Linear Controllers in Visual Studio .NET

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Linear Controllers
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FIG 4.28. Either a pulse train or a pulse duration signal may be used to direct the valve to its new position, at full speed.
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how the valve position would respond to the output of the computer. Note that position is the integral of computer output. It was found that a sampling controller increased in effectiveness as its control interval approached zero. For this reason, the DDC output program should move each control valve to its directed position at full speed. Because of the way the computer dictates a change in valve position, absence of an output signal means that valves will remain in their last position. This feature has two outstanding advantages: 1. Failure of the computer to produce an output signal will not disturb the plant. 2. Transfer from manual to automatic or automatic to manual cannot cause a sudden change in valve position, i.e., LLbumpless kansfer is inherent.
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Digital-control Algorithms
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Because the digital computer calculates a new value of output for a given loop only once each cycle, it cannot solve differential equations. Instead, digital-control algorithms are difference equations, whose time base is the sampling interval. The differential equation for an ideal noninteracting analog controller is
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Transformation to a difference equation yields
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e At + D en dterLpl
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The subscript n designates the present value of the indicated variable, while n - 1 is its value at the time of the last previous sample. Figure 4.28 indicated how the valve is incremented in a digital system. In practice, then, the control algorithm employed does not generate valve position, but rather its increment Am. The incremental equation is the difference between position equations from time n - 1 to n.
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1 Selecting
e, - enhl + 2 e, + g [(e, - endI) - (en-l - e,-2)1] (4.23)
Earlier in this chapt,er the disadvantage of derivative action on the set point was discussed. Although this presented something of a problem wit h analog devices, a digital comput,cr can be readily programmed for derivative only of the measurement : &IX = 7 e, - en-, + g e, + $ (%A-1 - G - CT&z) I Further simplification can be made if proportional action is applied only to the measurement, although set-point response suffers somewhat:
cnpl - c, + 2 (r, - c,) + g (2c,-l - Cn - G-2)
I (4.25)
This algorit~hm is a poor choice where set -point response is important, as in a cascade system (see Chap. 6). Therefore t he entire E:ach of the coefficients above is a pure number. ecpration could be reduced to a sum of the variables multiplied by their respective coeflicicnt~s:
Although this is effect,ively the same equation, the significance of the control parameters has been lost,. So single adjustment set s the damping of the loop, as did the proport ional band; none changes t,he phase In short, the experience angle in the same way as reset or dcrivat ive. and educat ion of the operating personnel is frustrated in an encounter So at the cost of complicating with such unfamiliar control paramekrs. the arithmetic operations somewhat, the value of ret aining t he familiar modes is inestimable.
Selection of the Sampling Interval
The discussion on uncert,ainty in sampled syst,ems concluded t hat the sampling interval should not exceed t,hc open-loop response t ime of the proccas. For best results with easy processes, t he sampling interval should be as short as prac%icable. But where dead time dominat es, the Unforsampling interval is best- keyed to the process response time. t~unatcly, practic*al c~onsidcrnt~ions take precedence over performance for most, applications.
Linear Controllers
Take the case of a process dominated by dead time, settling out about 1 hr after a disturbance. If the sampling interval were set for best performance, the control valve would be repositioned only once an hour. And if the valve program called for a maximum increment of 25 percent per sample, 4 hr would be needed t,o fully stroke the valve. From the viewpoint that an important function of automatic controls is to react to an emergency condition within the plant, a 4-hr valve stroke is intolerable. For this reason, seldom will sample intervals great er than a minute be encount ered. Intervals that are too short also pose a problem. The reset component in the incremental control algorithm is e, At/R. If At is very small with respect to reset time, t his component is subject to truncation or rounding-off . Suppose that t he word length in a given DDC computer is limited to .5 decimals. A deviation that when multiplied by At/R results in a product less than 5 X 10e6 will not be acted upon. As an example, let At = 1 see and R = 50 min. The minimum error causing reset action will be e 1L = (5 X 10+9(.5O)a9~ = 0.01.5 or 1.5%; In this case, a 1.5 percent offset could be expected. Reduction of this offset by a factor of 10 can be accomplished simply by increasing the sampling interval to 10 sec. But 10 set would be too seldom for a flow loop that would ot hcrwise oscillat e at a period of less than 10 sec. Therefore, more than one sampling interval should be used, whose sele&on is based on the speed of the loop in question. This is a reasonable approach, in light of the fact that a choice of at least two ranges of reset time is available in most analog controllers. Although an optimum value of At may exist for certain difficult processes, it is doubtful whether assigning the optimum interval to part,icuIar loops is generally warranted. Since the entire program of the digital computer is based on time usage, adjust ment of At is objectionable. As long as At is not more than twice the response time of a given process, A sampling interval of 1 set performance mill usually bc satisfactory.6 is acceptable for flow control, while somewhere between 10 and 30 see would be suita,ble for most ot,her loops. The Value of Derivative Because Dhe derivative component of output varies as the difference between two successive values of t,he controlled variable, it is fundamentally a measure of the average rate of change during a sampling interval. Put in other terms, it may be said to represent the rate of change of the controlled variable midway between sampling intervals. The effect is that derivative action is delayed by At. It can be seen that derivative
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