A simple time delay equation

In this chapler we present a technique for accurate computation of the stability exponent and other eigenvalue abscissas of a matrix delay equation. Previously the author introduced a finite dimensional linear operator, determined by the delay equation coefficients, having spectrum containing all possible imaginary axis eigenvalues of the delay system. Using the eigenvalues of this operator, and introducing a transmision in the equation’s characteristic function, we can make an accurate numerical determination of the system stability or growth exponent and other eigenvalue abscissas. Arter giving the basic theorems for the method, we give an example in which we go over essentials of implementation. Then we explore the method in some special cases, beginning with second order scalar delay equations and an interesting example of positive delay feedback. We proceed to a rather detailed examination of the effect of the delay parameter in some simple first order delay equations, finding that accurate computation of the system abscissa leads us to some interesting and unconventional conclusions on its behavior with respect to this parameter. We then give an example in which the method is adapted to a cenain distributed delay equation. We conclude with some comments on possible future research.

Many times a scientist is choosing a programming language or a software for a specific purpose. For the field of scientific computing, the methods for solving differential equations are one of the important areas. What I would like to do is take the time to compare and contrast between the most popular offerings. This is a good way to reflect upon what's available and find out where there is room for improvement. I hope that by giving you the details for how each suite was put together (and the "why", as gathered from software publications) you can come to your own conclusion as to which suites are right for you.

A simple time delay equation

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Rayleigh waves from underground nuclear explosions which are accompanied by high levels of tectonic strain release are observed to be reversed and time delayed relative to Rayleigh waves from “normal” underground explosions. The “time delays” can be explained as an interference effect between the explosion and tectonic source time functions which magnifies the true phase difference between the source functions. The phase of the explosion source time function is advanced relative to the phase of the tectonic source function, and the magnitude of the time advance is directly related to the amount of overshoot in the explosion source time function. The phase distortion is more complex than a simple time delay, and may appear as a constant phase shift with no linear trend. Apparent time advances are also possible; however because of small differences between the excitation functions of the explosion and tectonic sources at shallow depths, the phase shift will appear more frequently as an apparent time delay. Observations of Rayleigh wave phase shifts from underground explosions at the Soviet East Kazakh test site recorded at SRO stations are in very good agreement with simulated phase shifts derived from synthetic calculations.

In this paper, we report a common phenomenon observed in chaotic systems linked by time delay. Recently, the Lorenz chaotic system has been extended to the family of Lorenz systems which includes the Chen and Lü systems. These three chaotic systems, corresponding to different sets of system parameter values, are topologically different. With the aid of numerical simulations, we have surprisingly found that a simple time delay, directly applied to one or more state variables, transforms the Lorenz system to the generalized Chen system or the generalized Lü system without any parameter changes. The existence of this phenomenon has also been found in other known chaotic systems: the Rössler system, the Chua's circuit and the 4-Liu system. This finding has shown a common characteristic of chaotic systems: a new chaotic "branch" can be created from a chaotic attractor by simply adding a time delay.

In this paper the applications of acoustic surface wave device technology to electronic warfare systems have been reviewed. A variety of signal processing tasks such as simple time delay, frequency dependent time delay, programmable and discretely variable time delay, band pass filtering and generation and recognition of various coded waveforms can be performed by surface wave devices to improve the performance of EW system.

In this paper, we propose a simple in-service delay time estimation method and two adaptive transmission schemes, that is, adaptive GI length transmission scheme and adaptive modulation transmission scheme, based on the estimation method for broadband vehicular OFDM transmissions. The simple in-service delay time estimation method uses the characteristic that the symmetry of received signal point distribution collapses in multipath propagation environments, obtains the useful information about the delay spread and average BER from the accumulative average value of the detector outputs of sub-carriers of OFDM signals. The proposed adaptive transmission schemes utilize the measured delay time relative information to adaptively change the GI length or modulation method of OFDM signals, achieving high transmission quality and efficiency simultaneously. Assuming a two-ray Rayleigh fading multipath channel model, the effectiveness of these proposed schemes has been confirmed by computation simulations.

Active magnetic bearings are used to provide a long-life, low-loss suspension of a high-speed flywheel rotor. This paper describes a modeling effort used to understand the stability boundaries of the PD controller used to control the active magnetic bearings on a high speed test rig. Limits of stability are described in terms of allowable stiffness and damping values which result in stable levitation of the nonrotating rig. Small signal stability limits for the system is defined as a nongrowth in vibration amplitude of a small disturbance. A simple mass-force model was analyzed. The force resulting from the magnetic bearing was linearized to include negative displacement stiffness and a current stiffness. The current stiffness was then used in a PD controller. The phase lag of the control loop was modeled by a simple time delay. The stability limits and the associated vibration frequencies were measured and compared to the theoretical values. The results show a region on stiffness versus damping plot that have the same qualitative tendencies as experimental measurements. The resulting stability model was then extended to a flywheel system. The rotor dynamics of the flywheel was modeled using a rigid rotor supported on magnetic bearings. The equations of motion were written for the center of mass and a small angle linearization of the rotations about the center of mass. The stability limits and the associated vibration frequencies were found as a function of nondimensional magnetic bearing stiffness and damping and nondimensional parameters of flywheel speed and time delay.

