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FIG 5.14. Bidirectional programming requires two on-off controllers with separately adjustable switching points.
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points, the on-off controller will switch between 0 percent and q at the point eh:
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A control system designed for bidirectional optimal switching requires two on-off operators, as depicted in Fig. 5.14. The distance between t he two switching points, expressed in percent,, is the dead zone 2, analogous to the proportional band of a linear controller:
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If the same program is applied to a two-capacity process, the controlled variable will be more heavily damped than necessary. Therefore this program provides the minimum-time swikhing only for dead-time plus integrating processes. Referring back to Fig. 1.21, notice that the overshoot for a two-capacity process under on-off control was less than with dead time because of the reduced gain of the second capacity at, that period. It is possible to take advantage of that gain reduct,ion to save time. Figure 5.15 com-
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FIG 5.15. The minimum-time program (right) actually causes the intermediate variable to overshoot.
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FIG 5.16. The delay timer is actuated by the on-off controller.
pares the response of the previous program to that of the optimum program for a two-capacity process. The program consists of switching from 100 percent output at ei to 0 percent output for a specified time td, after which the steady-state value is selected. Values of el and t(, necessary for minimum-time control can be found by solving the following pair of equations:
td = -72 ln -
Q100
el =
&2 + td) 71
(5.10)
For t he particular case where p = 50 percent,, td = 0.69372 15.3572 el = 71
If the program designed for a dead-time plus integrating process were used, el would have been ~OT~/T~. With the minimum-time program, 100 percent output is retained 0.693r2 longer, which corresponds to the time required to dissipate that additional energy. In order to accommodate t,his more complicated program, a delay timer must be added to the system. The arrangement of the loop for an increasing set-point change is shown in Fig. 5.16. The on-off controller deenergizes when el is reached, sending 0 percent output to the process while the timer is operating. Again, processes do not fall into such neat classifications as twocapacity or single-capacity plus dead-time. The bulk of difficult processes lit between these limits. But the same control function dcscribcd by Fig. 5.16 and Eq. (5.10) can be adjusted to accommodate dead time in addition to two capacities. Equation (5.11) indicates the required settings for optimal switching:
td =
Q -72 In ioo
el = loo(T2
Td) - &Z + 71
Td +
(5.11)
This is one control function whose exact settings can be determined numerically for a process with three dynamic elements. But however difficult the process, settings for the switching parameters can be found which will provide absolute optimum set-point rcsponsc using this system.
1 Selecting the Feedback Controller
Adding a Linear Controller
Optimal switching programs were developed originally for positioning systems and vehicle control. The final state of these processes is generally quiescent, i.c., zero velocity, where no control is required over the steady state. But in fluid processes, control is needed to provide mass and energy balance in the steady state. As a result, programmed control, whose final state is open-loop, is incomplet e. The loop may be closed simply by adding a linear feedback controller to operate in conjunction with the programmed mode. When the programmed action is completed following a set-point change, the linear controller is switched into the loop. In effect, t wo controllers comprise the system, one for the steady st,ate, one for the unsteady state. The output of the control system should match the load, just as was done without the linear controller. If it does not, an error will develop after transfer is made to the linear controller. The addition of a linear controller should not be construed as license t o discount the settings of the program. Instead, the program should be designed just as if there were no linear controller. Every effort should be made to place the controlled variable exactly on the set point with zero velocity when transfer is made. In this way, the controller will have no work to do, and hence will not) disturb the process. To be sure, the linear controller will compensate as well as it can for inaccuracies in the program, and this is beneficial. The linear controller must, above all, be preloaded to the anticipated process conditions at the new set point. This was found to be advantageous whenever an Lantiwindup switch is used in the reset circuit. And because reset is normally required with a dual-mode arrangement, the control system must include an antiwindup switch into which the preload setting is introduced. The arrangement of the system for increasing set-point changes is shown in Fig. 5.17. The sequence of events is as follows: 1. While e > el, the on-off operator is energized, sending 100 percent output t,o the process. The preload setting q is sent to the proportionalplus-reset-plus-derivative controller.
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