Response inverse

It can easily accommodate difficult or unusual dynamic behavior such as large time delays and inverse responses. 

Repeat the time response simulation of inverse response in Section 3.5. Calculate the value of zero in each case. 

In the last chapter we used Laplace-domain techniques to study the dynamics and stability of simple closedloop control systems. In this chapter we want to apply these same methods to more complex systems cascade control, feedforward control, openloop unstable processes, and processes with inverse response. Finally we will discuss an alternative way to look at controller design that is called model-based control. 

Another interesting type of process is one that exhibits inverse response. This phenomenon, which occurs in a number of real systems, is sketched in Fig. 

Of). The initial response of tiw output variable is in the opposite direction to where it eventually ends up. Thus the process starts out in the wrong direction. You can imagine what this sort of behavior would do to a poor feedback controller in such a loop. We will show quantitatively how inverse response degrades control-loop performance. 

Thus the vapor-liquid hydraulics can produce inverse response in the effect of V on Xg (and also on the liquid holdup in the base). 

Mathematically, inverse response can be represented by a system that has a transfer function with a positive zero, a zero in the RHP. Consider the system sketched in Fig. ll.lOn. There are two parallel first-order lags with gains of opposite sign. The transfer function for the overall system is... 

Systems with inverse response and time delay... 

Luyben [14] has recently investigated a class of models that involve both inverse response and time delay. More specifically he has considered the following type of processes ... 

Control of integrating processes with time delay and inverse response (IPTD8t.IV)... 

W.L. Luyben, Tuning of proportional-integral controllers for processes with both inverse response and deadtime, Ind. Eng. Chem. Res. 39 (2000) 973-976. 

Dietary requirements for AAs and protein usually are stated as proportions of the diet. However, the level of feed consumption has to be taken into account to ensure that the total intake of protein and AAs is appropriate. The protein and AA requirements derived by the NRC (1994) relate to poultry kept in moderate temperatures (18-24°C). Ambient temperatures outside of this range cause an inverse response in feed consumption i.e. the lower the temperature, the greater is the feed intake and vice versa (NRC, 1994). Consequently, the dietary levels of protein and AAs to meet the requirements should be increased in warmer environments and decreased in cooler environments, in accordance with expected differences in feed intake. These adjustments are designed to help ensure the required daily intake of AAs. 

With batch reactors, it may be possible to add all reactants in their proper quantities initially, if the reaction rate can be controlled by injection of initiator or adjustment of temperature. In semibatch operation, one key ingredient is flow-controlled into the batch at a rate that sets the production. This ingredient should not be manipulated for temperature control of an exothermic reactor, as the loop includes two dominant lags—concentration of the reactant and heat capacity of the reaction mass—and can easily go unstable. It also presents the unfavorable dynamic of inverse response—increasing feed rate may lower temperature by its sensible heat before the increased reaction rate raises temperature. 

It can be argued that the differences between the compared schemes are mainly due to the different estimation accuracy of the quantity aq (Fig. 5.6). It can be seen that, after the initial transient phase in which the model-free observers present an inverse response, both the adaptive (model-based and model-free) approaches achieve very good estimates. As for the parameter estimate, since both the adaptive observers (0O) and the controller (0C) estimates converge to the true value of 0 (see Fig. 5.7), it is possible to argue that the persistency of excitation condition is fulfilled. 

Figure 3.9 gives results for a 20% step decrease in feed flowrate at time equal 20 min for the 350 K reactor design. The maximum deviation in reactor temperature is much smaller in the 95% conversion process than in the 85% case. Note the initial increase in the reactor temperature when the feed is decreased. This is caused by the feed being colder than the reactor liquid. This inverse response will be discussed in more detail later in this chapter. Figure 3.10 shows the responses for a 20% step increase in feed flow-rate. The 85% case is unstable. 

However, the exit temperature of the reactor with catalyst (rout)cat takes about 2 hours to attain the same steady state. And in fact, it initially actually decreases The inverse response or wrongway effect is caused by the thermal capacitance of the catalyst. 

These results illustrate the drastic effect of catalyst on the dynamics of tubular reactors. It is clear that a control structure that attempts to control reactor outlet temperature by adjusting reactor inlet temperature will exhibit poor performance because of the inverse response. 

Figures 7.23-7.27 show the closed-loop profiles for a 10% increase in the production rate at operating point I (attained by increasing Fo), and a decrease in the purity setpoint to Cb,Sp = 1.888 mol/1 - this reduction is necessary since the nominal purity is beyond the maximum attainable purity for the increased throughput. Although controller design was carried out to account for the inverse response exhibited by the system at operating points II and III, and in spite of the plant-model parameter mismatch, the proposed control structure clearly yields good performance at operating point I as well.

The magnitudes of various flowrates also come into consideration. For example, temperature (or bottoms product purity) in a distillation column is typically controlled by manipulating steam flow to the reboiler (column boilup) and base level is controlled with bottoms product flowrate. However, in columns with a large boilup ratio and small bottoms flowrate, these loops should be reversed because boilup has a larger effect on base level than bottoms flow (Richardson rule). However, inverse response problems in some columns may occur when base level is controlled by heat input. High reflux ratios at the top of a column require similar analysis in selecting reflux or distillate to control overhead product purity. 

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... 

Inverse response creates control difficulties. Assume, for example, that we wish to control the exit temperature of an adiabatic plug-flow reactor by manipulating the inlet temperature as shown in Fig. 4.13. From a steady-state viewpoint this is a perfectly reasonable thing to consider, since there are no issues of output multiplicity or open-loop instability, assuming the fluid is in perfect plug flow and there is no... 

To that end we have constructed a simulation of a fictitious system that has a severe inverse response. We show the design in Fig. 5.28 and give the design parameters in Table 5.1. The reactor has a large Lewis number (Le = 25), nearly complete per pass conversion of the reactant, and little axial dispersion. These are all factors necessary for wrong-way behavior. In fact the example plot of wrong-way behavior shown in Chap. 4 was generated from this reactor. 

TABLE 5.1 Design Data for Reactor with Inverse Response... 

We next verify that there exists a hot stable steady state when the system equations have three stationary solutions. With the temperature controller in manual, we lower the bypass rate from 12 percent to 5 percent. According to Fig. 5.21 this should create three steady states and we would expect the intermediate state to be unstable. Figure 5.30 shows what happens when we reduce the bypass rate. The reactor feed temperature goes up initially due to reduced bypassing. This makes the reactor exit temperature drop due to the inverse response. The drop in the reactor exit temperature causes a drop in the feed temperature below the intermediate steady state (point b in Fig. 5.21). However, this state is unstable so the temperatures continue to oscillate. Slowly, but surely, the temperatures trend toward the hot steady state (point c in Fig. 5.21) where the oscillations die out and the reactor remains stable. 

Tyreus, B. D.. and Luyben. W. L. Unusual Dynamics of a Reactor/Preheater Process with Deadtime, Inverse Response and Openloop Instability, J. Proc. Cont.. 3, 241-251,(1993). 

R-B When the boilup ratio is high bottoms flow should be used to control bottoms composition and heat input should control base level. However, in some columns potential inverse response may create problems in controlling base level with boilup. 

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