Parametric uncertainties

The determination of the electronic structure of lanthanide-doped materials and the prediction of the optical properties are not trivial tasks. The standard ligand field models lack predictive power and undergoes parametric uncertainty at low symmetry, while customary computation methods, such as DFT, cannot be used in a routine manner for ligand field on lanthanide accounts. The ligand field density functional theory (LFDFT) algorithm23-30 consists of a customized conduct of nonempirical DFT calculations, extracting reliable parameters that can be used in further numeric experiments, relevant for the prediction in luminescent materials science.31 These series of parameters, which have to be determined in order to analyze the problem of two-open-shell 4f and 5d electrons in lanthanide materials, are as follows. 

The model (DEP) covers the general case with parametric uncertainties in the objective function (q ), in the left-hand-side multipliers of x and yai (Tffl and W< , respectively) and in the right-hand-side parameters (h ). 

Parametric uncertainty A great number of bacterial species carry out the transformations of organic load and nutrients in wastewater treatment processes without direct or easily comprehensible relationships between the microbial populations and viability. The role of each bacterial species is fuzzy [30], and aspects such as cellular physiology and its modeling are not easily understood from external measurements [18], [68]. As a first consequence, the kinetics of these transformations is often poorly or inadequately known [66]. Extensive efforts to model the kinetics have been undertaken, but these have not been successful to elucidate how yield coefficients, kinetic parameters and the bacterial population distribution change as a function of both, the influent composition and the operating conditions. 

The use of feedback-control techniques to modulate combustion processes in propulsion systems has recently received extensive attention [1-3]. Most of the previous studies involved direct implementation of existing control methods designed for mechanical devices, with very limited effort devoted to the treatment of model and parametric uncertainties commonly associated with practical combustion problems. It is well established that the intrinsic coupling between flow oscillations and transient combustion responses prohibits detailed and precise modeling of the various phenomena in a combustion chamber, and, as such, the model may not accommodate all the essential processes involved due to the physical assumptions and mathematical approximations employed. The present effort attempts to develop a robust feedback controller for suppressing combustion instabilities in propulsion systems. Special attention is given to the treatment of model uncertainties. Various issues related to plant... 

The model and parametric uncertainties are represented by a differential operator A and can be properly treated as a disturbance to the plant, Ws = A(xp), which physically represents the energy amplification from input to output. Its global behavior is characterized by the L2 gain as follows ... 

With regards to the analysis of the quality of the various parts of the model, one may use the same methods as are used for practical identifiability analysis. Since the same methods are used, albeit with different objectives, one sometimes refers to this model quality analysis as a posteriori identifiability (and the previous analysis as a priori identifiability). Now, however, one is also interested in how the parametric uncertainty translates to an uncertainty in the various model predictions. For instance, it might be so that even though two individual parameters have a high uncertainty, they are correlated in such a manner that their effect on a specific (non-measured) model output is always the same. Such a translation may be obtained by simulations of the model using parameters within the determined confidence ellipsoids. A global alternative to this is to consider the outputs for all parameters that correspond to a cost function that is below a certain threshold, for example 2% above the found minimum. 

Optimal Design of Operating Procedures with Parametric Uncertainty... 

To date, the use of semiinfinite methods in worst-case control system design with parametric uncertainty has been extremely limited because of the above... 

Guay M. and Zang T., Adaptive extremum seeking control of nonlinear dynamic systems with parametric uncertainty , Automatica 39 1283-1294, 2003. 

DeHaan, D. Guay, M. Extremum seeking control of nonlinear systems with parametric uncertainties and state constraints. Proceedings of American Control Conference, Boston, MA, Jun 30 to Jul 2, 2004. 

Using this model, adaptive posi-cast controllers were designed, and detailed numerical simulation studies were carried out. These studies consisted of (i) the closed-loop performance of the adaptive controller, (ii) comparison of the adaptive controller with an empirical phase-shift controller, (iii) robustness with respect to parametric uncertainties, (w) robustness with respect to unmodeled dynamics and uncertain delays, (i/) performance in the presence of noise. 

Another purpose of model updating is to obtain a mathematical model to represent the underlying system for future prediction. Even though there are also parameters to be identified as in the previous case, these parameters may not necessarily be physical, e.g., coefficients of auto-regressive models. In this situation, the identified parameters are not necessarily as important as the previous case provided that the identified model provides an accurate prediction for the system output. It will be shown in the following chapters that there is no direct relationship between satisfactory model predictions and small posterior uncertainty of the parameters. This point will be further elaborated in Chapter 6. Nevertheless, no matter for which purpose, quantification of the parametric uncertainty is useful for further processing. For example, it can be utilized for comparison of the identified parameter values at different stages or for uncertainty analysis of the output of the identified model. Furthermore, it will be demonstrated in Chapter 6 that quantification of the posterior uncertainty allows for the selection of a suitable class of models for parametric identification. 

Example. Does Small Parametric Uncertainty Imply Good Data Fitting ... 

A reasonable approach to addressing epistemic uncertainties in a quantitative way would consist of evolving from completeness to parametric uncertainties as far as possible, with an aim at allowing the comparison of PRA numerical results including uncertainty with the appropriate decision guidelines. 

Prepare Prediction of an unobserved random variable is a fundamental problem in statistics. The aim of this paper is to construct lower (upper) prediction limits under parametric uncertainty that are exceeded with probability 1—a (a) by future observations or functions of observations. The prediction limits depend on early-failure data of the same sample from the two-parameter Weibull distribution, the shape and scale parameters of which are not known. 

The normalized sensitivity index of the residual to a parametric uncertainty Si is the ratio between effort (or flow) given by the uncertainty 5, and the effort (or flow) contributed by all the parameter uncertainties a. Thus, the sum of these indices gives... 

Campbell, M.E. Grocott, S.C.O. Parametric uncertainty model for control design and analysis. IEEE Trans. Control Systems Technol., 7, no. 1 (1999), pp. 85-96... 

The notion of spillover is important with respect to neglected structural modes. Other modelling errors include parametric uncertainties, which are more difficult to model and may have a substantial impact on the stability and performance of the closed-loop system. 

To reduce the estimation error caused by the temperature measurement and parametric uncertainties, a robust observer using the sliding mode technique by considering the NH3 dynamics is designed. As the ammonia sensor is not crosssensitive against NOx, such a feature can be beneficial for the observer design. Also, based on the sensitivity analysis of the observer, the observer is robust to NOx sensor uncertainty, which is preferable especially when the NOx sensor crosssensitivity is not completely compensated by the EKF correction approach. 

An integrated design and control approach that utilized dynamic optimization to select optimal equipment sizing and PID control configuration based on a defined dynamic requirement was introduced and developed by Perkins, et al. in a number of papers [6, 48]. It was further extended to incorporate parametric uncertainty [49, 50] and multivariable controllers [51]. 

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