Topics in Time Delay Systems

With the development of science and industry the rapid increase in time-delay systems has prompted more and more researchers to use intermittent control theory to analyze and synthesize the stability of the time-delay systems subject to actuator saturation and random perturbation. This note investigates the development status and research significance of time-delay systems, Intermittent control and actuator saturation, and that by method of Intermittent control analysis and discussion of the stability of time-delay systems with actuator saturation is of great theoretical significance and practical application value.

The processes which contain at least one pole at the origin are known as integrating systems. The process output varies continuously with time at certain speed when they are disturbed from the equilibrium operating point by any environment disturbance/change in input conditions and thus they are considered as non-self-regulating. In most occasions this phenomenon is very disadvantageous and dangerous. Therefore it is always a challenging task to efficient control such kind of processes. Depending upon the number of poles present at the origin and also on the location of other poles in transfer function different types of integrating systems exist. Stable first order plus time delay systems with an integrator (FOPTDI), unstable first order plus time delay systems with an integrator (UFOPTDI), pure integrating plus time delay (PIPTD) systems and double integrating plus time delay (DIPTD) systems are the classifications of integrating systems. By using a well-controlled positioning stage the advances in micro and nano metrology are inevitable in order satisfy the need to maintain the product quality of miniaturized components. As proportional-integral-derivative (PID) controllers are very simple to tune, easy to understand and robust in control they are widely implemented in many of the chemical process industries. In industries this PID control is the most common control algorithm used and also this has been universally accepted in industrial control. In a wide range of operating conditions the popularity of PID controllers can be attributed partly to their robust performance and partly to their functional simplicity which allows engineers to operate them in a simple, straight forward manner. One of the accepted control algorithms by the process industries is the PID control. However, in order to accomplish high precision positioning performance and to build a robust controller tuning of the key parameters in a PID controller is most inevitable. Therefore, for PID controllers many tuning methods are proposed. the main factors that lead to lifetime reduction in gain loss of PID parameters are described in This paper and also the main methods used for gain tuning based on optimization approach analysis is reviewed. The advantages and disadvantages of each one are outlined and some future directions for research are analyzed.

In the space of system parameters, the closedform stability chart is determined for the delayed Mathieu equation defined as (t)(cost)x(t) bx(t2). This stability chart makes the connection between t...

We introduce two deterministic models of time-delay gene transcriptional regulation system, and research the properties of the two systems by computational modeling. The time-delay positive feedback system has multistability, gene switch and staircases. The platform of staircases is narrower with shorter delay, in contrast, the platform is wider with longer delay. Moreover, the system has memory property and echo phenomena emerging. These properties are caused by stimulation responding to time delay. On the other hand, the time-delay positive-negative feedback system can elicit oscillations and staircases. Bifurcation diagram demonstrates oscillation region enlarging and amplitude increasing compare with no time delay system, furthermore, the transcription rate can adjust oscillation amplitude, and the platforms width are in accord with delay times. Oscillations and staircases can indicate that delay increases instability.

The paper proposes a time-delayed hyperchaotic system composed of multiscroll attractors with multiple positive Lyapunov exponents (LEs), which are described by a three-order nonlinear retarded type delay differential equation (DDE). The dynamical characteristics of the time-delayed system are far more complicated than those of the original system without time delay. The three-order time-delayed system not only generates hyperchaotic attractors with multiscroll but also has multiple positive LEs. We observe that the number of positive LEs increases with increasing time delay. Through numerical simulations, the time-delayed system exhibits a larger number of scrolls than the original system without time delay. Moreover, different numbers of scrolls with variable delay and coexistence of multiple attractors with a variable number of scrolls are also observed in the time-delayed system. Finally, we setup electronic circuit of the proposed system, and make Pspice simulations to it. The Pspice simulation results agree well with the numerical results.

We introduce the delayed sine/cosine-type matrix function and use the Laplace transform method to obtain a closed form solution to IVP for a second-order time-delayed linear system with noncommutative matrices A and Ω. We also introduce a delay Gramian matrix and examine a relative controllability linear/semi-linear time delay system. We have obtained the necessary and sufficient condition for the relative controllability of the linear time-delayed second-order system. In addition, we have obtained sufficient conditions for the relative controllability of the semi-linear second-order time-delay system. Finally, we investigate the Ulam–Hyers stability of a second-order semi-linear time-delayed system.

