System and Parameter Identification

With the conventional experimental design, information about spatial variations of the permeability is not available. With MRI, we can obtain information within the sample, so that we may determine the spatial distribution of the permeability. Clearly, the computational procedure required to estimate the entire distribution will not be as simple as that reflected by Eq. 4.1.7. We will use the principles of system and parameter identification, discussed in the following section, to determine the various macroscopic properties from experiments. 

We refer to system and parameter identification as the principles to determine the most appropriate equations, and properties within those equations, to describe physical phenomena. In particular, we refer to parameter identification as the estimation of properties within a specified model from observations of states or 

Suppose that we have a set of n measurements arranged within a vector Y. Further suppose that the errors associated with the measurements are random, and additive, so that we can represent the measurements as 

Statistical theory provides for the construction of a function incorporating differences between the measured and corresponding calculated values, with the best estimates being the properties that minimize that function. This procedure is shown in detail in the following section. Generally, the implementation is significantly impacted by the functionality of the properties. Three classes of problems are identified  

The properties are represented by a set of constants, P = C, a vector with m elements. 

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Our approach to determine the properties of heterogeneous media utilizes mathematical models of the measurement process and, as appropriate, the flow process itself. To determine the desired properties, we solve an associated system and parameter identification problem (also termed an inverse problem) to estimate the properties from the measured data. 

In this chapter, we describe the approaches used to mathematically model the flow of immiscible fluid phases through permeable media. We summarize the elements of system and parameter identification, and then describe our methods for determining properties of heterogeneous permeable media. 

We have developed a method to spatially resolve permeability distributions. We use MRI to determine spatially resolved velocity distributions, and solve an associated system and parameter identification problem to determine the permeability distribution. Not only is such information essential for investigating complex processes within permeable media, it can provide the means for determining improved correlations for predicting permeability from other measurements, such as porosity and NMR relaxation [17-19]. 

Displacement experiments can be relatively complex and time-consuming, so the experimental design can be a critical issue. Using suitable system and parameter identification methods, we obtain the best estimates of properties from the available data. It is most desirable to have some measures of the accuracy with which the properties are estimated. If that level of accuracy is less than desired, one can consider other ways of conducting the experiments so that additional information about the properties may be obtained. 

We presented a novel method to determine spatially resolved permeability distributions. We used MRI to measure spatially resolved flow velocities, and estimated the permeability from the solution of an associated system and parameter identification problem. 

The main idea of using an experimental-based model approach for damage calculation is discussed in the following. A brief introduction to known damage accumulation ideas is given. Additionally several aspects of system and parameter identification are described. Hence, the new approach is explained and the results of calculations are shown. This contribution closes with a discussion of the results and an outlook to future work. 

A program system called SIDYS using simulation techniques and parameter identification was applied to the absorption data. It includes random search techniques, Rosenbrock strategy, quasi-Newton methods, and lattice search. It runs on a mainframe computer and provides good agreement between simulated and experimental data. On the other hand, it proves that the data of this type of mechanism are not well conditioned and the quality of the evaluation drastically depends on the parameter set chosen [155]. 

J. Benz, J. Roister, R. BUr, G. Gauglitz, Program system SIDYS simulation and parameter identification of dynamic systems, Comput. Chem. 11 (1987) 41. 

Kemevez, J. P. Control, optimization and parameter identification in immobilized enz5me systems. In Proceedings of the International Sympositan on Analysis and Control of Immobilized Enzyme Systems. (Thomas, D. and Kernevez, J. P, eds.). North Holland/American Elsevier, 1976, pp. 199-225. 

Chapter 10 discusses state and parameter identification. Using uncertainty concepts, an optimal estimate of the state for a linear system is obtained based upon available measurements. The result is the Kalman filter. The Kalman filter is extended for nonlinear systems and discrete-time models. Kalman filtering is also shown to be effective for the estimation of model parameters. 

Simulation and modeling are tools that can identify, quantify and analyze maritime transportation phenomena and in the ITS context they can influence on development of maritime traffic control systems. Maritime traffic systems incorporate some intelligent systems technology to assist human operators. Therefore further research on dynamic models and parameter identification is still needed. 

