Delayed state feedback and chaos control for time-periodic systems via a symbolic approach

This paper presents a symbolic method for a delayed state feedback controller (DSFC) design for linear time-periodic delay (LTPD) systems that are open loop unstable and its extension to incorporate regulation and tracking of nonlinear time-periodic delay (NTPD) systems exhibiting chaos. By using shifted Chebyshev polynomials, the closed loop monodromy matrix of the LTPD system (or the linearized error dynamics of the NTPD system) is obtained symbolically in terms of controller parameters. The symbolic closed loop monodromy matrix, which is a finite dimensional approximation of an infinite dimensional operator, is used in conjunction with the Routh–Hurwitz criterion to design a DSFC to asymptotically stabilize the unstable dynamic system. Two controllers designs are presented. The first design is a constant gain DSFC and the second one is a periodic gain DSFC. The periodic gain DSFC has a larger region of stability in the parameter space than the constant gain DSFC. The asymptotic stability of the LTPD system obtained by the proposed method is illustrated by asymptotically stabilizing an open loop unstable delayed Mathieu equation. Control of a chaotic nonlinear system to any desired periodic orbit is achieved by rendering asymptotic stability to the error dynamics system. To accommodate large initial conditions, an open loop controller is also designed. This open loop controller is used first to control the error trajectories close to zero states and then the DSFC is switched on to achieve asymptotic stability of error states and consequently tracking of the original system states. The methodology is illustrated by two examples.     

The effect of time delay on the dynamics of an SEIR model with nonlinear incidence

In this paper, a SEIR epidemic model with nonlinear incidence rate and time delay is investigated in three cases. The local stability of an endemic equilibrium and a disease-free equilibrium are discussed using stability theory of delay differential equations. The conditions that guarantee the asymptotic stability of corresponding steady-states are investigated. The results show that the introduction of a time delay in the transmission term can destabilize the system and periodic solutions can arise through Hopf bifurcation when using the time delay as a bifurcation parameter. Applying the normal form theory and center manifold argument, the explicit formulas determining the properties of the bifurcating periodic solution are derived. In addition, the effect of the inhibitory effect on the properties of the bifurcating periodic solutions is studied. Numerical simulations are provided in order to illustrate the theoretical results and to gain further insight into the behaviors of delayed systems.     

Dynamics of the density dependent predator-prey system with Beddington-DeAngelis functional response

Two models of a density dependent predator-prey system with Beddington-DeAngelis functional response are systematically considered. One includes the time delay in the functional response and the other does not. The explorations involve the permanence, local asymptotic stability and global asymptotic stability of the positive equilibrium for the models by using stability theory of differential equations and Lyapunov functions. For the permanence, the density dependence for predators is shown to give some negative effect for the two models. Further the permanence implies the local asymptotic stability for a positive equilibrium point of the model without delay. Also the global asymptotic stability condition, which can be easily checked for the model is obtained. For the model with time delay, local and global asymptotic stability conditions are obtained.     

Stochastic stability of Duffing–Mathieu system with delayed feedback control under white noise excitation

The asymptotic Lyapunov stability with probability one of Duffing–Mathieu system with time-delayed feedback control under white-noise parametric excitation is studied. First, the time-delayed feedback control force is expressed approximately in terms of the system state variables without time delay. Then, the averaged Itô stochastic differential equations for the system are derived by using the stochastic averaging method and the expression for the Lyapunov exponent of the linearized averaged Itô equations is derived. Finally, the effects of time delay in feedback control on the Lyapunov exponent and the stability of the system are analyzed. Meanwhile, the stability conditions for the system with different time delays are also obtained. The theoretical results are well verified through digital simulation.     

