Time-delay Systems Research Articles

Non-singular control strategies for a class of strict-feedback nonlinear time-delay systems

This paper focuses on the non-singular control for strict-feedback nonlinear time-delay systems. The boundedness of all the signals of the closed-loop system is rigorously proved by introducing a class of Lyapunov-Krasovskii functionals, including exponential functions. Besides, novel auxiliary functions with continuous derivatives are proposed to construct the control law to avoid the discontinuity of the virtual controllers. Besides, the amount of computation is reduced by applying the new auxiliary functions when the other control methods require loads of calculations for handling singular problems. In the end, numerical examples are given to validate the effectiveness of the proposed control scheme.

Hidden Markov model–based asynchronous fault-tolerant control for mode-dependent time-delayed Markovian jump systems with uncertain conditional probability

In control engineering, time-delayed systems are quite common. Modern industrial systems often involve delays due to complicated network environments, the accuracy of sensors and actuators, or signal processing. Speaking of Markovian jump systems, such systems undergo discrete mode changes caused by a Markov process. Hence, it is reasonable to consider the mode-dependent time delays. In this paper, the asynchronous fault-tolerant controller is developed for mode-dependent time-delayed Markovian jump systems. First, the estimations of states and actuator fault are provided by designing an intermediate observer. Then, based on these estimations, the asynchronous fault-tolerant controller is designed by virtue of the hidden Markov process with uncertain conditional probability. Finally, the existence conditions of both the designed observer and fault-tolerant controller are given in terms of linear matrix inequalities. In the meantime, a numerical example is illustrated to validate the effectiveness and potential of the proposed method.

Numerical method for second order singularly perturbed delay differential equations with fractional order in time via fitted computational method

The numerical solution of time fractional parabolic differential equations with singular perturbations and delay is the subject of this article. An arbitrarily small perturbation parameter ɛ(0<ɛ<<1) is multiplied the highest order derivative term. The problem’s solution shows an exponential boundary layer on the right side of the spatial domain for small values of ɛ. The solution and its derivatives are discussed together with their properties and bounds. The Crank–Nicolson approach for time discretization and the fitted operator non-standard finite difference method for space discretization are used to solve the time fractional singularly perturbed time delay problem under consideration. The comparison principle and solution bound are used to examine and analyze the scheme’s stability. The scheme’s uniform convergence is examined and demonstrated. The proposed scheme has a uniform convergence order of Mx−1+(Δt)2−γ. Two numerical illustrations for various values of ɛ are taken into consideration to validate the theoretical analysis of the scheme. The constructed scheme provides a better order of convergence and more accurate results than methods found in the literature.

Dissipative control for linear time-delay systems based a modified equivalent-input-disturbance approach

Dissipative control for linear time-delay systems based a modified equivalent-input-disturbance approach

Periodic Solutions in a Simple Delay Differential Equation

A simple-form scalar differential equation with delay and nonlinear negative periodic feedback is considered. The existence of several types of slowly oscillating periodic solutions is shown with the same and double periods of the feedback coefficient. The periodic solutions are built explicitly in the case with piecewise constant nonlinearities involved. The periodic dynamics are shown to persist under small perturbations of the equation, which make it smooth. The theoretical results are verified through extensive numerical simulations.

Time-Delay Effects on the Collective Resonant Behavior in Two Coupled Fractional Oscillators with Frequency Fluctuations

In this study, we propose coupled time-delayed fractional oscillators with dichotomous fluctuating frequencies and investigate the collective resonant behavior. Firstly, we obtain the condition of complete synchronization between the average behavior of the two oscillators. Subsequently, we derive the precise analytical expression of the output amplitude gain. Based on the analytical results, we observe the collective resonant behavior of the coupled time-delayed system and further study its dependence on various system parameters. The observed results underscore that the coupling strength, fractional order, and time delay play significant roles in controlling the collective resonant behavior by facilitating the occurrence and optimizing the intensity. Finally, numerical simulations are also conducted and verify the accuracy of the analytical results.