Memristor, a controllable nonlinear element, is easy to generate a chaotic signal. More significantly, it can improve the complexity of the chaotic system and the randomness of signals. Although the memristor chaotic system is a hot spot of research currently, little attention has been paid to the memristive time-delayed chaotic system. Therefore, a new memristor-based time-delayed chaotic system is proposed in this paper. We construct the time-delayed chaotic system with single delay time by using the nonlinear relationship between the memristance and charge of memristor. The existence of time delay enhances the complexity of chaotic system, which makes the system produce richer and more complex dynamics. In order to study the complex dynamic characteristics of this memristive time-delayed system, we investigate the proposed system by theoretical derivation, numerical simulation, stabilization of equilibrium points, and power spectrum. In addition, the corresponding parameter region of the stable equilibrium point of the system is discussed in detail. Then, we discuss the effect of parameter variation on the dynamic behavior of the system, and a series of phase diagrams with different time-delayed parameters and system parameters is described by numerical simulation. We find that different combinations of parameters and slight changes of parameters can make the system a completely different phase diagram, which indicates that the proposed system has rich nonlinear characteristic. Moreover, the proposed time-delayed system is used to generate pseudo random sequences, and the experimental results show that the proposed system has good self-correlation, cross-correlation, and the significant approximate entropy. According to the theoretical analyses and experimental results, we conclude that the proposed new time-delayed chaotic system has complex dynamic behavior and good randomness, which can meet the needs of the applications in spread spectrum communication, image encryption and many other fields. This research provides a significant reference for further studying the usage of memristor.

Time-delay frequently occurs in many practical systems, such as chemical processes, manufacturing systems, long transmission lines, telecommunication and economic systems, etc. Since time-delay is a main source of instability and poor performance, the control problem of time-delay systems has received considerable attentions in literature, such as [1][9]. The design approaches adopt in these literatures can be divided into the delaydependent method [1]-[5] and the delay-independent method [6]-[9]. The delay-dependent method needs an exactly known delay, but the delay-independent method does not. In other words, the delay-independent method is more suitable for practical applications. Nevertheless, most literatures focus on linear time-delay systems due to the fact that the stability analysis developed in the two methods is usually based on linear matrix inequality techniques [10]. To deal with nonlinear time-delay systems, the Takagi-Sugeno (TS) fuzzy model-based approaches [11]-[12] extend the results of controlling linear time-delay systems to more general cases. In addition, some sliding-mode control (SMC) schemes have been applied to uncertain nonlinear time-delay systems in [13]-[15]. However, these SMC schemes still exist some limits as follows: i) specific form of the dynamical model and uncertainties [13]-[14]; ii) an exactly known delay time [15]; and iii) a complex gain design [13]-[15]. From the above, we are motivated to further improve SMC for nonlinear timedelay systems in the presence of matched and unmatched uncertainties. The fuzzy control and the neural network control have attractive features to keep the systems insensitive to the uncertainties, such that these two methods are usually used as a tool in control engineering. In the fuzzy control, the TS fuzzy model [16]-[18] provides an efficient and effective way to represent uncertain nonlinear systems and renders to some straightforward research based on linear control theory [11]-[12], [16]. On the other hand, the neural network has good capabilities in function approximation which is an indirect compensation of uncertainties. Recently, many fuzzy neural network (FNN) articles are proposed by combining the fuzzy concept and the configuration of neural network, e.g., [19]-[23]. There, the fuzzy logic system is constructed from a collection of fuzzy If-Then rules while the training algorithm adjusts adaptable parameters. Nevertheless, few results using FNN are proposed for time-delay nonlinear systems due to a large computational load and a vast amount of feedback data, for example, see [22]-[23]. Moreover, the training algorithm is difficultly found for time-delay systems.

