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DESIGN OFSAMPLED-DATA CONTROLLERS
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In this chapter, sampled-data control theory will be applied to the design of direct-digital control algorithms. In the most general terms, direct-digital control is the automatic control of a process by means of a digital computer. Today the single-station, analog-type continuous controller (pneumatic or electronic) has been almost completely replaced by an instrument that is essentially a small, self-contained digital computer. Such instruments are described as microprocessorbased controllers. This change, of course, was brought about by the great decrease in the cost of computing components and the tremendous increase in the speed of computation. In this chapter, the design philosophy for designing special purpose controllers will be developed and illustrated with some examples. The block diagram of the control system to be considered is shown in Fig. 27.1 The elements of the block that are implemented by the computer are enclosed by a dotted line and labeled computer. These elements, which consist of two impulse-modulation switches, the Z-transform of the digital control algorithm D(z), and the zero-order hold Gh( s), will be described later. For the moment, it is necessary to understand only the general operating features of the control system. To simplify the discussion, Gp(s) in Fig. 27.1 contains the valve, the currentto-pressure converter, and the process. The transfer function for the measuring element has been taken as one; for this reason, no measurement block is shown in the feedback path of Fig. 27.1. The output from the hold is a current (or voltage) signal. 405
SAMPLED-DATA
CONTROL
SYSTEMS
I - - - - -
Computer - - - - - - - - - - - - - - - - - - - - _
I 0 FIGURE 27-1
I 2T
I 3T
I 4T
Block Diagram for a computer control system.
Every T units of time, the computer reads and stores the measured value of the process variable C. The computer operates on this signal, according to the algorithm D(z) stored in it, to produce a signal to the valve M,. It is assumed that the computation of M, is instantaneous, relative to the sampling period of the process. For many chemical processes that am slow, this is a reasonable assumption. By means of the hold, the signal to the valve, M,, is held constant (i.e., clamped) between sampling instants. Consequently, the valve response during transient operation of the process will resemble a stair-step function. The control algorithm is simply a mathematical description that tells the computer how to calculate the signal to the valve each sampling instant. The digital computer implements an algorithm of the form (27.1) m(nT) = -&ieK n - i)T] - Ahjm[(n - j)T] i=o j=l This equation gives the value at which me(t) is to be held constant during the following sampling period, that is, m&) = m(nT) for nT I t < (n + l)T The term T is the sampling period and gi and h j are constants. The set of constants (gi, hj) constitutes the control algorithm. In the following pages, methods will be developed for finding these constants for a specific design of a controller. To understand Eq. (27.1) more readily, consider the case where k = 2 and p = 2. If we want to compute m at the one-hundredth sampling instant, Eq. (27.1) is written m(lOOT) = gae(lOOT) + gle(99T) + gze(98T) - hlm(99T) - hzm(98T)
DESIGN
SAMPLED-DATA
CONTROLLERS
97T 98T 99T 100T 102T 101T 103T t FIGURE 27-2 Qpical relationship between m,(t) and e(r) for a computer control system.
Figure 27.2 illustrates the nature of the signals used in this expression for m(lOOT). Notice that m(lOOT) is computed instantaneously at t = 1OOT. For this particular example, the computation requires the present value of error, two past values of error, and two past values of manipulated variable. The more constants (gi, hj) in the algorithm, the more complex it becomes in terms of computer storage and computer time needed to solve the algorithm. To illustrate how an algorithm of the fotm of Eq. (27.1) is derived, several algorithms will be derived for a process consisting of a first-order process with transport lag. ALGORITHMS FOR A FIRST-ORDER WITH TRANSPORT LAG MODEL A variety of algorithms will be derived for a process with a transfer function that is first-order with transport lag, that is,
G,(s) = 5
(27.2)
Figure 27.1 is redrawn as Fig. 27.3 with GP(s) expressed as a first-order process with transport lag and G&) expressed as a zero-order hold. In Fig. 27.3 the clamped signal M, is obtained from the zero-order hold, which obtains its input
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