Linear transfer functions

Neural networks can also be classified by their neuron transfer function, which typically are either linear or nonlinear models. The earliest models used linear transfer functions wherein the output values were continuous. Linear functions are not very useful for many applications because most problems are too complex to be manipulated by simple multiplication. In a nonlinear model, the output of the neuron is a nonlinear function of the sum of the inputs. The output of a nonlinear neuron can have a very complicated relationship with the activation value. 

A number of techniques have been proposed. We will discuss only the more conventional methods that are widely used in the chemical and petroleum industries. Only the identification of linear transfer-function models will be discussed. Nonlinear identification is beyond the scope of this book. 

Figure 17.3 gives some comparisons of the performance of the multivariable DMC stmctuie with the diagonal stmcture. Three linear transfer-function models are presented, varying from the 2 x 2 Wood and Berry column to the 4 x 4 sidestream column/stripper complex configuration. The DMC tuning constants used for these three examples are NP = 40 and NC = 15. See Chap. 8, Sec. 8.9. 

Before training the net, the transfer functions of the neurons must be established. Here, different assays can be made (as detailed in the previous sections), but most often the hyperbolic tangent function tansig function in Table 5.1) is selected for the hidden layer. We set the linear transfer function purelin in Table 5.1) for the output layer. In all cases the output function was the identity function i.e. no further operations were made on the net signal given by the transfer function). 

In the multivariate calibration field, Khayatzadeh et al. [65] compared ANNs with PLS to determine U, Ta, Mn, Zr and W by ICP in the presence of spectral interferences. For ANN modelling, a PCA preprocessing was found to be very effective and, therefore, the scores of the first five dominant PCs were input to an ANN. The network had a linear transfer function on both the hidden and output layers. They used only 20 samples for training. 

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) ... 

Of all the possible non-linear transfer functions used in ANNs, the sigmoid function (Equation 8.39) has the desirable property that it is approximately linear for small deviations of X around zero, and non-linear for larger deviations of X around zero. As a result, the degree of non-linearity of the transfer function can actually be altered through scaling of the inputs to the transfer function. 

This result could be an indicator of the improved ability of the ANN method to model non-linear relationships between the X-data and the 7-data. It could be the case that one of the four PLS latent variables is used primarily to account for such non-linearities, whereas the ANN method can more efficiently account for these non-linearities through the non-linear transfer function in its hidden layer. 

In some situations where one or more of the latex properties are measured either directly or indirectly through their correlation with surrogate variables and where extreme nonlinearities such as the periodic generation of polymer particles does not occur, one can use much simpler modehng and control techniques. Linear transfer function-type models can he identified directly from the plant reactor data. Conventional control devices such as PID controllers or PID controllers with dead-time compensation can then be designed. If process data is also used to identify... 

FIGURE 23.6 Schematic representation of a 5-2-1 BP network. The network consists of an input layer containing five nodes into which values for descriptors D1-D5 are input. This is connected by weighted non-linear transfer functions to a hidden layer of two nodes, which is connected by weighted non-linear transfer functions, the final output layer of one node which is the activity value. The network is trained in an iterative fashion by adjusting the weights until the predicted activity values best match the measured activity values. 

Therefore, for measurements with noise excitation, the linear transfer function K (co) (cf. Fig. 4,1.1 (a)) is obtained after cross-correlation of excitation and response and subsequent Fourier transformation of the cross-correlation function Ci (tr) (cf. Fig. 4.1.1 (c)). 

Sensor linearity may concern primary measurand (concentration of analyte) or refractive index and defines the extent to which the relationship between the measurand and sensor output is linear over the working range. Linearity is usually specified in terms of the maximum deviation from a hn-ear transfer function over the specified dynamic range, hi general, sensors with linear transfer functions are desirable as they require fewer calibration points to produce an accurate sensor calibration. However, response of SPR biosensors is usually a non-linear function of the analyte concentration and therefore calibration needs to be carefully considered. 

For the sake of simplicity, let us now set up the case of a second-order bandpass filter and a comparator with saturation levels This closed-loop system verifies the required premises the system is autonomous, the nonlinearity is both separable and frequency-independent, and the linear transfer function contains enough low-pass filtering to neglect the higher harmonics at the comparator output. Choosing adequately the band-pass filter, it can be forced that the first-order characteristic equation for the closed-loop system of Fig. 4 has an oscillation solution being and the oscillation frequency and... 

Although this bio-process example is not described quantitatively and is to some extent unrealistic, it is discussed in some detail here to provide a perspective on the kinds of industrial problems we encounter (different fi om a t3q)ical problem where we are given a process with known linear transfer functions and asked to design a control algorithm). It should help industrial and academic researchers understand what realistic challenges are faced when looking at the interaction between design and control of industrial chemical processes. 

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