一类不确定长时延网络控制系统的H∞状态反馈控制

本文针对一类不确定的长时延网络控制系统，在系统方程右端加上扰动项，利用李亚普诺夫函数，借助线性矩阵不等式，设计出了使系统具有H∞性格指标的闭环系统渐近稳定充分条件，并设计出了状态反馈控制器，仿真实例表明结论具有可行性与有效性。     

Stability and chaos control of regularized Prabhakar fractional dynamical systems without and with delay

In this paper, we studied the stabilization of nonlinear regularized Prabhakar fractional dynamical systems without and with time delay. We establish a Lyapunov stabiliy theorem for these systems and study the asymptotic stability of these systems without design a positive definite function V (without considering the fractional derivative of function V is negative). We design a linear feedback controller to control and stabilize the nonautonomous and autonomous chaotic regularized Prabhakar fractional dynamical systems without and with time delay. By means of the Lyapunov stability, we obtain the control parameters for these type of systems. We further present a numerical method to solve and analyze regularized Prabhakar fractional systems. Furthermore, by employing numerical simulation, we reveal chaotic attractors and asymptotic stability behaviors for four systems to illustrate the presented theorem.     

Some new results on an allelopathic competition model with quorum sensing and delayed toxicant production

A 4-equation delay differential system representing a bacterial allelopathic competition is analyzed. A distributed delay term models a linear quorum-sensing mechanism which regulates the delayed allelochemicals’ production process. The proved qualitative properties of the solutions are positivity, boundedness, global existence in the future, and uniqueness. Sufficient conditions for local asymptotic stability properties of biologically meaningful steady-state solutions are given in terms of the parameters of the system. The global asymptotic stability of a biologically meaningful steady-state solution is proved by constructing a suitable Lyapunov functional.     

Terminal constrained robust hybrid iterative learning model predictive control for complex time-delayed batch processes

This work mainly addresses terminal constrained robust hybrid iterative learning model predictive control against time delay and uncertainties in a class of complex batch processes with input and output constraints. In this work, an equivalently novel extended two-dimensional switched system is first constructed to represent the process model by introducing state difference, output error and new relaxation variable information. Then, a hybrid predictive updating controller is proposed and an optimal performance index function including terminal constraints is designed. Under the condition that the switching signal meets certain conditions, the solvable problem of model predictive control is realized by Lyapunov stability theory. Meanwhile, the design scheme of controller parameters is also given. In addition, the robust constraint set is adopted to overcome the disadvantage that the traditional asymptotic stability cannot converge to the origin when it involves disturbances, such that the system state converges to the constraint set and meets its expected value. Finally, the effectiveness of the proposed algorithm is verified by controlling the speed and pressure parameters of the injection molding process.     

Stability switches and bifurcation in a two-degrees-of-freedom nonlinear quarter-car with small time-delayed feedback control

In this paper, we investigate a two-degrees-of-freedom nonlinear quarter-car model with time-delayed feedback control. It is well known that a time delay has destabilizing effects in mathematical models. However, delays are not necessarily destabilizing. In this work we explore a system where a time delay can be both stabilizing and destabilizing. Using the generalized Sturm criterion, the critical control gain for the delay-independent stability region and critical time delays for stability switches are derived. It is shown that there is a small parameter region for delay-independently stability of the system. Once the controlled system with time delay is not delay-independently stable, the system may undergo stability switches with the variation of the time delay. These stability switches correspond to Hopf bifurcations that occur when the time delays cross critical values. Properties of Hopf bifurcation such as direction and stability of bifurcating periodic solutions are determined by using the normal form theory and centre manifold theorem. Numerical simulations are provided to support the theoretical analysis. The critical conditions can provide a theoretical guidance for the design of vehicles with significant reduction of vibration in order to increase passengers ride comfort.     

Non-fragile observer-based passive control for uncertain time delay systems subjected to input nonlinearity

The problem of non-fragile observer-based passive control for uncertain time delay systems subjected to input nonlinearity is investigated by using sliding mode control. A novel control law is established such that the sliding surface in the state-estimation space can be reached in a finite time and chattering reduction is obtained. A sufficient condition for passivity and asymptotic stability of the combined system is derived via linear matrix inequality (LMI). Finally, a simulation example is presented to show the validity and advantages of the proposed method.     

