Observer model-free

Another way to describe deviations from the simple BPP spectral density is the so-called model-free approach of Lipari and Szabo [10]. This takes account of the reduction of the spectral density usually observed in NMR relaxation experiments. Although the model-free approach was first applied mainly to the interpretation of relaxation data of macromolecules, it is now also used for fast internal dynamics of small and middle-sized molecules. For very fast internal motions the spectral density is given by ... 

The elastic free energy of the constrained-junction model, similar to that of the slip-link model, is the sum of the phantom network free energy and that due to the constraints. Both the slip-link and the constrained-junction model free energies reduce to that of the phantom network model when the effect of entanglements diminishes to zero. One important difference between the two models, however, is that the constrained-junction model free energy equates to that of the affine network model in the limit of infinitely strong constraints, whereas the slip-link model free energy may exceed that for an affine deformation, as may be observed from Equation (41). 

To indicate to the reader that the theory is based on the observable distribution of charge and its theoretical consequences. It is thus not only model-free but relates directly to the measurable properties of a system. 

In the following, the model-based controller-observer adaptive scheme in [15] is presented. Namely, an observer is designed to estimate the effect of the heat released by the reaction on the reactor temperature dynamics then, this estimate is used by a cascade temperature control scheme, based on the closure of two temperature feedback loops, where the output of the reactor temperature controller becomes the setpoint of the cooling jacket temperature controller. Model-free variants of this control scheme are developed as well. The convergence of the overall controller-observer scheme, in terms of observer estimation errors and controller tracking errors, is proven via a Lyapunov-like argument. Noticeably, the scheme is developed for the general class of irreversible nonchain reactions presented in Sect. 2.5. 

When an accurate model of the reaction kinetics is not available (e.g., due to the lack of reliable data for identification), the previously developed approach may be ineffective and model-free strategies for the estimation of the effect of the heat released by the reaction, aq, must be adopted. In detail, the approach in [27] can be considered, where aq is estimated, together with the heat transfer coefficient, via a suitably designed nonlinear observer [24], Other model-free approaches can be adopted, e.g., those based on the adoption of universal interpolators (neural networks, polynomials) for the direct online estimation of the heat [16] and purely neural approaches [11], Approaches based on the combination of neural and model-based paradigms [2] or on tendency models [25] can be considered as well. 

In the following, two model-free approaches based on adaptive observer are presented the first one is based on the results in [51], and the second one is the well-established observer proposed by [24] and applied to batch reactors in [27],... 

Remark 5.5 The stability analysis has been developed for the controller in conjunction with the observer (5.23), (5.24). It can be verified that, if the model-free observer defined by (5.29), (5.30), and (5.34) is considered, the stability of the overall scheme can be proven by using similar arguments. 

It can be argued that the differences between the compared schemes are mainly due to the different estimation accuracy of the quantity aq (Fig. 5.6). It can be seen that, after the initial transient phase in which the model-free observers present an inverse response, both the adaptive (model-based and model-free) approaches achieve very good estimates. As for the parameter estimate, since both the adaptive observers (0O) and the controller (0C) estimates converge to the true value of 0 (see Fig. 5.7), it is possible to argue that the persistency of excitation condition is fulfilled. 

As expected, the effect of the singularity in the model-free observer (5.36) is visible in the last part of the batch, where / r and 7] tends to be equal, and thus the... 

The effect of measurement noise can be observed for all variables (especially the control input, as shown in Fig. 5.5). In particular, the model-free observer based on RBFI is the most sensitive to measurement noise. 

In the metal refining of ores, the metal is solubilized in an aqueous solution. The optical control of metal refining requires quick, accurate analysis of the major chemical species present in solution. Raman spectroscopy and resonance Raman are used to identify the amine complexes of Co, Ni, and Cu species, as well as ammonia sulfate and sulfamate, present in these industrial solutions. The Raman spectra of an industrial plant solution from mine tailings are shown in Fig. 7-15. Each solution contains one or more metal species, sulfate, sulfamate, ammonia, ammonium sulfate, and water. From a comparison with model ammine complexes, the vibrations in the spectra are identifiable. Bands were observed at 615, 980 and 1,110 cm-1 and were assigned to the sulfate ion. No bands were observed for free ammonia. A band at 376 cm-1 was assigned to the Ni(NH3)j + specie. A band at 490 cm-1 was assigned to the Co(NH3)g+ specie. 

Experimentalists often rely on motional models, based on hydrodynamics, in order to interpret their liquid state spectra. MD simulations, can be considered as model-free in the sense that they do not assume the molecular motion to be in any specific regime. MD can be used to evaluate the motional models and even replace them. MD simulations can be used to calculate both the correlation times and the whole correlation functions. This is useful in those cases when correlation times cannot be deduced from measurements of other isotopes in the same molecule or when there is no method available at all. Correlation functions give information about intermolecular interactions and reveal cases when several motional modes are contributing to relaxation mechanisms at slightly different time scales. This can be observed as multiple decay rates. Time correlation functions from MD simulations can be Fourier transformed to power spectra if needed to provide line shapes and frequencies. 

Given knowledge of the covalent geometry and fundamental constants underlying relaxation, the spectral densities, J(co), remain to be defined. Here, we adopt the so-called model free treatment due to Lipari and Szabo (1982a,b) which seeks to encapsulate the unique motional character expressed in observed relaxation behavior by defining the spectral density in terms of two local parameters ... 

In order to eliminate both dividing and non-dividing bacteria, a fluoroquinolone antibiotic, ciprofloxacin, has been associated with PIBCA and PIHCA nanospheres. In an animal model of persisting Salmonella infection, although an effect on the early phase of the infection was observed, neither free nor nanosphere-bound ciprofloxacin was able to eradicate truly persisting bacteria. ... 

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