Convolution model

The step-response model is also referred to as a finite impulse response (FIR) model or a discrete convolution model. 

Dynamic matrix control uses time-domain step-response models (called convolution models). As sketched in Fig. 8.18, the response (x) of a process to a unit step change in the input (Ami = ) made at time equal zero can be described by the values of x at discrete points in time (the fc, s shown on the figure). At r nTJ, the value of X is h r,. If Affii is not equal to one, the value of x at f = n7 is b j Aibi, The complete response can be described using a finite number (NP) values of b coefficients. NP is typically chosen such that the response has reached 90 to 95 percent of its final value. 

For PSDs measured by GPC, we expect a greater degree of success with the simple model for retention (eq. 5). Halasz noted that the PSDs he measured were always broader than corresponding PSDs from porisimetry and capillary condensation. This is in keeping with the convolution model (eq. 7) and indicates that the PSDs measured by GPC already contain the convolution between Kqpq and the classical PSD. If this is the case, then the "effective PSDs" provided by the GPC method should be useful for the direct prediction of calibration curves. 

Deconvolution, the inverse operation of recovering the original function o from the convolution model as given in Eq. (1), employs procedures that almost always result in an increase in resolution of the various components of interest in the data. However, there are many broadening and degrading effects that cannot be explicitly expressed as a convolution integral. To consider resolution improvement alone, it is instructive to consider other viewpoints. The uncertainty principle of Fourier analysis provides an interesting perspective on this question. 

Fig. 11 a, b. Comparison of the growth rates of crazes in two diblock copolymers at 20 °C and —20 °C. The solid lines and dotted tines show the predictions of the systematic domain cavitation model and interface convolution model, respectively... 

Fig. 12. Craze growth rate as a function of applied stress for SB5/S4 blend containing 18 vol.% PB (O) quenched sample, (O) slow-cooled sample, ( ) sample isothermally aged 7 days at 87 °C and ( ) sample isothermally aged 5 days at 87 °C solid line and dashed line show predictions of the cavitation model and the meniscus convolution model, respectively...

When the dUuent has precipitated out into PB-2.76K pools for diluent concentrations v > Vg then much more concentrated plasticization is possible in the craze borders in the manner illustrated in Fig. 31. Now the free PB, coating the surfaces of the craze subject to deformation induced negative pressures can be sorbed into a fringing craze surface layer to a depth of od, where d is the craze tuft diameter. However, the tuft diameter is itself dependent on the local plastic resistance by a product expression that is a principal finding of the meniscus interface convolution model, and is usually given in terms of the craze flow stress in the form... 

Noting the possibility that a variant of the meniscus instability of Taylor (1950) could be the mechanism of craze advance, Argon and Salama (1977) proposed a continually repeating interface-convolution model shown in Fig. 11.16 as the... 

This section illustrates a set of case studies in which root-finding plays an important role in chemical engineering including the calculation of the volume of a nonideal gas, bubble point, and zero-crossing. However, these scenarios also crop up in several other areas. For instance, the calculation of the volume of a nonideal gas is a typical problem in fiuid dynamics, whereas the zero-crossing problem is very common in all disciplines involving differential and differential-algebraic systems as convolutions models, such as the optimal control for electrical and electronic purposes. 

Nonclassical space-time. More generally, the convolution models any relation-ship between two variables linked by a time-dependent operator. (Naturally, the convolution is not limited to the time dependence and can be used for space... 

Mahon, P.J. and Oldham, K.B. (1999) Convolutive modelling of electrochemical processes based on the relationship between the current and the surface concentration. Journal of Electroanalytical Chemistry, 464, 1-13. 

Another type of discrete-time model, the finite impulse response (FIR) or convolution model, has become important in computer control. This model can be written as... 

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