A lot of studies of control highlight fractional calculus in modeling systems and designing controllers have been carried out. More recently, a lot of chaotic behaviors have been found in fractional-order systems. Then, controlling the fractional-order systems, especially controlling nonlinear fractional-order systems has become a hot research subject. The design of state estimators is one of the essential points in control theory. Time delays are often considered as the sources of complex behaviors in dynamical systems. A lot progress has been made in the research of time delay systems with real variables. In recent years, fractional-order time-delay chaotic synchronization and chaotic secure communication have received ever-increasing attention. In this paper we focus our study on the synchronization of fractional-order time-delay chaotic systems and its application in secure communication. Firstly, based on the Lipschitz condition, the nonlinear fractional-order time-delay system is proposed. Secondly, the fractional-order time-delay observer for the system is constructed. The necessary and sufficient conditions for the existence of the fractional-order observer are given by some lemmas. Thirdly, the synchronous controller is designed based on the state observer and the stability theory of fractional-order system. Instead of the state variables, the output variables of drive system and response system are used to design the synchronous controller, which makes the design much more simple and practical. With the Lyapunov stability theory and fractional order matrix inequalities, the method of how to obtain the parameters of the controller is presented. The sufficient conditions for asymptotical stability of the state error dynamical system are derived. After that, with the Chen fractional-order time-delay chaotic system, the synchronous controller is designed to make the system run synchronously. Finally, the proposed approach is then applied to secure communications, where the information signal is injected into the transmitter and simultaneously transmitted to the receiver. With the observer design technique, a chaotic receiver is then derived to recover the information signal at the receiving end of the communication. In the conventional chaotic masking method, the receiver is driven by the sum of the information signal and the output of the transmitter, whose dynamics is autonomous. The simulation results show that the design of the synchronous controller works effectively and efficiently, which implies that the proposed fractional order time-delay observer in this paper runs effectively. The proposed method is able to be applied to other fractional order time-delay chaos systems, and also to chaotic secure communication system.

In practical applications, system time delay (STD) of a VHF lightning mapping system, resulting from system manufacturing, installation, site differences, and signal channel inconsistency etc., is unavoidable and will affect the localization accuracy of the direction finding algorithm. In this article, a multiantenna VHF radiation continuous observation system (MARCOS) consisting of 7 channels is used as an example to make a quantitative analysis for the relationship between STD and localization error. The array is in an “L” shape, with a 9 m baseline. And its STD is obtained through actual measurement, using sweep-frequency method. The analysis results show that STD reduces the localization accuracy significantly. Especially, emissions with lower elevation are more vulnerable to STD. In order to correct localization error caused by STD, we utilize the actual measured STD of MARCOS to correct the error during the implementation of the algorithm. Field experiments are designed to verify the effectiveness of this correction method, with an unmanned aerial vehicle (UAV) and a portable radiation source. For MARCOS used in this article, the UAV experimental results show that the error caused by STD exceeds at least 2°, and even worse for the low elevation. However, after the STD is compensated, the localization error is alleviated within 1°. Finally, STD correction is applied to a triggered lightning data. Comparisons and analyses of the results are presented before and after STD compensation.

Many processes of Chemical Engineering importance are represented by unstable first/second order plus time delay systems for the design PID controllers. It is proposed to design PID controller with lead lag filter for unstable First Order Plus Time Delay (FOPTD), Second Order Plus Time Delay (SOPTD) systems using synthesis method. The novelty of the proposed work is the selection of desired closed loop transfer function. For FOPTD systems the closed loop transfer function is selected as third order with second order lead term with time delay and for the SOPTD systems the closed loop transfer function is selected as fourth order with second order lead term with time delay. The closed loop time delay is same as process time delay. Simulation results on various transfer function models and on the non-linear model equations of Jacketed CSTR carrying out first order exothermic reaction show that the controllers designed by the proposed method perform better than the recently reported methods. Robustness of the controller is related to maximum magnitude of the sensitivity function. The proposed controllers are robust to parametric uncertainties as they give low Ms value compared to the recently reported methods.

