Time-Varying Models

The fact that some kinetic profiles are fitted by sums of exponentials, and others are fitted by power functions, suggests that different types of basic mechanisms are at work. In fact, as concluded in Chapter 7, while kinetics from homogeneous media can be fitted by sums of exponentials, heterogeneity shapes kinetic profiles best represented by empirical power-law models. Conversely, when power laws fit the observed data, they suggest that the rate at which a material leaves the site of a process is itself a function of time in the process, i.e., age of material in the process. 

The stochastic formulation would be the most appropriate choice to capture the structural and functional heterogeneity in these biological media. When the process is heterogeneous, one frequently observes chaos-like behaviors. Heterogeneity is at the origin of fluctuations, and fluctuations are the prelude of instability and chaotic behavior. Stochastic modeling is able to  

The usual deterministic approach is incapable of accurately describing all these features. However  

More likely, we think that time-varying parameters are expressions of feedback regulation mechanisms involving the states of the process. This is our fundamental working hypothesis in the subsequent procedure. To unveil the dependence of the time-varying parameters on the states of the process, we propose the following procedure  

Given a set of experimental data, we look for the time profile of A (t) and b(t) parameters in (C.l). To perform this key operation in the procedure, it is necessary to estimate the model on-line at the same time as the input-output data are received [600]. Identification techniques that comply with this context are called recursive identification methods, since the measured input-output data are processed recursively (sequentially) as they become available. Other commonly used terms for such techniques are on-line or real-time identification, or sequential parameter estimation [352]. Using these techniques, it may be possible to investigate time variations in the process in a real-time context. However, tools for recursive estimation are available for discrete-time models. If the input r (t) is piecewise constant over time intervals (this condition is fulfilled in our context), then the conversion of (C.l) to a discrete-time model is possible without any approximation or additional hypothesis. Most common discrete-time models are difference equation descriptions, such as the Auto-.Regression with eXtra inputs (ARX) model. The basic relationship is the linear difference equation  

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Sun Y. and Chiaramida S. 1992. Simulation of hemodynamics and regulatory mechanisms in the cardiovascular system based on a nonlinear and time-varying model. Simulation 59 28. 

Table 2.7 Performance of time-varying models and ANN model...

Hoi, K.-I., Yuen, K.-V. and Mok, K.-M. Prediction of daily averaged PMIO concentrations by statistical time-varying model. Atmospheric Environment 43(16) (2009), 2579-2581. 

A dynamic tubular reactor model, comprising a set of partial differential equations, has been used to test the computational efficiency and the data handling capabilities of the various software packages. Experimental data of three time-varying model inputs, i.e. the reactor temperature, the fluid velocity and the reactant inlet concentration, are used to estimate the model parameters fix)m experimental data of the reactor temperature measured at several fixed reactor locations as a function of time. This problem was originally published in 1992 [3]. 

Hoi KI, Yuen KV, Mok KM (2011) Iterative probabilistic approach for selection of time-varying model classes. Procedia Eng 14 2585-2592. doi 10.1016/j.proeng. 2011.07.325... 

This work focused on non linear time invariant processes. An approach to deal with time varying model was proposed by Genceli and Nikolau [16] using a finite impulse response (FIR) model that was updated in real time. Recently, Alexandridis and Sarimveis [17] proposed an adaptive model predictive control using a non linear... 

EMGRESP is overly conservative for passive gas dispersion applications. No time-varying releases may be modeled. Dense gas dispersion may be computed for only "instantaneous" releases conditions. 

Figure 2. Experimental trial used to Identify transfer function. In this experiment, the reactant flow rate was deliberately varied and the reactant temperature measured on-line in the pilot plant. This allowed us to identify the proper time series model.

There are several examples in the literature of GFC now being utilized for small molecule analysis (17). However, in this case, attempts to obtain monomer concentrations for kinetic modelling were frustrated by irreproducible impurity peak interference with monomer peaks, time varying refractometer responses and insufficient resolution for utilization of a reference peak. This last point meant that injected concentration would have to be extremely reproducible. 

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