Inverse response compensation

There are two very popular ways to control systems with inverse response the first uses a PID feedback controller with Ziegler-Nichols tuning and the second uses an inverse response compensator. 

The inverse response compensator predicts the inverse behavior of the process and provides a corrective signal to eliminate it. The prediction is based on a model for the process. The ideal prediction comes if the transfer function of the process is completely known. In such case the compensator is given by... 

Modeling inaccuracies in terms of X and t2 will deteriorate the performance of an inverse response compensator (i.e., they will cause increased inverse shoots and sluggish responses). 

Show that the dead-time and inverse response compensators are based on the same logic. What are their implementational difficulties ... 

What is our goal when designing a controller for a system with inverse response Describe what an inverse response compensator does. 

The inverse response compensator was first proposed by Iinoya and Altpeter in the following paper ... 

Perhaps the most common system that has given trouble is base level control via steam. This is particularly true if a thermosyphon reboiler is employed or if the column has valve trays, or both. It is the result of inverse response (see Chapter 13). At low boilup rates, sieve trays give the same trouble. To rninirnize difficulties the design recommendations of Chapter 16, Section 7, should be followed. One of the authors has shown, in an unpublished study, that an inverse response compensator can be designed and implemented on a computer or with some microprocessor controls. 

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]. 

Adding the signal y (s) to the main feedback signal y(s) means the creation of the local loop around the controller as shown in Figure 19.5b. The system in this local loop is the modified Smith predictor and the actual compensator of the inverse response. As can be seen from eq. (19.6), its transfer function is... 

V.6 Design a perfect compensator for the system with inverse response described in Problem III.59. 

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. 

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

Consider the system with inverse response described in Section 19.3. Identify the transfer function of the compensator. Notice that it is a function of the parameter k which must satisfy condition (19.7). Do you have any ideas on how k would affect the quality of the controlled response (See also the numerical example in Ref. 5.)... 

If the control loop has to be tightly tuned because holdup is small, we can design a compensator for inverse response called an inverse response predictor. It is analogous to the Smith predictor for dead-time compensation. ... 

Some of the practical problems in feeding forward from feed compensation have been discussed by Luyben. Luyben has also discussed the problems caused by inverse response in the design of feedforward compensators as well as feedback controllers for composition. 

Figure 23 Calculation of the shape of the actively compensated pulse can be carried out on the software. (A) shows the real (red line) and the imaginary (green line) component of an example of the target pulse shape t>,(f). Its leading and the trailing edges have a cosine shape with a transition time of 1.25 xs in 50 steps, and the width of the plateau is 5 ps. (B) Laplace transformation B(s) multiplied by the Laplace transformed step function U(s). (C) It was then divided by the Laplace transformation Y(s) of the measured step response y(t) of the proton channel of a 3.2-mm Varian T3 probe tuned at 400.244 MHz to obtain V(s). (D) Finally, inverse Laplace transformation was performed on V(s) to obtain the compensated pulse that results in the RF pulse with the target shape. Time resolution was 25 ns, and o = 20 was used for the Laplace and inverse Laplace transformations.

Next, bi(t) was Laplace transformed into B(s), and then multiplied by the Laplace transformation U(s) of the step function u(t). The result B(s)U(s) is displayed in Figure 23B. In this example, the step response y(t) was measured for the 1H channel of a Varian 3.2 mm T3 probe tuned at 400.244 MHz with a time resolution of 25 ns, and Laplace transformed into Y(s). By dividing B(s)U(s) by Y(s), the function plotted in Figure 23C was obtained, from which, by performing inverse Laplace transformation, the programming pulse shape v(t) was finally obtained, as shown in Figure 23D. The amplitude and the phase of the complex function v(t) give the intensity and the phase of the transient-compensated shaped pulse. 

A consequence of incorporating the absorptive filters into the lossless prototype is that the frequency response envelope of the reverberator will no longer be flat. For exponentially decaying reverberation, the frequency response envelope is proportional to the reverberation time at all frequencies. We can compensate for this effect by associating a correction filter t(z) in series with the reference filter, whose squared magnitude is inversely proportional to the reverberation time [Jot, 1992b] ... 

It is seen that in the case of flat top sampling, the sampled signal spectrum is a distorted version of X f) because of the factor T/sSinc(T/). If r < T, this distortion is small and the original signal can be recovered almost exactly. If not, the factor r/j sinc(T/) must be compensated for by an inverse filter having a frequency response of the form [sinc(r/)] before recovery of x t) takes place. 

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