Decay ratio

Foxboro developed a self-tuning PID controller that is based on a so-called expert system approach for adjustment of the controller parameters. The on-line tuning of K, Xi, and Xo is based on the closed-loop transient response to a step change in set point. By evaluating the salient characteristics of the response (e.g., the decay ratio, overshoot, and closed-loop period), the controller parameters can be updated without actually finding a new process model. The details of the algorithm, however, are proprietary... 

Use MATLAB to make plots of overshoot and decay ratio as functions of the damping ratio. 

In terms of design specification, it is not uncommon to use decay ratio as the design criterion. Repeating Eq. (3-29), the decay ratio DR (or the overshoot OS) is a function of the damping ratio ... 

If we assume that an oscillatory system response can be fitted to a second order underdamped function. With Eq. (3-29), we can calculate that with a decay ratio of 0.25, the damping ratio f is 0.215, and the maximum percent overshoot is 50%, which is not insignificant. (These values came from Revew Problem 4 back in Chapter 5.)... 

By and large, a quarter decay ratio response is acceptable for disturbances but not desirable for set point changes. Theoretically, we can pick any decay ratio of our liking. Recall Section 2.7 (p. 2-17) that the position of the closed-loop pole lies on a line governed by 0 = cos C In the next chapter, we will locate the pole position on a root locus plot based on a given damping ratio. 

MATLAB calculation details and plots can be found on our Web Support. You should observe that Cohen-Coon and Ziegler-Nichols tuning relations lead to roughly 74% and 64% overshoot, respectively, which are more significant than what we expect with a quarter decay ratio criterion. 

If we repeat the time response simulations as in Example 5.7C, we should find that the settings for the quarter decay leads to a 51% overshoot (a roughly 0.26 decay ratio), the little overshoot settings have a 27% overshoot, and the so-called no overshoot settings still have about 8% overshoot. 

Usually the proportional gain. Limited to second order systems. No unique answer other than a P-controller. Theoretically can use other transient response criteria. 1/4 decay ratio provides a 50% overshoot. 

Proportional gain, integral and derivative time constants to PI and PID controllers. Cohen-Coon was designed to handle disturbances by preventing a large initial deviation from the set point. The one-quarter decay ratio response is generally too underdamped for set point changes. 

Proportional gain, integral and derivative time constants of PID controllers. Experimental analog of the s = jco substitution calculation. Not necessarily feasible with chemical systems in practice. Tuning relations allow for choices from 1/4 decay ratio to little oscillations. 

The value of the pss calculated from Eq. (9.24) is 2.74 and the measured value<7> is 2.64. While this agreement could be taken as evidence in favor of a triplet mechanism, it could also mean that the twist singlet and twist triplet have similar geometries (see Figure 9.2), and therefore similar decay ratios. In this mechanism we have neglected fluorescence from fraws-stilbene and dihydrophenanthrene formation from cw-stilbene. (See Problem 4.)... 

Evidence that eliminates the triplet mechanism as the mode for the cis-trans isomerization of stilbene upon direct photolysis has been provided by azulene quenching studies.(48) Using the experimentally determined decay ratio a/(l — a) and the triplet mechanism, it is possible to calculate what the effect of azulene is upon the pss. The predicted and observed azulene effects on the direct photoisomerization are shown in Figure 9.6. The failure of the triplet mechanism in predicting the very small changes observed in the pss provides a crucial test that is the basis for rejecting the triplet mechanism. 

For simple alkenes, Saltiel has proposed that the deviation of the decay ratio from unity is a measure of the involvement of Schenck type intermediates (22) in the cis-trans photosensitized isomerization. Table 9.4 gives pss concentration ratios for the 2-pentenes at 30°C ... 

A forcing function, whose transform is a constant K is applied to an under-damped second-order system having a time constant of 0.5 min and a damping coefficient of 0.5. Show that the decay ratio for the resulting response is the same as that due to the application of a unit step function to the same system. 

The decay ratio for the impulse response from equation (iv) =... 

Decay ratio the ratio of maximum amplitudes of successive oscillations... 

Similar observations were made by Saltiel in a study of the isomerization of 2-pentene.14S In contrast to the stilbene system, a single decay ratio could not account for the results with different sensitizers (Table VI). This implies that different intermediates were produced with different sensitizers. 

Cundall has done extensive work on benzene231,237 and acetone243 sensitized isomerizations of the 2-butenes, and in every case reported a photostationary or radiostationary trans/cis ratio of 1.27-1.37. Sato, however, has measured a value of unity for the benzene photosensitized isomerization.510 With higher homologs, from 2-pentene to 2-octene, benzene-sensitized isomerizations yield trans/cis ratios of 1.0,238 while acetone-sensitized isomerization of the 2-pentenes in solution yields a ratio of 1.65.244 At present no explanation is possible for the differences between 2-butene and 2-pentene. Until much more information is gathered relating to rates of triplet energy transfer as functions of olefin structure, sensitizer, and medium, the natural decay ratios of each olefin s common triplet cannot be deduced from photostationary trans/cis ratios. 

Cohen and C00N(27) determined the relationships in Table 7.4 so as to give responses having large decay ratios, minimum offset and minimum area under the closed-loop response curve. 

Murrill<5) has pointed out that the /t decay ratio constraint upon which the previous methods are based has disadvantages, e.g. the decay ratio is determined from the first two peaks and a comparison of the second and third peaks may yield a different result. Since a perfect response would correspond to there being no error at any time, some criterion based upon the minimisation of some function of the total error under the response curve would be logical. Three such functions are commonly employed, viz. ... 

Generally, an objective function is required for the adaptation strategy which guides the adaptation mechanism to produce the best settings of the controller parameters. For example, the A decay ratio specification could be employed, or the ISE criterion (Section 7.11). For instance, if the A decay ratio criterion is used, then, if any change in process parameters leads to decay ratios other than A, the adaptation mechanism adjusts the controller parameters until a A decay ratio in the controlled response is achieved once again. 

In non-viscous organic solvents the fluorescence efficiency of most SNA derivatives is comparable to that of TS as are the yields of trans to ois photoisomerization (33,3A). The double chain derivatives, MSNA and MSN, show substantially higher fluorescence yields (0.2 0.02) in non-viscous solvents with a corresponding decrease in the trans to ois isomerization efficiency (3A). In the sensitized isomerization, the MSN derivatives investigated thus far show only photoisomerization but there is an increase in the triplet lifetime by a factor of 2 however there is no change in the decay ratio of the isomerization precursor and the sum of the benzophenone sensitized isomerization efficiencies of ASA is approximately unity. When water insoluble SNA or MSNA molecules are spread as a film, either pure or in mixtures with insoluble fatty acids, at the air-water interface, it is found that the films show comparable behavior on compression to those of pure fatty acid with an indicated area per molecule of the surfactant stilbenes very close to that of a linear fatty acid (oa. 202 /molecule) (35). This suggests strongly that the TS chromophore in these molecules offers suitable molecular dimensions to pack into a crystalline-like array of linear parafin chains. 

Decay Ratios Representing the to cis Isomers for SNA Fraction of Twisted "P" Intermediates Decaying and MSNA in Different Media at 25°C. ... 

Responses to equal maximum deviation when the controllers are tuned for the noted three decay ratios. 

The loop gain is the product of the gains of the loop component. In a properly tuned loop (decay ratio of 1 4), this gain product should be constant at 0.5. 

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