Numerical methods least mean square method

Nonmodel-based controllers, such as the least mean square (LMS) and artificial neural network back-propagation adaptive controllers, employ iterative approaches to update control parameters in real time [14-17]. However, those methods may encounter difficulties of numerical divergence and local optimiza-... 

The full set of equations was used to model experiments from the literature using numerical methods. In one of these experiments [3], a clay sample in a flexible wall permeameter was subjected to a salt concentration gradient and salinity and pressure profiles were measured. In [4], a scripted finite element solver was used to provide numerical simulations. Using a least mean squares fit, the storage parameter and the reflection coefficient were inferred from the experimental data. Relevant parameters for this experiment are shown in Table 2. 

For common adsorbates the equilibrium constants of reactions involving only solution species are available from literature for less common adsorbates they can be determined in separate experiments that do not involve the adsorbent. The equilibrium constants of (hypothetical) surface reactions are the adjustable parameters of the model, and they are determined from the adsorption data by means of appropriate fitting procedure. With simple models (e.g. the model leading to Langmuir equation which has two adjustable parameters) the analytical equations exist for least-square best-fit model parameters as the function of directly measured quantities, but more complicated models require numerical methods to calculate their parameters. 

The values of the silanol number, aon, of 100 silica samples, with a completely hydroxylated surface, were established [3-5]. The average silanol number (arithmetical mean) was found to be aoH,av = 4.9 OH/nm. Calculations by the least-squares method yielded aon,av = 4.6 OH/nm. These values are in agreement with those reported by De Boer and Vleeskens [11] as well as with results reported by other researchers. To sum up, the magnitude of the silanol number, which is independent of the origin and structural characteristics of amorphous silicas is considered to be a physicochemical constant. The results fully confirmed the idea predicted earlier by Kiselev and co-workers [13,14] on the constancy of the silanol number for a completely hydroxylated silica surface. This constant now has a numerical value cioH,av = 4.6 0.5 OH/nm [3-5] and is known in literature as the Kiselev-Zhuravlev constant. 

The quantities AUMC and AUSC can be regarded as the first and second statistical moments of the plasma concentration curve. These two moments have an equivalent in descriptive statistics, where they define the mean and variance, respectively, in the case of a stochastic distribution of frequencies (Section 3.2). From the above considerations it appears that the statistical moment method strongly depends on numerical integration of the plasma concentration curve Cp(r) and its product with t and (r-MRT). Multiplication by t and (r-MRT) tends to amplify the errors in the plasma concentration Cp(r) at larger values of t. As a consequence, the estimation of the statistical moments critically depends on the precision of the measurement process that is used in the determination of the plasma concentration values. This contrasts with compartmental analysis, where the parameters of the model are estimated by means of least squares regression. 

Multiple parameter fault isolation by means of minimisation of least squares of ARR residuals needs residual parameter sensitivity functions if a gradient search based method is used. If ARRs can be derived in closed symbolic form from a bond graph, their analytical expressions can be used in the formulation of the least squares cost function and can be differentiated with respect to the vector of targeted parameters either numerically or residuals as functions of the targeted parameters can be differentiated symbolically. If ARRs are not available in symbolic form, they can be numerically computed by solving the equations of a DBG. 

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