Model inversion

Dold, J.W. and Joulin, G., An evolution equation modeling inversion of tulip flames. Combustion and Flame, 100, 450-456, 1995. 

In a very recent publication [1], we have presented a new model for the rotation-vibration motion of pyramidal XY3 molecules, based on the Hougen-Bunker-Johns (henceforth HBJ) approach [2] (see also Chapter 15, in particular Section 15.2, of Ref. [3]). In this model, inversion is treated as a large-amplitude motion in the HBJ sense, while the other vibrations are assumed to be of small amplitude they are described by linearized stretching and bending coordinates. The rotation-vibration Schrddinger equation is solved variationally to determine rotation-vibration energies. The reader is referred to Ref. [1] for a complete description of the theoretical and computational details. 

Aside from univariate linear regression models, inverse MLR models are probably the simplest types of models to construct for a process analytical application. Simplicity is of very high value in PAC, where ease of automation and long-term reliability are critical. Another advantage of MLR models is that they are rather easy to communicate to the customers of the process analytical technology, since each individual X-variable used in the equation refers to a single wavelength (in the case of NIR) that can often be related to a specific chemical or physical property of the process sample. 

Instrumented impact tests, three-point bending, bending force, analogical model, inverse problem. 

Scheffd + reciprocal iiiiiiW Canonical polynomial model + inverse terms as proposed by Box and Draper... 

A. P. d. Weijer, C. B. Lucasius, L. Buydens, and G. Kateman, Chemometrics Intell. Lab. Syst., 20, 45 (1993). Using Genetic Algorithms for an Artificial Neural Network Model Inversion. 

Regarding handling of model responses, process inversion (calculation of u°p with the help of the model) can be performed implicitly with the help of numerical procedures (the model provides process responses y as functions of inputs u and initial states x), or can be performed explicitly, by developing empirical and/or hybrid neural models off-line (the model provides inputs u as functions of process responses y and initial states x) [ 196, 203-206]. In the first case, model responses are more robust, although model inversion is much faster in the second case. Besides, if the process model can be fairly described by linear or bilinear models, then analytical results can be provided for the optimization problem [40,193,207,208], which makes the real-time implementation of predictive controllers much easier. 

In other words, given known parameters 0, model inversion means to determine the input u in terms of the state, the output y, and time derivatives of y. 

This section presents the theoretical material required for the methodology concepts and the proof of its effectiveness. A very brief review of model inversion is first recalled. Then the definitions of relative orders, orders of zeros at infinity, and essential orders are presented. These notions are also reviewed in the bond graph language for defining structural analysis in this framework. In particular the concepts of power lines and causal paths are defined. They will be used for checking the structural criteria of invertibility and differentiability. 

This section defines the criteria that will be used in the bond graph sizing methodology based on model inversion. Then bicausality is presented as a tool for determining the inverse model directly from a bond graph representation. As seen in Procedure 4, bicausality also enables the essential orders to be determined. 

In fact if the specitications in a sizing problem based on the approach of model inversion do not verity this criterion, unit pulses may appear when inverting the equations which is not physically feasible. 

On the basis of the concepts developed in the former sections, the latter section showed a series of design problems. The used approach can also be interesting for problems like system architecture synthesis and comparison [28], parameter synthesis [16], equilibrium or steady-state position determination [4], or the coupling of model inversion with dynamic optimization [24, 26, 27, 32], Finally, the approach was used in the domain of active systems [31], in industrial applications like in aeronautics for electro-hydraulic actuators [17] or in automotive for electric power steering and suspension systems [29, 30], and for classic and hybrid power trains [3,28]. 

Perfect control is thus limited by factors that prohibit the use of the plant model inverse as the IMC controller, Gc- These are time delays, which result in prediction in G right-halfplane transmission (RHPT) zeros, which result in unstable and input constraints, since for strictly proper G, Gc = G would require infinite controller power. A fourth limitation to perfect control is model uncertainty, which requires the controller to be de-tuned in order to avoid instability in the face of plant-model mismatch. 

E. Tomba, N. Meneghetti, P. Facco, T. Zelenkova, A.A. Barresi, D.L. Marchisio, F. Bezzo, and M. Barolo, Transfer of a nanoparticle product between different mixers using latent variable model inversion, A.I.Ch.E. Journal, 60 (1), 123-135,2014. 

E. Tomba, M. Barolo, and S. Garcia-Munoz, General framework for latent variable model inversion for the design and manufacturing of new products. Industrial and Engineering Chemistry Research, 51 (39), 12886-12900, 2012. 

Budko, N. V., and Remis, R. F. (2004) Electromagnetic inversion using a reduced-order three-dimensional homogeneous model. Inverse Probl, 20, S17-S26. 

For now, let us return to reviewing some fields of application of genetic algorithms and then have a more detailed look at the problem of model inversion as an example of an abundant and hard optimization problem. 

Model Inversion The problem of model inversion arises in many fields, applied and scientific, where one tries to find a model that phenomenologically approximates observational data. This can range from (again) financial applications... 

In the following we will take a closer look at the model inversion with genetic algorithms. Three applications of such an inverse modeling procedure will be reviewed, two applications in petroleum geology and one in physics. 

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