Postprocessing technique of the discontinuous Galerkin method for solving delay differential equations

Postprocessing technique of the discontinuous Galerkin method for solving delay differential equations

Stabilization and destabilisation of non-autonomous stochastic nonlinear delay differential equations

This paper focuses on a class of non-autonomous stochastic nonlinear delay differential equations, where the time delay functions are no longer required to be bounded or differentiable. The key aim is to study the stabilisation and destabilisation of the non-autonomous nonlinear stochastic delay differential equations by using the function ln ⁡ | x ( t ) | 2 , the improved LaSalle-type theorem, and non-negative semimartingale convergence theorem. In comparison with recent works on stabilisation for stochastic time-varying delay nonlinear systems (see, e.g. [Dong, H., & Mao, X. (2022). Advances in stabilization of highly nonlinear hybrid delay systems. Automatica, 136, 110086. https://doi.org/10.1016/j.automatica.2021.110086; Hu, J., Mao,W., & Mao, X. (2023). Advances in nonlinear hybrid stochastic differential delay equations: Existence, boundness and stability. Automatica, 147, 110682. https://doi.org/10.1016/j.automatica.2022.110682; Xu, H., & Mao, X. (2023). Improved delay-dependent stability of superlinear hybrid stochastic systems with general time-varying delays. Nonlinear Analysis: Hybrid Systems, 50, 101413. https://doi.org/10.1016/j.nahs.2023.101413]), the proposed results exhibit significant improvements and can be applied to a broader class of non-autonomous equations.

Stochastic analysis and vibration suppression of a time-delayed system with nonlinear energy sink

Nonlinear energy sink (NES) is of great significance in attenuating structural vibration. Time-delay phenomenon in a mechanical system is inevitable. Different from existing works, the influences of time-delayed factors in a nonlinear oscillator attached with NES are considered in this paper under Gaussian white noise (GWN). Firstly, the time-delayed term is transformed into a linear combination of stiffness and damping terms, and an equivalent system without the time-delayed term is obtained. Stationary probability density function (SPDF) is deduced by leveraging stochastic averaging method. Secondly, good agreement is observed between analytical and numerical results, which verifies the effectiveness of the method. Furthermore, bifurcation conditions for the amplitude of SPDF of the primary system are obtained, indicating that delayed time, feedback gain and damping coefficients can cause stochastic P-bifurcation of the primary system. Finally, the vibration suppression performance of NES is analyzed via energy variation and the time–history diagram of displacement.

Application of generalized Haar wavelet technique on simultaneous delay differential equations

In this paper, the Haar wavelet technique is applied to a system of simultaneous linear and nonlinear proportional delay differential equations. By implementing this method, the delay differential equations are transformed into a series of linear and nonlinear algebraic equations, respectively. These equations are then solved using numerical code constructed in MATLAB. The obtained approximate solutions are compared with exact solutions, demonstrating the accuracy and efficacy of the Haar wavelet method. Using the fixed point theorem, the existence and uniqueness of the solutions is established. The reliability and efficiency of the proposed technique are illustrated through two numerical examples. Additionally, a comprehensive error analysis is presented and the convergence result is discussed, providing a thorough examination of the robustness of the method. This study not only shows that the Haar wavelet technique is a powerful tool for solving delay differential equations but also sets the groundwork for future research in this area.

Quantized impact of rod on plate

In this work, we attacked the problem of elastic impact of a rod on a plate. By considering wave propagation on the plate and along the rod, we established a mathematic model with delay differential equation that governs such impact. The proposed model could consolidate classical impact problems from bead-on-wall, bead-on-plate, to rod-on-wall impact. Solving the governing equation, we obtained a universal loading and unloading history of such impact: the unloading history exhibits quantized plateaus, the values of which are quantitatively provided in the present study. Also, the contact duration also shows a band structure while tuning the compliance of plate. In addition, we analyzed the overshoots leading the plateaus during unloading, which cause the final detachment between the rod and the plate.

Periodic solutions in reversible symmetric second order systems with multiple distributed delays

In this paper, we study a class of nonlinear differential equations with multiple distributed delays. We reduced the problem of search for periodic solutions of the distributed delay differential equations to that of finding zeros of an operator on some Banach space. Utilizing the equivariant degree theory, we obtain some sufficient conditions to guarantee the existence of multiplicity of periodic solutions. Also, we give some information about the symmetry of the periodic solutions. At last, we give an example to illustrate our result.

