Discrete-time processes

Simple and straightforward qualitative and quantitative analysis of how discrete-time processes react to external input changes... 

Deller, j. R., and Proakis, J. Discrete-Time Processing of Speech Signals. John Wiley and Sons, 2000. 

For a dynamic system, we assume that the discrete-time process input sequence u(A ) and the discrete-time measured process output sequence y(ft) ) where /s = 1,2,..., M enumerates the sampling intervals, are related by the linear regression model given by... 

One other point worth making is the relationship between the FSF model parameters and the entire process frequency response. As stated earlier, the parameters of the FSF model are the values of the discrete-time process... 

Discrete time signals may be generated by a discrete time process or may be the result of a sampling process on a continuous time signal. 

Dynamic simulation with discrete-time events and constraints. In an effort to go beyond the integer (logical) states of process variables and include quantitative descriptions of temporal profiles of process variables one must develop robust numerical algorithms for the simulation of dynamic systems in the presence of discrete-time events. Research in this area is presently in full bloom and the results would significantly expand the capabilities of the approaches, discussed in this chapter. 

Design modifications for the accommodation of feasible operating procedures. Section V introduced some early ideas on how the ideas of planning operating procedures could be used to identify modifications to a process flowsheet, which are necessary to render feasible operating procedures. More work is needed in this direction. Clearly, any advances in dynamic simulation with discrete-time events would have beneficial effects on this problem. 

Having completed the decomposition and reconstruction of a function at a finite number of discrete values of scale, let us turn our attention to the discretization of the translation parameter, u, dictated by the discrete-time character of all measured process variables. The classical approach, suggested by Meyer (1985-1986), is to discretize time over dyadic intervals, using the sampling interval, t, as the base. Thus, the translation parameter, u, can be expressed as... 

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 ... 

An exhaustive statistical description of living copolymers is provided in the literature [25]. There, proceeding from kinetic equations of the ideal model, the type of stochastic process which describes the probability measure on the set of macromolecules has been rigorously established. To the state Sa(x) of this process monomeric unit Ma corresponds formed at the instant r by addition of monomer Ma to the macroradical. To the statistical ensemble of macromolecules marked by the label x there corresponds a Markovian stochastic process with discrete time but with the set of transient states Sa(x) constituting continuum. Here the fundamental distinction from the Markov chain (where the number of states is discrete) is quite evident. The role of the probability transition matrix in characterizing this chain is now played by the integral operator kernel ... 

It may be useful to point out a few topics that go beyond a first course in control. With certain processes, we cannot take data continuously, but rather in certain selected slow intervals (c.f. titration in freshmen chemistry). These are called sampled-data systems. With computers, the analysis evolves into a new area of its own—discrete-time or digital control systems. Here, differential equations and Laplace transform do not work anymore. The mathematical techniques to handle discrete-time systems are difference equations and z-transform. Furthermore, there are multivariable and state space control, which we will encounter a brief introduction. Beyond the introductory level are optimal control, nonlinear control, adaptive control, stochastic control, and fuzzy logic control. Do not lose the perspective that control is an immense field. Classical control appears insignificant, but we have to start some where and onward we crawl. 

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. 

We can then conclude that while the discrete time STN and RTN models are quite general and effective in monitoring the level of limited resources at the fixed times, their major weakness in terms of capability is the handling of relatively small processing and changeover times. Regarding the objective function, these models can easily handle profit maximization (cost minimization) for a fixed time horizon. Intermediate due dates can be easily modeled. Other objectives such as makespan minimization are more complex to implement since the time horizon and, in consequence, the number of time intervals, are unknown a priori (see [11]). 

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... 

Lee, K Park, H. and Lee, I. (2001) A novel nonuniform discrete time formulation for short-term scheduling of batch and continuous processes. Ind. Eng. Chem. Res., 40, 4902—4911. 

A MILP model of the aggregated scheduling problem of the EPS process was proposed by Sand and Engell [16]. The model is formulated as a discrete time multi-period model where each period i e 1,..., 1 corresponds to two days. The degrees of freedom of the aggregated problem are the following discrete production decisions ... 

Floudas, C.A. and Lin, X. (2004) Continuous-time versus discrete-time approaches for scheduling of chemical processes a review. Computers and Chemical Engineering, 28, 2109-2129. 

Third, processing times may require special modeling in chemical industry. While in discrete manufacturing processing times for a certain lot are usually dependent on the lot size, i.e., the number of units to be produced, this is often not true in the chemical industry. Here, processing times are often constant, irrespective of whether a reactor is filled to 70% or 90% of its capacity. This is often referred to as batch production [5], On the other hand, the quality of the material produced may depend on resource utilization. Certain reactions may not even be feasible, if a minimum bound of the procured material is not exceeded. This implies additional restrictions regarding the resource utilization level on the planning situation. 

Minimal bounds on the production quantity are most often process dependent. Typically, a minimal campaign length is required if for example a critical mass is necessary to initiate a chemical reaction. The same is valid for maximal bounds on the production quantity. The rationale here is that a cleaning operation may be required every time a certain amount has been produced. Finally, batch size restrictions often arise in the chemical industry, if for example the batch size is determined by a reactor load or, as discussed above, the processing time for a certain production step is independent of the amount of material processed. In these scenarios, when working with model formulations using a discrete time scale, it is important that the model formulation takes into account that lot sizes may comprise of production in several adjacent periods. 

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

As is the case for Dic, complete autohesion and complete bonding correspond to Dau = 1 and Db = 1, respectively. To account for nonisothermal autohesion, which occurs in on-line consolidation processes such as filament winding, the degree of bonding must be calculated at discrete time steps and summed [21]. 

An ancient but still instructive example is the discrete-time random walk. A drunkard moves along a line by making each second a step to the right or to the left with equal probability. Thus his possible positions are the integers — oo < n < oo, and one asks for the probability pn(r) for him to be at n after r steps, starting from n = 0. While we shall treat this example in IV.5 as a stochastic process, we shall here regard it as a problem of adding variables. 

Exercise. A gambler plays heads and tails. Let Yt be the amout of his capital after t throws. Show that Yt is a discrete-time Markov process and find its transition probability. 

Exercise. In the discrete-time random walk the steps were taken at fixed time points. Suppose now that the times of the steps are randomly distributed as given by the Poisson process. Show that this is identical to the situation described by (2.1). )... 

The misnormalized data of Lee et al.16) was interpreted in terms of two discrete relaxation processes. It was proposed that the relaxation function should be represented as the sum of two Williams-Watts functions. The slope at short times was claimed to be equal to the / for the faster of the two processes. Numerical calculations and graphical representations of exact relaxation functions with parameters equal to those reported by Lee et al.16) were carried out. They did not look even qualitatively similar to their reported data. The slope at the shortest times must be related to a weighted sum of both of the /3 values for the sum of two WW functions. If it was desired to fit the data to a sum of two WW functions, then this could easily be carried out with a nonlinear least squares routine. In most cases it would not be possible to obtain statistically independent values of all six parameters, but at least no further errors would be introduced by faulty manipulations of the data. The graphical procedure of Lee et al.16) cannot be recommended as of any worth in this problem. 

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