This questions includes parameter optimization, optimization of the structure, optimal control problems and parameter identification problems. The latter problem arises if some of the system parameters are unknown and measurements are taken to determine them. 

Additionally, we assume that we have measurements of the system s absolute acceleration, via the use of appropriate accelerometer sensors. Since the aim is the joint state and parameter identification, as the nonlinear dof z(t) cannot be measured, the system is cast into the following state-space form ... 

This reference entry describes the use of particle-based methods for joint state and parameter identification of a structural system for SHM purposes. This class of methods is adopted in order to tackle the difficulties arising due to the nonlinear nature of the physical system and the uncertainty related to our knowledge of the system characteristics. The workings of each method are described, and the advantages, limitations, and enhancements of the presented approaches... 

Table 2.3 is used to classify the differing systems of equations, encountered in chemical reactor applications and the normal method of parameter identification. As shown, the optimal values of the system parameters can be estimated using a suitable error criterion, such as the methods of least squares, maximum likelihood or probability density function. 

The design procedures depend heavily on the dynamic model of the process to be controlled. In more advanced model-based control systems, the action taken by the controller actually depends on the model. Under circumstances where we do not have a precise model, we perform our analysis with approximate models. This is the basis of a field called "system identification and parameter estimation." Physical insight that we may acquire in the act of model building is invaluable in problem solving. 

Identification of a process involves formulating a mathematical model which properly describes the characteristics of the real system. Initial model forms are developed from first principles and a priori knowledge of the system. Model parameters are typically estimated in accordance with experimental observations. The method in which these parameters are evaluated is critical in judging the reliability and accuracy of the model. 

Hence the flow of each chapter of this book will lead from a description of specific chemical/biological processes and systems to the identification of the main state variables and processes occurring within the boundaries of the system, as well as the interaction between the system and its surrounding environment. The necessary system processes and interactions are then expressed mathematically in terms of state variables and parameters in the form of equations. These equations may most simply be algebraic or transcendental, or they may involve functional, differential, or matrix equations in finitely many variables. 

Chapter 3 provides an introduction to the identification of mathematical models for reactive systems and an extensive review of the methods for estimating the relevant adjustable parameters. The chapter is initiated with a comparison between Bayesian approach and Poppers falsificationism. The aim is to establish a few fundamental ideas on the reliability of scientific knowledge, which is based on the comparison between alternative models and the experimental results, and is limited by the nonexhaustive nature of the available theories and by the unavoidable experimental errors. 

Process identification and parameter estimation has been applied in water quality and wastewater treatment systems (7-9). The overall oxygen transfer coefficient can be determined on-line. The hydraulic dispersion has been identified by manipulation of the influent flow rate or the return sludge flow rate (9). 

Beck, M.B. "Identification and parameter estimation of biological process models" In "System Simulation in Water Resources", Vansteenkiste, G.C., Ed. North Holland, 1975, 19-44. 

Generally, software sensors are typical solutions of so-called inverse problems. A so-called forward problem is one in which the parameters and starting conditions of a system, and the kinetic or other equations which govern its behavior, are known. In a complex biological system, in particular, the things which are normally easiest to measure are the variables, not the parameters. In the case of metabolism, the usual parameters of interest are the enzymatic rate and affinity constants, which are difficult to measure accurately in vitro and virtually impossible in vivo [93,118,275,384]. Yet to describe, understand, and simulate the system of interest we need knowledge of the parameters. In other words, one must go backwards from variables such as fluxes and metabolite concentrations, which are relatively easy to measure, to the parameters. Such problems, in which the inputs are the variables and the outputs the parameters, are known as system identification problems or as so-called inverse problems. 

I expect that SA of stochastic and multiscale models will be important in traditional tasks such as the identification of rate-determining steps and parameter estimation. I propose that SA will also be a key tool in controlling errors in information passing between scales. For example, within a multiscale framework, one could identify what features of a coarse-level model are affected from a finer scale model and need higher-level theory to improve accuracy of the overall multiscale simulation. Next a brief overview of SA for deterministic systems is given followed by recent work on SA of stochastic and multiscale systems. 

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