1 Multiple-loop Systems in .NET framework

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example 7.5
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Consider a process wherein two pairs of variables interact such that X1, = 0.7 and XIZ = 0.3. Yet c, responds to m, with a dead time of 10 min, but to m2 with a dead time of only 1 min. In this case, 701 would be 40 min and 702, 4 min. If c1 is paired with m,, it will be disturbed by m2 by the amount 0.3, that is, XIZ, because the dynamic coupling factor is 1.0.
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But if cl were paired with m2, it would be disturbed by ml by only about 0.15, because the ratio of the periods is 10. But this arrangement would allow a higher controller gain for cl, reducing its sensitivity to disturbances of other periods.
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198 1 Multiple-loop Systems
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Unfortunately, however, the pairing of variables also affects the other loop, which must be considered. So some judgment is necessary to determine which of the variables is more deserving of precise control.
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Controlling a single-variable process is comparatively easy even if the dynamics in the loop are unfavorable. There is only one way to close the loop. When a second pair of variables appears, however, the picture is entirely different-not only must a choice be made between pairs used for control, but coupling can exist. And if there is coupling, the ease of control that was found with independent loops disappears. This facility can be restored, however, by decoupling the variables through a computing system. Just as a single valve can affect several controlled variables through the natural coupling of the process, a single controller can be made to adjust several valves through a decoupling system. Consider the problem that an engineer is faced with when he starts out to design a system to control a complex process like a distillation column. He may have five pairs of variables to connect: Throughput Distillate composition Bottoms composition Bottoms level Accumulator level Feed flow Distillate flow Reflux flow Steam flow Bottoms flow
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There are factorial five or 120 different ways to connect these variables. If coupling exists, which is often the case, even the best way of arranging these loops may of itself be inadequate. Yet it is possible to design one control system which will surpass all others in performance by completely decoupling its process. This is the mirror image of the steady-state process model. It is unique, and as such, it can never be found through trial and error or by accident. T h e balance of this chapter will be devoted to its pursuit. The balance of the book will demonstrate its application to the more commonly encountered multivariable processes.
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Reversing the Process Model
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In the process, the values of all the controlled variables are related to all the manipulated variables through a series of equations known as the process model. An ideal control system would be one which correctly positioned all the manipulated variables so as to satisfy all the set points. In this sense, the control system could have the same mathematical
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Multivariable Process Control
structure as the process, but would solve for manipulated variables. Since each process has its own individual mathematical model, there exists only one control system which can solve the same equations. The procedure for arriving at this system is not evolutionary or intuitive-it is as exact and as well defined as the process model. Assuming that the process model is well defined, a computing system can be designed which will solve these same equations for the manipulated variables. The computing system will ordinarily be made up of multipliers and dividers as well as linear operators, because these functions exist in the process. (Decoupling systems cannot be expected to be linear when their processes are not.) The computing system must be fed input data if it is to generate any output. Three items of information are available from each loop: set point, measurement, and controller output. (The error signal cannot be used in the steady-state model because it has no steady-state value.) Of this information, the set points are most useful because they represent the exact demands on the control system and are free from feedback transients, lying outside the loops. But if a measurement is used to decouple fully coupled loops, a positive feedback path is formed through the process, cancelling the effect of control action. As a result, the system has no direction and the controlled variable tends to float. Measurements may be used in systems with halfcoupled loops, however, because there is no feedback from one loop to another through the process. The output of a feedback controller is an unknown variable. If it were known, or predictable with sufficient accuracy, the controller would not be needed. But the output eventually finds its correct level, and in so doing, solves for all the unknowns in the process. Thus the controller output contains more information regarding the manipulated variables than either measurements or set points. The set point will not indicate the presence of a load change, for example. So for the most part, controller outputs will be combined to perform decoupling. The general configuration of a coupled process and decoupling control system would appear as shown in Fig. 7.8.
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