Effects of time delays on bifurcation and chaos in a non-autonomous system with multiple time delays

Time delays are often sources of complex behavior in dynamic systems. Yet its complexity needs to be further explored, particularly when multiple time delays are present. As a purpose to gain insight into such complexity under multiple time delays, we investigate the mechanism for the action of multiple time delays on a particular non-autonomous system in this paper. The original mathematical model under consideration is a Duffing oscillator with harmonic excitation. A delayed system is obtained by adding delayed feedbacks to the original system. Two time delays are involved in such system, one of which in the displacement feedback and the other in the velocity feedback. The time delays are taken as adjustable parameters to study their effects on the dynamics of the system. Firstly, the stability of the trivial equilibrium of the linearized system is discussed and the condition under which the equilibrium loses its stability is obtained. This leads to a critical stability boundary where Hopf bifurcation or double Hopf bifurcation may occur. Then, the chaotic behavior of such system is investigated in detail. Particular emphasis is laid on the effect of delay difference between two time delays on the chaotic properties. A Melnikov’s analysis is employed to obtain the necessary condition for onset of chaos resulting from homoclinic bifurcation. And numerical analyses via the bifurcation diagram and the top Lyapunov exponent are carried out to show the actual time delay effect. Both the results obtained by the two analyses show that the delay difference between two time delays plays a very important role in inducing or suppressing chaos, so that it can be taken as a simple but efficient “switch” to control the motion of a system: either from order to chaos or from chaos to order.     

Effect of time delay in an autotroph-herbivore system with nutrient cycling

In the present study we consider a mathematical model of a non-interactive type autotroph-herbivore system in which the amount of autotroph biomass consumed by the herbivore is assumed to follow a Holling type II functional response. We have also incorporated discrete time delays in the numerical response term to represent a delay due to gestation, and in the recycling term which represents the time required for bacterial decomposition. We have derived conditions for global asymptotic stability of the model in the absence of delays. Conditions for delay-induced asymptotic stability of the steady state are also derived. The length of the delay preserving stability has been estimated and interpreted ecologically.     

An adaptive robust controller for time delay maglev transportation systems

For engineering systems, uncertainties and time delays are two important issues that must be considered in control design. Uncertainties are often encountered in various dynamical systems due to modeling errors, measurement noises, linearization and approximations. Time delays have always been among the most difficult problems encountered in process control. In practical applications of feedback control, time delay arises frequently and can severely degrade closed-loop system performance and in some cases, drives the system to instability. Therefore, stability analysis and controller synthesis for uncertain nonlinear time-delay systems are important both in theory and in practice and many analytical techniques have been developed using delay-dependent Lyapunov function. In the past decade the magnetic and levitation (maglev) transportation system as a new system with high functionality has been the focus of numerous studies. However, maglev transportation systems are highly nonlinear and thus designing controller for those are challenging. The main topic of this paper is to design an adaptive robust controller for maglev transportation systems with time-delay, parametric uncertainties and external disturbances. In this paper, an adaptive robust control (ARC) is designed for this purpose. It should be noted that the adaptive gain is derived from Lyapunov–Krasovskii synthesis method, therefore asymptotic stability is guaranteed.     

On Delay-Dependent Global Asymptotic Stability for Pendulum-Like Systems

This paper deals with global asymptotic stability for the delayed nonlinear pendulum-like systems with polytopic uncertainties. The delay-dependent criteria, guaranteeing the global asymptotic stability for the pendulum-like systems with state delay for the first time, are established in terms of linear matrix inequalities (LMIs) which can be checked by resorting to recently developed algorithms solving LMIs. Furthermore, based on the derived delay-dependent global asymptotic stability results, LMI characterizations are developed to ensure the robust global asymptotic stability for delayed pendulum-like systems under convex polytopic uncertainties. The new extended LMIs do not involve the product of the Lyapunov matrix and the system matrices. It enables one to check the global asymptotic stability by using parameter-dependent Lyapunov methods. Finally, a concrete application to phase-locked loop (PLL) shows the validity of the proposed approach.     

Application of nonlocal continuum theory to the primary resonance analysis of an axially loaded nano beam under time delay control

The paper is concerned with the nonlinear primary resonance of nano beams with axial load under the velocity time delay control. In order to have a deep insight into the system, the amplitude frequency response curve of the system is firstly obtained using the multiple scales method. The effects of the control gains and time delays on the system stability are then investigated. The analyses illustrated that both delay feedback gain coefficient and velocity time delay control can mitigate the system vibrations properties (e.g. hardening nonlinearity, resonance amplitude and the corresponding width) to an excellent level. The nonlinear primary resonance of nano beam is also discussed with the influences of small scale effect, axial initial load, wave number, Winkler foundation modulus and the ratio of the length to the diameter. This paper establishes the relationship between the time delay controller and vibration properties of a nano beam system which provides the guidance for applying time delayed active control for different types of nano devices in engineering practices.     