Self-synchronization of time delay implies that the synchronization between the time-delay system and the original system keeps the structure and parameters of systems unchanged, thus these various problems produced by time-delay in practice are avoided. Taking a time-delay complex Lorenz system for example, we investigate its dynamic characteristics and the influence of of time lag factor. A nonlinear feedback controller is designed to realize the self-synchronization of time delay of the complex Lorenz system. Numerical simulations verify the effectiveness of the presented controller. The controller adopts some states to realize the synchronization of all states. It is simple in principle and easy to implement in engineering.

Time delay in dynamic systems has long been considered a complex phenomenon. One can see increased research activity on this topic since the early 1980s, and primarily from the mathematics research community. Most investigations are directed toward the issue of stability, which is considerably different from those of delay free dynamics. Stability research is still alive and progressing at an increased pace after four or five decades of brilliant work by prominent researchers. In the past decade, we observe increased participation by the engineering and applied science researchers. This is the primary reason for this Special Issue on Time Delayed Systems (TDS). We have collected over 65 papers around the main theme, and selected 20 after a careful review process involving active researchers in the area. The papers in this issue present a variety of interesting research results in the Time Delayed Systems (TDS) area. The reader will find three categories among these papers dealing with: a) Stability and control of TDS, b) Practical application of TDS, and c) Modeling and identification of system dynamics with time delays. Stability is, without dispute, the hottest area in TDS research today. It spans from Linear Time Invariant (LTI) single delay TDS to multiple and unrelated time delay cases, and further to nonlinear and uncertain time varying time delayed structures. A common objective is to find the stability interval(s) of time delay for a given dynamics, another is to find the ranges of some structural parameters which render stability while the time delay remains fixed. The capabilities of most current methods are often limited due to some inherent assumptions for the methods to be viable. And they offer many valuable directions for future research. It is obvious that our community is still looking for more precise, universal and practicable procedures in order for the TDS to be a part of the every-day arsenal. In many practical applications, time delay may appear due to two main reasons: either the dynamics is inherently time delayed (such as the machine tool chatter problem), or the feedback control structure introduces the delay. Although the source of the delay may be different, these two groups render mathematically identical representations and ultimately the same tools for analysis apply to both. In modeling, if there is a time delay, then its proper representation in a dynamic model is a very critical step. Many interesting problems are encountered in this class especially as computerized control systems improve rapidly yielding better and faster system identification. We are at a very exciting period. Both the reviewers and I were mindful, in selecting the final papers for this Special Issue, that it provide a source for inspiration. I hope our readers will be pleased with the content. As I complete the editorial work, I would like to acknowledge the support of my graduate students in my lab at UCONN (ALARM Lab), without which the task of the Guest Editor would have been an impossible mission. I extend my deep appreciation to the reviewers of the papers. The value of the assistance I received from the Journal Office (G. Ulsoy and T. Marion) is not measurable. I also thank wholeheartedly, all the authors who participated in this activity, including those who did not make the final selection. I hope to handle their future work through regular issues of our Journal. Best wishes to you all.

This paper presents an investigation on the phenomenon of delayed bifurcation in time-delayed slow-fast differential systems. Here the two delayed’s have different meanings. The delayed bifurcation means that the bifurcation does not happen immediately at the bifurcation point as the bifurcation parameter passes through some bifurcation point, but at some other point which is above the bifurcation point by an obvious distance. In a time-delayed system, the evolution of the system depends not only on the present state but also on past states. In this paper, the time-delayed slow-fast system is firstly simplified to a slow-fast system without time delay by means of the center manifold reduction, and then the so-called entry-exit function is defined to characterize the delayed bifurcation on the basis of Neishtadt’s theory. It shows that delayed Hopf bifurcation exists in time-delayed slow-fast systems, and the theoretical prediction on the exit-point is in good agreement with the numerical calculation, as illustrated in the two illustrative examples.

This paper studies stability problems of a class of impulsive systems with time delay whose linear parts contain unstable system matrices. By using the method of variation of parameters, Lyapunov functions and inequalities, several stability criteria are established for both linear and nonlinear impulsive systems with time delay. It is shown that the time delay systems can be stabilized by impulses even if the system matrices are unstable. Several numerical examples are given to illustrate the results.