Dominant subspaces of high-fidelity polynomial structured parametric dynamical systems and model reduction

In this work, we investigate a model order reduction scheme for high-fidelity nonlinear structured parametric dynamical systems. More specifically, we consider a class of nonlinear dynamical systems whose nonlinear terms are polynomial functions, and the linear part corresponds to a linear structured model, such as second-order, time-delay, or fractional-order systems. Our approach relies on the Volterra series representation of these dynamical systems. Using this representation, we identify the kernels and, thus, the generalized multivariate transfer functions associated with these systems. Consequently, we present results allowing the construction of reduced-order models whose generalized transfer functions interpolate these of the original system at pre-defined frequency points. For efficient calculations, we also need the concept of a symmetric Kronecker product representation of a tensor and derive particular properties of them. Moreover, we propose an algorithm that extracts dominant subspaces from the prescribed interpolation conditions. This allows the construction of reduced-order models that preserve the structure. We also extend these results to parametric systems and a special case (delay in input/output). We demonstrate the efficiency of the proposed method by means of various numerical benchmarks.

Basin entropy as an indicator of a bifurcation in a time-delayed system.

The basin entropy is a measure that quantifies, in a system that has two or more attractors, the predictability of a final state, as a function of the initial conditions. While the basin entropy has been demonstrated on a variety of multistable dynamical systems, to the best of our knowledge, it has not yet been tested in systems with a time delay, whose phase space is infinite dimensional because the initial conditions are functions defined in a time interval [-τ,0], where τ is the delay time. Here, we consider a simple time-delayed system consisting of a bistable system with a linear delayed feedback term. We show that the basin entropy captures relevant properties of the basins of attraction of the two coexisting attractors. Moreover, we show that the basin entropy can give an indication of the proximity of a Hopf bifurcation, but fails to capture the proximity of a pitchfork bifurcation. The Hopf bifurcation is detected because before the fixed points become unstable, a oscillatory, limit-cycle behavior appears that coexists with the fixed points. The new limit cycle modifies the structure of the basins of attraction, and this change is captured by basin entropy that reaches a maximum before the Hopf bifurcation. In contrast, the pitchfork bifurcation is not detected because the basins of attraction do not change as the bifurcation is approached. Our results suggest that the basin entropy can yield useful insights into the long-term predictability of time-delayed systems, which often have coexisting attractors.

Stability analysis and stabilization control of discrete-time impulsive switched time-delay systems with all unstable subsystems

n this paper, stability analysis and stabilization control of discrete-time impulsive switched time-delay systems with all unstable subsystems are discussed. By utilizing a switching time-varying Lyapunov–Krasovskii functional and the mode-dependent interval dwell-time switching rule, we derive some more general stability theorems for the considered time-delay system with all subsystems being unstable. Moreover, we design a time-varying state feedback controller to ensure the stabilization of the resulting closed-loop system. Eventually, the theoretical findings are demonstrated utilizing numerical examples.

Decentralized Implicit Inverse Control for Large-Scale Hysteretic Nonlinear Time-Delay Systems and Its Application on Triple-Axis Giant Magnetostrictive Actuators.

This article proposes fuzzy-logic systems (FLSs)-based decentralized adaptive implicit inverse control scheme for a class of large-scale nonlinear systems with time delays and multihysteretic loops. Our novel algorithms feature hysteretic implicit inverse compensators designed to effectively mitigate multihysteretic loops in large-scale systems. In this article, hysteretic implicit inverse compensators can replace the traditional hysteretic inverse models, which are exceedingly difficult to construct, and no longer necessary. The authors provide three contributions: 1) a searching mechanism to obtain the approximate value of the practical input signal from the so-called hysteretic temporary control law; 2) the arbitrarily small L∞ norm of the tracking error attained by utilizing the proposed initializing technique, which applies the combination of FLSs and a finite covering lemma to deal with time delays; and 3) the construction of a triple-axis giant magnetostrictive motion control platform, which validates the effectiveness of the proposed control scheme and algorithms.

A New Grey Wolf Optimizer Tuned Extended Generalized Predictive Control for Distillation Process.