Sampled‐data reliable stabilization of T‐S fuzzy systems and its application

In this article, based on sampled‐data approach, a new robust state feedback reliable controller design for a class of Takagi–Sugeno fuzzy systems is presented. Different from the existing fault models for reliable controller, a novel generalized actuator fault model is proposed. In particular, the implemented fault model consists of both linear and nonlinear components. Consequently, by employing input‐delay approach, the sampled‐data system is equivalently transformed into a continuous‐time system with a variable time delay. The main objective is to design a suitable reliable sampled‐data state feedback controller guaranteeing the asymptotic stability of the resulting closed‐loop fuzzy system. For this purpose, using Lyapunov stability theory together with Wirtinger‐based double integral inequality, some new delay‐dependent stabilization conditions in terms of linear matrix inequalities are established to determine the underlying system's stability and to achieve the desired control performance. Finally, to show the advantages and effectiveness of the developed control method, numerical simulations are carried out on two practical models. © 2016 Wiley Periodicals, Inc. Complexity 21: 518–529, 2016     

一类具有非线性扩散和时滞的捕食系统的持续性与周期解

研究了一类具有非线性扩散和Beddington-Deangelis功能性反应,且同时具有连续时滞和离散时滞的非自治两食饵一捕食者系统,证明了在适当条件下该系统是一致持久的,并且得到了系统正周期解全局渐近稳定的充分条件.最后,给出一个例子以说明得到的结果.     

Controlling the nonlinear dynamics of a beam system

Control based on linear error feedback is applied to reduce vibration amplitudes in a piecewise linear beam system. Hereto small amplitude 1-periodic solutions are stabilized wherever they coexist with two or more long-term solutions. In theory, no control effort is required to maintain the 1-periodic response once it has been stabilized. For the beam system, 1-periodic solutions are stabilized by feedback at one location along the beam. Feedback is represented by servo-stiffness or servo-damping which results from increasing two corresponding control parameters. At appropriate levels of these parameters local, or global, asymptotic stability (of the zero-equilibrium) of the error dynamics, i.e. stability of the underlying 1-periodic solutions, can be guaranteed. Local asymptotic stability can be guaranteed for a large range of actuator locations and excitation frequencies and is indicated by bifurcations. Global asymptotic stability can only be guaranteed for a limited range of actuator locations on the basis of the well-known circle criterion. The difference between local and global asymptotic stability in terms of the required values for the control parameters can be significant, and may result in large differences in control performance.     

On the asymptotic stability of solutions of nonlinear systems with delay

Under study are systems of homogeneous differential equations with delay. We assume that in the absence of delay the trivial solutions to the systems under consideration are asymptotically stable. Using the direct Lyapunov method and Razumikhin??s approach, we show that if the order of homogeneity of the right-hand sides is greater than 1 then asymptotic stability persists for all values of delay. We estimate the time of transitions, study the influence of perturbations on the stability of the trivial solution, and prove a theorem on the asymptotic stability of a complex system describing the interaction of two nonlinear subsystems.     

Analysis of a viral infection model with delayed immune response

It is well known that the immune response plays an important role in eliminating or controlling the disease after human body is infected by virus. In this paper, we investigate the dynamical behavior of a viral infection model with retarded immune response. The effect of time delay on stability of the equilibria of the system has been studied and sufficient condition for local asymptotic stability of the infected equilibrium and global asymptotic stability of the infection-free equilibrium and the immune-exhausted equilibrium are given. By numerical simulating,we observe that the stationary solution becomes unstable at some critical immune response time, while the delay time and birth rate of susceptible host cells increase, and we also discover the occurrence of stable periodic solutions and chaotic dynamical behavior. The results can be used to explain the complexity of the immune state of patients.