Describing functions

S. N. Floerger, "Polymer Soil Conditioners," undated product Hterature, Saint Etienne, Erance describes functions and uses of cross-linked anionic polyacrylamides in agriculture. 

N. Noiray, D. Durox, T. Schuller, and S. Candel. A unified framework for nonlinear combustion instability analysis based on the flame describing function. /. Fluid Mech., 2008 (In press). 

The cumulative molar concentration of polymeric species P 0 may be evaluated from the population density distribution, Equation 16. The first two rate expressions represent monomeric additions which do not change the molar concentration of polymeric molecules. A rate constant that describes functionality as a separable function of the molecule s degree of polymerization satisfies this constraint. The simplest, realistic function is the linear expression... 

In this chapter we will focus on pathophysiological aspects of iron deficiency and iron overload, describing functional relations between cells and organs as a link between biochemistry and medicine. Clinical, diagnostic and therapeutic aspects will only be mentioned if particularly relevant for pathophysiology. 

These examples demonstrate that nucleolin could be a therapeutic target for the development of innovative molecules against cancer and virus infection. The newly described function of nucleolin in chromatin dynamics and gene expression is potentially another function of nucleolin that could be targeted for the search of specific inhibitors of cell proliferation. 

Remark 5. In order to construct the controller (22), (30), (36) and (54), it is not necessary to know neither the continuous steady state Xss nor the discrete one Xdss, but only the continuous steady state input Ugs (t). However, for nonlinear systems, in the particular case of polynomials describing function (such as triangular systems describing by polynomial terms), ... 

The triethyloxonium cation is an effective oxidant for the production of aromatic cation-radicals, but only as the hexachloroantimonate salt. The analogous EtjO BE or Bu4N SbClg cannot be used to prepare any cation-radical. This is consistent with the previously described function of SbClg as an oxidant. The slow release of SbClj forms the basis for the efficacy of EtjO SbClg. Kochi s group describes this slow release as follows (Rathore et al. 1998a) ... 

The twin goals of 58.47 are to prevent cross-contamination and mix-ups. Facility management must provide the necessary degree of separation to meet these goals. Separate rooms are not required for each of the described functions if adequate separation can be provided by spatial arrangements within a room, by special air-handhng techniques, and/or by strictly enforced procedural requirements. 

Figure 11.8 is plotted semilogarithmically to include the wide range of pore radii. The function D (r), as is true for surface area and all subsequently described functions, is more conveniently obtained by computer-aided data reduction, as mentioned previously. 

The wide array of plant protein products, methods, and terms used to evaluate and describe functional properties makes it difficult to compare products and results of separate studies. In high protein products, water and fat absorption is attributable primarily to the protein. In other products, additional ingredient components have an effect. Hydrophilic polysaccharides greatly affect water absorption. Water absorption also is affected by product concentration, pH, temperature, and ionic strength. Fat absorption has been studied... 

If btb= 1, then b is said to be normalized. If a set of column vectors has every member normalized and every member orthogonal to every other member, the set is called orthonormal. (Orthonormality was previously used to describe functions we shall see in the next section that orthonormal functions can be represented by orthonormal column vectors.)... 

The control of a stirred tank reactor with hysteresis in the control element-II Describing function analysis (with J.C. Hyun). Chem. Eng. Sci 27, 1361-1370 (1972). 

Although there is no universal procedure for analysing non-linear systems, there are several methods which can be applied to particular types or classes of system. One of these (viz. the describing function technique) is discussed in this section. First, however, the linearisation procedure employed in Section 7.5.2 is expanded to include relationships containing more than one independent variable. 

The describing function method is applicable to any non-linearity which has the characteristic that if the input is a sinusoidal signal then the output is a periodic function 38 401. Because of its simplicity and wide range of applicability, the describing function technique is one of the most versatile procedures for analysing non-linear effects. 

The describing function approach will now be applied to two common non-linear elements which fulfil these conditions... 

N is called the describing function for the non-linear element. N can be considered as a non-linear equivalent of the linear system transfer function. 

The characteristic of this describing function is conveniently plotted as N/k versus M/S in Fig. 7.83. 

It is possible to determine the characteristics of more complex non-linear functions and to derive their corresponding describing functions by adding and subtracting the describing functions of simpler functions. For example, the characteristic of... 

Fig. 7.83. Characteristic of describing function for linear element with dead-zone...

The describing function for the element with the dead zone N2 is given by equation 7.201. Hence, the describing function for the saturating element Nj is (Fig. 7.84b) ... 

Stability Analysis Using the Describing Function Approach... 

The analysis of the stability of a control system which contains a non-linear element that can be characterised by a describing function N is facilitated by considering the non-linear element as a variable gain for which N depends upon the amplitude M of the input signal(40). The behaviour (in terms of frequency response) of linear elements in the control loop will be a function of frequency only. 

Consider a control loop containing a non-linear component represented by its describing function N and a number of linear components which can be combined together to be described by a linear transfer function G ( ) (Fig- 7.85). The closed-loop transfer function for this system is (from equation 7.111) ... 

Fig. 7.84. Graphical method of obtaining describing function from known describing functions (a) operating characteristics (b) describing function characteristics...

Number of stages or total number of samples Describing function... 

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