In the present paper we deal with the approximate disturbance decoupling problem with measurement (DDPM) for a class of nonlinear systems with a simple time delay at the input. The analysis is based on a standard singularly perturbed form free of delay which is an approximation of the original system.

A novel active disturbance rejection control (ADRC) solution and a particular tuning method are presented for a class of time delay system (TDS) with uncertainty. First, the complicated process dynamics is modeled as a simple first order plus large time delay (FOPTD) plant, with the difference between the actual dynamics and its model treated as disturbances to be rejected. Then the reduced order linear extended state observer (RLESO) with input delay is proposed to estimate the time delay state and disturbance. It is shown how the time delay could be eliminated from the characteristic equation of the closed-loop system by manipulations of controller parameters. Secondly, the one parameter tuning (OPT) technique is developed where all controller parameters are made function of a single coefficient. In comparison with optimal proportional-integral-derivative (PID) controller and twice optimum controller (TOC), the simulation results show that the proposed method not only has better accuracy and faster response, but also ensures better robustness and adaptability against uncertain model parameters and external disturbances, especially for the plant with very large time delays.

As an important part of the smart grid, a wide-area measurement system (WAMS) provides the key technical support for power system monitoring, protection and control. But 20 uncertainties in system parameters and signal transmission time delay could worsen the damping effect and deteriorate the system stability. In the presented study, the subspace system identification technique (SIT) is used to firstly derive a low-order linear model of a power system from the measurements. Then, a novel adaptive wide-area damping control scheme for online tuning of the wide-area damping controller (WADC) parameters using the residue method is proposed. In order to eliminate the effects of the time delay to the signal transmission, a simple and practical time delay compensation algorithm is proposed to compensate the time delay in each wide-area control signal. Detailed examples, inspired by the IEEE test system under various disturbance scenarios, have been used to verify the effectiveness of the proposed adaptive wide-area damping control scheme.

<div class="section abstract"><div class="htmlview paragraph">The autoignition resistance of a practical gasoline is best characterized by the Octane Index, OI, defined as RON-KS, where RON and MON are respectively, Research and Motor Octane Numbers, S is the sensitivity (RON-MON) and K is a constant depending on the pressure and temperature history of the fuel/air mixture in an engine. Experiments in knocking SI engines, HCCI engines and in premixed compression ignition (PCI) engines have shown that if two fuels of different composition have the same OI and experience the same pressure/temperature history, they will have the same autoignition phasing. A practical gasoline is a complex mixture of hydrocarbons and a simple surrogate is needed to describe its autoignition chemistry. A mixture of toluene and PRF (iso-octane + n-heptane), TPRF, can have the same RON and S as a target gasoline and so will have the same OI at any given K value and will be a very good surrogate for the gasoline.</div><div class="htmlview paragraph">In this paper, a method to define the composition of a TPRF to match both RON and MON of a target gasoline is presented. The appropriate TPRF as a surrogate for a particular gasoline, which has been extensively tested in a knocking SI engine, is identified using this method. A chemical kinetic model is used to calculate ignition delays at different pressures and temperatures for this surrogate TPRF. From these data, a simple Arrhenius type equation with a pressure correction to predict ignition delays is identified. This equation is used to find the ignition delay as a function of crank angle and calculate the Livengood-Wu integral, I, for a number of individual knocking cycles covering a wide range of operating conditions using the gasoline in a single cylinder engine. Knock is predicted to occur at the crank angle when the integral, I, reaches unity. The crank angle at which knock is predicted to occur using the simple equation for ignition delay for the surrogate TPRF agrees very well with the experimentally observed value for the gasoline for all the cases considered. Finally, using the chemical kinetic model for TPRF, simple equations which can be used to estimate ignition delay are presented for a range of RON and sensitivity. Such equations can be used to predict when knock occurs during the cycle for these gasolines if the pressure and temperature development with crank angle is known.</div></div>

IMPROVED PI CONTROL VIA DYNAMIC INVERSION

A series of simulations have been carried out to compare the PI and PID controller tunings of different tuning methods, namely Ziegler-Nichols(ZN), Cohen-Coon (CC), Chien-Hrones-Reswick(CHR) and minimum error criteria ISE, ISTE, ISTSE methods for various process models. Using simple first order plus time delay (FOPTD) and second order plus time delay (SOPTD) processes in a feedback control loop, simulations of system responses to setpoint changes were plotted and analyzed for speed of response, stability, and robustness of these tuning methods. It was found that minimum error criteria ISE, ISTE, ISTSE tuning methods, although they give a relatively slow response, were superior in stability and robustness in almost all cases. The minimum error tuning method is easy to implement and gives the desired results using MATLAB/Simulink effortlessly. It gives faster responses with less oscillation. This superiority is observed both for PI and PID controllers. On the other hand, ZN and CHR gave larger overshoots with longer settling time for PI and PID, while CC gives very sluggish response for PI controller. The ZN and CHR tuning methods have higher proportional gains and smaller integral time constants leading to very poor damping, thus they are only suitable for processes that operate deep within the stable region while for processes operating on the periphery of the stable region will be unsafe to tune using ZN and CHR tuning methods.