Zero order hold element

Theoretically there is no output from a sampler between sampling times (Fig. 7.88c). In practice the sampler output is controlled between samples by a filter or hold element. The most common type of filter is the zero-order hold element (ZOH) in which the value of the previous sample is retained until the next sample is taken (Fig. 7.92a). 

Fig. 7.92. Control loop containing a sampler and a zero-order hold element (a) block diagram (b) output of hold element (filter)...

As f (s) is the Laplace transform of the output of the sampler without the ZOH, the remaining term must represent the Laplace transform for the zero-order hold element, i.e. ... 

Gzoh Transfer function of zero order hold element — —... 

Supervisory control and data acquisition Solid-state relay Self-tuning regulator Temperature recorder controller Transistor-transistor logic Zero order hold element... 

Equation (27.4) yields the first-order hold, and the continuous signal it produces is shown in Figure 27.5c. Notice that the first-order hold element needs at least two values to start construction of the continuous signal, whereas the zero-order hold needs only one. 

It is possible to develop second-, third-, or higher-order hold elements. They need three, four, or more initial discrete-time values before they can start the construction of a continuous signal. As the order of a hold element increases, the computational load increases and becomes more complex, with marginal improvements in the quality of the reconstructed signal. Therefore, for most process control applications the zero-order hold element provides satisfactory results with low computational load and is normally used. To improve the quality of a reconstructed signal, it is better to decrease the period between two successive discrete-time values rather than increase the order of the hold element. 

Example 27.4 Comparing the Results of Zero- and First-Order Hold Elements... 

If we retain only the zero-order term (i.e., the constant) we take the zero-order hold element [eq. (27.3)],... 

The output of a zero-order hold element is like a pulse, having a constant height equal to m(nT) and duration T. After recalling that the Laplace transform of a unit pulse is given by eq. (7.12),... 

The output of the hold element to the impulse c (0) depends on the order of the hold element. For a zero-order hold element its output is a pulse of height c(0) and duration T (Figure 29.5a). This pulse is the input to the process and produces the output MO shown in Figure 29.5b. How can we compute this output ... 

VII.25 Compute the discrete-time, closed-loop response to unit step changes in the load for each of the systems described in Problem VII.24. Assume zero-order hold element and sampling period T = 1 sec. 

The last equation implies that the transfer function of a zero-order hold element is given by... 

Discuss the mathematical basis for the construction of various orders of hold elements. Develop the time-domain expressions for zero- and first-order hold elements. Describe their functions in physical terms. Can you construct simple electrical circuits that function as zero- and first-order hold elements ... 

For hold we have used the zero-order hold element with... 

Compute the pulse transfer function of the following second-order process with and without a zero-hold element ... 

VII.5 Reconstruct the continuous signal from the following sampled data using zero- and first-order hold elements. 

VII.21 Find the pulse transfer functions in the z-domain of the continuous systems with the following transfer functions in the Laplace domain. Assume a zero-order hold element. 

Assume a zero-order hold element and approximate the process dead time by the nearest larger integer multiple of sampling periods. 

Several kinds of hold elements exist. The best known and most used is the zero-order hold with the transfer function ... 

In (17-59), Gc(z) is the discrete transfer function for the digital controller. A digital controller is inherently a discrete-time device, but with the zero-order hold, the discrete-time controller output is converted to a continuous signal that is sent to the final control element. So the individual elements of G are inherently continuous, but by conversion to discrete-time we compute their values at each sampling instant. The discrete closed-loop transfer function in (17-59) provides a framework to perform closed-loop analysis and controller design, as discussed in the next section. Additional material on closed-loop analysis for discrete-time systems is available elsewhere (Ogata, 1994 Seborg et al., 1989). 

Therefore, zero, one, two or all three coordinates change their signs, but this only holds for symmetry elements of the first and second order when they are aligned with one of the three major crystallographic axes. Symmetry operations describing both diagonal symmetry elements and symmetry elements with higher order (i.e. three-, four- and six-fold rotations) may cause permutations and more complex relationships between the coordinates. For example ... 

The important conclusions, therefore, to be drawn from the Debye theory are that at low temperatures the atomic heat capacity of an element should be proportional to T, and that it should become zero at the absolute zero of temperature. In order for equation (17.4) to hold, it is necessary that the temperature should be less than about 9/10 this means that for most... 

We see therefore that no part of the Coulson-Rushbrooke Theorem on alternant hydrocarbons depends on having all non-zero Hamiltonian matrix-elements, Hrs, equal. In order for the reader to be quite clear which assumptions, in the context of the simple Hiickel-method, are necessary for the Theorem to hold, we summarise them again below. We require... 

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