Nonlinear Time-varying Delay Systems Research Articles

Adaptive multi-dimensional Taylor network tracking control of time-varying delay nonlinear systems subject to input saturation

For the tracking control problem of a class of nonlinear systems with input saturation constraint and time-varying delay, an adaptive tracking control scheme based on multi-dimensional Taylor network (MTN) is designed, and the proposed control scheme has the advantages of simple structure and small computation. A Gaussian error function is introduced to transform the input saturation into a linear model with bounded error. The Lyapunov-Krasovskii function is constructed, and the nonlinearity of the system is approximated using MTN, and an adaptive control strategy is constructed based on the backstepping method. The results show that the proposed control method can both ensure that all signals in the closed-loop system are bounded and achieve convergence of the tracking error to a small neighborhood near the origin. Numerical simulations verify the effectiveness of the method.

The finite-time boundedness and control analysis of switched nonlinear time-varying delay systems based on auxiliary matrices

This paper explores the finite-time boundedness and controller problem of switching nonlinear delay systems. The time delay is considered as time-varying delay rather than constant delay. First of all, an appropriate Lyapunov–Krasovskii functional (LKF) which contains three integral terms is selected. The upper and lower bounds of these integral terms are given by segmenting the time delay interval. Meanwhile, the auxiliary matrix method is used to separate the Lyapunov matrix from the system matrix. Moreover, a new state-dependent switching law is designed to switch between subsystems according to the change of the magnitude of the LKF. Based on the above methods, the first step is to research the finite-time bounded analysis of delayed system without control input. Then, a sufficient condition is obtained to ensure the finite-time boundedness of switched systems with control input. Further, by designing an observer-based H∞ controller, the closed-loop system with time-varying delay is bounded and has H∞ performance index ν by using the Finsler’s Lemma and Singular Value Decomposition (SVD). Simultaneously, the controller and observer gains are designed. Finally, two simulations are used to verify the effectiveness and feasibility of the proposed methods.

Mathematical Model for Fault Handling of Singular Nonlinear Time-Varying Delay Systems Based on T-S Fuzzy Model

In this paper, a mathematical model based on the T-S fuzzy model is proposed to solve the fault estimation (FE) and fault-tolerant control (FTC) problem for singular nonlinear time-varying delay (TVD) systems with sensor fault. TVD is is extremely difficult to solve and the Laplace transform is devised to build an equal system free of TVD. Additionally, the sensor fault is changed to actuator fault by the developed coordinate transformation. A fuzzy learning fault estimator is first built to estimate the detailed sensor fault information. Then, a PI FTC scheme is suggested aiming at minimizing the damage caused by the fault. Simulation results from multiple faults reveal that the FE and FTC algorithms are able to estimate the fault and guarantee the system performance properly.

Unknown input observers for time-varying delay Takagi–Sugeno fuzzy systems with unmeasured nonlinear consequents

This paper investigates the problem of unknown input observer (UIO) design for Takagi–Sugeno (TS) fuzzy systems with time-varying delay. For observer design, all the unmeasured nonlinearities are isolated in the nonlinear consequent parts of TS fuzzy systems. This allows dealing with the well-known issue related to unmeasured premise variables for TS fuzzy observer design in a more effective fashion via the mean value theorem. Using the unknown input (UI) decoupling technique, no a priori information on the UI is needed for the proposed TS fuzzy observer design, which can guarantee the asymptotic estimations of both the system state and the unknown input. Employing a fuzzy Lyapunov–Krasovskii functional with a suitable augmented vector, the effect of time-varying delay on the estimation error dynamics is taken into account in the observer design. Numerical examples are presented to demonstrate the effectiveness of the proposed fuzzy UIO design method for nonlinear time-varying delay systems.

Event-triggered tracking control for nonlinear time-varying delay systems with full state constraints and unknown control coefficients

This paper addresses the problem of event-triggered control for a class of nonlinear systems with unknown control coefficients, time-varying delays, full state constraints and external disturbances, simultaneously. Firstly, based on integral Barrier-Lyapunov Functionals, an adaptive event-triggered controller is designed to account for the impacts of complete state constraints, unknown time-varying delays and external disturbances. Using fuzzy logic systems, the unknown functions of systems are then roughly estimated. The system redundancy can be significantly reduced by implementing the event-triggered control technique in the meanwhile. By introducing the separation technique and Lyapunov–Krasovskii functionals, it is shown that the strategy can ensure tracking performance greatly and all the closed-loop signals of the system are bounded. Compared with the existing results, the proposed design scheme is less conservative and has a wilder application range. Finally, the simulation results show the effectiveness of the proposed approach.

