Processes With Inverse Response

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. 

Cooling-water valve size Twice normal design flow rate 

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. 

Which of these two opposing effects dominates depends on the tray design and operating level. The pressure drops through valve trays change little with vapor rates unless the valves are completely lifted. Therefore the second effect is sometimes larger than the first. If this occurs, an increase in vapor boiiup produces a transient increase in liquid rates down the column. This increase in liquid 

Eventually, of course, the liquid rates will return to normal when the liquid inventory on the trays has dropped to the new steadystate levels. Then the eifect of the increase in vapor boilup will drive down. 

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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. 

The most common case of a process with inverse response is that resulting from the conflict of two first-order systems with opposing effects (Figure 12.5). In this section we limit our attention to the regulation of such processes. Extensions to more complex systems such as those of Table 12.1 are easy and straightforward. 

Figure 19.5 Feedback control of process with inverse response (a) conventional (b) with compensator.

Process with inverse response, (n) Block diagram, (b) Step response, (c) Root locus plot. 

Processes with Inverse Response Model-Based Control... 

Fig. 5a. Part of the spectra obtained with the ID multiple selective INADEQUATE experiment III applied to cembrene 3 dissolved in CDCI3. The spectra were obtained after appropriate processing, showing the responses in the olefinic and the aliphatic regions after selective inversion of carbons...

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. 

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). 

In Section 19.2 we discussed how we can develop a Smith predictor (dead-time compensator) which cancels the effect of dead time. The same general concept of the predictor (compensator) can be used to cope with the inverse response of a process and was proposed by Iinoya and Altpeter [Ref. 5]. 

Part V (Chapters 19 through 22) deals with the description, analysis, and design of more complex control systems, with one controlled output. In particular, Chapter 19 introduces the concept of feedback compensation with Smith s predictor, to cope with systems possessing large dead times or inverse response. Chapter 20 describes and analyzes a variety of multiloop control systems (with one controlled output) often encountered in chemical processes, such as cascade, selective, and split-range. Chapter 21 is devoted exclusively to the analysis and design of feedforward and ratio control systems, while Chapter 22 makes a rather descriptive presentation of adaptive and inferential control schemes why they are needed and how they can be used. 

Compensatory control for processes with large dead time or inverse response... 

All the chapters of Part IV were devoted to the analysis and design of feedback control systems for rather simple processes. In this chapter we are concerned with the feedback control of two special types of systems those with large dead times or inverse responses. We will see that for such systems, conventional P, PI, or PID controllers may not be sufficient to yield the desired response. 

An important example of a physical process that shows inverse response is the base of a distillation column with the reaction of bottoms composition and base level to a change in vapor boilup. In a binary distillation column, we know that an increase in vapor boilup V must drive more low-boiling material up the column and therefore decrease the mole If action of light component in the bottoms xg. However, the tray hydraulics can produce some unexpected results. When the vapor rate through a tray is increased, it tends to (1) back up more liquid in the downcomer to overcome the increase in pressure drop through the tray and (2) reduce the density of the liquid and vapor Ifoth on the active part of the tray. The first effect momentarily reduces the liquid flow rates through the column while the liquid holdup in the downcomer is... 

The flowsheet is decomposed into component parts. These are MIMO subsections of the flowsheet that are approximated by matrices of low-order transfer functions (usually first order with dead time). This decomposition permits process units to be modeled in sufficient detail, allowing inverse response and overshoot phenomena to be represented. 

More complex equations can be used to identify higher order models. For example Equation (2.23) was developed by applying a z-transform to the Laplace form of a second order process (with lags tj and X2, and lead x ) as shown in Equation (2.68). Lead is required if there is PV overshoot (xj, > 0) or inverse response (T3 < 0). 

As shown in the example in this chapter, inverse response is caused by two competing processes - the faster of which takes the process first in a direction opposite to the steady state. We can approximate this as two first-order processes with gains of opposite sign, so that the combined effect is given by... 

The process with an inverse response as well as the dead time process have an amplitude ratio which is frequency independent, they only introduce a phase shift. 

The response is the sum of two parallel processes with different time constant and different gain. The first term in Eqn. (11.39) is a fast first-order response the second term is a slower first-order response in the opposite direction. The values used for pA compared with p are physically not realistic however, it shows mathematically what the response would do. If pA = 2/7, the transfer function is exactly a first-order process, since the second tern is equal to zero, if /7 > 3/7, an inverse response is obtained. In that case, the slowest transfer function has a contribution in the steady state situation, which is opposite the contribution of the fastest transfer function. 

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