Discrete time model

We consider here the discrete-time transport process corresponding to (5.1). After the hyperbolic scaling (4.33), the equation governing the rescaled field reads 

We seek a solution in the exponential form (4.35), so that G x, t) obeys, to leading order in e, the following equation  

Taking logarithms of both sides, we find that G x, t) satisfies the nonlinear partial 

This result is of particular importance. It shows that the front dynamics for (5.1) must be different from that of the classical RD equation. This follows from the 

If the jumps can only take the values a or —a with the same probability, i.e., the kernel w(z) is a superposition of two delta-functions as in (5.4), the Hamilton-Jacobi equation takes the form 

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When experimental data are collected over time or distance there is always a chance of having autocorrelated residuals. Box et al. (1994) provide an extensive treatment of correlated disturbances in discrete time models. The structure of the disturbance term is often moving average or autoregressive models. Detection of autocorrelation in the residuals can be established either from a time series plot of the residuals versus time (or experiment number) or from a lag plot. If we can see a pattern in the residuals over time, it probably means that there is correlation between the disturbances. 

Recursive estimation methods are routinely used in many applications where process measurements become available continuously and we wish to re-estimate or better update on-line the various process or controller parameters as the data become available. Let us consider the linear discrete-time model having the general structure ... 

The literature example results for the scenario without heat integration are summarized in Table 10.2. The discrete-time model proposed by Papageorgiou et al. 

The most relevant contribution for global discrete time models is the State Task Network representation proposed by Kondili et al. [7] and Shah et al. [8] (see also [9]). The model involves 0-1 variables for allocating tasks to processing units at the beginning of the postulated time intervals. Most important equations comprise mass balances over the states, constraints on batch sizes and resource constraints. The STN model covers all the features that are included at the column on discrete time in Table 8.1. 

The computational results for the case studies allow the comparison and study of the efficiency and limitations of specific modeling approaches. However, it is worth mentioning that problem data involves only integer processing times, which represents a fortunate situation for discrete time models since no special provisions... 

Although the usefulness and performance of continuous and discrete time models strongly depends on the particular problem and solution characteristics, our... 

Perhaps the biggest gap in terms of effective models is the capability of simultaneously handling changeovers, inventories and resource constraints. Sequential methods can handle well the first, while discrete time models (e.g., STN, RTN), can handle well the last two. While continuous-time models with global time intervals can theoretically handle all of the three issues, they are at this point still much less efficient than discrete time models, and therefore require further research. 

Suerie, C. (2005) Time Continuity in Discrete Time Models. New Approaches for Production Planning in Process Industries, Springer, Berlin. 

Several identification methods result in a state space model, eithejp by direct identification in the state space structure or by identjLfication in a structure that can be transformed into a state space model. In system identification, discrete-time models are used. The discrete-time state-space model is given by... 

To determine the state space model with system Identification, responses of the nonlinear model to positive and negative steps on the Inputs as depicted in Figure 4 were used. Amplitudes were 20 kW for P,, . 4 1/s for and. 035 1/s for Q. The sample interval for the discrete-time model was chosen to be 18 minutes. The software described In ( 2 ) was used for the estimation of the ARX model, the singular value analysis and the estimation of the approximate... 

We consider a discrete-time model with periods 1, —, T, wherein we consider both T < < and T=°°.3 We vastly simplify the model by assuming that in each period, l. consumption of an addictive product, a is either 0 or 1 Each period people can either take a hit or not take a hit, wherein , 1 if they take a hit and a, = 0 if they refrain. Furthermore, we assume that the good is free. Our focus on free products helps highlight the fact that people may avoid addictive products because they lead to unpleasant long-run consequences, rather than because of the purchase price per se. It also simplifies notation and analysis. 

Valimaki, 1995) Valimaki, V. (1995). Discrete-Time Modeling of Acoustic Tubes Using Fractional Delay Filters. PhD thesis, Report no. 37, Helsinki University of Technology Faculty of Elec. Eng., Lab. of Acoustic and Audio Signal Processing, Espoo, Finland... 

Pattern recognition self-adaptive controllers exist that do not explicitly require the modeling or estimation of discrete time models. These controllers adjust their tuning based on the evaluation of the system s closed-loop response characteristics (i.e., rise time, overshoot, settling time, loop damp-... 