Adaptive Compensation Tracking Control for Time-Varying Delay Nonlinear Systems with Unknown Actuator Dead Zone

This paper concerns the problem of adaptive compensation tracking control for a class of time-varying delay nonlinear systems with unknown structures and unknown actuator dead zones where time-varying delays are unknown. First, a variable separation approach is used to overcome the difficulty in dealing with a nonstrict-feedback structure, and multilayer neural networks are used to approximate the unknown nonlinear structures with time-varying delays. On this basis, we designed an adaptive multilayer neural-network compensation controller to reduce the error of multilayer neural networks. Furthermore, for unknown actuator dead zones, this paper separates the controller and adopts multilayer neural networks to deal with unknown actuator dead zones. In order to reduce the error of the dead-zone controller, wer designed an adaptive compensation controller for the dead zones. Lastly, this paper proves the stability of the systems with the Lyapunov method, and simulation results demonstrate the effectiveness of the scheme.

Global Practical Output Tracking for a Class of Uncertain Inherently Time-Varying Delay Nonlinear Systems by Output Feedback

This article addresses the problem of global practical output tracking by output feedback for a class of uncertain inherently time-varying delay nonlinear systems. Firstly, a homogeneous output-feedback controller is designed for the nominal uncertain inherently system by virtue of adding a power integrator technique. Then, with the help of an appropriate Lyapunov–Krasovskii functional and reduced-order observer, by using the homogeneous domination approach and adding a power integrator method, an output-feedback controller is successfully developed to guarantee all the states of the closed-loop system remain bounded and simultaneously making the tracking error arbitrarily small. The simulation results of an example verify the proposed approach.

RBF Neural Network Sliding Mode Control for Passification of Nonlinear Time-Varying Delay Systems with Application to Offshore Cranes.

This paper is devoted to studying the passivity-based sliding mode control for nonlinear systems and its application to dock cranes through an adaptive neural network approach, where the system suffers from time-varying delay, external disturbance and unknown nonlinearity. First, relying on the generalized Lagrange formula, the mathematical model for the crane system is established. Second, by virtue of an integral-type sliding surface function and the equivalent control theory, a sliding mode dynamic system can be obtained with a satisfactory dynamic property. Third, based on the RBF neural network approach, an adaptive control law is designed to ensure the finite-time existence of sliding motion in the face of unknown nonlinearity. Fourth, feasible easy-checking linear matrix inequality conditions are developed to analyze passification performance of the resulting sliding motion. Finally, a simulation study is provided to confirm the validity of the proposed method.

Global stabilization via output feedback for a class of uncertainty nonlinear systems with time-varying delay and zero dynamics

This paper proposes a new control strategy to deal with three different types of uncertainties for a class of time-varying delay nonlinear systems. Quite different from related celebrated works, the considered systems allow the existence of parameter uncertainties in the output function and nonlinearities, dynamic uncertainties of the unmeasurable state, and continuous external disturbances, which invokes the motivation of the paper. The objective is to construct a continuous output feedback controller by utilizing a new restructured integral function and the double-domination method. The novelty control design is the capability of tackling zero-dynamic and unknown output function simultaneously, thereby guaranteeing the state convergence of the closed-loop system. Finally, the proposed scheme is applied to a practical example and a numerical one simultaneously to demonstrate the effectiveness and the superiority.

Periodic event-triggered modified repetitive control with equivalent-input-disturbance estimator based on T-S fuzzy model for nonlinear systems

In this paper, the periodic signal tracking and the disturbance rejection problems are considered for a class of time-varying delay nonlinear systems with unknown exogenous disturbances under limited communication resources. The Takagi–Sugeno (T-S) fuzzy model is used to approximate the nonlinear system. The developed scheme achieves periodic reference tracking and improves the performance of periodic and aperiodic unknown disturbances rejection effectiveley. This can be operated by incorporating the equivalent-input-disturbance (EID) estimator with the modified repetitive controller (MRC) scheme. Moreover, a fuzzy periodic event-triggered feedback observer (FPETFO) is proposed for the purpose of reducing the computational burden, energy consumption and saving communication resources. The periodic event-triggered technique is designed to observe the occurrence of an event which is described by an error signal. When this error signal exceeds a prescribed threshold, the event occurs and the current data are transmitted; otherwise, there is a zero-order hold to keep data unchanged. The overall system consists of MRC, EID and FPETFO based on a T-S fuzzy model. Then, some sufficient conditions are derived to gurantee the asymptotic stability of the overall system subjected to unknown disturbances using the Lyapunov–Krasovskii functional (LKF) stability theory and linear matrix inequalities (LMIs). The fuzzy state feedback controller and observer gains are designed using the LMI and matrix decomposition approaches. Simulation results illustrate the effectiveness and feasibility of the proposed scheme with comparative study.

