Uncertainty estimation, robustness

R. Femat, J. Alvarez-Ramirez, and M. Rosales-Torres. Robust asymptotic linearization via uncertainty estimation Regulation of temperature in a fluidized bed reactor. Comput. Chem. Eng., 23 697-708, 1999. 

It has been proposed that the addition of other uncertainty components not taken into consideration by PT should also be performed. However, unlike a well-designed inter-laboratory test, confidentiality is usually an integral part of PT. It is unlikely that an individual laboratory would have access to the information required to determine the nature of the other uncertainty components. Additionally, given that participant methods are not identical and that robust statistics are used, uncertainty estimates may already be over-estimated. It is therefore assumed that the information derived from PT data are adequate. ... 

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. 

In addition to the insightful use of existing data, the acquisition of new chemical and physical property data continues to grow in importance— as does the need to retrieve data for future needs. Such efforts require careful experimental measurements as well as skilled evaluation of related data from multiple sources. It will be necessary to assess confidence with robust uncertainty estimates validate data with experimentally or calculated benchmark data of known accuracy and document the metadata needed for intapretation. 

Estimates of the variance and uncertainty intervals in robust calibration can be taken from the literature (Huber [1981] Rousseeuw and Leroy [1987]). 

In robustness tests, usually the factors are examined at two extreme levels.For mixture-related and quantitative factors, these levels usually are chosen symmetrically around the nominal. The range between the extreme levels is selected so that it represents the variability that can occur when transferring the method.However, specifications to estimate such variability are not given in the ICH guidelines. Often the levels are chosen based on personal experience, knowledge, or intuition. Some define the extreme levels as nominal level +x%. However, this relative variation in factor levels is not an appropriate approach, since the absolute variation then depends on the value of the nominal level. Another possibility is to define the levels based on the precision or the uncertainty, with which... 

The formulation described above provides a useful framework for treating feedback control of combustion instability. However, direct application of the model to practical problems must be exercised with caution due to uncertainties associated with system parameters such as and Eni in Eq. (22.12), and time delays and spatial distribution parameters bk in Eq. (22.13). The intrinsic complexities in combustor flows prohibit precise estimates of those parameters without considerable errors, except for some simple well-defined configurations. Furthermore, the model may not accommodate all the essential processes involved because of the physical assumptions and mathematical approximations employed. These model and parameter uncertainties must be carefully treated in the development of a robust controller. To this end, the system dynamics equations, Eqs. (22.12)-(22.14), are extended to include uncertainties, and can be represented with the following state-space model ... 

The robust controller consists of two main components the first is an observer, which estimates the states of the generalized plant described by Eq. (22.29), and consequently the dynamics in the combustion chamber. It is capable of treating exogenous inputs and uncertainty-induced disturbances. The second is a state-feedback control gain, which determines the control action based on the estimated states x. The final configuration of the controller is plotted in Fig. 22.3. 

A comprehensive framework of robust feedback control of combustion instabilities in propulsion systems has been established. The model appears to be the most complete of its kind to date, and accommodates various unique phenomena commonly observed in practical combustion devices. Several important aspects of distributed control process (including time delay, plant disturbance, sensor noise, model uncertainty, and performance specification) are treated systematically, with emphasis placed on the optimization of control robustness and system performance. In addition, a robust observer is established to estimate the instantaneous plant dynamics and consequently to determine control gains. Implementation of the controller in a generic dump combustor has been successfully demonstrated. 

Single observables. Block averaging is a simple, relatively robust procedure for estimating statistical uncertainty. Visual and correlation analyses should also be performed. 

Such uncertainty about the future is a major reason why agencies such as NICE determine a future point in time when their guidance will be reviewed and specify the additional research they would like to see duringthe intervening period. From a methodological standpoint, analysts should not present just one set of cost-effectiveness estimates using a single method of extrapolation. Rather, a series of scenarios should be presented based on different extrapolation techniques. This will provide an indication of how robust the cost-effectiveness results are to the extrapolation approach. 

Risk assessment starts with risk identification, a systematic use of available information to identify hazards (i.e., events or other conditions that have the potential to cause harm). Information can be from a variety of sources including stakeholders, historical data, information from the literature, and mathematical or scientific analyses. Risk analysis is then conducted to estimate the degree of risk associated with the identified hazards. This is estimated based on the likelihood of occurrence and resultant severity of harm. In some risk management tools, the ability to detect the hazard may also be considered. If the hazard is readily detectable, this may be considered a factor in the overall risk assessment. Risk evaluation determines if the risk is acceptable based on specified criteria. In a quality system environment, criteria would include impact on the overall performance of the quality system and the quality attributes of the finished product. The value of the risk assessment depends on how robust the data used in the assessment process is judged to be. The risk assessment process should take into account assumptions and reasonable sources of uncertainty. Risk assessment activities should be documented. 

From the CLAMP data and associated mean annual climate data, Forest et al. (1999) obtained estimates of enthalpy, temperature, relative humidity, and specific humidity (Fig. 6). The data set has been reduced by removing the outliers as indicated by scores along the third and fourth axes (see Wolfe 1995 for a description). The axis eigenvalues from CANOCO indicate that significant information is contained in the first 6 axes and implies that the use of the axes three and four as an outlier indicator should be robust. The estimates of the climate data indicate that mean annual enthalpy can be predicted from fossil leaf physiognomy with an uncertainty of aH = + 5.5 kJ/kg. Additionally, the standard errors for the estimates of temperature, specific humidity, and relative humidity are respectively, aT= 1.8 °C, aq= 1.7 g/kg, and = 13%. 

In order to tackle the problem of uncertainties in the available model, nonlinear robust and adaptive strategies have been developed, while, in the absence of full state measurements, output-feedback control schemes can be adopted, where the unmeasurable state variables can be estimated by resorting to state observers. The development of model-based nonlinear strategies has been fostered by the development of efficient experimental identification methods for nonlinear models and by significantly improved capabilities of computer-control hardware and software. 

The above remark is of the utmost importance for evaluating the potential of the proposed observer in a real setup. In fact, exponential stability would ensure robustness of the state estimation against bounded and/or vanishing model uncertainties and disturbances [35], due to inaccurate and/or incomplete knowledge of reaction kinetics and to usual simplifying assumptions adopted for the model derivation (e.g., perfect mixing). 

Remarks 5.1 and 5.2 on the exponential stability of the estimation error dynamics can be extended to the overall controller-observer scheme as well. Hence, robustness with respect to effects due to modeling uncertainties and/or disturbances is guaranteed. Moreover, the following remarks can be stated. 

Although different fitting methods may produce estimates of parameters with the same uncertainties, the methods may differ greatly in reliability. It is thus of prime importance to choose an accurate method that has the greatest robustness for the application being considered. 

The essential criteria for a good fit are that the returned parameters should be as accurate as permitted by the data and that the fitting process should be robust. The theoretical limit on the uncertainties of each of the parameters in the model is given by the Cramer Rao lower bound. The Cramer Rao bound applies to any unbiased estimator 0(y) of a parameter vector, 9, using measurements, y. The measurements are described by their joint probability density function p(y 6), which is influenced by 9. 

The importance of this result is that it leads to an overall objective criterion for sample size determination that averages criteria based on specific model assumptions. Thus it provides a solution that is robust to model uncertainty. Closed-form calculations of (8) are intractable, so we have developed numerical approximations to the conditional entropies Ent(6k n, yk, MLk) and Ent(9k n, yk, MGk). The computations of the expected Bayes risk are performed via stochastic simulations and the exact objective function is estimated by curve fitting as suggested by Miiller and Parmigiani (1995). These details are available on request from the authors. 

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