Sampled-Data Systems

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... 

A first-order sampled-data system is shown in Figure 7.10. 

Consider the characteristic equation of a sampled-data system... 

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. 

The category of algebraic equation models is quite general and it encompasses many types of engineering models. For example, any discrete dsmamic model described by a set of difference equations falls in this category for parameter estimation purposes. These models could be either deterministic or stochastic in nature or even combinations of the two. Although on-line techniques are available for the estimation of parameters in sampled data systems, off-line techniques... 

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. 

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. 

The impulse sampler is, of course, a mathematical fiction an impulse sampler is not physically realizable. But the behavior of a real sampler and hold circuit is practically identical to that of the idealized impulse sampler and hold circuit. The impulse sampler is used in the analysis of sampled-data systems and in the design of sampled-data controllers because it greatly simplifies these calculations. 

A very important theorem of sampled-data systems is ... 

Deadtime in a sampled-data system is very easily handled, particularly if the deadtime D is an integer multiple of the sampling period T,. Let us assume that... 

We are now ready to use the concepts of impulse-sampled functions, pulse transfer functions, and holds to study the dynamics of sampled-data systems. 

Consider the sampled-data system shown in Fig. 18.11a in the Laplace domain. The input enters through an impulse sampler. The continuous output of the process is... 

Figure 18.13 sketches a sampled-data system of input m, y and output x, . Knowing gives us knowledge about Xj, only at the sampling instants. Therefore we only know X( j-,). 

We developed the mathematical tool of z transformation in the last chapter. Now we are ready to apply it to analyze the dynamics of sampled-data systems. Our primary task is to design sampled-data feedback controllers for these systems. We will explore the very important impact of sampling period 7 on these designs. 

First we will look at the question of stability in the z plane. Then root locus and frequency response methods will be used to analyze sampled-data systems. Various types of processes and controllers will be studied. 

The stability of a sampled-data system is determined by the location of the roots of a characteristic equation that is a polynomial in the complex variable z. This characteristic equation is the denominator of the system transfer function set equal to zero. The roots of this polynomial (the poles of the system transfer function) are plotted in the z plane. The ordinate is the imaginary part of z, and the abscissa is the real part of z. 

A sampled-data system is stable if all the roots of its characteristic equation (the poles of its transfer function) lie inside the unit circle in the z plane. 

It is sometimes convenient to write the closedloop characteristic equation of a general sampled-data system as... 

With sampled-data systems, root locus plots can be made in the z plane in almost exactly the same way. Controller gain is varied from zero to infinity, and the roots of the closedloop characteristic equation 1 - - = 0 are plotted. When... 

Notice the very significant result that the damping coefficient is less than one on the negative real axis. This means that in sampled-data systems a real root can give underdamped response. This can never happen in a continuous system the roots must be complex to give underdamped response. 

So we can design sampled-data controllers for a desired closedloop damping coefficient by adjusting the controller gain to position the roots on the desired damping line. Some examples will illustrate the method and will point out the differences and the similarities between continuous systems and sampled-data systems. 

For gains less than this, the system is overdamped. For gains greater than this, the system will be underdamped This is a distinct difiference between sampled-data and continuous systems. A first-order continuous system can never by underdamped. But this is not true for a sampled-data system. 

This little example has demonstrated several extremely important facts about sampled-data control. This simple first-order system, which could never be made elosedloop unstable in a continuous control system, can become closedloop unstable in a sampled-data system. This is an extremely important difference between continuous control and sampled-data control. It points out the fact that continuous control is almost always better than sampled-data control ... 

The examples below illustrate the use of the bilinear transformation to analyze the stability of sampled-data systems. We can use all the classical methods that we are used to employing in the s plane. The price that we pay is the additional algebra to convert to ID from z. 

Thus the Nyquist plot of the sampled-data system does not end at the origin as the Nyquist plot of the continuous system does. It ends on the negative real axis at the value given in Eq. (19.66). 

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