The distillation process plays an essential role in the petrochemical industry. However, the high-purity distillation column has complicated dynamic characteristics such as strong coupling and large time delay. To control the distillation column accurately, we proposed an extended generalized predictive control (EGPC) method inspired by the principles of extended state observer and proportional-integral-type generalized predictive control method; the proposed EGPC can adaptively compensate the system for the effects of coupling and model mismatch online and performs well in controlling time-delay systems. The strong coupling of the distillation column needs fast control, and the large time delay requires soft control. To balance the requirement for fast and soft control at the same time, a grey wolf optimizer with reverse learning and adaptive leaders number strategies (RAGWO) was proposed to tune the parameters of EGPC, and these strategies enable RAGWO to have a better initial population and improve its exploitation and exploration ability. The benchmark test results indicate that the RAGWO outperforms the existing optimizers for most of the selected benchmark functions. Extensive simulations show that the proposed method in terms of fluctuation and response time is superior to other methods for controlling the distillation process.

Output Feedback Stabilization of High-Order Nonlinear Time-Delay Systems With Low-Order and High-Order Nonlinearities

In this paper, the output feedback stabilization of a class of high-order nonlinear time-delay systems with more general low-order and high-order nonlinearities is investigated. By constructing the new Lyapunov-Krasovskii functional and reduced-order observer, based on homogeneous domination theory together with the adding a power integrator method, an output feedback controller is developed to guarantee the equilibrium of the closed system globally uniformly asymptotically stable.

Spiking Neural Membrane Systems with Adaptive Synaptic Time Delay.

Spiking neural membrane systems (or spiking neural P systems, SNP systems) are a new type of computation model which have attracted the attention of plentiful scholars for parallelism, time encoding, interpretability and extensibility. The original SNP systems only consider the time delay caused by the execution of rules within neurons, but not caused by the transmission of spikes via synapses between neurons and its adaptive adjustment. In view of the importance of time delay for SNP systems, which are a time encoding computation model, this study proposes SNP systems with adaptive synaptic time delay (ADSNP systems) based on the dynamic regulation mechanism of synaptic transmission delay in neural systems. In ADSNP systems, besides neurons, astrocytes that can generate adenosine triphosphate (ATP) are introduced. After receiving spikes, astrocytes convert spikes into ATP and send ATP to the synapses controlled by them to change the synaptic time delays. The Turing universality of ADSNP systems in number generating and accepting modes is proved. In addition, a small universal ADSNP system using 93 neurons and astrocytes is given. The superiority of the ADSNP system is demonstrated by comparison with the six variants. Finally, an ADSNP system is constructed for credit card fraud detection, which verifies the feasibility of the ADSNP system for solving real-world problems. By considering the adaptive synaptic delay, ADSNP systems better restore the process of information transmission in biological neural networks, and enhance the adaptability of SNP systems, making the control of time more accurate.

Novel Robust Estimation-Based Control of One-Sided Lipschitz Nonlinear Systems Subject to Output and Input Delays

This paper highlights the design of a controller established on estimated states for one-sided Lipschitz (OSL) nonlinear systems subject to output and input delays. The controller has been devised by involving Luenberger-like estimated states. The stability of the time-delayed nonlinear system is reckoned by assuming a Lyapunov functional for delayed dynamics and for which a delay-range dependent criterion is posed with a delay ranging between known upper and lower bounds. The time derivative of the functional is further exploited with linear matrix inequality (LMI) procedures, and employing Wirtinger’s inequality for the integral terms instead of the traditional and more conservative Jensen’s condition. Moreover, a sufficient and necessary solution is derived for the proposed design by involving the tedious decoupling technique to attain controller and observer gain simultaneously. The proposed methodology validates the observer error stability between observers and states asymptotically. The solution of matrix inequalities was obtained by employing cone-complementary linearization techniques to solve the tiresome constraints through simulation tools by convex optimization. Additionally, a novel scheme of an observer-based controller for the linear counterpart is also derived for one-sided Lipschitz nonlinear systems with multiple delays. Finally, the effectualness of the presented observer-based controller under input and output delays for one-sided Lipschitz nonlinear systems is validated by considering a numerical simulation of mobile systems in Cartesian coordinates.