and Y. Liang, Stability of highly nonlinear neutral stochastic differential delay equations, Systems & Control Letters, 115 (2018), pp. 1–8.], etc. is generalized to that described with Lyapunov function. Besides, to provide an approach for the solvability of the implicit scheme, the generalized monotonicity of the vectorial functions is introduced. An application is given for the scheme and the stability conclusion, by virtue of the generalized monotonicity condition, the scheme is solvable also without restriction to the step size. At the end of the paper, a high order example is proposed to illustrate the theory of this paper, and a further research direction from this work is pointed out.

Sliding Mode Multiobserver for Time-Varying Delay Nonlinear Systems Based on Discrete Uncoupled Multimodel

The present paper deals with the design of sliding mode multiobserver for nonlinear systems with delayed measurements. The discrete uncoupled state multimodel is exploited to describe the global behavior of nonlinear systems. Therefore, the complexity of the latter can be reduced by its decomposition into a finite number of partial models. The proposed sliding mode multiobserver is designed based on this uncoupled multimodel approach. Sufficient conditions are formulated in terms of linear matrix inequalities in order to ensure the asymptotic stability of the designed multiobserver. Illustrative examples are included to show the effectiveness of the proposed strategy.