Loop gain, variable

Requires K12K21 = 0. Open-loop gain is the same as the closed-loop gain. The controlled variable (or loop) / is not subject to interaction from other manipulated variables (or other loops). Of course, we know nothing about whether other manipulated variables may interact and affect other controlled variables. Nevertheless, pairing the i-th controlled variable to they-th manipulated variable is desirable. 

The open-loop gain is zero. The manipulated variablej has no effect on the controlled variable i. Of course nij may still influence other... 

The dead time of a heat exchanger equals its volume divided by the flow rate through it. As process flow increases, the process dead time is reduced and the loop gain is also decreased. If the controlled variable (Th2 in Figure 2.105) is differentiated with respect to the coolant flow (manipulated variable Fc), the steady-state gain of the process is given by ... 

The closed-loop gain A" is the gain in the i-j loop when all the other loops are closed, that is, their controlled variables are held at their set points ... 

The relative gains may be evaluated from open-loop gains obtained from a steady-state simulator, and from derived closed-loop gains. To determine the open-loop gains from the simulator, the manipulated variables are perturbed one at a time, and the variation in each controlled variable is recorded. If a manipulated variable Mj is perturbed by AMj, resulting in a variation AQ in controlled variable Cj, the gain is approximated as... 

The closed-loop gains can be derived mathematically from the open-loop gains. For a two by two process, the variations AC, and ACj resulting from changes in both manipulated variables are given by the equations... 

The open-loop gains are determined from steady-state simulation. The responses of the controlled to manipulated variables are expected, in general, to be nonlinear, and therefore the gains are best determined by perturbing each manipulated variable in both directions around the design conditions. For the specified compositions, the design reflux rate is 99.91 kmol/h and the design reboiler duty is 3.491 X 10 kJ/h. Each variable is perturbed by 0.25% while the other is held constant. The results are tabulated below. 

For measurement of quartz crystal damping the amplitude limiting can be replaced by an automatic level control (ALC). For this purpose the oscillator, Fig. 21, must be modified by opening fhe feedback loop and inserting a variable gain amplifier. The confrol variable effecfing the loop gain is proportional to the series resonance resistance Rs. 

Figure 12.16-left illustrates the open loop gain for manipulation, L and V, on the controlled variables and xb. Figure 12.16-right shows the scaled gains produced by a disturbance in feed. On the range where Gd is less than one the rejection do not need control. In this case the disturbance in feed Gd.i l seems not to affect the distillate composition, but Gd,2l l indicates that control for purity of bottoms is needed. Indeed, from the control problem definition the distillate purity is loose, while the purity of bottoms is much more strict. 

Although a pH process is nonlinear, its characteristic curve cannot be corrected with an equal-percentage valve, because the valve acts on the output of the controller, not on the input. The valve characteristic, in fact, made matters worse. Sot only did the loop gain become variable, but it was higher than it would have been witli an cquivnlcnt linear valve. 

Elements with nonlinear properties appear both in processes and in their control systems. Up to this point an effort has been made to compensate for severe nonlinear elements naturally occurring in the process, so as to maintain a constant loop gain. But in this chapter the effects of variable loop gain will be thoroughly explored in a search to improve performance and economy. 

Because gain and phase both change with the amplitude of the con-troller-output oscillation, gain and period of the closed loop are both variable in the presence of hysteresis. A condition like this is cause for concern, in that stability is also variable. Loop gain should be checked for selected values of amplitude to determine whether stability is conditional. 

The problem of variable dead time was discussed in Chap. 2. This problem was resolved by using an equai-percentage control valve. If a flow loop were placed around the valve, however, its characteristic would be lost. Furthermore, if the flow loop were of the differential type, its nonlinearity would be in the wrong direction, making the primary-loop gain vary inversely as the square of flow. These factors deserve careful consideration before deciding on a cascade flow loop. 

Figure 6.5 has a divider within the closed loop, regardless of which variable is affected by the output of the controller. If X is manipulated, loop gain changes with the wild variable Y ... 

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