Z-transforms

See Refs. 198, 218, and 256. The z -transform is useful when data is available at only discrete points. Let... 

The z -transform is used in process control when the signals are at intervals of At. A brief table (Table 3-2) is provided here. 

The z -transform can also be used to solve difference equations, just like the Laplace transform can be used to solve differential equations. 

Figure 8-5 illustrates the concept of samphng a continuous function. At integer values of the saiTmling rate. At, the value of the variable to be sampled is measured and held until the next sampling instant. To deal with sampled data systems, the z transform has been developed. The z transform of the function given in Fig. 8-5 is defined as... 

The z-transform is the prineipal analytieal tool for single-input-single-output dis-erete-time systems, and is analogous to the Laplaee transform for eontinuous systems. 

Table 7.1 gives Laplaee and z-transforms of eommon funetions. z-transform Theorems ... 

The discrete-time solution of the state equation may be considered to be the vector equivalent of the scalar difference equation method developed from a z-transform approach in Chapter 7. 

To obtain the z-transform of a first-order sampled data system in cascade with a zero-order hold (zoh), as shown in Figure 7.10. 

Example 7.3 Transfer Function to z-Transform %Continuous and Discrete Step Response num=[1] den=[1 1 ] ... 

G. Doetsch, Anleitung zum praktischen gebrauch der Laplace-transformation und der Z-transformation. R. Oldenbourg, Munchen 1989. 

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. 

See Ogunnaike, Babatunde A., and W. Harmon Ray, Process Dynamics, Modeling, and Control, Oxford University Press (1994) Seborg, D., T. F. Edgar, and D. A. Mellichamp, Process Dynamics and Control, 2d ed., Wiley, New York (2003). The z-transform is useful when data is available at only discrete points. Let... 

A graphic or tabular data display can be generated for any z-statistic value given a population correlation coefficient, p. This is accomplished by using the Fisher s Z transformation (i.e., the Z-statistic) computation as (equation 60-20)... 

An electric dipole operator, of importance in electronic (visible and uv) and in vibrational spectroscopy (infrared) has the same symmetry properties as Ta. Magnetic dipoles, of importance in rotational (microwave), nmr (radio frequency) and epr (microwave) spectroscopies, have an operator with symmetry properties of Ra. Raman (visible) spectra relate to polarizability and the operator has the same symmetry properties as terms such as x2, xy, etc. In the study of optically active species, that cause helical movement of charge density, the important symmetry property of a helix to note, is that it corresponds to simultaneous translation and rotation. Optically active molecules must therefore have a symmetry such that Ta and Ra (a = x, y, z) transform as the same i.r. It only occurs for molecules with an alternating or improper rotation axis, Sn. 

Equation (9.41) constitutes a fundamental solution for purely convective mass burning flux in a stagnant layer. Sorting through the S-Z transformation will allow us to obtain specific stagnant layer solutions for T and Yr However, the introduction of a new variable - the mixture fraction - will allow us to express these profiles in mixture fraction space where they are universal. They only require a spatial and temporal determination of the mixture fraction/. The mixture fraction is defined as the mass fraction of original fuel atoms. It is as if the fuel atoms are all painted red in their evolved state, and as they are transported and chemically recombined, we track their mass relative to the gas phase mixture mass. Since these fuel atoms cannot be destroyed, the governing equation for their mass conservation must be... 

Data values x following a normal distribution N(p, a2) can be transformed to a standard normal distribution by the so-called z-transformation... 

Variance scaling standardizes each variable j by its standard deviation s/, usually, it is combined with mean-centering and is then called autoscaling (or z-transformation). 

Figure 5. Movements from a two-minute interaction. Greydensity changes of a person throughout a whole film sequence of two minutes. In (a), greydensities obtained by averaging all greyvalue differences of a difference-picture are plotted (end of step 5, cf. text for details). In (b), the data are z-transformed (step 6), smoothed (step 7), and a threshold has been calculated (step 8). AMA identified 15 bursts.

The last decade has seen the development a several control concepts that are based on using a model of the process as part of the controller. Most of these methods use Laplace or z-transform representations of the process, which we are not yet ready to handle. After our Russian lessons have been completed, we will discuss some of these. 

To analyze systems with discontinuous control elements we will need to learn another new language. The mathematical tool of z transformation is used to desigii control systems for discrete systems, z transforms are to sampled-data systems what Laplace transforms are to continuous systems. The mathematics in the z domain and in the Laplace domain are very similar. We have to learn how to translate our small list of words from English and Russian into the language of z transforms, which we will call German. 

In Chap. 18 we will define mathematically the sampling process, derive the z transforms of common functions (learn our German vocabulary) and develop transfer functions in the z domain. These fundamentals are then applied to basic controller design in Chap. 19 and to advanced controllers in Chap. 20. We will find that practically all the stability-analysis and controller-design techniques that we used in the Laplace and frequency domains can be directly applied in the z domain for sampled-data systems. 

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