Differential equations time-delay

The simplest model of time-dependent behavior of a neutron population in a reactor consists of the point kinetics differential equations, where the space-dependence of neutrons is disregarded. The safety of reactors is greatly enhanced inherently by the existence of delayed neutrons, which come from radioactive decay rather than fission. The differential equations for the neutron population, n, and delayed neutron emitters, are... 

This is identical to the first order Fade power series and gives a crude time delay approximation, when transformed back into differential equation form. 

The limitation of transfer function representation becomes plain obvious as we tackle more complex problems. For complex systems with multiple inputs and outputs, transfer function matrices can become very clumsy. In the so-called modem control, the method of choice is state space or state variables in time domain—essentially a matrix representation of the model equations. The formulation allows us to make use of theories in linear algebra and differential equations. It is always a mistake to tackle modem control without a firm background in these mathematical topics. For this reason, we will not overreach by doing both the mathematical background and the control together. Without a formal mathematical framework, we will put the explanation in examples as much as possible. The actual state space control has to be delayed until after tackling classical transfer function feedback systems. 

Since there is no axial mixing, clearly any input is delayed at the outlet by the period of the residence time. This result also can demonstrated formally by solving the pertinent differential equations. 

Dynamic problems expressed in transfer function form are often very easily reformulated back into sets of differential equation and associated time delay functions. An example of this is shown in the simulation example TRANSIM. 

The outline of this paper is as follows. First, a theoretical model of unsteady motions in a combustion chamber with feedback control is constructed. The formulation is based on a generalized wave equation which accommodates all influences of acoustic wave motions and combustion responses. Control actions are achieved by injecting secondary fuel into the chamber, with its instantaneous mass flow rate determined by a robust controller. Physically, the reaction of the injected fuel with the primary combustion flow produces a modulated distribution of external forcing to the oscillatory flowfield, and it can be modeled conveniently by an assembly of point actuators. After a procedure equivalent to the Galerkin method, the governing wave equation reduces to a system of ordinary differential equations with time-delayed inputs for the amplitude of each acoustic mode, serving as the basis for the controller design. 

This set of equations is a nonlinear eigenvalue time delay differential equation. Such equations, even for one variable, often have periodic or chaotic solutions and, from the physics of the problem are also certain of having pulse-like solutions in some systems. 

Time delays can also be handled with the LQP, although the discrete-time formulation (46) of the LQP is better suited to the time delay problem (especially when there are only a few such elements in the differential equations). ... 

In the above definitions, 9 represents a set of parameters of the system, having constant values. These parameters are also called control parameters. The set of the system s variables forms a representation space called the phase space [32]. A point in the phase space represents a unique state of the dynamic system. Thus, the evolution of the system in time is represented by a curve in the phase space called trajectory or orbit for the flow or the map, respectively. The number of variables needed to describe the system s state, which is the number of initial conditions needed to determine a unique trajectory, is the dimension of the system. There are also dynamic systems that have infinite dimension. In these cases, the processes are usually described by differential equations with partial derivatives or time-delay differential equations, which can be considered as a set of infinite in number ordinary differential equations. The fundamental property of the phase space is that trajectories can never intersect themselves or each other. The phase space is a valuable tool in dynamic systems analysis since it is easier to analyze the properties of a dynamic system by determining... 

Cortisol concentration is described by a nonlinear time-delay differential equation [47,519] with two terms, i.e., a secretion rate term that adheres to the negative feedback mechanism [520, 521] and drives the pulsatile secretion, and a first-order output term with rate constant ka ... 

For the last 20 to 25 years, computer modelling has been used increasingly to interpret combustion phenomena. When the chemistry is the main interest, for example, in predicting ignition delay times or predicting product profiles, large comprehensive mechanisms can be used which in theory cover every species possible in the total oxidation. In most treatments differential equations are written for the reactants, intermediate species and for the final products. These are solved by standard mathematical treatments and profiles of concentration - time produced for all species. This approach will be discussed in detail in Chapter 4. 

In Refs. 244 and 248 the possibility to control friction has been discussed in model systems described by differential equations. Usually, in realistic systems, time series of dynamical variables rather than governing equations are experimentally available. In this case the time-delay embedding method [258] can be applied in order to transform a scalar time series into a trajectory in phase space. This procedure allows one to find the desired unstable periodic orbits and to calculate variations of parameters required to control friction. 

Model representations in Laplace transform form are mainly used in control theory. This approach is limited to linear differential equation systems or their linearized approximations and is achieved by a combination of first-order lag function and time delays. This limitation together with additional complications of modelling procedures are the main reasons for not using this method here. Specialized books in control theory as mentioned above use this approach and are available to the interested reader. 

Using the transfer function concept, Koppel (1967) derived the optimal control policy for a heat exchanger system described by hyperbolic partial differential equations using the lumped system approach. Koppel and Shih (1968) also presented a feedback interior control for a class of hyperbolic differential equations with distributed control. In an earlier paper Koppel e/ al. (1968) discussed the necessary conditions for the system with linear hyperbolic partial differential equations having a control which is independent of spatial coordinates. The optimal feedback-feedforward control law for linear hyperbolic systems, whose dynamical response to input variations is characterized by an initial pure time delay, was derived by Denn... 

The model used in this section neglects the time delays due to recycles and the capacity of the mixing vessels. Consequently, the model is obtained by the combination of the differential equations describing the dynamics of the reactor and the closed-loop separation. F and c are molar flow rate and reactant concentration, respectively. Dimensionless values are denoted by /=c/c and z=F/Fo with reference to process inlet. Subscripts follow the numbering explained in Fig. 13.18. When two reactants are involved, a second subscript is used. Because high purity product C4 = Z4 = 0. 

Time delays also play a significant role in the onset of oscillations in the cascade. Such time delays result here in a natural way from the thresholds in the activation curves of cdc2 kinase and cyclin protease, and are by no means inserted in an ad hoc manner into the kinetic equations (10.1), as is often done for systems governed by time-delay differential equations. 

Chemical reaction network is a typical example of complexity, where the reactants can interact in a variety of ways depending on the nature of interaction (chemical as well as non-chemical). Oscillatory reactions involve a number of steps, including positive and negative feedbacks. The complexity leads to periodic as well as aperiodic oscillations (multi-periodic, bursting/intermittency sequential oscillations separated by a time pause, relaxation and chaotic oscillations). The mechanism is usually determined by non-linear kinetics and computer modelling. Once the reaction mechanism has been postulated, the non-linear time-dependent kinetic equation can be formulated in terms of concentrations of different reactants, which would yield a multi-variable equation. Delay differential equations are sometimes used to characterize oscillatory behaviour as in economics (Chapter 14). 

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