Open-Loop Behavior

In Chaps. 4 and 5 we discussed the open-loop behavior of reactors with process feedback. We showed how nonlinearities combined with 

So far we have been discussing output multiplicity as the primary source of open-loop instability. It is also possible to have input multiplicity in distillation systems. Input multiplicity means that we can get the same output for different levels of the input variables. 

Another phenomenon of highly nonlinear systems is parametric sensitivity. We illustrated this behavior for the temperature profile in the plug-flow reactor. Nonideal distillation systems can also show this sensitivity. For example, in Fig. 6.5 a small change in the feed composition or organic reflux flow can dramatically change the composition ( and t emperature) profile in the column. Instead of a vinyl acetate-rich profile in the top section, a water-rich profile can be present. 

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Open-loop behavior of multicomponent distillation may be studied by solving modifications of the multicomponent equations of Distefano [Am. Inst. Chem. Eng. J., 14, 190 (1968)] as presented in the subsection Batch Distillation. One frequent modification is to include an equation, such as the Francis weir formula, to relate liquid holdup on a tray to liquid flow rate leaving the tray. Applications to azeotropic-distillation towers are particularly interesting because, as discussed by and ihustrated in the Following example from Prokopalds and Seider... 

Most chemical processes exhibit stable open-loop behavior. However, there are some important processes, such as chemical and biological reactors, that might be operated around an unstable steady state. The tuning of PID controller forunstable systems has recently attracted attention [16-19], Lee et al. [18] have presented a powerful IMC-based tuning methodology that appears to surpass most of available tuning techniques for unstable systems with low order dy-... 

Figure 4.9 shows the results of a dynamic simulation we performed featuring the open-loop behavior of a backmixed reactor that satisfies the slope condition for steady-state stability but has dynamically unstable roots. Table 4.1 contains the reactor parameters and operating conditions used in the model, as listed by Vleeschhouwer et al. (1992). 

The basic concepts of modeling, open-loop behavior, reactor control, and plantwide control apply7 to polymer processes. It would be folly to attempt a comprehensive treatment of the subject in this text. We refer... 

Here we have dealt with the control of chemical reactors. We covered some of the fundamentals about kinetics, reactor types, reactor models, and open-loop behavior. In particular we have shown that reactors with recycle or backmixing can exhibit multiple steady states, some of which are unstable. Nonlinearities in reactor systems also frequently give rise to open-loop parametric sensitivity. 

A given process has unknown detailed dynamics that is, it exhibits overdamped open-loop behavior but its exact order and parameter values are poorly known. From the process reaction curve (see Section 16.5) we have approximated its transfer function by the following first-order system with dead time ... 

It has already been said in the introduction, that the core problem for the control engineer is not the nonlinearity of the plant, but that the plant nonlinearity may or may not require a nonlinear controller. The consequent idea of control-relevant nonlinearity as introduced in Ref 24 is to examine the nonlinearity of suitable controllers for a given plant instead of just analyzing the plant s open-loop behavior. The controller nonlinearity will be determined by the three factors... 

The key is to recognize that the system may exhibit underdamped behavior even though the open-loop process is overdamped. The closed-loop characteristic polynomial can have either real or complex roots, depending on our choice of Kc. (This is much easier to see when we work with... 

A set of experiments on gas-liquid motion in a vertical column has been carried out to study its d3mamical behavior. Fluctuations volume fraction of the fluid were indirectly measured as time series. Similar techniques that previous section were used to study the system. Time-delay coordinates were used to reconstruct the underl3ung attractor. The characterization of such attractor was carried out via Lyapunov exponents, Poincare map and spectral analysis. The d3mamical behavior of gas-liquid bubbling flow was interpreted in terms of the interactions between bubbles. An important difference between this study case and former is that gas-liquid column is controlled in open-loop by manipulating the superficial velocity. The gas-liquid has been traditionally studied in the chaos (turbulence) context [24]. 

Choose the best value of Kp and compare the behavior with the behavior for the open loop system without controls, i.e., for Kp = 0.0. Show that it is possible to remove the offset by using a proportional plus integral (PI) controller and find the best value of Kj that can be used with the best value for Kp as obtained above. Plot the change of height with time and compare it with the results of the open-loop system in the case without control and also when the system has only proportional control. 

