Nonlinearity measurements

The interesting (and important) difference is in the values for the ratio of sums-of-squares, which is the nonlinearity measure. As we see, at small values of nonlinearity (i.e., k — 0,1, 2) the values for the nonlinearity are almost the same. As k increases, however, the value of the nonlinearity measure decreases for the case of Normally distributed data, as compared to the uniformly distributed data, and the discrepancy between the two gets greater as k continues to increase. In retrospect, this should also not be surprising, since in the Normally distributed case, more data is near the center of the plot, and therefore in a region where the local nonlinearity is smaller than the nonlinearity over the full range. Therefore the Normally distributed data is less subject to the effects of the nonlinearity at the wings, since less of the data is there. 

As a quantification of the amount of nonlinearity, we see that when we compare the values of the nonlinearity measure between Tables 67-1 and 67-3, they differ. This indicates that the test is sensitive to the distribution of the data. Furthermore, the disparity increases as the amount of curvature increases. Thus this test, as it stands, is not completely satisfactory since the test value does not depend solely on the amount of nonlinearity, but also on the data distribution. 

In Chapters 63 through 67 [1-5], we devised a test for the amount of nonlinearity present in a set of comparative data (e.g., as are created by any of the standard methods of calibration for spectroscopic analysis), and then discovered a flaw in the method. The concept of a measure of nonlinearity that is independent of the units that the X and Y data have is a good one. The flaw is that the nonlinearity measurement depends on the distribution of the data uniformly distributed data will provide one value, Normally distributed data will provide a different value, randomly distributed (i.e., what is commonly found in real data sets) will give still a different value, and so forth, even if the underlying relationship between the pairs of values is the same in all cases. 

Thus, the user must make a trade-off between the amount of computation performed and the exactness of the calculated nonlinearity measure, taking into account the actual amount of nonlinearity in the data. However, if sufficient points are used, the results are stable and depend only on the amount of nonlinearity in the original data set. 

As we recall from the previous chapter [5], the nonlinearity measure we devised equals the first divided by the second. Let us now develop the formula for this. We will use a subscripted small a for the coefficients of the quadratic equation, and a subscripted small k for those of the linear equation. Thus the equation describing the quadratic function fitted to the data is... 

The nonlinearities measured by DFWM reveal a smooth saturation of the second-order hyperpolarizability around ten monomer units as well (Fig. 26). Below the saturation, the experiments show a power law dependence of the second-order hyperpolarizabilities with an exponent a=2.64 0.20. Comparing the exponents for DFWM and THG, the difference is small and within the experimental error. The absolute values of the second-order hyperpolarizabilities are similar with an increasing deviation for longer PTAs. 

Barzoukas and Blanchard-Desce proposed an approach of molecular engineering using multivalence-bond state models [55]. Push-pull polyenes were shown also to present an enhancement of the TPA response and a loss of transparency of molecules, as a function of the increase of the polyenic chain length [56,57]. Trends observed in these polyenic systems are supported by the large third-order optical nonlinearities measured in asymmetric carotenoids, in which the role of the large value of dipole moment difference A/z was shown [58]. 

Carroll, R.J., Juchenhodd, H., Lombard, F., and Stefanki, L.A. Asymptotics for the SIMEX estimator in nonlinear measurement error models. Journal of the American Statistical Association 1996 91 242-250. 

The second order optical nonlinearity shown in these figures has been measured by second harmonic generation as previously described. Relevant optical data and measured values of optical nonlinearity (measured for the polymer systems reported in this communication) are summarized in Table 1. 

Improvements in these different techniques may come from either improved diodes or detection schemes. The nonlinearities measured in the current-VOItage curves of our MIM diodes are extremely small and conversion efficiencies could be 100 times larger. We are optimistic that better materials which will result in larger FIR powers may be found. Differential detection schemes would also significantly improve the sensitivity and permit the detection of weaker lines. The sensitivity, however, is still only about 1% of that of laser magnetic resonance. Laser magnetic resonance is useful only for paramagnetic... 

This is qualitatively similar to Eq. (6.13) which describes the behavior of a micropore-controlled system, except that the concentration dependence of the effective diffusivity is stronger. The solution for the uptake curve shows similar general features. The fractional uptake is a function of the dimensionless time variable D t/R and the step size or nonlinearity, measured by the parameter X = small values of X the system approaches linearity. 

Finally, methods of verification of obtained impedances and the modeling of experimental data are discussed. The last two chapters deal with applications of nonlinear measurements and instrumental limitations. 

Schweickhardt and Allgower in Chapter A3 mainly concentrate on the nonlinearity assessment of processes. A comprehensive overview of general nonlinearity measures and a thorough investigation of the predictive and computational dimension of open loop measures are presented. As the main objective becomes the development of a tool to judge whether a nonlinear controller should be benefieial or needed for a particular process with specific nonlinear characteristics, the controller relevant nonlinearity is quantified. The selected measure is based on the relative differences between the output of nonlinear state feedback law and that of an equivalent linear state feedback law. The controller relevant nonlinearity measure depends not only on the plant dynamics and region of operation but also on the performance criterion used in the derivation of the controller law. 

Figure 7 is a plot of the nonlinearity measure (5) as a function of reactor feedrate. The results show severe nonlinearity in the region of steady-state gain change seen in Figure 5. The results further suggest essentially linear behavior at high flowrates and increasingly nonlinear behavior at low flowrates.

Figure 7 Nonlinearity in yi as a function of reactor feedrate as characterized using the lower bound of the nonlinearity measure.

The proposed metrics in this chapter should be considered as purely starting points for use in characterization of the three process attributes of extent of interaction, dynamic character and nonlinearity and not as the final solution. Further theoretical development of a joint metric of the three quantities should be considered keeping in mind the need for the metric to be controlrelevant, as described in the discussion of the nonlinearity measure. Only once a clear definition of the mappings between the process characterization and controller design cube exists will these techniques be able to be used to their fullest potential. Work on clarifying these mappings is on-going (e.g., nonlinearity Ref [25]). 

Quantitative nonlinearity assessment -An introduction to nonlinearity measures... 

Nonlinearity measures represent an approach to systematically quantify the degree of nonlinearity of a system. However, it is for example easy to formulate an observer for a Hammer-stein-type system (i.e. a static nonlinearity followed by a linear system) because the problem is linear, while it may be more involved to design a controller for the same system. The goal of nonlinearity measures is therefore to reveal whether the system s nonlinearity is crucial or not in the context of the given task. 

This chapter on nonlinearity quantification introduces the basic concepts of nonlinearity measures and shows the insights into a system s behaviour and structure they can deliver. After introducing the concept of nonlinearity measures for general I/O-systems, the presentation will focus on the control-relevant nonlinearity characterization, i.e. the relevance of the system nonlinearity with respect to controller design. 

The exposition is structured as follows Sec. 2 introduces the fundamentals of nonlinearity measures. First, an introduction to nonlinearity measures based on signal norms is given. We then take a closer look at one specific, particularly useful formulation of a nonlinearity measure. An efficient computational scheme to derive numerical values of the measure is presented and an example is given. 

Sec. 3 then shifts the focus to control-relevant analysis and the analysis of the closed loop. The nonlinearity measure introduced beforehand is applied to quantify the nonlinearity of an optimal state feedback controller during closed-loop operation. Other approaches to controlrelevant nonlinearity assessment are discussed. An example shows the need of dedicated measures for control-relevant analysis (as opposed to the previously considered general open-loop measures) if conclusions concerning the necessary controller structure are to be drawn. 

Finally, Sec. 4 summarizes the merits of nonlinearity measures and its applications. 

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