Open-loop control definition

After definition of the optimum operation policies, different strategies can be used for actual implementation at the polymerization plant. The simplest strategy is the implementation of u°P at the plant, without using any sort of feedback signal for x and/or y, for evaluation of the operation performance. In mathematical terms, one might define the implementation of open-loop control strategies in the form ... 

Next, we show that feedback reduces sensitivity by comparing the relative sensitivities for open-loop control and closed-loop control. By definition, open-loop control occurs when the feedback control loop in Fig. J.l is disconnected from the comparator. For this condition ... 

As stated in Chapter 1, control refers to a closed-loop system where the desired operating point is compared to an actual operating point and a knowledge of the error is fed back to the system to drive the actual operating point towards the desired one. However, the optimal control problems we consider here do not fall under this definition of control. Because the decision variables that will result in optimal performance are time-dependent, the control problems described here are referred to as optimal control problems. Thus, use of the control function here provides an open-loop control. The dynamic nature of these decision variables makes these problems much more difficult to solve as compared to normal optimization where the decision variables are scalar. 

The second issue for cooled tubular reactors is how to introduce the coolant. One option is to provide a large flowrate of nearly constant temperature, as in a recirculation loop for a jacketed CSTR. Another option is to use a moderate coolant flowrate in countercurrent operation as in a regular heat exchanger. A third choice is to introduce the coolant cocurrently with the reacting fluids (Borio et al., 1989). This option has some definite benefits for control as shown by Bucala et al. (1992). One of the reasons cocurrent flow is advantageous is that it does not introduce thermal feedback through the coolant. It is always good to avoid positive feedback since it creates nonmonotonic exit temperature responses and the possibility for open-loop unstable steady states. 

RGA provides a measure of interactions caused by decentralized control. Let assume that the input My controls the output y,. Let the corresponding gain be g,j when the other loops are open, and when the other loops are closed, so that the other outputs are constant, except for the considered loop. By definition the element Ay of the RGA is given by ... 

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. 

It should also be noted that the open-loop transfer functions given in equations (18.43) and (18.44) are valid, strictly speaking, ftx onty one coinhiiiation of holdup volumes in the condensate receiver and the column base. Studies to date, however, suggest that if these holdups are no larger than the total holdup on the trays, then the open-loop transfer functions are relatively insensitive to variations in terminal volumes. This topic definitely needs more study. In the meantime we should determine terming volumes by material-balance and protective-control calculations before calculating column composition dynamics. 

The NI [4] is a useful tool to analyse the stability of the control loop pairings determined using the relative gain analysis. If a manipulated variable is to be used to control an output variable, then the loop must not become unstable in dynamic situations. The NI can be used to prove that a 2 x 2 matrix is stable however, when n > 2 (if there are more than two input-output variables being paired) the NI can only be used to prove that the control loop is definitely not stable. Then, for the steady-state matrix G described in Equation 9.7, where each element in 6 is rational and open-loop stable [3], the system will definitely be imstable if the NI is negative, i.e. if... 

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