Since FCC units are usually operated at their middle unstable steady state, extensive efforts are needed to analyze the design and dynamic behavior of open loop and closed loop control systems to stabilize the desirable middle steady state. 

The dynamic behavior of industrial unit 1 is studied in this section. The steady-state behavior of unit 1 has been studied previously. Both open-loop and closed-loop feedback controlled configurations will be presented. 

General Second-Order Element Figure 8-3 illustrates the fact that closed-loop systems can exhibit oscillatory behavior. A general second-order transfer function that can exhibit oscillatory behavior is important for the study of automatic control systems. Such a transfer function is given in Fig. 8-15. For a unit step input, the transient responses shown in Fig. 8-16 result. As can be seen, when t, < 1, the response oscillates and when t, < 1, the response is S-shaped. Few open-loop chemical processes exhibit an oscillating response most exhibit an S-shaped step response. 

The final open-loop reactor issue we discuss is the problem of inverse response or wrong-way behavior as it is called in the reactor engineering literature. The inverse response refers to the temporary increase in the exit temperature in some packed, plug-flow reactors following a decrease in the feed temperature (Fig. 4.12). The wrong-way behavior stems from the difference in propagation speed between concentration... 

Figure 5.29 Open-loop dynamic behavior of packed adiabatic plug-flow reactor with FEHE.

State feedback control is commonly used in control systems, due to its simple structure and powerful functions. Data-driven methods such as neural networks are useful only for situations with fully measured state variables. For this system in which state variables are not measurable and measurement function is nonlinear, we are dependant on system model for state estimation. On the other hand, as shown in figure 2, in open-loop situations, system has limit cycle behavior and measurements do not give any information of system dynamics. Therefore, we use model-based approach. 

Control systems may be classified from their signal flow diagrams as either open-loop systems or closed-loop systems depending on whether the output of the primary control circuit is fed back to the controlling component. As Fig. 2 suggests, the typical control circuit consists of sequential arrays of components deployed about the process under control. If the controller is not apprised of the behavior of the controlled variable, the control system is an open-loop one. Conversely, if the measuring means on the controlled variable sends its signals back to the controller so that the behavior of the controlled variable is always under the scrutiny of the controller, the system is a closed-loop or feedback control system. 

A simple example of an open-loop control system would be a steam-jacketed resin kettle very much like that in Fig. 1 except that the steam pressure is regulated automatically by the behavior of the measured jacket pressure but not by the actual temperature of the resin batch in the kettle. In the corresponding closed-loop system the steam pressure is regulated by the temperature of the resin batch as in Figs. 1 and 2. The only way open-loop control can be precise is through a close calibration between steam pressure and batch temperature. Since this close calibration can be maintained inexpensively only in the absence of load changes of any kind, it is obvious that the field of application of open-loop control is limited. In the example of Fig. 1, load changes would result... 

An elaboration of open-loop control which sometimes can be competitive with closed-loop control is feed-forward control, in which the controller is apprised of factors which affect the key process variable to be controlled, but is not directly apprised of the behavior of this variable. Thus for the kettle of Fig. 1, it would be possible to measure the flow rate and temperature of the feed and have a simple computer analyze the information and on the basis of this information set the steam pressure to be maintained by the controller. So long as the computer considered all pertinent factors, the control would be satisfactory, but an unaccounted factor, such as a change in agitator speed, might make the control ineffectual. 

A generally used set of criteria for good control is that the controlled variable in response to a unit step change in set point (a) overshoot by not more than 20 per cent of the step and (b) damp out with a subsidence ratio of about one-third. This behavior is approximated by many systems if the closed-loop frequency response and the corresponding open-loop frequency response have certain simple characteristics. Since the closed-loop frequency response characteristics can be determined readily from the open-loop frequency response, the latter characteristics of simple control systems can be used as a convenient basis for design. 

There are a couple of simple empirical approaches for estimating the optimum controller settings for a particular process. Both approaches require data on the response of the existing process to simple stimuli one the open-loop response to a step the other the behavior of the closed-loop at the condition of ultimate gain. 

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