Sampled data discrete time systems

Consider the sampler switch shown in Fig. 7.88a where f(t) is the continuous input. Suppose the switch closes every ST units of time (the sampling time) and opens again instantaneously. Then the output of the sampler (generally written as f (0) will be a sequence of impulses each of magnitude equal to that of f(r) at that particular sampling time (Figs 7.886 and 7.88c). [In practice this will not be so as a 

It can be seen from the above that the output of a sampler and its transform can both be represented by infinite series. The use of the z-transform simplifies the treatment of such systems, and relationships for sampled data processes can be derived in terms of z-transforms which are similar to those obtained for equivalent continuous systems employing the Laplace transform. 

For simple functions the z-transforms can be obtained easily from their equivalent Laplace transforms by the use of partial fractions and Appendix 7.1. 

Determine the z-transform equivalent of the transfer function of the heat exchanger system described in Example 7.3. 

A combination of Appendix 7.1 and partial fractions may also be used to derive the inverse of a z-transform (i.e. to determine the corresponding time domain function) in a similar manner to that for Laplace transforms. This is illustrated by the following example. 

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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. 

Optical devices or optical systems have provided most of the available strong shock data and were the primary tools used in the early shock-compression investigations. They are still the most widely used systems in fundamental studies of high explosives. The earliest systems, the flash gap and mirror systems on samples, provided discrete or continuous measurements of displacement versus time. 

Many industrial applications employ discrete time controllers. These operate on the basis of discrete signals rather than on a continuous signal as with analog controllers. They are ideally suited to the digital environment produced by computer control and/or sampled data systems. 

MPC is traditionally formulated directly in the discrete-time domain, i.e., for sampled data systems. In detail, it is assumed that the system s input is computed (and the system s output is measured) only at discrete time steps tk = kAt, where k is an integer variable, and At is the sampling time. Hereafter, for the sake of compactness, the discrete-time variable is denoted simply by the integer k. 

Chapters 27,28,29. The material in these chapters has by now become classic and the reader can consult several good references for more details. The following two texts provide an excellent treatment on the mathematical analysis of discrete-time (sampled-data) systems ... 

Sampled-data systems have signals that are discontinuous or discrete. Figure 14.1 shows a continuous analog signal or function /(,) being fed into a sampler. Every Ts minutes the sampler closes for a brief instant. The output of the sampler fs t) is therefore an intermittent series of pulses. Between sampling times, the sampler output is zero. At the instant of sampling the output of the sampler is equal to the input function. 

This section considers the response measurement of a single-degree-of-freedom system or a single-channel measurement of a multi-degree-of-freedom system. Discrete data is sampled with a time step Ar and y denotes the measured response at time t = nAt. The measurement is different from the model response ... 

This chapter will focus on practicable methods to perform both the model specification and model estimation tasks for systems/modek that are static or dynamic and linear or nonlinear. Only the stationary case will be detailed here, although the potential use of nonstationary methods will be also discussed briefly when appropriate. In aU cases, the models will take deterministic form, except for the presence of additive error terms (model residuak). Note that stochastic experimental inputs (and, consequently, outputs) may stiU be used in connection with deterministic models. The cases of multiple inputs and/or outputs (including multidimensional inputs/outputs, e.g., spatio-temporal) as well as lumped or distributed systems, will not be addressed in the interest of brevity. It will also be assumed that the data (single input and single output) are in the form of evenly sampled time-series, and the employed modek are in discrete-time form (e.g., difference equations instead of differential equations, discrete summations instead of integrak). 

To minimize data acquisition time, the overall magnification of this system was made as small as possible, consistent with the spatial resolution desired. Since the image is sampled by the discrete pixels of the CCD camera, the Nyquist sampling criterion must be obeyed to avoid the generation of artifacts. There is an explicit relationship describing how the number of counts per pixel depends on the optical design. We know from Equation (1.2) that the radius of the Airy disk r is equal to 0.61A/NA (since 2r). If the size of the CCD pixel is p, then the Nyquist sam-... 

The strategy depends on the situation and how we measure the concentration. If we can rely on pH or absorbance (UV, visible, or Infrared spectrometer), the sensor response time can be reasonably fast, and we can make our decision based on the actual process dynamics. Most likely we would be thinking along the lines of PI or PID controllers. If we can only use gas chromatography (GC) or other slow analytical methods to measure concentration, we must consider discrete data sampling control. Indeed, prevalent time delay makes chemical process control unique and, in a sense, more difficult than many mechanical or electrical systems. 

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