Given a set of experimental data, we look for the time profile of A (t) and b(t) parameters in (C.l). To perform this key operation in the procedure, it is necessary to estimate the model on-line at the same time as the input-output data are received [600]. Identification techniques that comply with this context are called recursive identification methods, since the measured input-output data are processed recursively (sequentially) as they become available. Other commonly used terms for such techniques are on-line or real-time identification, or sequential parameter estimation [352]. Using these techniques, it may be possible to investigate time variations in the process in a real-time context. However, tools for recursive estimation are available for discrete-time models. If the input r (t) is piecewise constant over time intervals (this condition is fulfilled in our context), then the conversion of (C.l) to a discrete-time model is possible without any approximation or additional hypothesis. Most common discrete-time models are difference equation descriptions, such as the Auto-.Regression with eXtra inputs (ARX) model. The basic relationship is the linear difference equation ... 

Discretization for a sampling period of 4 min yields a corresponding discrete-time model that is used in the MPC on-line objective function... 

For consequence analysis, we have developed a dynamic simulation model of the refinery SC, called Integrated Refinery In-Silico (IRIS) (Pitty et al., 2007). It is implemented in Matlab/Simulink (MathWorks, 1996). Four types of entities are incorporated in the model external SC entities (e.g. suppliers), refinery functional departments (e.g. procurement), refinery units (e.g. crude distillation), and refinery economics. Some of these entities, such as the refinery units, operate continuously while others embody discrete events such as arrival of a VLCC, delivery of products, etc. Both are considered here using a unified discrete-time model. The model explicitly considers the various SC activities such as crude oil supply and transportation, along with intra-refinery SC activities such as procurement planning, scheduling, and operations management. Stochastic variations in transportation, yields, prices, and operational problems are considered. The economics of the refinery SC includes consideration of different crude slates, product prices, operation costs, transportation, etc. The impact of any disruptions or risks such as demand uncertainties on the profit and customer satisfaction level of the refinery can be simulated through IRIS. 

Where Xk are state variables, Wk and Vk are the model and process disturbances, respectively, Lk is a stage cost function and Zt-n (z) is the arrival cost function. A discrete-time model is adopted in the above formulation for illustrative purposes. A continuous-time model can also be used. 

While the distinction between discrete time and continuous time is mathematically clear-cut, we may in applied work use discrete time approximation to a continuous time phenomena and vice versa. However, discrete time models are usually easier for numerical analysis, whereas simple analytical solutions are more likely to emerge in continuous time. 

A general linear discrete-time model for a variable y k) can be written... 

The continuous models (e.g., differential equations in the time domain, or input-output models in the Laplace domain) are not convenient to use to analyze the dynamic behavior of loops with computer control discrete-time models are needed. 

Example 27.5 Discrete-Time Model of a Digital PID Controller... 

Consequently, the control action of a digital PID controller is determined by the following discrete-time model ... 

Example (27.5) is very instructive on how to develop a discrete-time model from its equivalent continuous one. The procedure can be generalized for any continuous dynamic model as follows ... 

The discrete-time modeling equation(s) resulting from the procedure above is known as the difference equation(s), in contrast to the term differential equation (s) used for the continuous model. 

Example 27.6 Discrete-Time Model of a First-Order Process... 

With the first-order difference approximation for the derivatives, we take the following difference equations, which represent the discrete-time model of the multivariable process ... 

Let td = kT that is, the dead time is an integer multiple of the period, T. Then the discrete-time model is easily found to be... 

Why is a discrete-time model an approximation of its continuous counterpart ... 

With a discretization time interval T = 1 second, which one of the two systems will have a better discrete-time representation Explain why. Also, show how you can improve the quality of the other (worst) discrete-time model. 

Bode diagram, 330-31, 334-37 frequency response, 323-24 interacting capacities, 197-200 noninteracting capacities, 194-96 pulse transfer function, 619 Multiple-input multiple-output system, 20 discrete-time model, 586 discrete transfer function, 612 input-output model, 83-85, 163-68 linearization, 121-26 transfer-function matrix, 164, 166 Multiple loop control systems, 394-409 Multiplexer, 560, 564 Multivariable control systems, 461-62 alternative configurations, 467-84 decoupling of loops, 503-8 design questions, 461-62 interaction of loops, 487-94 selection of loops, 494-503 Multivariable process (see Multiple-input multiple-output system)... 

Saturation of controllers, 247, 257, 637 Scheduling computer control, 33 Secondary loop, cascade control, 395, 397, 398-99, 400-2 Secondary measurements, 16, 16-18 Second-order system, 186-87 Bode diagrams, 328-30 with dead time, 215, 216 discrete-time model, 585-86 dynamic characteristics, 187-93 experimental parameter identification, 233,668... 

Equation (27.11) represents the discrete-time model of a second-order process. Notice that in order to compute the next value (yn+2) of y, we... 

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