Estimates for Solutions to a Class of Nonlinear Time-Varying Delay Systems

Estimates for Solutions to a Class of Nonlinear Time-Varying Delay Systems

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.

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.

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.

Pole assignment stabilization for a class of switched nonlinear time-varying delay systems

This paper investigates the pole assignment stabilization with memory state feedback controllers for switched nonlinear time-varying delay systems. More general time-varying delays which depend on the subsystem number are considered. Based on novel common Lyapunov functions (CLFs), the aggregation techniques, the proprieties and the Borne–Gentina criterion, new algebraic stabilization criteria are established. The proposed design can guarantee the stability of the closed-loop systems under arbitrary switching. It allows to avoid researching a common Lyapunov function (CLF), considered a difficult matter. Four illustrate examples were finally utilized to exhibit the effectiveness of the obtained results.

Adaptive Fuzzy Fault-Tolerant Tracking Control of Uncertain Nonlinear Time-Varying Delay Systems

This paper investigates the problem of fuzzy-approximation-based adaptive fault-tolerant (FT) tracking control for uncertain nonlinear time-varying delay systems with nonaffine nonlinear faults. Compared with the existing results, the proposed control scheme avoids the use of the Lyapunov–Krasovskii functional, which relaxes the restriction on the derivative of unknown time-varying delay ( ${\dot \tau (t) ). By utilizing the finite covering lemma and fuzzy logic systems (FLSs), the considered time-varying delay system is transformed into novel switched nonlinear system with known constant delays. In addition, the nonaffine nonlinear faults make the control design more complicated. To overcome the difficulty, the fault function is decoupled by utilizing the mean-value theorem. In the design procedure, the problem of unknown control gain and control direction is produced. To solve this problem, the Nussbaum-type function is introduced. Combined backstepping technique with FLSs, an adaptive approximation FT tracking control scheme is proposed. Finally, the developed control method ensures that all the closed-loop signals are uniformly ultimately bounded and the tracking error can converge to a small neighborhood of the origin. A simulation example is presented to illustrate the effectiveness of the developed control scheme.

Tracking control and dynamic regulation of time-varying delay nonlinear systems with actuator saturation via multi-dimensional Taylor networks

For desirable tracking control and transient optimization of time-varying nonlinear delay systems with actuator saturation, a controller with a fixed structure, i.e., multi-dimensional Taylor networks (MTN) controller, is proposed here. Based on the extension of the Lie derivative, a series of assumptions for most nonlinear time-varying delays systems are identified. Based on the filter variable and Lyapunov–Razumikhin theorem, it is proven that the MTN controller can track the reference satisfactorily, which conforms to the assumption. Furthermore, the dynamic regulation to improve system transient response using the MTN controller is realized through polynomial approximation. Parameters of the controller are determined by means of the appropriate optimization method, and its general design is explored and detailed. As verified by the results from two numerical examples, the proposed approach is feasible and effective. Depending on the conclusion of the interconnected control, the application of MTN controller in MIMO time-varying delays nonlinear system with actuator saturation is discussed.

Exponential stability of solutions to nonlinear time-varying delay systems of neutral type equations with periodic coefficients

We consider a class of nonlinear time-varying delay systems of neutral type differential equations with periodic coefficients in the linear terms, $$\begin{aligned} \frac{d}{dt} y(t) &= A(t) y(t) + B(t) y(t-\tau(t)) + C(t) \frac{d}{dt} y(t-\tau(t)) \cr &\quad + F\Big(t, y(t), y(t-\tau(t)), \frac{d}{dt} y(t-\tau(t)) \Big), \end{aligned}$$ where A(t), B(t), C(t) are T-periodic matrices, and $$ \|F(t,u,v,w)\| \le q_1\|u\| + q_2\|v\| + q_3 \|w\|, \quad q_1, q_2, q_3 \ge 0, \quad t > 0. $$ We obtain conditions for the exponential stability of the zero solution and estimates for the exponential decay of the solutions at infinity.
 For more information see https://ejde.math.txstate.edu/Volumes/2020/20/abstr.html