Identifiability problem nonlinear systems

In his paper On Governors , Maxwell (1868) developed the differential equations for a governor, linearized about an equilibrium point, and demonstrated that stability of the system depended upon the roots of a eharaeteristie equation having negative real parts. The problem of identifying stability eriteria for linear systems was studied by Hurwitz (1875) and Routh (1905). This was extended to eonsider the stability of nonlinear systems by a Russian mathematieian Lyapunov (1893). The essential mathematieal framework for theoretieal analysis was developed by Laplaee (1749-1827) and Fourier (1758-1830). 

The completely reliable computational technique that we have developed is based on interval analysis. The interval Newton/generalized bisection technique can guarantee the identification of a global optimum of a nonlinear objective function, or can identify all solutions to a set of nonlinear equations. Since the phase equilibrium problem (i.e., particularly the phase stability problem) can be formulated in either fashion, we can guarantee the correct solution to the high-pressure flash calculation. A detailed description of the interval Newton/generalized bisection technique and its application to thermodynamic systems described by cubic equations of state can be found... 

The kinetic parameters in this form of the rate law can be identified with the slopes (gii, Si2> and g,3) and intercqit (In a,) in a linear coordinate system relating the logarithm of V, to the logarithms of the X,. Estimating the values for these kinetic parameters from appropriate experimental data is a solvable problem in linear-regression (see above). This is in sharp contrast to most other nonlinear formalisms for which there are no general methods that are practical for extracting kinetic parameters from experimental data (see above). 

Adsorption kinetics of a single particle (activated carbon type) is dealt with in Chapter 9, where we show a number of adsorption / desorption problems for a single particle. Mathematical models are presented, and their parameters are carefully identified and explained. We first start with simple examples such as adsorption of one component in a single particle under isothermal conditions. This simple example will bring out many important features that an adsorption engineer will need to know, such as the dependence of adsorption kinetics behaviour on many important parameters such as particle size, bulk concentration, temperature, pressure, pore size and adsorption affinity. We then discuss the complexity in the dealing with multicomponent systems whereby governing equations are usually coupled nonlinear differential equations. The only tool to solve these equations is... 

In order to find all solutions of the above system of nonlinear equations the algorithm outlined in the previous section was applied. The variables lower and upper bounds are -10 and 1, respectively, which was found by a fast prerun of the algorithm using a larger tolerance of 0.001. The problem was solved utilizing a GAMS [23] implementation of the algorithm. A total of 31 iterations and 1.1 CPU sec were required to identify the solutions within a tolerance of lOE-5. The optimal solution obtained was ... 

The neural network approach is an alternative way of solving the problem. Unlike multiple linear or nonlinear regression techniques, which require a predefined empirical form, the neural network can identify and learn the correlative patterns between the input and the corresponding output values once a training set is provided. This approach avoids some of the shortcomings encountered in more traditional correlative methods, and with modem software it can provide useful models in a relatively short time for both linear and non-linear systems. 

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