First-order sampled-data system

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

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. 

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 frequency response of sampled-data systems can be easily calculated using MATLAB software. Table 15.2 gives a program that generates a Nyquist plot for the first-order process with a deadtfme of one sampling period. 

Some important properties of sampled-data systems can be obtained from long division of their z-transforms. For example, for the first-order z-transform,... 

These results are plotted in Fig. 18.8. The system is closedloop unstable for this value of gain K = 15), as we will see in Chap. 19. Notice that this example shows that a first-order process which is controlled by a sampled-data proportional controller can be made unstable if the gain is high enough. If an analog controller had been used, the first-order process could never be made closedloop unstable. Thus... 

Several important features should be noted. The first-order process considered in Example 19.1 gave a pulse transfer function that was also first-order, i.e., the denominator of the transfer function was first-order in z. The second-order process considered in this example gave a sampled-data pulse transfer function that had a second-order denominator polynomial. These results can be generalized to an Nth-order system. The order of s in the continuous transfer function is the same as the order of z in the corresponding sampled-data transfer function. 

Example 19.7. The first-order lag process, zero-order hold, and proportional sampled-data controller from Example 19.1 gave an openloop system transfer function... 

Design a minimal-prototype sampled-data controller for a first-order system with a deadtime that is three sampling periods. The input is a unit step change in setpoint. 20.6. Design minimal prototype controllers for step changes in setpoint and load for a process that is a pure integrator. 

Because reactions in solids tend to be heterogeneous, they are generally described by rate laws that are quite different from those encountered in solution chemistry. Concentration has no meaning in a heterogeneous system. Consequently, rate laws for solid-phase reactions are described in terms of a, the fraction of reaction (a = quantity reacted -r- original quantity in sample). The most commonly encountered rate laws are given in Table 1. These rate laws and their application to solid-phase reactions are described elsewhere. 1 4 10-12 Unfortunately, it is often merely assumed that solid-phase reactions are first order. This uncritical analysis of kinetic data produces results that must be accepted only with great caution. 

There are two competing and equivalent nomenclature systems encountered in the chemical literature. The description of data in terms of ways is derived from the statistical literature. Here a way is constituted by each independent, nontrivial factor that is manipulated with the data collection system. To continue with the example of excitation-emission matrix fluorescence spectra, the three-way data is constructed by manipulating the excitation-way, emission-way, and the sample-way for multiple samples. Implicit in this definition is a fully blocked experimental design where the collected data forms a cube with no missing values. Equivalently, hyphenated data is often referred to in terms of orders as derived from the mathematical literature. In tensor notation, a scalar is a zeroth-order tensor, a vector is first order, a matrix is second order, a cube is third order, etc. Hence, the collection of excitation-emission data discussed previously would form a third-order tensor. However, it should be mentioned that the way-based and order-based nomenclature are not directly interchangeable. By convention, order notation is based on the structure of the data collected from each sample. Analysis of collected excitation-emission fluorescence, forming a second-order tensor of data per sample, is referred to as second-order analysis, as compared with the three-way analysis just described. In this chapter, the way-based notation will be arbitrarily adopted to be consistent with previous work. 

A sampling of the type of data obtained from this experiment is given in Figure 4.3.1b. Kinetic constants can be calculated from these data using analyses like those presented above for the simple reversible, first-order system [Equation (4.3.1)]. 

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